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| | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| | | | |
| − | Version : May-Jun 2004 | + | Version : May-Jun 2004 [Draft 11.00] |
| | | | |
| | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| Line 21: |
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| | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| | </pre> | | </pre> |
| | + | |
| | + | <div class="nonumtoc">__TOC__</div> |
| | | | |
| | ==1. Introduction== | | ==1. Introduction== |
| Line 385: |
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| | </pre> | | </pre> |
| | | | |
| − | ====1.3.1. Initial Analysis of Inquiry -- Allegro Aperto==== | + | ====1.3.1. Initial Analysis of Inquiry — Allegro Aperto==== |
| | | | |
| | <pre> | | <pre> |
| Line 4,221: |
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| | </pre> | | </pre> |
| | | | |
| − | ====1.3.8. Rondeau — Tempo di Menuetto==== | + | ====1.3.8. Rondeau — Tempo di Menuetto==== |
| | | | |
| | <pre> | | <pre> |
| Line 5,470: |
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| | =====1.3.9.3. The Formative Tension===== | | =====1.3.9.3. The Formative Tension===== |
| | | | |
| − | <pre>
| + | The incidental arena or the informal context is presently described in casual, derivative, and negative terms, simply as the "not yet formal", and so this admittedly unruly region is currently depicted in ways that suggest a purely unformed and a wholly formless chaos, which it is not. But increasing experience with the formalization process can help one to develop a better appreciation of the informal context, and in time one can argue for a more positive characterization of this realm as a truly "formative context". The formal domain is where risks are contemplated, but the formative context is where risks are taken. |
| − | The incidental arena or the informal context is presently described in | |
| − | casual, derivative, and negative terms, simply as the "not yet formal", | |
| − | and so this admittedly unruly region is currently depicted in ways that | |
| − | suggest a purely unformed and a wholly formless chaos, which it is not. | |
| − | But increasing experience with the formalization process can help one | |
| − | to develop a better appreciation of the informal context, and in time | |
| − | one can argue for a more positive characterization of this realm as | |
| − | a truly "formative context". The formal domain is where risks are | |
| − | contemplated, but the formative context is where risks are taken. | |
| | | | |
| − | In this view, the informal context is more clearly seen as the off-stage | + | In this view, the informal context is more clearly seen as the off-stage staging ground where everything that appears on the formal scene is first assembled for a formal presentation. In taking this view, one steps back a bit in one's imagination from the scene that presses on one's attention, gets a sense of its frame and its stage, and becomes accustomed to see what appears in ever dimmer lights, in effect, one is learning to reflect on the more obvious actions, to read their pretexts, and to detect the motives that end in them. |
| − | staging ground where everything that appears on the formal scene is first | |
| − | assembled for a formal presentation. In taking this view, one steps back | |
| − | a bit in one's imagination from the scene that presses on one's attention, | |
| − | gets a sense of its frame and its stage, and becomes accustomed to see what | |
| − | appears in ever dimmer lights, in effect, one is learning to reflect on the | |
| − | more obvious actions, to read their pretexts, and to detect the motives that | |
| − | end in them. | |
| | | | |
| − | It is fair to assume that an agent of inquiry possesses a faculty of inquiry | + | It is fair to assume that an agent of inquiry possesses a faculty of inquiry that is available for exercise in the informal context, that is, without the agent being required to formalize its properties prior to their initial use. If this faculty of inquiry is a unity, then it appears as a whole on both sides of the "glass", that is, on both sides of the imaginary line that one pretends to draw between a formal arena and its informal context. |
| − | that is available for exercise in the informal context, that is, without the | |
| − | agent being required to formalize its properties prior to their initial use. | |
| − | If this faculty of inquiry is a unity, then it appears as a whole on both | |
| − | sides of the "glass", that is, on both sides of the imaginary line that | |
| − | one pretends to draw between a formal arena and its informal context. | |
| | | | |
| − | Recognizing the positive value of an informal context as | + | Recognizing the positive value of an informal context as an open forum or a formative space, it is possible to form the alignments of capacities that are indicated in Table 5. |
| − | an open forum or a formative space, it is possible to form | |
| − | the alignments of capacities that are indicated in Table 5. | |
| | | | |
| | + | <pre> |
| | Table 5. Alignments of Capacities | | Table 5. Alignments of Capacities |
| | o-------------------o-----------------------------o | | o-------------------o-----------------------------o |
| Line 5,510: |
Line 5,490: |
| | | Afforded | Possessed | Exercised | | | | Afforded | Possessed | Exercised | |
| | o-------------------o--------------o--------------o | | o-------------------o--------------o--------------o |
| | + | </pre> |
| | | | |
| − | This arrangement of capacities, based on the distinction between | + | This arrangement of capacities, based on the distinction between possession and exercise that arises so naturally in this context, stems from a root that is old indeed. In this connection, it is instructive to compare these alignments with those that we find in Aristotle's treatise ''On the Soul'', a germinal textbook of psychology that ventures to analyze the concept of the mind, psyche, or soul to the point of arriving at a definition. The alignments of capacites, analogous correspondences, and illustrative materials outlined by Aristotle are summarized in Table 6. |
| − | possession and exercise that arises so naturally in this context, | |
| − | stems from a root that is old indeed. In this connection, it is | |
| − | instructive to compare these alignments with those that we find | |
| − | in Aristotle's treatise 'On the Soul', a germinal textbook of | |
| − | psychology that ventures to analyze the concept of the mind, | |
| − | psyche, or soul to the point of arriving at a definition. | |
| − | The alignments of capacites, analogous correspondences, | |
| − | and illustrative materials outlined by Aristotle are | |
| − | summarized in Table 6. | |
| | | | |
| | + | <pre> |
| | Table 6. Alignments of Capacities in Aristotle | | Table 6. Alignments of Capacities in Aristotle |
| | o-------------------o-----------------------------o | | o-------------------o-----------------------------o |
| Line 5,536: |
Line 5,509: |
| | | Ship? | Sailor? | | | | Ship? | Sailor? | |
| | o-------------------o-----------------------------o | | o-------------------o-----------------------------o |
| | + | </pre> |
| | | | |
| − | An attempt to synthesize the materials and the schemes that are given | + | An attempt to synthesize the materials and the schemes that are given in Tables 5 and 6 leads to the alignments of capacities that are shown in Table 7. I do not pretend that the resulting alignments are perfect, since there is clearly some sort of twist taking place between the top and the bottom of this synthetic arrangement. Perhaps this is due to the modifications of case, tense, and grammatical category that occur throughout the paradigm, or perhaps it has to do with the fact that the relations through the middle of the Table are more analogical than categorical. For the moment I am content to leave all of these paradoxes intact, taking the pattern of tensions and torsions as a puzzle for future study. |
| − | in Tables 5 and 6 leads to the alignments of capacities that are shown | |
| − | in Table 7. I do not pretend that the resulting alignments are perfect, | |
| − | since there is clearly some sort of twist taking place between the top | |
| − | and the bottom of this synthetic arrangement. Perhaps this is due to | |
| − | the modifications of case, tense, and grammatical category that occur | |
| − | throughout the paradigm, or perhaps it has to do with the fact that | |
| − | the relations through the middle of the Table are more analogical | |
| − | than categorical. For the moment I am content to leave all of | |
| − | these paradoxes intact, taking the pattern of tensions and | |
| − | torsions as a puzzle for future study. | |
| | | | |
| | + | <pre> |
| | Table 7. Synthesis of Alignments | | Table 7. Synthesis of Alignments |
| | o-------------------o-----------------------------o | | o-------------------o-----------------------------o |
| Line 5,561: |
Line 5,526: |
| | | Matter | Form | | | | Matter | Form | |
| | o-------------------o-----------------------------o | | o-------------------o-----------------------------o |
| | + | </pre> |
| | | | |
| − | Due to the importance of Aristotle's account for every discussion that | + | Due to the importance of Aristotle's account for every discussion that follows it, not to mention for those that follow it without knowing it, and because the issues that it raises arise repeatedly throughout this project, I am going to cite an extended extract from the relevant text (Aristotle, ''Peri Psyche'', 2.1), breaking up the argument into a number of individual premisses, stages, and examples. |
| − | follows it, not to mention for those that follow it without knowing it, | |
| − | and because the issues that it raises arise repeatedly throughout this | |
| − | project, I am going to cite an extended extract from the relevant text | |
| − | (Aristotle, 'Peri Psyche', 2.1), breaking up the argument into a number | |
| − | of individual premisses, stages, and examples. | |
| | | | |
| | Aristotle wrote (W.S. Hett translation): | | Aristotle wrote (W.S. Hett translation): |
| | | | |
| | + | <pre> |
| | | a. The theories of the soul (psyche) | | | a. The theories of the soul (psyche) |
| | | handed down by our predecessors have | | | handed down by our predecessors have |
| Line 5,763: |
Line 5,725: |
| | how sign relations can be used to clarify the very languages that | | how sign relations can be used to clarify the very languages that |
| | are used to talk about them. | | are used to talk about them. |
| | + | </pre> |
| | + | |
| | + | '''1.3.10. Recurring Themes (CFR Version)''' |
| | + | |
| | + | <pre> |
| | + | The overall purpose of the next sixteen Subsections is threefold: |
| | + | |
| | + | 1. To continue to illustrate the salient properties of sign relations |
| | + | in the medium of selected examples. |
| | + | |
| | + | 2. To demonstrate the use of sign relations to analyze and to clarify |
| | + | a particular order of difficult symbols and complex texts, namely, |
| | + | those that involve recursive, reflective, or reflexive features. |
| | + | |
| | + | 3. To begin to suggest the implausibility of understanding this order |
| | + | of phenomena without using sign relations or something like them, |
| | + | namely, concepts with the power of 3-adic relations. |
| | + | |
| | + | The prospective lines of an inquiry into inquiry cannot help but meet at |
| | + | various points, where a certain entanglement of the subjects of interest |
| | + | repeatedly has to be faced. The present discussion of sign relations is |
| | + | currently approaching one of these points. As the work progresses, the |
| | + | formal tools of logic and set theory become more and more indispensable |
| | + | to say anything significant or to produce any meaningful results in the |
| | + | study of sign relations. And yet it appears, at least from the vantage |
| | + | of the pragmatic perspective, that the best way to formalize, to justify, |
| | + | and to sharpen the use of these tools is by means of the sign relations |
| | + | that they involve. And so the investigation shuffles forward on two or |
| | + | more feet, shifting from a stance that fixes on a certain level of logic |
| | + | and set theory, using it to advance the understanding of sign relations, |
| | + | and then exploits the leverage of this new pivot to consider variations, |
| | + | and hopefully improvements, in the very language of concepts and terms |
| | + | that one uses to express questions about logic and sets, in all of its |
| | + | aspects, from syntax, to semantics, to the pragmatics of both human and |
| | + | computational interpreters. |
| | + | |
| | + | The main goals of this Section are as follows: |
| | + | |
| | + | 1. To introduce a basic logical notation and a naive theory of sets, |
| | + | just enough to treat sign relations as the set-theoretic extensions |
| | + | of logically expressible concepts. |
| | + | |
| | + | 2. To use this modicum of formalism to define a number of conceptual |
| | + | constructs, useful in the analysis of more general sign relations. |
| | + | |
| | + | 3. To develop a proof format that is amenable to deriving facts about |
| | + | these constructs in careful and potentially computational fashions. |
| | + | |
| | + | 4. More incidentally, but increasingly effectively, to get a sense |
| | + | of how sign relations can be used to clarify the very languages |
| | + | that are used to talk about them. |
| | </pre> | | </pre> |
| | | | |
| Line 5,850: |
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| | </pre> | | </pre> |
| | | | |
| − | =====1.3.10.2. Intermediary Notions=====
| + | '''1.3.10.1. Preliminary Notions (CFR Version)''' |
| | | | |
| | <pre> | | <pre> |
| − | A number of additional definitions are relevant to sign relations whose
| + | The discussion in this Subsection proceeds by recalling a series of basic |
| − | connotative components constitute equivalence relations, if only in part.
| + | definitions, refining them to deal with more specialized situations, and |
| | + | refitting them as necessary to cover larger families of sign relations. |
| | | | |
| − | A "dyadic relation on a single set" (DROSS) is a non-empty set of points
| + | In this discussion the word "semantic" is being used as a generic |
| − | plus a set of ordered pairs on these points. Until further notice, any
| + | adjective to describe anything concerned with or related to meaning, |
| − | reference to a "dyadic relation" is intended to be taken in this sense,
| + | whether denotative, connotative, or pragmatic, and without regard to |
| − | in other words, as a reference to a DROSS.
| + | how these different aspects of meaning are correlated with each other. |
| | + | The word "semiotic" is being used, more specifically, to indicate the |
| | + | connotative relationships that exist between signs, in particular, to |
| | + | stress the aspects of process and of potential for progress that are |
| | + | involved in the transitions between signs and their interpretants. |
| | + | Whenever the focus fails to be clear from the context of discussion, |
| | + | the modifiers "denotative" and "referential" are available to pinpoint |
| | + | the relationships that exist between signs and their objects. Finally, |
| | + | there is a common usage of the term "pragmatic" to highlight aspects of |
| | + | meaning that have to do with the context of use and the language user, |
| | + | but I reserve the use of this term to refer to the interpreter as an |
| | + | agent with a purpose, and thus to imply that all three aspects of |
| | + | sign relations are involved in the subject under discussion. |
| | | | |
| − | When the maximum precision of notation is needed, a dyadic relation !G!
| + | Recall the definitions of "semiotic equivalence classes" (SEC's), |
| − | will be given in the form !G! = <G(1), G(2)>, where G(1) is a non-empty
| + | "semiotic partitions" (SEP's), "semiotic equations" (SEQ's), and |
| − | set of points and G(2) c G(1) x G(1) is a set of ordered pairs from G(1).
| + | "semiotic equivalence relations" (SER's) from Subsection 1.3.4.3. |
| | | | |
| − | At other times, a dyadic relation may be specified in the form <X, G>, where
| + | The discussion of sign relations up to this point has been centered around |
| − | X is the set of points and where G c X x X is the set of ordered pairs that
| + | and remained partial to examples of sign relations that enjoy especially |
| − | go together to define the relation. This option is often used in contexts
| + | nice properties, in particular, its focus has been on sign relations |
| − | where the set of points is understood, and thus it becomes convenient to
| + | whose connotative components form equivalence relations and whose |
| − | call the whole relation <X, G> by the name of its second set G c X x X.
| + | denotative components conform to these equivalences, in the sense |
| | + | that all of the signs in each semiotic equivalence class always |
| | + | denote one and the same object. By way of liberalizing the |
| | + | discussion to more general cases of sign relations, this |
| | + | Subsection develops a number of additional concepts for |
| | + | describing the internal structures of sign relations |
| | + | and it lays out a set of definitions that do not |
| | + | take the aforementioned features for granted. |
| | | | |
| − | A "subrelation" of a dyadic relation !G! = <X, G> = <G(1), G(2)>
| + | The complete sign relation involved in a given situation encompasses |
| − | is a dyadic relation !H! = <Y, H> = <H(1), H(2)> that has all of
| + | all of the things that one thinks about and all of the thoughts that |
| − | its points and pairs in !G!, more precisely, that has all of its
| + | one thinks about them while engaged in that particular situation, in |
| − | point-set Y c X and all of its pair-set H c G.
| + | other words, all of the signs and ideas that flit through one's mind |
| | + | in relation to a given domain of objects. Only a rarefied sample of |
| | + | this rarely completed sign relation is bound or even likely to avail |
| | + | itself to any reflective awareness, still less of it has much chance |
| | + | to inspire a concerted interest in the community of inquiry at large, |
| | + | and only bits and pieces of it can be expected to suit themselves to |
| | + | a formal analysis. In view of these considerations, it is useful to |
| | + | have a general idea of the "sampling relation" that any investigator, |
| | + | oneself in particular, is likely to forge between two sign relations: |
| | | | |
| − | The "induced subrelation on a subset" (ISOS), taken with respect to
| + | 1. The whole sign relation that one intends to study. |
| − | the dyadic relation G c X x X and the subset Y c X, is the maximal
| |
| − | subrelation of G whose points belong to Y. In other words, it is
| |
| − | the dyadic relation on Y whose extension contains all of the pairs
| |
| − | of Y x Y that appear in G. Since the construction of an ISOS is
| |
| − | uniquely determined by the data of G and Y, it can be represented
| |
| − | as a function of those arguments, as in the notation ISOS(G, Y),
| |
| − | which can be denoted more briefly as !G!_Y. Using the symbol
| |
| − | "|^|" to indicate the intersection of sets, the construction
| |
| − | of !G!_Y = ISOS(G, Y) can be defined as follows:
| |
| | | | |
| − | !G!_Y = <Y, G_Y> = <G_Y (1), G_Y (2)> | + | 2. The selective portion of it that one is able to |
| − | | + | pin down for the sake of a formal investigation. |
| − | = <Y, {<x, y> in Y x Y : <x, y> in G(2)}> | + | |
| | + | It is important to realize that a "sampling relation", to express it |
| | + | roughly, is a special case of a sign relation. Aside from acting on |
| | + | sign relations and creating an association between sign relations, a |
| | + | sampling relation is also involved in a larger sign relation, at least, |
| | + | it can be subsumed within a general order of sign relations that allows |
| | + | sign relations themselves to be taken as the objects, the signs, and the |
| | + | interpretants of what can be called a "higher order" (HO) sign relation. |
| | + | Considered with respect to its full potential, its use, and its purpose, |
| | + | a sampling relation does not fall outside the closure of sign relations. |
| | + | To be precise, a sampling relation falls within the denotative component |
| | + | of a HO sign relation, since the sign relation sampled is the object of |
| | + | study and the sample is taken as a sign of it. |
| | + | |
| | + | Out of the general variety of sampling relations one can pick out |
| | + | a number of specific conceptions that are likely to be useful in |
| | + | our study, a few of which can now be discussed. I close out the |
| | + | current Subsection with a concept of very general application in |
| | + | the world of sign relations, and dedicate the next Subsection to |
| | + | a collection of more specialized concepts. |
| | + | |
| | + | A "piece" of a sign relation is defined to be any subset of its extension, |
| | + | that is, a wholly arbitrary selection from the set of its ordered 3-tuples. |
| | + | |
| | + | Described in relation to sampling relations, a piece of a sign relation |
| | + | is just the most arbitrary possible sample of it, and thus its occurring |
| | + | to mind implies the most general form of sampling relation to be in effect. |
| | + | In essence, it is just as if a piece of a sign relation, by virtue of its |
| | + | appearing in evidence, can always be interpreted as a piece of evidence |
| | + | that some sort of sampling relation is being applied. |
| | + | </pre> |
| | + | |
| | + | =====1.3.10.2. Intermediary Notions===== |
| | + | |
| | + | <pre> |
| | + | A number of additional definitions are relevant to sign relations whose |
| | + | connotative components constitute equivalence relations, if only in part. |
| | + | |
| | + | A "dyadic relation on a single set" (DROSS) is a non-empty set of points |
| | + | plus a set of ordered pairs on these points. Until further notice, any |
| | + | reference to a "dyadic relation" is intended to be taken in this sense, |
| | + | in other words, as a reference to a DROSS. |
| | + | |
| | + | When the maximum precision of notation is needed, a dyadic relation !G! |
| | + | will be given in the form !G! = <G(1), G(2)>, where G(1) is a non-empty |
| | + | set of points and G(2) c G(1) x G(1) is a set of ordered pairs from G(1). |
| | + | |
| | + | At other times, a dyadic relation may be specified in the form <X, G>, where |
| | + | X is the set of points and where G c X x X is the set of ordered pairs that |
| | + | go together to define the relation. This option is often used in contexts |
| | + | where the set of points is understood, and thus it becomes convenient to |
| | + | call the whole relation <X, G> by the name of its second set G c X x X. |
| | + | |
| | + | A "subrelation" of a dyadic relation !G! = <X, G> = <G(1), G(2)> |
| | + | is a dyadic relation !H! = <Y, H> = <H(1), H(2)> that has all of |
| | + | its points and pairs in !G!, more precisely, that has all of its |
| | + | point-set Y c X and all of its pair-set H c G. |
| | + | |
| | + | The "induced subrelation on a subset" (ISOS), taken with respect to |
| | + | the dyadic relation G c X x X and the subset Y c X, is the maximal |
| | + | subrelation of G whose points belong to Y. In other words, it is |
| | + | the dyadic relation on Y whose extension contains all of the pairs |
| | + | of Y x Y that appear in G. Since the construction of an ISOS is |
| | + | uniquely determined by the data of G and Y, it can be represented |
| | + | as a function of those arguments, as in the notation ISOS(G, Y), |
| | + | which can be denoted more briefly as !G!_Y. Using the symbol |
| | + | "|^|" to indicate the intersection of sets, the construction |
| | + | of !G!_Y = ISOS(G, Y) can be defined as follows: |
| | + | |
| | + | !G!_Y = <Y, G_Y> = <G_Y (1), G_Y (2)> |
| | + | |
| | + | = <Y, {<x, y> in Y x Y : <x, y> in G(2)}> |
| | | | |
| | = <Y, Y x Y |^| G(2)> | | = <Y, Y x Y |^| G(2)> |
| Line 5,918: |
Line 6,021: |
| | </pre> | | </pre> |
| | | | |
| − | ====1.3.10. Recurring Themes (CFR Version)====
| + | '''1.3.10.2. Intermediary Notions (CFR Version)''' |
| | | | |
| | <pre> | | <pre> |
| − | The overall purpose of the next sixteen Subsections is threefold:
| + | A number of additional definitions are relevant to sign relations whose |
| | + | connotative components constitute equivalence relations, if only in part. |
| | | | |
| − | 1. To continue to illustrate the salient properties of sign relations
| + | A "dyadic relation on a single set" (DROSS) is a non-empty set of points |
| − | in the medium of selected examples.
| + | plus a set of ordered pairs on these points. Until further notice, any |
| | + | reference to a "dyadic relation" or to a "2-adic relation" is intended |
| | + | to be taken in this sense, in other words, as a reference to a DROSS. |
| | | | |
| − | 2. To demonstrate the use of sign relations to analyze and to clarify | + | In a typical notation, the 2-adic relation !G! = <Y, G> = <!G!^(1), !G!^(2)> |
| − | a particular order of difficult symbols and complex texts, namely,
| + | is given by the set Y = G^(1) of its points and the set G = !G!^(2) c YxY of |
| − | those that involve recursive, reflective, or reflexive features.
| + | its ordered pairs that go together to define the relation. In contexts where |
| | + | the underlying set of points is understood, it is customary to call the entire |
| | + | 2-adic relation !G! by the name of the set G, that is, the set of its 2-tuples. |
| | | | |
| − | 3. To begin to suggest the implausibility of understanding this order
| + | A "subrelation" of a 2-adic relation !G! = <Y, G> = <!G!^(1), !G!^(2)> |
| − | of phenomena without using sign relations or something like them,
| + | is a 2-adic relation !H! = <Z, H> = <!H!^(1), !H!^(2)> that has all of |
| − | namely, concepts with the power of 3-adic relations.
| + | its points and all of its pairs in !G!, more precisely, that has all of |
| | + | its points Z c Y and all of its pairs H c G. |
| | | | |
| − | The prospective lines of an inquiry into inquiry cannot help but meet at | + | The "induced subrelation on a subset" (ISOS), taken with respect to the |
| − | various points, where a certain entanglement of the subjects of interest
| + | 2-adic relation G c YxY and the subset Z c Y, is the maximal subrelation |
| − | repeatedly has to be faced. The present discussion of sign relations is
| + | of G whose points belong to Z. In other words, it is the 2-adic relation |
| − | currently approaching one of these points. As the work progresses, the
| + | on Z whose extension contains all of the pairs of ZxZ that appear in G. |
| − | formal tools of logic and set theory become more and more indispensable
| + | Since the construction of an ISOS is uniquely determined by the data of |
| − | to say anything significant or to produce any meaningful results in the | + | G and Z, it can be represented as a function of these arguments, as in |
| − | study of sign relations. And yet it appears, at least from the vantage
| + | the notation "Isos(G, Z)", which can be written more briefly as "!G!_Z". |
| − | of the pragmatic perspective, that the best way to formalize, to justify, | + | Using the symbol "|^|" to indicate the intersection of a pair of sets, |
| − | and to sharpen the use of these tools is by means of the sign relations
| + | the construction of !G!_Z = Isos(G, Z) can be defined as follows: |
| − | that they involve. And so the investigation shuffles forward on two or
| |
| − | more feet, shifting from a stance that fixes on a certain level of logic
| |
| − | and set theory, using it to advance the understanding of sign relations, | |
| − | and then exploits the leverage of this new pivot to consider variations,
| |
| − | and hopefully improvements, in the very language of concepts and terms
| |
| − | that one uses to express questions about logic and sets, in all of its
| |
| − | aspects, from syntax, to semantics, to the pragmatics of both human and
| |
| − | computational interpreters.
| |
| | | | |
| − | The main goals of this Section are as follows:
| + | | !G!_Z = <Z, G_Z> = <(!G!_Z)^(1), (!G!_Z)^(2)> |
| | + | | |
| | + | | = <Z, {<z, z'> in ZxZ : <z, z'> in !G!^(2)}> |
| | + | | |
| | + | | = <Z, ZxZ |^| !G!^(2)>. |
| | | | |
| − | 1. To introduce a basic logical notation and a naive theory of sets,
| + | These definitions for 2-adic relations can now be applied in a context |
| − | just enough to treat sign relations as the set-theoretic extensions
| + | where each piece of a sign relation that is being considered satisfies |
| − | of logically expressible concepts.
| + | a special set of conditions, to wit, if M is the piece under the scope: |
| | | | |
| − | 2. To use this modicum of formalism to define a number of conceptual
| + | | Syntactic Domain !Y! = Sign Domain !S! = Interpretant Domain !I! |
| − | constructs, useful in the analysis of more general sign relations.
| + | | |
| | + | | Connotative Component = M_YY = M_SI = Equivalence Relation E |
| | | | |
| − | 3. To develop a proof format that is amenable to deriving facts about
| + | Under these assumptions, and with regard to pieces of sign relations that |
| − | these constructs in careful and potentially computational fashions.
| + | satisfy these conditions, it is useful to consider further selections of |
| | + | a specialized sort, namely, those that keep equivalent signs synonymous. |
| | | | |
| − | 4. More incidentally, but increasingly effectively, to get a sense
| + | An "arbit" of a sign relation is a decidedly more judicious piece of it, |
| − | of how sign relations can be used to clarify the very languages
| + | preserving a semblance of whatever SEP actually and objectively happens |
| − | that are used to talk about them.
| + | to rule over its signs and respecting the semiotic parts of the sampled |
| | + | sign relation, when and if it has such parts. In regard to its effects, |
| | + | an arbit suggests a deliberate act of selection that fairly represents |
| | + | the parts of the sampled SEP by means of the parts of the sample SEP, |
| | + | that extracts an ISOS of each clique in the SER from which it exerts |
| | + | to select any points at all, and that manages to portray in at least |
| | + | this partial fashion either all or none of every SEC that appears in |
| | + | the initial, sampled source, or soi-disant "objective" sign relation. |
| | + | </pre> |
| | + | |
| | + | =====1.3.10.3. Propositions and Sentences===== |
| | | | |
| − | 1.3.10.1. Preliminary Notions
| + | <pre> |
| | + | The concept of a sign relation is typically extended as a set L c O x S x I. |
| | + | Because this extensional representation of a sign relation is one of the most |
| | + | natural forms that it can take up, along with being one of the most important |
| | + | forms that it is likely to be encountered in, a good amount of set-theoretic |
| | + | machinery is necessary in order to carry out a reasonably detailed analysis |
| | + | of sign relations in general. |
| | | | |
| − | The discussion in this Subsection proceeds by recalling a series of basic
| + | For the purposes of this discussion, let it be supposed that each set Q, |
| − | definitions, refining them to deal with more specialized situations, and
| + | that comprises a subject of interest in a particular discussion or that |
| − | refitting them as necessary to cover larger families of sign relations.
| + | constitutes a topic of interest in a particular moment of discussion, |
| | + | is a subset of a set X, one that is sufficiently universal relative |
| | + | to that discussion or big enough to cover everything that is being |
| | + | talked about in that moment. In this setting it is possible to |
| | + | make a number of useful definitions, to which I now turn. |
| | | | |
| − | In this discussion the word "semantic" is being used as a generic
| + | The "negation" of a sentence z, written as "(z)" and read as "not z", |
| − | adjective to describe anything concerned with or related to meaning,
| + | is a sentence that is true when z is false, and false when z is true. |
| − | whether denotative, connotative, or pragmatic, and without regard to
| |
| − | how these different aspects of meaning are correlated with each other.
| |
| − | The word "semiotic" is being used, more specifically, to indicate the | |
| − | connotative relationships that exist between signs, in particular, to
| |
| − | stress the aspects of process and of potential for progress that are
| |
| − | involved in the transitions between signs and their interpretants.
| |
| − | Whenever the focus fails to be clear from the context of discussion,
| |
| − | the modifiers "denotative" and "referential" are available to pinpoint
| |
| − | the relationships that exist between signs and their objects. Finally,
| |
| − | there is a common usage of the term "pragmatic" to highlight aspects of
| |
| − | meaning that have to do with the context of use and the language user,
| |
| − | but I reserve the use of this term to refer to the interpreter as an
| |
| − | agent with a purpose, and thus to imply that all three aspects of
| |
| − | sign relations are involved in the subject under discussion.
| |
| | | | |
| − | Recall the definitions of "semiotic equivalence classes" (SEC's),
| + | The "complement" of a set Q with respect to the universe X |
| − | "semiotic partitions" (SEP's), "semiotic equations" (SEQ's), and | + | is denoted by "X - Q", or simply by "~Q" if the universe X |
| − | "semiotic equivalence relations" (SER's) from Subsection 1.3.4.3.
| + | is understood from context, and it is defined as the set of |
| | + | elements in X that do not belong to Q. In symbols, we have: |
| | | | |
| − | The discussion of sign relations up to this point has been centered around
| + | ~Q = X - Q = {x in X : (x in Q)}. |
| − | and remained partial to examples of sign relations that enjoy especially
| |
| − | nice properties, in particular, its focus has been on sign relations
| |
| − | whose connotative components form equivalence relations and whose
| |
| − | denotative components conform to these equivalences, in the sense
| |
| − | that all of the signs in each semiotic equivalence class always
| |
| − | denote one and the same object. By way of liberalizing the
| |
| − | discussion to more general cases of sign relations, this
| |
| − | Subsection develops a number of additional concepts for
| |
| − | describing the internal structures of sign relations
| |
| − | and it lays out a set of definitions that do not
| |
| − | take the aforementioned features for granted.
| |
| | | | |
| − | The complete sign relation involved in a given situation encompasses | + | The "relative complement" of P in Q, for two sets P, Q c X, |
| − | all of the things that one thinks about and all of the thoughts that
| + | is denoted by "Q - P" and defined as the set of elements in |
| − | one thinks about them while engaged in that particular situation, in
| + | Q that do not belong to P. In symbols: |
| − | other words, all of the signs and ideas that flit through one's mind
| |
| − | in relation to a given domain of objects. Only a rarefied sample of
| |
| − | this rarely completed sign relation is bound or even likely to avail
| |
| − | itself to any reflective awareness, still less of it has much chance
| |
| − | to inspire a concerted interest in the community of inquiry at large,
| |
| − | and only bits and pieces of it can be expected to suit themselves to
| |
| − | a formal analysis. In view of these considerations, it is useful to
| |
| − | have a general idea of the "sampling relation" that any investigator,
| |
| − | oneself in particular, is likely to forge between two sign relations:
| |
| | | | |
| − | 1. The whole sign relation that one intends to study.
| + | Q - P = {x in X : x in Q and (x in P)}. |
| | | | |
| − | 2. The selective portion of it that one is able to
| + | The "intersection" of P and Q, for two sets P, Q c X, is denoted |
| − | pin down for the sake of a formal investigation.
| + | by "P |^| Q" and defined as the set of elements in X that belong |
| | + | to both P and Q. In symbols: |
| | | | |
| − | It is important to realize that a "sampling relation", to express it
| + | P |^| Q = {x in X : x in P and x in Q}. |
| − | roughly, is a special case of a sign relation. Aside from acting on
| |
| − | sign relations and creating an association between sign relations, a
| |
| − | sampling relation is also involved in a larger sign relation, at least,
| |
| − | it can be subsumed within a general order of sign relations that allows
| |
| − | sign relations themselves to be taken as the objects, the signs, and the
| |
| − | interpretants of what can be called a "higher order" (HO) sign relation.
| |
| − | Considered with respect to its full potential, its use, and its purpose,
| |
| − | a sampling relation does not fall outside the closure of sign relations.
| |
| − | To be precise, a sampling relation falls within the denotative component
| |
| − | of a HO sign relation, since the sign relation sampled is the object of
| |
| − | study and the sample is taken as a sign of it.
| |
| | | | |
| − | Out of the general variety of sampling relations one can pick out
| + | The "union" of P and Q, for two sets P, Q c X, is denoted |
| − | a number of specific conceptions that are likely to be useful in
| + | by "P |_| Q" and defined as the set of elements in X that |
| − | our study, a few of which can now be discussed. I close out the
| + | belong to at least one of P or Q. In symbols: |
| − | current Subsection with a concept of very general application in
| |
| − | the world of sign relations, and dedicate the next Subsection to
| |
| − | a collection of more specialized concepts.
| |
| | | | |
| − | A "piece" of a sign relation is defined to be any subset of its extension,
| + | P |_| Q = {x in X : x in P or x in Q}. |
| − | that is, a wholly arbitrary selection from the set of its ordered 3-tuples.
| |
| | | | |
| − | Described in relation to sampling relations, a piece of a sign relation
| + | The "symmetric difference" of P and Q, for two sets P, Q c X, |
| − | is just the most arbitrary possible sample of it, and thus its occurring | + | is denoted by "P + Q" and defined as the set of elements in X |
| − | to mind implies the most general form of sampling relation to be in effect.
| + | that belong to just one of P or Q. In symbols: |
| − | In essence, it is just as if a piece of a sign relation, by virtue of its
| |
| − | appearing in evidence, can always be interpreted as a piece of evidence
| |
| − | that some sort of sampling relation is being applied.
| |
| | | | |
| − | 1.3.10.2. Intermediary Notions
| + | P + Q = {x in X : x in P - Q or x in Q - P}. |
| | | | |
| − | A number of additional definitions are relevant to sign relations whose
| + | The preceding "definitions" are the bare essentials that are needed to |
| − | connotative components constitute equivalence relations, if only in part.
| + | get the rest of this discussion moving, but they have to be regarded as |
| | + | almost purely informal in character, at least, at this stage of the game. |
| | + | In particular, these definitions all invoke the undefined notion of what |
| | + | a "sentence" is, they all rely on the reader's native intuition of what |
| | + | a "set" is, and they all derive their coherence and their meaning from |
| | + | the common understanding, but the equally casual use and unreflective |
| | + | acquaintance that just about everybody has of the logical connectives |
| | + | "and", "or", "not", as these are expressed in natural language terms. |
| | | | |
| − | A "dyadic relation on a single set" (DROSS) is a non-empty set of points
| + | As formative definitions, these initial postulations neither acquire |
| − | plus a set of ordered pairs on these points. Until further notice, any
| + | the privileged status of untouchable axioms and infallible intuitions |
| − | reference to a "dyadic relation" or to a "2-adic relation" is intended
| + | nor do they deserve any special suspicion, at least, nothing over and |
| − | to be taken in this sense, in other words, as a reference to a DROSS.
| + | above the reflective critique that one ought to apply to all important |
| | + | definitions. As the dim beginnings of anything that approaches genuine |
| | + | definitions they also serve to accustom the mind's eye to one particular |
| | + | style of observation, namely, that of seeing informal concepts presented |
| | + | in a formal frame, in a way that demands their increasing clarification. |
| | + | In this style of examination, the frame of the set-builder expression |
| | + | "{x in X : ... }" functions like the eye of the needle through which |
| | + | one is trying to transport a suitably rich import of mathematics. |
| | | | |
| − | In a typical notation, the 2-adic relation !G! = <Y, G> = <!G!^(1), !G!^(2)>
| + | Much of the task of the remaining discussion is to formalize the promissory notes |
| − | is given by the set Y = G^(1) of its points and the set G = !G!^(2) c YxY of
| + | that are represented by the foregoing terms and stipulations and to see whether |
| − | its ordered pairs that go together to define the relation. In contexts where
| + | their informal comprehension can be converted into an explicit subject matter, |
| − | the underlying set of points is understood, it is customary to call the entire | + | one that depends on grasping an array of increasingly formalized concepts. |
| − | 2-adic relation !G! by the name of the set G, that is, the set of its 2-tuples.
| |
| | | | |
| − | A "subrelation" of a 2-adic relation !G! = <Y, G> = <!G!^(1), !G!^(2)>
| + | | NB. In the following asciification of a pre-existing text, |
| − | is a 2-adic relation !H! = <Z, H> = <!H!^(1), !H!^(2)> that has all of
| + | | markups like "!...!" indicate singly-underlined text, and |
| − | its points and all of its pairs in !G!, more precisely, that has all of
| + | | markups like "%...%" indicate doubly-underlined text. |
| − | its points Z c Y and all of its pairs H c G.
| |
| | | | |
| − | The "induced subrelation on a subset" (ISOS), taken with respect to the | + | The "binary domain" is the set !B! = {!0!, !1!} of two algebraic values, |
| − | 2-adic relation G c YxY and the subset Z c Y, is the maximal subrelation
| + | whose arithmetic operations obey the rules of GF(2), the "galois field" |
| − | of G whose points belong to Z. In other words, it is the 2-adic relation | + | of exactly two elements, whose addition and multiplication tables are |
| − | on Z whose extension contains all of the pairs of ZxZ that appear in G.
| + | tantamount to addition and multiplication of integers "modulo 2". |
| − | Since the construction of an ISOS is uniquely determined by the data of
| |
| − | G and Z, it can be represented as a function of these arguments, as in
| |
| − | the notation "Isos(G, Z)", which can be written more briefly as "!G!_Z".
| |
| − | Using the symbol "|^|" to indicate the intersection of a pair of sets,
| |
| − | the construction of !G!_Z = Isos(G, Z) can be defined as follows:
| |
| | | | |
| − | | !G!_Z = <Z, G_Z> = <(!G!_Z)^(1), (!G!_Z)^(2)>
| + | The "boolean domain" is the set %B% = {%0%, %1%} of two logical values, |
| − | |
| + | whose elements are read as "false" and "true", or as "falsity" and "truth", |
| − | | = <Z, {<z, z'> in ZxZ : <z, z'> in !G!^(2)}>
| + | respectively. |
| − | |
| |
| − | | = <Z, ZxZ |^| !G!^(2)>.
| |
| | | | |
| − | These definitions for 2-adic relations can now be applied in a context
| + | At this point, I cannot tell whether the distinction between these two |
| − | where each piece of a sign relation that is being considered satisfies | + | domains is slight or significant, and so this question must evolve its |
| − | a special set of conditions, to wit, if M is the piece under the scope: | + | own answer, while I pursue a larger inquiry by means of its hypothesis. |
| | + | The weight of the matter appears to increase as the investigation moves |
| | + | from abstract, algebraic, and formal settings to contexts where logical |
| | + | semantics, natural language syntax, and concrete categories of grammar |
| | + | are compelling considerations. Speaking abstractly and roughly enough, |
| | + | it is often acceptable to identify these two domains, and up until this |
| | + | point there has rarely appeared to be a sufficient reason to keep their |
| | + | concepts separately in mind. The boolean domain %B% comes with at least |
| | + | two operations, though often under different names and always included |
| | + | in a number of others, that are analogous to the field operations of the |
| | + | binary domain !B!, and operations that are isomorphic to the rest of the |
| | + | boolean operations in %B% can always be built on the binary basis of !B!. |
| | | | |
| − | | Syntactic Domain !Y! = Sign Domain !S! = Interpretant Domain !I!
| + | As sets of the same cardinality, the domains !B! and %B%, along with |
| − | |
| + | all of the structures that can be built on them, are isomorphic at a |
| − | | Connotative Component = M_YY = M_SI = Equivalence Relation E
| + | high enough level of abstraction. But the main reason for preserving |
| | + | their distinction in the present context appears to be more a matter |
| | + | of natural language grammar than an issue of logical or mathematical |
| | + | substance, namely, just so that the signs "%0%" and "%1%" can appear |
| | + | with a semblance of syntactic legitimacy in linguistic contexts that |
| | + | call for grammatical sentences to represent the classes of sentences |
| | + | that are "always false" and "always true", respectively. The signs |
| | + | "0" and "1", that are customarily read as nouns but not as sentences, |
| | + | fail to be suitable for this purpose. Whether these scruples, that |
| | + | are needed to conform to a natural language context, are ultimately |
| | + | important or not, is a thing that I just do not know at this point. |
| | | | |
| − | Under these assumptions, and with regard to pieces of sign relations that
| + | The "negation" of x, for x in %B%, written as "(x)" |
| − | satisfy these conditions, it is useful to consider further selections of
| + | and read as "not x", is the boolean value (x) in %B% |
| − | a specialized sort, namely, those that keep equivalent signs synonymous.
| + | that is %1% when x is %0%, and %0% when x is %1%. |
| | | | |
| − | An "arbit" of a sign relation is a decidedly more judicious piece of it,
| + | Thus, negation is a monadic operation on boolean |
| − | preserving a semblance of whatever SEP actually and objectively happens
| + | values, a function of the form (_) : %B% -> %B%. |
| − | to rule over its signs and respecting the semiotic parts of the sampled
| |
| − | sign relation, when and if it has such parts. In regard to its effects,
| |
| − | an arbit suggests a deliberate act of selection that fairly represents
| |
| − | the parts of the sampled SEP by means of the parts of the sample SEP,
| |
| − | that extracts an ISOS of each clique in the SER from which it exerts
| |
| − | to select any points at all, and that manages to portray in at least
| |
| − | this partial fashion either all or none of every SEC that appears in
| |
| − | the initial, sampled source, or soi-disant "objective" sign relation.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | It is convenient to transport the product and the sum operations of !B! |
| | + | into the logical setting of %B%, where they can be symbolized by signs |
| | + | of the same character, doubly underlined as necessary to avoid confusion. |
| | + | This yields the following definitions of a "product" and a "sum" in %B% |
| | + | and leads to the following forms of multiplication and addition tables. |
| | | | |
| − | IDS. Note 120
| + | The "product" of x and y, for x, y in %B%, is given by Table 8. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | + | Table 8. Boolean Product |
| | + | o---------o---------o---------o |
| | + | | %*% % %0% | %1% | |
| | + | o=========o=========o=========o |
| | + | | %0% % %0% | %0% | |
| | + | o---------o---------o---------o |
| | + | | %1% % %0% | %1% | |
| | + | o---------o---------o---------o |
| | | | |
| − | 1.3.10.3. Propositions and Sentences
| + | Viewed as a function on logical values, %*% : %B% x %B% -> %B%, the |
| | + | product corresponds to the logical operation that is commonly called |
| | + | "conjunction" and that is otherwise expressed as "x and y". In accord |
| | + | with common practice, the multiplication sign "*", doubly underlined or |
| | + | otherwise, is frequently omitted from written expressions of the product. |
| | | | |
| − | The concept of a sign relation is typically extended as a set L c O x S x I. | + | The "sum" of x and y, for x, y in %B%, is given by Table 9. |
| − | Because this extensional representation of a sign relation is one of the most
| |
| − | natural forms that it can take up, along with being one of the most important
| |
| − | forms that it is likely to be encountered in, a good amount of set-theoretic
| |
| − | machinery is necessary in order to carry out a reasonably detailed analysis
| |
| − | of sign relations in general.
| |
| | | | |
| − | For the purposes of this discussion, let it be supposed that each set Q,
| + | Table 9. Boolean Sum |
| − | that comprises a subject of interest in a particular discussion or that
| + | o---------o---------o---------o |
| − | constitutes a topic of interest in a particular moment of discussion,
| + | | %+% % %0% | %1% | |
| − | is a subset of a set X, one that is sufficiently universal relative
| + | o=========o=========o=========o |
| − | to that discussion or big enough to cover everything that is being
| + | | %0% % %0% | %1% | |
| − | talked about in that moment. In this setting it is possible to
| + | o---------o---------o---------o |
| − | make a number of useful definitions, to which I now turn.
| + | | %1% % %1% | %0% | |
| | + | o---------o---------o---------o |
| | | | |
| − | The "negation" of a sentence z, written as "(z)" and read as "not z",
| + | Viewed as a function on logical values, %+% : %B% x %B% -> %B%, |
| − | is a sentence that is true when z is false, and false when z is true.
| + | the sum corresponds to the logical operation that is generally |
| | + | called "exclusive disjunction" and that is otherwise expressed |
| | + | as "x or y, but not both". Depending on the context, a couple |
| | + | of other signs and readings that can invoke this operation are: |
| | | | |
| − | The "complement" of a set Q with respect to the universe X
| + | 1. "x =/= y", read "x is not equal to y", or "exactly one of x and y". |
| − | is denoted by "X - Q", or simply by "~Q" if the universe X
| |
| − | is understood from context, and it is defined as the set of
| |
| − | elements in X that do not belong to Q. In symbols, we have:
| |
| | | | |
| − | ~Q = X - Q = {x in X : (x in Q)}. | + | 2. "x <=/=> y", read "x is not equivalent to y", or "x opposes y". |
| | | | |
| − | The "relative complement" of P in Q, for two sets P, Q c X,
| + | For sentences, the signs of equality ("=") and inequality ("=/=") |
| − | is denoted by "Q - P" and defined as the set of elements in
| + | are reserved to signify the syntactic identity and the syntactic |
| − | Q that do not belong to P. In symbols:
| + | non-identity, respectively, of the literal strings of characters |
| | + | that make up the sentences, while the signs of equivalence ("<=>") |
| | + | and inequivalence ("<=/=>") refer to the logical values, if any, |
| | + | that these strings may conceivably bear, and thus they serve to |
| | + | signify the equality or the inequality, respectively, of their |
| | + | conceivable boolean values. For the logical values themselves, |
| | + | the two pairs of symbols collapse in their meanings to a single |
| | + | pair, signifying a single form of coincidence or a single form |
| | + | of distinction, respectively, between the boolean values of the |
| | + | entities in question. |
| | | | |
| − | Q - P = {x in X : x in Q and (x in P)}.
| + | In logical studies, one tends to be interested in all of the |
| | + | operations or all of the functions of a given type, at least, |
| | + | to the extent that their totalities and their individualities |
| | + | can be comprehended, and not just the specialized collections |
| | + | that define particular algebraic structures. |
| | | | |
| − | The "intersection" of P and Q, for two sets P, Q c X, is denoted
| + | Although the rest of the conceivably possible dyadic operations |
| − | by "P |^| Q" and defined as the set of elements in X that belong
| + | on boolean values, in other words, the remainder of the sixteen |
| − | to both P and Q. In symbols: | + | functions f : %B% x %B% -> %B%, could be presented in the same |
| | + | way as the multiplication and addition tables, it is better to |
| | + | look for a more efficient style of representation, one that is |
| | + | able to express all of the boolean functions on the same number |
| | + | of variables on a roughly equal basis, and with a bit of luck, |
| | + | affords us with a calculus for computing with these functions. |
| | | | |
| − | P |^| Q = {x in X : x in P and x in Q}.
| + | The utility of a suitable calculus would involve, among other things: |
| | | | |
| − | The "union" of P and Q, for two sets P, Q c X, is denoted
| + | 1. Finding the values of given functions for given arguments. |
| − | by "P |_| Q" and defined as the set of elements in X that
| |
| − | belong to at least one of P or Q. In symbols:
| |
| | | | |
| − | P |_| Q = {x in X : x in P or x in Q}. | + | 2. Inverting boolean functions, that is, "finding the fibers" |
| | + | of boolean functions, or solving logical equations that |
| | + | are expressed in terms of boolean functions. |
| | | | |
| − | The "symmetric difference" of P and Q, for two sets P, Q c X,
| + | 3. Facilitating the recognition of invariant forms that |
| − | is denoted by "P + Q" and defined as the set of elements in X
| + | take boolean functions as their functional components. |
| − | that belong to just one of P or Q. In symbols:
| |
| | | | |
| − | P + Q = {x in X : x in P - Q or x in Q - P}.
| + | The whole point of formal logic, the reason for doing logic formally and |
| − | | + | the measure that determines how far it is possible to reason abstractly, |
| − | The preceding "definitions" are the bare essentials that are needed to
| + | is to discover functions that do not vary as much as their variables do, |
| − | get the rest of this discussion moving, but they have to be regarded as
| + | in other words, to identify forms of logical functions that, though they |
| − | almost purely informal in character, at least, at this stage of the game.
| + | express a dependence on the values of their constituent arguments, do not |
| − | In particular, these definitions all invoke the undefined notion of what
| + | vary as much as possible, but approach the way of being a function that |
| − | a "sentence" is, they all rely on the reader's native intuition of what | + | constant functions enjoy. Thus, the recognition of a logical law amounts |
| − | a "set" is, and they all derive their coherence and their meaning from
| + | to identifying a logical function, that, though it ostensibly depends on |
| − | the common understanding, but the equally casual use and unreflective
| + | the values of its putative arguments, is not as variable in its values as |
| − | acquaintance that just about everybody has of the logical connectives
| + | the values of its variables are allowed to be. |
| − | "and", "or", "not", as these are expressed in natural language terms.
| |
| | | | |
| − | As formative definitions, these initial postulations neither acquire
| + | The "indicator function" or the "characteristic function" of a set Q c X, |
| − | the privileged status of untouchable axioms and infallible intuitions | + | written "f_Q", is the map from X to the boolean domain %B% = {%0%, %1%} |
| − | nor do they deserve any special suspicion, at least, nothing over and
| + | that is defined in the following ways: |
| − | above the reflective critique that one ought to apply to all important
| |
| − | definitions. As the dim beginnings of anything that approaches genuine
| |
| − | definitions they also serve to accustom the mind's eye to one particular
| |
| − | style of observation, namely, that of seeing informal concepts presented
| |
| − | in a formal frame, in a way that demands their increasing clarification.
| |
| − | In this style of examination, the frame of the set-builder expression
| |
| − | "{x in X : ... }" functions like the eye of the needle through which
| |
| − | one is trying to transport a suitably rich import of mathematics.
| |
| | | | |
| − | Much of the task of the remaining discussion is to formalize the promissory notes
| + | 1. Considered in extensional form, f_Q is the subset of X x %B% |
| − | that are represented by the foregoing terms and stipulations and to see whether | + | that is given by the following formula: |
| − | their informal comprehension can be converted into an explicit subject matter,
| |
| − | one that depends on grasping an array of increasingly formalized concepts.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | f_Q = {<x, b> in X x %B% : b = %1% <=> x in Q}. |
| | | | |
| − | IDS. Note 121
| + | 2. Considered in functional form, f_Q is the map from X to %B% |
| | + | that is given by the following condition: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | f_Q (x) = %1% <=> x in Q. |
| | + | |
| | + | A "proposition about things in the universe", for short, a "proposition", |
| | + | is the same thing as an indicator function, that is, a function of the |
| | + | form f : X -> %B%. The convenience of this seemingly redundant usage |
| | + | is that it allows one to refer to an indicator function without having |
| | + | to specify right away, as a part of its designated subscript, exactly |
| | + | what set it indicates, even though a proposition always indicates some |
| | + | subset of its designated universe, and even though one will eventually, |
| | + | most likely, want to know exactly what subset that is. |
| | | | |
| − | 1.3.10.3. Propositions and Sentences (cont.)
| + | According to the stated understandings, a proposition is a function that |
| | + | indicates a set, in the sense that a function associates values with the |
| | + | elements of a domain, some of which values can be interpreted to mark out |
| | + | for special consideration a subset of that domain. The way in which an |
| | + | indicator function is interpreted to "indicate" a set can be expressed |
| | + | in terms of the following concepts. |
| | | | |
| − | | NB. In the following asciification of a pre-existing text,
| + | The "fiber" of a codomain element y in Y under a function f : X -> Y is the |
| − | | markups like "!...!" indicate singly-underlined text, and
| + | subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X. |
| − | | markups like "%...%" indicate doubly-underlined text.
| |
| | | | |
| − | The "binary domain" is the set !B! = {!0!, !1!} of two algebraic values,
| + | In other language that is often used, the fiber of y under f is |
| − | whose arithmetic operations obey the rules of GF(2), the "galois field"
| + | called the "antecedent set", the "inverse image", the "level set", |
| − | of exactly two elements, whose addition and multiplication tables are
| + | or the "pre-image" of y under f. All of these equivalent concepts |
| − | tantamount to addition and multiplication of integers "modulo 2".
| + | can be defined as follows: |
| | | | |
| − | The "boolean domain" is the set %B% = {%0%, %1%} of two logical values,
| + | Fiber of y under f = f^(-1)(y) = {x in X : f(x) = y}. |
| − | whose elements are read as "false" and "true", or as "falsity" and "truth",
| |
| − | respectively.
| |
| | | | |
| − | At this point, I cannot tell whether the distinction between these two
| + | In the special case where f is the indicator function f_Q of the set Q c X, |
| − | domains is slight or significant, and so this question must evolve its
| + | the fiber of %1% under f_Q is just the set Q back again: |
| − | own answer, while I pursue a larger inquiry by means of its hypothesis.
| |
| − | The weight of the matter appears to increase as the investigation moves
| |
| − | from abstract, algebraic, and formal settings to contexts where logical
| |
| − | semantics, natural language syntax, and concrete categories of grammar
| |
| − | are compelling considerations. Speaking abstractly and roughly enough,
| |
| − | it is often acceptable to identify these two domains, and up until this
| |
| − | point there has rarely appeared to be a sufficient reason to keep their
| |
| − | concepts separately in mind. The boolean domain %B% comes with at least
| |
| − | two operations, though often under different names and always included
| |
| − | in a number of others, that are analogous to the field operations of the
| |
| − | binary domain !B!, and operations that are isomorphic to the rest of the
| |
| − | boolean operations in %B% can always be built on the binary basis of !B!.
| |
| | | | |
| − | As sets of the same cardinality, the domains !B! and %B%, along with
| + | Fiber of %1% under f_Q = (f_Q)^(-1)(%1%) = {x in X : f_Q (x) = %1%} = Q. |
| − | all of the structures that can be built on them, are isomorphic at a
| |
| − | high enough level of abstraction. But the main reason for preserving
| |
| − | their distinction in the present context appears to be more a matter
| |
| − | of natural language grammar than an issue of logical or mathematical
| |
| − | substance, namely, just so that the signs "%0%" and "%1%" can appear
| |
| − | with a semblance of syntactic legitimacy in linguistic contexts that
| |
| − | call for grammatical sentences to represent the classes of sentences
| |
| − | that are "always false" and "always true", respectively. The signs
| |
| − | "0" and "1", that are customarily read as nouns but not as sentences,
| |
| − | fail to be suitable for this purpose. Whether these scruples, that
| |
| − | are needed to conform to a natural language context, are ultimately
| |
| − | important or not, is a thing that I just do not know at this point.
| |
| | | | |
| − | The "negation" of x, for x in %B%, written as "(x)"
| + | In this specifically boolean setting, as in the more generally logical |
| − | and read as "not x", is the boolean value (x) in %B%
| + | context, where "truth" under any name is especially valued, it is worth |
| − | that is %1% when x is %0%, and %0% when x is %1%.
| + | devoting a specialized notation to the "fiber of truth" in a proposition. |
| | + | For this purpose, I introduce the use of "fiber bars" or "ground signs", |
| | + | written as "[| ... |]" around a sentence, in other words, around the |
| | + | sign of a proposition, and whose application is defined as follows: |
| | | | |
| − | Thus, negation is a monadic operation on boolean
| + | Given f : X -> %B%, define [|f|] c X as follows: |
| − | values, a function of the form (_) : %B% -> %B%.
| |
| | | | |
| − | It is convenient to transport the product and the sum operations of !B!
| + | [| f |] = f^(-1)(%1%) = {x in X : f(x) = %1%}. |
| − | into the logical setting of %B%, where they can be symbolized by signs
| |
| − | of the same character, doubly underlined as necessary to avoid confusion.
| |
| − | This yields the following definitions of a "product" and a "sum" in %B%
| |
| − | and leads to the following forms of multiplication and addition tables.
| |
| | | | |
| − | The "product" of x and y, for x, y in %B%, is given by Table 8. | + | The definition of a fiber, in either the general or the boolean case, |
| | + | is a purely nominal convenience for referring to the antecedent subset, |
| | + | the inverse image under a function, or the pre-image of a functional value. |
| | + | The definition of the fiber operator on propositions, signified by framing |
| | + | the signs of propositions with fiber bars or ground signs, remains a purely |
| | + | notational device, and yet the use of the fiber concept in a logical context |
| | + | raises a number of problematic issues. By way of example, consider the fact |
| | + | that it is legitimate to rewrite the above definition in the following form: |
| | | | |
| − | Table 8. Boolean Product
| + | Given f : X -> %B%, define [|f|] c X as follows: |
| − | o---------o---------o---------o
| |
| − | | %*% % %0% | %1% |
| |
| − | o=========o=========o=========o
| |
| − | | %0% % %0% | %0% |
| |
| − | o---------o---------o---------o
| |
| − | | %1% % %0% | %1% |
| |
| − | o---------o---------o---------o
| |
| | | | |
| − | Viewed as a function on logical values, %*% : %B% x %B% -> %B%, the
| + | [| f |] = f^(-1)(%1%) = {x in X : f(x)}. |
| − | product corresponds to the logical operation that is commonly called
| |
| − | "conjunction" and that is otherwise expressed as "x and y". In accord
| |
| − | with common practice, the multiplication sign "*", doubly underlined or
| |
| − | otherwise, is frequently omitted from written expressions of the product.
| |
| | | | |
| − | The "sum" of x and y, for x, y in %B%, is given by Table 9. | + | The set-builder frame "{x in X : ... }" requires a grammatical sentence or |
| | + | a sentential clause to fill in the blank, as with the sentence "f(x) = %1%" |
| | + | that serves to fill the frame in the initial definition of a logical fiber. |
| | + | And what is a sentence but the expression of a proposition, in other words, |
| | + | the name of an indicator function? As it happens, the sign "f(x)" and the |
| | + | sentence "f(x) = %1%" represent the very same value to this context, for |
| | + | all x in X. That is to say, the two expressions will appear to be equal |
| | + | in their truth or falsity to any reasonable interpreter of sentences in |
| | + | this context, and so either one of them can be tendered for the other, |
| | + | in effect, exchanged for the other, within this context. |
| | | | |
| − | Table 9. Boolean Sum
| + | Given f : X -> %B%, the sign "f(x)" manifestly names the value f(x). |
| − | o---------o---------o---------o
| + | The value f(x) can in turn be interpreted in many different lights. |
| − | | %+% % %0% | %1% |
| + | Just to enumerate a few of them, the value f(x) can be taken as: |
| − | o=========o=========o=========o
| |
| − | | %0% % %0% | %1% |
| |
| − | o---------o---------o---------o
| |
| − | | %1% % %1% | %0% |
| |
| − | o---------o---------o---------o
| |
| | | | |
| − | Viewed as a function on logical values, %+% : %B% x %B% -> %B%,
| + | 1. The value that the proposition f has at the point x, |
| − | the sum corresponds to the logical operation that is generally | + | in other words, the value that f bears at the point x |
| − | called "exclusive disjunction" and that is otherwise expressed
| + | where f is being evaluated, the value that f takes on |
| − | as "x or y, but not both". Depending on the context, a couple
| + | with respect to the argument or the object x that the |
| − | of other signs and readings that can invoke this operation are:
| + | whole proposition f is taken to be about. |
| | | | |
| − | 1. "x =/= y", read "x is not equal to y", or "exactly one of x and y". | + | 2. The value that the proposition f not only takes up at |
| | + | the point x, but that it carries, conveys, transfers, |
| | + | or transports into the setting "{x in X : ... }", or |
| | + | into any other context of discourse where f is meant |
| | + | to be evaluated. |
| | | | |
| − | 2. "x <=/=> y", read "x is not equivalent to y", or "x opposes y". | + | 3. The value that the sign "f(x)" has in the context where it is |
| | + | placed, that it stands for in the context where it stands, and |
| | + | that it continues to stand for in this context just so long as |
| | + | the same proposition f and the same object x are borne in mind. |
| | | | |
| − | For sentences, the signs of equality ("=") and inequality ("=/=")
| + | 4. The value that the sign "f(x)" represents to its complete |
| − | are reserved to signify the syntactic identity and the syntactic
| + | interpretive context as being its own logical interpretant, |
| − | non-identity, respectively, of the literal strings of characters
| + | in other words, the value that it signifies as its canonical |
| − | that make up the sentences, while the signs of equivalence ("<=>") | + | connotation to any interpreter who is cognizant of the context |
| − | and inequivalence ("<=/=>") refer to the logical values, if any,
| + | in which the sign "f(x)" appears. |
| − | that these strings may conceivably bear, and thus they serve to
| |
| − | signify the equality or the inequality, respectively, of their
| |
| − | conceivable boolean values. For the logical values themselves,
| |
| − | the two pairs of symbols collapse in their meanings to a single
| |
| − | pair, signifying a single form of coincidence or a single form
| |
| − | of distinction, respectively, between the boolean values of the
| |
| − | entities in question.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | The sentence "f(x) = %1%" indirectly names what the sign "f(x)" |
| | + | more directly names, that is, the value f(x). In other words, |
| | + | the sentence "f(x) = %1%" has the same value to its interpretive |
| | + | context that the sign "f(x)" imparts to any comparable context, |
| | + | each by way of its respective evaluation for the same x in X. |
| | | | |
| − | IDS. Note 122
| + | What is the relation among connoting, denoting, and "evaluing", where |
| − | | + | the last term is coined to describe all the ways of bearing, conveying, |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | developing, or evolving a value in, to, or into an interpretive context? |
| − | | + | In other words, when a sign is evaluated to a particular value, one can |
| − | 1.3.10.3. Propositions and Sentences (cont.)
| + | say that the sign "evalues" that value, using the verb in a way that is |
| | + | categorically analogous or grammatically conjugate to the times when one |
| | + | says that a sign "connotes" an idea or that a sign "denotes" an object. |
| | + | This does little more than provide the discussion with a "weasel word", |
| | + | a term that is designed to avoid the main issue, to put off deciding the |
| | + | exact relation between formal signs and formal values, and ultimately to |
| | + | finesse the question about the nature of formal values, whether they are |
| | + | more akin to conceptual signs and figurative ideas or to the kinds of |
| | + | literal objects and platonic ideas that are independent of the mind. |
| | + | |
| | + | These questions are confounded by the presence of certain peculiarities in |
| | + | formal discussions, especially by the fact that an equivalence class of signs |
| | + | is tantamount to a formal object. This has the effect of allowing an abstract |
| | + | connotation to work as a formal denotation. In other words, if the purpose of |
| | + | a sign is merely to lead its interpreter up to a sign in an equivalence class |
| | + | of signs, then it follows that this equivalence class is the object of the |
| | + | sign, that connotation can achieve denotation, at least, to some degree, |
| | + | and that the interpretant domain collapses with the object domain, |
| | + | at least, in some respect, all things being relative to the |
| | + | sign relation that embeds the discussion. |
| | | | |
| − | In logical studies, one tends to be interested in all of the
| + | Introducing the realm of "values" is a stopgap measure that temporarily |
| − | operations or all of the functions of a given type, at least,
| + | permits the discussion to avoid certain singularities in the embedding |
| − | to the extent that their totalities and their individualities | + | sign relation, and allowing the process of "evaluation" as a compromise |
| − | can be comprehended, and not just the specialized collections
| + | mode of signification between connotation and denotation only manages to |
| − | that define particular algebraic structures.
| + | steer around a topic that eventually has to be mapped in full, but these |
| | + | strategies do allow the discussion to proceed a little further without |
| | + | having to answer questions that are too difficult to be settled fully |
| | + | or even tackled directly at this point. As far as the relations among |
| | + | connoting, denoting, and evaluing are concerned, it is possible that |
| | + | all of these constitute independent dimensions of significance that |
| | + | a sign might be able to enjoy, but since the notion of connotation |
| | + | is already generic enough to contain multitudes of subspecies, I am |
| | + | going to subsume, on a tentative basis, all of the conceivable modes |
| | + | of "evaluing" within the broader concept of connotation. |
| | | | |
| − | Although the rest of the conceivably possible dyadic operations
| + | With this degree of flexibility in mind, one can say that the sentence |
| − | on boolean values, in other words, the remainder of the sixteen
| + | "f(x) = %1%" latently connotes what the sign "f(x)" patently connotes. |
| − | functions f : %B% x %B% -> %B%, could be presented in the same
| + | Taken in abstraction, both syntactic entities fall into an equivalence |
| − | way as the multiplication and addition tables, it is better to
| + | class of signs that constitutes an abstract object, a thing of value |
| − | look for a more efficient style of representation, one that is
| + | that is identified by the sign "f(x)", and thus an object that might |
| − | able to express all of the boolean functions on the same number
| + | as well be identified with the value f(x). |
| − | of variables on a roughly equal basis, and with a bit of luck,
| |
| − | affords us with a calculus for computing with these functions.
| |
| | | | |
| − | The utility of a suitable calculus would involve, among other things: | + | The upshot of this whole discussion of evaluation is that it allows one to |
| | + | rewrite the definitions of indicator functions and their fibers as follows: |
| | | | |
| − | 1. Finding the values of given functions for given arguments.
| + | The "indicator function" or the "characteristic function" of a set Q c X, |
| | + | written "f_Q", is the map from X to the boolean domain %B% = {%0%, %1%} |
| | + | that is defined in the following ways: |
| | | | |
| − | 2. Inverting boolean functions, that is, "finding the fibers" | + | 1. Considered in its extensional form, f_Q is the subset of X x %B% |
| − | of boolean functions, or solving logical equations that | + | that is given by the following formula: |
| − | are expressed in terms of boolean functions.
| |
| | | | |
| − | 3. Facilitating the recognition of invariant forms that
| + | f_Q = {<x, b> in X x %B% : b <=> x in Q}. |
| − | take boolean functions as their functional components.
| |
| | | | |
| − | The whole point of formal logic, the reason for doing logic formally and
| + | 2. Considered in its functional form, f_Q is the map from X to %B% |
| − | the measure that determines how far it is possible to reason abstractly,
| + | that is given by the following condition: |
| − | is to discover functions that do not vary as much as their variables do, | + | |
| − | in other words, to identify forms of logical functions that, though they
| + | f_Q (x) <=> x in Q. |
| − | express a dependence on the values of their constituent arguments, do not
| |
| − | vary as much as possible, but approach the way of being a function that
| |
| − | constant functions enjoy. Thus, the recognition of a logical law amounts
| |
| − | to identifying a logical function, that, though it ostensibly depends on
| |
| − | the values of its putative arguments, is not as variable in its values as
| |
| − | the values of its variables are allowed to be.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | The "fibers" of truth and falsity under a proposition f : X -> %B% |
| | + | are subsets of X that are variously described as follows: |
| | | | |
| − | IDS. Note 123
| + | 1. The fiber of %1% under f = [| f |] = f^(-1)(%1%) |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | = {x in X : f(x) = %1%} |
| | | | |
| − | 1.3.10.3. Propositions and Sentences (cont.)
| + | = {x in X : f(x) }. |
| | | | |
| − | The "indicator function" or the "characteristic function" of a set Q c X, | + | 2. The fiber of %0% under f = ~[| f |] = f^(-1)(%0%) |
| − | written "f_Q", is the map from X to the boolean domain %B% = {%0%, %1%}
| |
| − | that is defined in the following ways:
| |
| | | | |
| − | 1. Considered in extensional form, f_Q is the subset of X x %B%
| + | = {x in X : f(x) = %0%} |
| − | that is given by the following formula:
| |
| | | | |
| − | f_Q = {<x, b> in X x %B% : b = %1% <=> x in Q}.
| + | = {x in X : (f(x)) }. |
| | | | |
| − | 2. Considered in functional form, f_Q is the map from X to %B%
| + | Perhaps this looks like a lot of work for the sake of what seems to be |
| − | that is given by the following condition:
| + | such a trivial form of syntactic transformation, but it is an important |
| | + | step in loosening up the syntactic privileges that are held by the sign |
| | + | of logical equivalence "<=>", as written between logical sentences, and |
| | + | by the sign of equality "=", as written between their logical values, or |
| | + | else between propositions and their boolean values. Doing this removes |
| | + | a longstanding but wholly unnecessary conceptual confound between the |
| | + | idea of an "assertion" and notion of an "equation", and it allows one |
| | + | to treat logical equality on a par with the other logical operations. |
| | | | |
| − | f_Q (x) = %1% <=> x in Q.
| + | As a purely informal aid to interpretation, I frequently use the letters |
| | + | "p", "q" to denote propositions. This can serve to tip off the reader |
| | + | that a function is intended as the indicator function of a set, and |
| | + | it saves us the trouble of declaring the type f : X -> %B% each |
| | + | time that a function is introduced as a proposition. |
| | | | |
| − | A "proposition about things in the universe", for short, a "proposition",
| + | Another convention of use in this context is to let boldface letters |
| − | is the same thing as an indicator function, that is, a function of the | + | stand for k-tuples, lists, or sequences of objects. Typically, the |
| − | form f : X -> %B%. The convenience of this seemingly redundant usage
| + | elements of the k-tuple, list, or sequence are all of one type, and |
| − | is that it allows one to refer to an indicator function without having | + | typically the boldface letter is of the same basic character as the |
| − | to specify right away, as a part of its designated subscript, exactly
| + | indexed or subscripted letters that are used denote the components |
| − | what set it indicates, even though a proposition always indicates some
| + | of the k-tuple, list, or sequence. When the dimension of elements |
| − | subset of its designated universe, and even though one will eventually,
| + | and functions is clear from the context, we may elect to drop the |
| − | most likely, want to know exactly what subset that is.
| + | bolding of characters that name k-tuples, lists, and sequences. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | For example: |
| | | | |
| − | IDS. Note 124
| + | 1. If x_1, ..., x_k in X, then #x# = <x_1, ..., x_k> in X' = X^k. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | 2. If x_1, ..., x_k : X, then #x# = <x_1, ..., x_k> : X' = X^k. |
| | | | |
| − | 1.3.10.3. Propositions and Sentences (cont.)
| + | 3. If f_1, ..., f_k : X -> Y, then #f# = <f_1, ..., f_k> : (X -> Y)^k. |
| | | | |
| − | According to the stated understandings, a proposition is a function that
| + | There is usually felt to be a slight but significant distinction between |
| − | indicates a set, in the sense that a function associates values with the
| + | the "membership statement" that uses the sign "in" as in Example (1) and |
| − | elements of a domain, some of which values can be interpreted to mark out
| + | the "type statement" that uses the sign ":" as in examples (2) and (3). |
| − | for special consideration a subset of that domain. The way in which an
| + | The difference that appears to be perceived in categorical statements, |
| − | indicator function is interpreted to "indicate" a set can be expressed
| + | when those of the form "x in X" and those of the form "x : X" are set |
| − | in terms of the following concepts. | + | in side by side comparisons with each other, is that a multitude of |
| | + | objects can be said to have the same type without having to posit |
| | + | the existence of a set to which they all belong. Without trying |
| | + | to decide whether I share this feeling or even fully understand |
| | + | the distinction in question, I can only try to maintain a style |
| | + | of notation that respects it to some degree. It is conceivable |
| | + | that the question of belonging to a set is rightly sensed to be |
| | + | the more serious matter, one that has to do with the reality of |
| | + | an object and the substance of a predicate, than the question of |
| | + | falling under a type, that may have more to do with the way that |
| | + | a sign is interpreted and the way that information about an object |
| | + | is organized. When it comes to the kinds of hypothetical statements |
| | + | that appear in these Examples, those of the form "x in X => #x# in X'" |
| | + | and "x : X => #x# : X'", these are usually read as implying some order |
| | + | of synthetic construction, one whose contingent consequences involve the |
| | + | constitution of a new space to contain the elements being compounded and |
| | + | the recognition of a new type to characterize the elements being moulded, |
| | + | respectively. In these applications, the statement about types is again |
| | + | taken to be less presumptive than the corresponding statement about sets, |
| | + | since the apodosis is intended to do nothing more than to abbreviate and |
| | + | to summarize what is already stated in the protasis. |
| | | | |
| − | The "fiber" of a codomain element y in Y under a function f : X -> Y is the
| + | A "boolean connection" of degree k, also known as a "boolean function" |
| − | subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X.
| + | on k variables, is a map of the form F : %B%^k -> %B%. In other words, |
| | + | a boolean connection of degree k is a proposition about things in the |
| | + | universe X = %B%^k. |
| | | | |
| − | In other language that is often used, the fiber of y under f is
| + | An "imagination" of degree k on X is a k-tuple of propositions about things |
| − | called the "antecedent set", the "inverse image", the "level set",
| + | in the universe X. By way of displaying the various kinds of notation that |
| − | or the "pre-image" of y under f. All of these equivalent concepts
| + | are used to express this idea, the imagination #f# = <f_1, ..., f_k> is given |
| − | can be defined as follows: | + | as a sequence of indicator functions f_j : X -> %B%, for j = 1 to k. All of |
| | + | these features of the typical imagination #f# can be summed up in either one |
| | + | of two ways: either in the form of a membership statement, to the effect that |
| | + | #f# is in (X -> %B%)^k, or in the form of a type statement, to the effect that |
| | + | #f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise than |
| | + | the former. |
| | | | |
| − | Fiber of y under f = f^(-1)(y) = {x in X : f(x) = y}.
| + | The "play of images" that is determined by #f# and x, more specifically, |
| | + | the play of the imagination #f# = <f_1, ..., f_k> that has to with the |
| | + | element x in X, is the k-tuple #b# = <b_1, ..., b_k> of values in %B% |
| | + | that satisfies the equations b_j = f_j (x), for all j = 1 to k. |
| | | | |
| − | In the special case where f is the indicator function f_Q of the set Q c X,
| + | A "projection" of %B%^k, typically denoted by "p_j" or "pr_j", |
| − | the fiber of %1% under f_Q is just the set Q back again: | + | is one of the maps p_j : %B%^k -> %B%, for j = 1 to k, that is |
| | + | defined as follows: |
| | | | |
| − | Fiber of %1% under f_Q = (f_Q)^(-1)(%1%) = {x in X : f_Q (x) = %1%} = Q. | + | For all #b# = <b_1, ..., b_k> in %B%^k we have: |
| | | | |
| − | In this specifically boolean setting, as in the more generally logical
| + | p_j (#b#) = p_j (<b_1, ..., b_k>) = b_j in %B%. |
| − | context, where "truth" under any name is especially valued, it is worth
| |
| − | devoting a specialized notation to the "fiber of truth" in a proposition.
| |
| − | For this purpose, I introduce the use of "fiber bars" or "ground signs",
| |
| − | written as "[| ... |]" around a sentence, in other words, around the
| |
| − | sign of a proposition, and whose application is defined as follows:
| |
| | | | |
| − | Given f : X -> %B%, define [|f|] c X as follows:
| + | The "projective imagination" of %B%^k is the imagination <p_1, ..., p_k>. |
| | | | |
| − | [| f |] = f^(-1)(%1%) = {x in X : f(x) = %1%}.
| + | A "sentence about things in the universe", for short, a "sentence", |
| | + | is a sign that denotes a proposition. In other words, a sentence is |
| | + | any sign that denotes an indicator function, any sign whose object is |
| | + | a function of the form f : X -> %B%. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | To emphasize the empirical contingency of this definition, one can say |
| | + | that a sentence is any sign that is interpreted as naming a proposition, |
| | + | any sign that is taken to denote an indicator function, or any sign whose |
| | + | object happens to be a function of the form f : X -> %B%. |
| | | | |
| − | IDS. Note 125
| + | An "expression" is a type of sign, for instance, a term or a sentence, |
| | + | that has a value. In forming this conception of an expression, I am |
| | + | deliberately leaving a number of options open, for example, whether |
| | + | the expression amounts to a term or to a sentence and whether it |
| | + | ought to be accounted as denoting a value or as connoting a value. |
| | + | Perhaps the expression has different values under different lights, |
| | + | and perhaps it relates to them differently in different respects. |
| | + | In the end, what one calls an expression matters less than where |
| | + | its value lies. Of course, no matter whether one chooses to call |
| | + | an expression a "term" or a "sentence", if the value is an element |
| | + | of %B%, then the expression affords the option of being treated as |
| | + | a sentence, meaning that it is subject to assertion and composition |
| | + | in the same way that any sentence is, having its value figure into |
| | + | the values of larger expressions through the linkages of sentential |
| | + | connectives, and affording us the consideration of what things in |
| | + | what universe the corresponding proposition happens to indicate. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | Expressions with this degree of flexibility in the types under |
| | + | which they can be interpreted are difficult to translate from |
| | + | their formal settings into more natural contexts. Indeed, |
| | + | the whole issue can be difficult to talk about, or even |
| | + | to think about, since the grammatical categories of |
| | + | sentential clauses and noun phrases are rarely so |
| | + | fluid in natural language settings as they can |
| | + | be rendered in artificially formal arenas. |
| | | | |
| − | 1.3.10.3. Propositions and Sentences (cont.)
| + | To finesse the issue of whether an expression denotes or connotes its value, |
| | + | or else to create a general term that covers what both possibilities have |
| | + | in common, one can say that an expression "evalues" its value. |
| | | | |
| − | The definition of a fiber, in either the general or the boolean case,
| + | An "assertion" is just a sentence that is being used in a certain way, |
| − | is a purely nominal convenience for referring to the antecedent subset, | + | namely, to indicate the indication of the indicator function that the |
| − | the inverse image under a function, or the pre-image of a functional value.
| + | sentence is usually used to denote. In other words, an assertion is |
| − | The definition of the fiber operator on propositions, signified by framing
| + | a sentence that is being converted to a certain use or that is being |
| − | the signs of propositions with fiber bars or ground signs, remains a purely
| + | interpreted in a certain role, and one whose immediate denotation is |
| − | notational device, and yet the use of the fiber concept in a logical context
| + | being pursued to its substantive indication, specifically, the fiber |
| − | raises a number of problematic issues. By way of example, consider the fact
| + | of truth of the proposition that the sentence potentially denotes. |
| − | that it is legitimate to rewrite the above definition in the following form: | + | Thus, an assertion is a sentence that is held to denote the set of |
| | + | things in the universe for which the sentence is held to be true. |
| | | | |
| − | Given f : X -> %B%, define [|f|] c X as follows:
| + | Taken in a context of communication, an assertion is basically a request |
| | + | that the interpreter consider the things for which the sentence is true, |
| | + | in other words, to find the fiber of truth in the associated proposition, |
| | + | or to invert the indicator function that is denoted by the sentence with |
| | + | respect to its possible value of truth. |
| | | | |
| − | [| f |] = f^(-1)(%1%) = {x in X : f(x)}.
| + | A "denial" of a sentence z is an assertion of its negation -(z)-. |
| | + | The denial acts as a request to think about the things for which the |
| | + | sentence is false, in other words, to find the fiber of falsity in the |
| | + | indicted proposition, or to invert the indicator function that is being |
| | + | denoted by the sentence with respect to its possible value of falsity. |
| | | | |
| − | The set-builder frame "{x in X : ... }" requires a grammatical sentence or
| + | According to this manner of definition, any sign that happens to denote |
| − | a sentential clause to fill in the blank, as with the sentence "f(x) = %1%" | + | a proposition, any sign that is taken as denoting an indicator function, |
| − | that serves to fill the frame in the initial definition of a logical fiber. | + | by that very fact alone successfully qualifies as a sentence. That is, |
| − | And what is a sentence but the expression of a proposition, in other words,
| + | a sentence is any sign that actually succeeds in denoting a proposition, |
| − | the name of an indicator function? As it happens, the sign "f(x)" and the
| + | any sign that one way or another brings to mind, as its actual object, |
| − | sentence "f(x) = %1%" represent the very same value to this context, for
| + | a function of the form f : X -> %B%. |
| − | all x in X. That is to say, the two expressions will appear to be equal
| + | |
| − | in their truth or falsity to any reasonable interpreter of sentences in | + | There are many features of this definition that need to be understood. |
| − | this context, and so either one of them can be tendered for the other,
| + | Indeed, there are problems involved in this whole style of definition |
| − | in effect, exchanged for the other, within this context.
| + | that need to be discussed, and doing this requires a slight excursion. |
| | + | </pre> |
| | | | |
| − | Given f : X -> %B%, the sign "f(x)" manifestly names the value f(x).
| + | =====1.3.10.4. Empirical Types and Rational Types===== |
| − | The value f(x) can in turn be interpreted in many different lights.
| |
| − | Just to enumerate a few of them, the value f(x) can be taken as:
| |
| | | | |
| − | 1. The value that the proposition f has at the point x,
| + | <pre> |
| − | in other words, the value that f bears at the point x
| + | In this Subsection, I want to examine the style of definition that I used |
| − | where f is being evaluated, the value that f takes on
| + | to define a sentence as a type of sign, to adapt its application to other |
| − | with respect to the argument or the object x that the
| + | problems of defining types, and to draw a lesson of general significance. |
| − | whole proposition f is taken to be about.
| |
| | | | |
| − | 2. The value that the proposition f not only takes up at
| + | Notice that I am defining a sentence in terms of what it denotes, and not |
| − | the point x, but that it carries, conveys, transfers,
| + | in terms of its structure as a sign. In this way of reckoning, a sign is |
| − | or transports into the setting "{x in X : ... }", or
| + | not a sentence on account of any property that it has in itself, but only |
| − | into any other context of discourse where f is meant
| + | due to the sign relation that actually works to interpret it. This makes |
| − | to be evaluated.
| + | the property of being a sentence a question of actualities and contingent |
| | + | relations, not merely a question of potentialities and absolute categories. |
| | + | This does nothing to alter the level of interest that one is bound to have |
| | + | in the structures of signs, it merely shifts the axis of the question from |
| | + | the logical plane of definition to the pragmatic plane of effective action. |
| | + | As a practical matter, of course, some signs are better for a given purpose |
| | + | than others, more conducive to a particular result than others, and turn out |
| | + | to be more effective in achieving an assigned objective than others, and the |
| | + | reasons for this are at least partly explained by the relationships that can |
| | + | be found to exist among a sign's structure, its object, and the sign relation |
| | + | that fits the sign and its object to each other. |
| | | | |
| − | 3. The value that the sign "f(x)" has in the context where it is
| + | Notice the general character of this development. I start by |
| − | placed, that it stands for in the context where it stands, and
| + | defining a type of sign according to the type of object that it |
| − | that it continues to stand for in this context just so long as
| + | happens to denote, ignoring at first the structural potential that |
| − | the same proposition f and the same object x are borne in mind.
| + | it brings to the task. According to this mode of definition, a type |
| | + | of sign is singled out from other signs in terms of the type of object |
| | + | that it actually denotes and not according to the type of object that it |
| | + | is designed or destined to denote, nor in terms of the type of structure |
| | + | that it possesses in itself. This puts the empirical categories, the |
| | + | classes based on actualities, at odds with the rational categories, |
| | + | the classes based on intentionalities. In hopes that this much |
| | + | explanation is enough to rationalize the account of types that |
| | + | I am using, I break off the digression at this point and |
| | + | return to the main discussion. |
| | + | </pre> |
| | | | |
| − | 4. The value that the sign "f(x)" represents to its complete
| + | =====1.3.10.5. Articulate Sentences===== |
| − | interpretive context as being its own logical interpretant,
| |
| − | in other words, the value that it signifies as its canonical
| |
| − | connotation to any interpreter who is cognizant of the context
| |
| − | in which the sign "f(x)" appears.
| |
| | | | |
| − | The sentence "f(x) = %1%" indirectly names what the sign "f(x)"
| + | <pre> |
| − | more directly names, that is, the value f(x). In other words,
| + | A sentence is called "articulate" if: |
| − | the sentence "f(x) = %1%" has the same value to its interpretive
| |
| − | context that the sign "f(x)" imparts to any comparable context,
| |
| − | each by way of its respective evaluation for the same x in X.
| |
| | | | |
| − | What is the relation among connoting, denoting, and "evaluing", where
| + | 1. It has a significant form, a compound construction, |
| − | the last term is coined to describe all the ways of bearing, conveying,
| + | a multi-part constitution, a well-developed composition, |
| − | developing, or evolving a value in, to, or into an interpretive context?
| + | or a non-trivial structure as a sign. |
| − | In other words, when a sign is evaluated to a particular value, one can
| |
| − | say that the sign "evalues" that value, using the verb in a way that is
| |
| − | categorically analogous or grammatically conjugate to the times when one
| |
| − | says that a sign "connotes" an idea or that a sign "denotes" an object.
| |
| − | This does little more than provide the discussion with a "weasel word",
| |
| − | a term that is designed to avoid the main issue, to put off deciding the
| |
| − | exact relation between formal signs and formal values, and ultimately to
| |
| − | finesse the question about the nature of formal values, whether they are
| |
| − | more akin to conceptual signs and figurative ideas or to the kinds of
| |
| − | literal objects and platonic ideas that are independent of the mind.
| |
| | | | |
| − | These questions are confounded by the presence of certain peculiarities in
| + | 2. There is an informative relationship that exists |
| − | formal discussions, especially by the fact that an equivalence class of signs
| + | between its structure as a sign and the content |
| − | is tantamount to a formal object. This has the effect of allowing an abstract
| + | of the proposition that it happens to denote. |
| − | connotation to work as a formal denotation. In other words, if the purpose of
| |
| − | a sign is merely to lead its interpreter up to a sign in an equivalence class
| |
| − | of signs, then it follows that this equivalence class is the object of the
| |
| − | sign, that connotation can achieve denotation, at least, to some degree,
| |
| − | and that the interpretant domain collapses with the object domain,
| |
| − | at least, in some respect, all things being relative to the
| |
| − | sign relation that embeds the discussion.
| |
| | | | |
| − | Introducing the realm of "values" is a stopgap measure that temporarily
| + | A sentence of the articulate kind is typically given in the form of |
| − | permits the discussion to avoid certain singularities in the embedding
| + | a "description", an "expression", or a "formula", in other words, as |
| − | sign relation, and allowing the process of "evaluation" as a compromise
| + | an articulated sign or a well-structured element of a formal language. |
| − | mode of signification between connotation and denotation only manages to
| + | As a general rule, the category of sentences that one will be willing to |
| − | steer around a topic that eventually has to be mapped in full, but these
| + | contemplate is compiled from a particular selection of complex signs and |
| − | strategies do allow the discussion to proceed a little further without
| + | syntactic strings, those that are assembled from the basic building blocks |
| − | having to answer questions that are too difficult to be settled fully
| + | of a formal language and held in especial esteem for the roles that they |
| − | or even tackled directly at this point. As far as the relations among
| + | play within its grammar. Still, even if the typical sentence is a sign |
| − | connoting, denoting, and evaluing are concerned, it is possible that
| + | that is generated by a formal regimen, having its form, its meaning, |
| − | all of these constitute independent dimensions of significance that
| + | and its use governed by the principles of a comprehensive grammar, |
| − | a sign might be able to enjoy, but since the notion of connotation | + | the class of sentences that one has a mind to contemplate can also |
| − | is already generic enough to contain multitudes of subspecies, I am
| + | include among its number many other signs of an arbitrary nature. |
| − | going to subsume, on a tentative basis, all of the conceivable modes
| |
| − | of "evaluing" within the broader concept of connotation.
| |
| | | | |
| − | With this degree of flexibility in mind, one can say that the sentence
| + | Frequently this "formula" has a "variable" in it that "ranges over" the |
| − | "f(x) = %1%" latently connotes what the sign "f(x)" patently connotes. | + | universe X. A "variable" is an ambiguous or equivocal sign that can be |
| − | Taken in abstraction, both syntactic entities fall into an equivalence
| + | interpreted as denoting any element of the set that it "ranges over". |
| − | class of signs that constitutes an abstract object, a thing of value
| |
| − | that is identified by the sign "f(x)", and thus an object that might
| |
| − | as well be identified with the value f(x). | |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | If a sentence denotes a proposition f : X -> %B%, then the "value" of the |
| | + | sentence with regard to x in X is the value f(x) of the proposition at x, |
| | + | where "%0%" is interpreted as "false" and "%1%" is interpreted as "true". |
| | | | |
| − | IDS. Note 126
| + | Since the value of a sentence or a proposition depends on the universe of discourse |
| | + | to which it is "referred", and since it also depends on the element of the universe |
| | + | with regard to which it is evaluated, it is conventional to say that a sentence or |
| | + | a proposition "refers" to a universe of discourse and to its elements, though often |
| | + | in a variety of different senses. Furthermore, a proposition, acting in the guise |
| | + | of an indicator function, "refers" to the elements that it "indicates", namely, the |
| | + | elements on which it takes a positive value. In order to sort out the potential |
| | + | confusions that are capable of arising here, I need to examine how these various |
| | + | notions of reference are related to the notion of denotation that is used in the |
| | + | pragmatic theory of sign relations. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | One way to resolve the various and sundry senses of "reference" that arise |
| | + | in this setting is to make the following brands of distinctions among them: |
| | | | |
| − | 1.3.10.3. Propositions and Sentences (cont.) | + | 1. Let the reference of a sentence or a proposition to a universe of discourse, |
| | + | the one that it acquires by way of taking on any interpretation at all, be |
| | + | taken as its "general reference", the kind of reference that one can safely |
| | + | ignore as irrelevant, at least, so long as one stays immersed in only one |
| | + | context of discourse or only one moment of discussion. |
| | | | |
| − | The upshot of this whole discussion of evaluation is that it allows one to
| + | 2. Let the references that an indicator function f has to the elements |
| − | rewrite the definitions of indicator functions and their fibers as follows:
| + | on which it evaluates to %0% be called its "negative references". |
| | | | |
| − | The "indicator function" or the "characteristic function" of a set Q c X,
| + | 3. Let the references that an indicator function f has to the elements |
| − | written "f_Q", is the map from X to the boolean domain %B% = {%0%, %1%}
| + | on which it evaluates to %1% be called its "positive references" |
| − | that is defined in the following ways:
| + | or its "indications". |
| | | | |
| − | 1. Considered in its extensional form, f_Q is the subset of X x %B%
| + | Finally, unspecified references to the "references" of a sentence, |
| − | that is given by the following formula:
| + | a proposition, or an indicator function can all be taken by default |
| | + | as references to their specific, positive references. |
| | | | |
| − | f_Q = {<x, b> in X x %B% : b <=> x in Q}.
| + | The universe of discourse for a sentence, the set whose elements the |
| | + | sentence is interpreted to be about, is not a property of the sentence |
| | + | by itself, but of the sentence in the presence of its interpretation. |
| | + | Independently of how many explicit variables a sentence contains, its |
| | + | value can always be interpreted as depending on any number of implicit |
| | + | variables. For instance, even a sentence with no explicit variable, |
| | + | a constant expression like "%0%" or "%1%", can be taken to denote |
| | + | a constant proposition of the form c : X -> %B%. Whether or not it |
| | + | has an explicit variable, I always take a sentence as referring to |
| | + | a proposition, one whose values refer to elements of a universe X. |
| | | | |
| − | 2. Considered in its functional form, f_Q is the map from X to %B%
| + | Notice that the letters "p" and "q", interpreted as signs that denote |
| − | that is given by the following condition:
| + | the indicator functions p, q : X -> %B%, have the character of sentences |
| | + | in relation to propositions, at least, they have the same status in this |
| | + | abstract discussion as genuine sentences have in concrete applications. |
| | + | This illustrates the relation between sentences and propositions as |
| | + | a special case of the relation between signs and objects. |
| | | | |
| − | f_Q (x) <=> x in Q.
| + | To assist the reading of informal examples, I frequently use the letters |
| | + | "t", "u", "v", "z" to denote sentences. Thus, it is conceivable to have |
| | + | a situation where z = "q" and where q : X -> %B%. Altogether, this means |
| | + | that the sign "z" denotes the sentence z, that the sentence z is the same |
| | + | thing as the sentence "q", and that the sentence "q" denotes the proposition, |
| | + | characteristic function, or indicator function q : X -> %B%. In settings where |
| | + | it is necessary to keep track of a large number of sentences, I use subscripted |
| | + | letters like "e_1", ..., "e_n" to refer to the various expressions in question. |
| | | | |
| − | The "fibers" of truth and falsity under a proposition f : X -> %B%
| + | A "sentential connective" is a sign, a coordinated sequence of signs, |
| − | are subsets of X that are variously described as follows:
| + | a syntactic pattern of contextual arrangement, or any other syntactic |
| − | | + | device that can be used to connect a number of sentences together in |
| − | 1. The fiber of %1% under f = [| f |] = f^(-1)(%1%)
| + | order to form a single sentence. If k is the number of sentences that |
| | + | are thereby connected, then the connective is said to be of "order k". |
| | + | If the sentences acquire a logical relationship through this mechanism, |
| | + | and are not just strung together by this device, then the connective |
| | + | is called a "logical connective". If the value of the constructed |
| | + | sentence depends on the values of the component sentences in such |
| | + | a way that the value of the whole is a boolean function of the |
| | + | values of the parts, then the connective earns the title of |
| | + | a "propositional connective". |
| | + | </pre> |
| | | | |
| − | = {x in X : f(x) = %1%}
| + | =====1.3.10.6. Stretching Principles===== |
| | | | |
| − | = {x in X : f(x) }.
| + | <pre> |
| | + | There is a principle of constant use in this work that needs |
| | + | to be made explicit at this point. In order to give it a name, |
| | + | I will refer to it as the "stretching principle". Expressed in |
| | + | a variety of different ways, it can be taken to say any one of |
| | + | the following things: |
| | | | |
| − | 2. The fiber of %0% under f = ~[| f |] = f^(-1)(%0%) | + | 1. Any relation of values extends |
| | + | to a relation of what is valued. |
| | | | |
| − | = {x in X : f(x) = %0%}
| + | 2. Any statement about values says something |
| | + | about the things that are given these values. |
| | | | |
| − | = {x in X : (f(x)) }.
| + | 3. Any association among a range of values |
| | + | establishes an association among the |
| | + | domains of things that these values |
| | + | are the values of. |
| | | | |
| − | Perhaps this looks like a lot of work for the sake of what seems to be
| + | 4. Any connection between two values can be stretched |
| − | such a trivial form of syntactic transformation, but it is an important
| + | to create a connection, of analogous form, between |
| − | step in loosening up the syntactic privileges that are held by the sign
| + | the objects, persons, qualities, or relationships |
| − | of logical equivalence "<=>", as written between logical sentences, and | + | that are valued in these connections. |
| − | by the sign of equality "=", as written between their logical values, or
| |
| − | else between propositions and their boolean values. Doing this removes
| |
| − | a longstanding but wholly unnecessary conceptual confound between the
| |
| − | idea of an "assertion" and notion of an "equation", and it allows one
| |
| − | to treat logical equality on a par with the other logical operations.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | 5. For every operation on values, there is a corresponding operation |
| | + | on the actions, conducts, functions, procedures, or processes that |
| | + | lead to these values, as well as there being analogous operations |
| | + | on the objects that instigate all of these various proceedings. |
| | | | |
| − | IDS. Note 127
| + | Nothing about the application of the stretching principle guarantees that |
| | + | the analogues it generates will be as useful as the material it works on. |
| | + | It is another question entirely whether the links that are forged in this |
| | + | fashion are equal in their strength and apposite in their bearing to the |
| | + | tried and true utilities of the original ties, but in principle they are |
| | + | always there. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | The purpose of this exercise is to illuminate how a sentence, |
| | + | a sign constituted as a string of characters, can be enfused |
| | + | with a proposition, an object of no slight abstraction, in a |
| | + | way that can speak about an external universe of discourse X. |
| | | | |
| − | 1.3.10.3. Propositions and Sentences (cont.)
| + | To complete the general discussion of stretching principles, |
| | + | we will need to call back to mind the following definitions: |
| | | | |
| − | As a purely informal aid to interpretation, I frequently use the letters
| + | A "boolean connection" of degree k, also known as a "boolean function" |
| − | "p", "q" to denote propositions. This can serve to tip off the reader | + | on k variables, is a map of the form F : %B%^k -> %B%. In other words, |
| − | that a function is intended as the indicator function of a set, and
| + | a boolean connection of degree k is a proposition about things in the |
| − | it saves us the trouble of declaring the type f : X -> %B% each
| + | universe of discourse X = %B%^k. |
| − | time that a function is introduced as a proposition.
| |
| | | | |
| − | Another convention of use in this context is to let boldface letters
| + | An "imagination" of degree k on X is a k-tuple of propositions about things |
| − | stand for k-tuples, lists, or sequences of objects. Typically, the
| + | in the universe X. By way of displaying the various brands of notation that |
| − | elements of the k-tuple, list, or sequence are all of one type, and
| |
| − | typically the boldface letter is of the same basic character as the
| |
| − | indexed or subscripted letters that are used denote the components
| |
| − | of the k-tuple, list, or sequence. When the dimension of elements
| |
| − | and functions is clear from the context, we may elect to drop the
| |
| − | bolding of characters that name k-tuples, lists, and sequences.
| |
| − | | |
| − | For example:
| |
| − | | |
| − | 1. If x_1, ..., x_k in X, then #x# = <x_1, ..., x_k> in X' = X^k.
| |
| − | | |
| − | 2. If x_1, ..., x_k : X, then #x# = <x_1, ..., x_k> : X' = X^k.
| |
| − | | |
| − | 3. If f_1, ..., f_k : X -> Y, then #f# = <f_1, ..., f_k> : (X -> Y)^k.
| |
| − | | |
| − | There is usually felt to be a slight but significant distinction between
| |
| − | the "membership statement" that uses the sign "in" as in Example (1) and
| |
| − | the "type statement" that uses the sign ":" as in examples (2) and (3).
| |
| − | The difference that appears to be perceived in categorical statements,
| |
| − | when those of the form "x in X" and those of the form "x : X" are set
| |
| − | in side by side comparisons with each other, is that a multitude of
| |
| − | objects can be said to have the same type without having to posit
| |
| − | the existence of a set to which they all belong. Without trying
| |
| − | to decide whether I share this feeling or even fully understand
| |
| − | the distinction in question, I can only try to maintain a style
| |
| − | of notation that respects it to some degree. It is conceivable
| |
| − | that the question of belonging to a set is rightly sensed to be
| |
| − | the more serious matter, one that has to do with the reality of
| |
| − | an object and the substance of a predicate, than the question of
| |
| − | falling under a type, that may have more to do with the way that
| |
| − | a sign is interpreted and the way that information about an object
| |
| − | is organized. When it comes to the kinds of hypothetical statements
| |
| − | that appear in these Examples, those of the form "x in X => #x# in X'"
| |
| − | and "x : X => #x# : X'", these are usually read as implying some order
| |
| − | of synthetic construction, one whose contingent consequences involve the
| |
| − | constitution of a new space to contain the elements being compounded and
| |
| − | the recognition of a new type to characterize the elements being moulded,
| |
| − | respectively. In these applications, the statement about types is again
| |
| − | taken to be less presumptive than the corresponding statement about sets,
| |
| − | since the apodosis is intended to do nothing more than to abbreviate and
| |
| − | to summarize what is already stated in the protasis.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Note 128
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | 1.3.10.3. Propositions and Sentences (cont.)
| |
| − | | |
| − | A "boolean connection" of degree k, also known as a "boolean function"
| |
| − | on k variables, is a map of the form F : %B%^k -> %B%. In other words,
| |
| − | a boolean connection of degree k is a proposition about things in the
| |
| − | universe X = %B%^k.
| |
| − | | |
| − | An "imagination" of degree k on X is a k-tuple of propositions about things | |
| − | in the universe X. By way of displaying the various kinds of notation that | |
| | are used to express this idea, the imagination #f# = <f_1, ..., f_k> is given | | are used to express this idea, the imagination #f# = <f_1, ..., f_k> is given |
| | as a sequence of indicator functions f_j : X -> %B%, for j = 1 to k. All of | | as a sequence of indicator functions f_j : X -> %B%, for j = 1 to k. All of |
| Line 6,671: |
Line 6,862: |
| | of two ways: either in the form of a membership statement, to the effect that | | of two ways: either in the form of a membership statement, to the effect that |
| | #f# is in (X -> %B%)^k, or in the form of a type statement, to the effect that | | #f# is in (X -> %B%)^k, or in the form of a type statement, to the effect that |
| − | #f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise than | + | #f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise |
| − | the former. | + | than the former. |
| | | | |
| | The "play of images" that is determined by #f# and x, more specifically, | | The "play of images" that is determined by #f# and x, more specifically, |
| Line 6,683: |
Line 6,874: |
| | defined as follows: | | defined as follows: |
| | | | |
| − | For all #b# = <b_1, ..., b_k> in %B%^k we have: | + | If #b# = <b_1, ..., b_k> in %B%^k, |
| | | | |
| − | p_j (#b#) = p_j (<b_1, ..., b_k>) = b_j in %B%. | + | then p_j (#b#) = p_j (<b_1, ..., b_k>) = b_j in %B%. |
| | | | |
| | The "projective imagination" of %B%^k is the imagination <p_1, ..., p_k>. | | The "projective imagination" of %B%^k is the imagination <p_1, ..., p_k>. |
| | | | |
| − | A "sentence about things in the universe", for short, a "sentence",
| + | As an application of the stretching principle, a connection F : %B%^k -> %B% |
| − | is a sign that denotes a proposition. In other words, a sentence is
| + | can be understood to indicate a relation among boolean values, namely, the |
| − | any sign that denotes an indicator function, any sign whose object is
| + | k-ary relation L = F^(-1)(%1%) c %B%^k. If these k values are values of |
| − | a function of the form f : X -> %B%.
| + | things in a universe X, that is, if one imagines each value in a k-tuple |
| | + | of values to be the functional image that results from evaluating an |
| | + | element of X under one of its possible aspects of value, then one |
| | + | has in mind the k propositions f_j : X -> %B%, for j = 1 to k, |
| | + | in sum, one embodies the imagination #f# = <f_1, ..., f_k>. |
| | + | Together, the imagination #f# in (X -> %B%)^k and the |
| | + | connection F : %B%^k -> %B% stretch each other to |
| | + | cover the universe X, yielding a new proposition |
| | + | q : X -> %B%. |
| | | | |
| − | To emphasize the empirical contingency of this definition, one can say | + | To encapsulate the form of this general result, I define a scheme of composition |
| − | that a sentence is any sign that is interpreted as naming a proposition,
| + | that takes an imagination #f# = <f_1, ..., f_k> in (X -> %B%)^k and a boolean |
| − | any sign that is taken to denote an indicator function, or any sign whose
| + | connection F : %B%^k -> %B% and gives a proposition q : X -> %B%. Depending |
| − | object happens to be a function of the form f : X -> %B%.
| + | on the situation, specifically, according to whether many F and many #f#, |
| | + | a single F and many #f#, or many F and a single #f# are being considered, |
| | + | I refer to the resultant q under one of three descriptions, respectively: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | 1. In a general setting, where the connection F and the imagination #f# |
| | + | are both permitted to take up a variety of concrete possibilities, |
| | + | call q the "stretch of F and #f# from X to %B%", and write it in |
| | + | the style of a composition as "F $ #f#". This is meant to suggest |
| | + | that the symbol "$", here read as "stretch", denotes an operator |
| | + | of the form $ : (%B%^k -> %B%) x (X -> %B%)^k -> (X -> %B%). |
| | | | |
| − | IDS. Note 129
| + | 2. In a setting where the connection F is fixed but the imagination #f# |
| | + | is allowed to vary over a wide range of possibilities, call q the |
| | + | "stretch of F to #f# on X", and write it in the style "F^$ #f#", |
| | + | as if "F^$" denotes an operator F^$ : (X -> %B%)^k -> (X -> %B%) |
| | + | that is derived from F and applied to #f#, ultimately yielding |
| | + | a proposition F^$ #f# : X -> %B%. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | 3. In a setting where the imagination #f# is fixed but the connection F |
| | + | is allowed to range over a wide variety of possibilities, call q the |
| | + | "stretch of #f# by F to %B%", and write it in the fashion "#f#^$ F", |
| | + | as if "#f#^$" denotes an operator #f#^$ : (%B%^k -> %B%) -> (X -> %B%) |
| | + | that is derived from #f# and applied to F, ultimately yielding |
| | + | a proposition #f#^$ F : X -> %B%. |
| | | | |
| − | 1.3.10.3. Propositions and Sentences (concl.)
| + | Because the stretch notation is used only in settings |
| | + | where the imagination #f# : (X -> %B%)^k and the |
| | + | connection F : %B%^k -> %B% are distinguished |
| | + | by their types, it does not really matter |
| | + | whether one writes "F $ #f#" or "#f# $ F" |
| | + | for the initial form of composition. |
| | | | |
| − | An "expression" is a type of sign, for instance, a term or a sentence,
| + | Just as a sentence is a sign that denotes a proposition, |
| − | that has a value. In forming this conception of an expression, I am
| + | which thereby serves to indicate a set, a propositional |
| − | deliberately leaving a number of options open, for example, whether
| + | connective is a provision of syntax whose mediate effect |
| − | the expression amounts to a term or to a sentence and whether it
| + | is to denote an operation on propositions, which thereby |
| − | ought to be accounted as denoting a value or as connoting a value.
| + | manages to indicate the result of an operation on sets. |
| − | Perhaps the expression has different values under different lights,
| + | In order to see how these compound forms of indication |
| − | and perhaps it relates to them differently in different respects.
| + | can be defined, it is useful to go through the steps |
| − | In the end, what one calls an expression matters less than where | + | that are needed to construct them. In general terms, |
| − | its value lies. Of course, no matter whether one chooses to call
| + | the ingredients of the construction are as follows: |
| − | an expression a "term" or a "sentence", if the value is an element
| |
| − | of %B%, then the expression affords the option of being treated as
| |
| − | a sentence, meaning that it is subject to assertion and composition
| |
| − | in the same way that any sentence is, having its value figure into
| |
| − | the values of larger expressions through the linkages of sentential | |
| − | connectives, and affording us the consideration of what things in
| |
| − | what universe the corresponding proposition happens to indicate.
| |
| | | | |
| − | Expressions with this degree of flexibility in the types under
| + | 1. An imagination of degree k on X, in other words, a k-tuple |
| − | which they can be interpreted are difficult to translate from
| + | of propositions f_j : X -> %B%, for j = 1 to k, or an object |
| − | their formal settings into more natural contexts. Indeed,
| + | of the form #f# = <f_1, ..., f_k> : (X -> %B%)^k. |
| − | the whole issue can be difficult to talk about, or even | |
| − | to think about, since the grammatical categories of
| |
| − | sentential clauses and noun phrases are rarely so
| |
| − | fluid in natural language settings as they can
| |
| − | be rendered in artificially formal arenas.
| |
| | | | |
| − | To finesse the issue of whether an expression denotes or connotes its value,
| + | 2. A connection of degree k, in other words, a proposition |
| − | or else to create a general term that covers what both possibilities have | + | about things in %B%^k, or a boolean function of the form |
| − | in common, one can say that an expression "evalues" its value.
| + | F : %B%^k -> %B%. |
| | | | |
| − | An "assertion" is just a sentence that is being used in a certain way,
| + | From this 2-ply of material, it is required to construct a proposition |
| − | namely, to indicate the indication of the indicator function that the
| + | q : X -> %B% such that q(x) = F(f_1(x), ..., f_k(x)), for all x in X. |
| − | sentence is usually used to denote. In other words, an assertion is
| + | The desired construction can be developed in the following manner: |
| − | a sentence that is being converted to a certain use or that is being
| |
| − | interpreted in a certain role, and one whose immediate denotation is
| |
| − | being pursued to its substantive indication, specifically, the fiber
| |
| − | of truth of the proposition that the sentence potentially denotes.
| |
| − | Thus, an assertion is a sentence that is held to denote the set of
| |
| − | things in the universe for which the sentence is held to be true.
| |
| | | | |
| − | Taken in a context of communication, an assertion is basically a request
| + | The cartesian power %B%^k, as a cartesian product, is characterized |
| − | that the interpreter consider the things for which the sentence is true, | + | by the possession of a projective imagination #p# = <p_1, ..., p_k> |
| − | in other words, to find the fiber of truth in the associated proposition,
| + | of degree k on %B%^k, along with the property that any imagination |
| − | or to invert the indicator function that is denoted by the sentence with
| + | #f# = <f_1, ..., f_k> of degree k on an arbitrary set W determines |
| − | respect to its possible value of truth.
| + | a unique map !f! : W -> %B%^k, the play of whose projective images |
| | + | <p_1(!f!(w)), ..., p_k(!f!(w))> on the functional image !f!(w) |
| | + | matches the play of images <f_1(w), ..., f_k(w)> under #f#, |
| | + | term for term and at every element w in W. |
| | | | |
| − | A "denial" of a sentence z is an assertion of its negation -(z)-.
| + | Just to be on the safe side, I state this again in more standard terms. |
| − | The denial acts as a request to think about the things for which the | + | The cartesian power %B%^k, as a cartesian product, is characterized by |
| − | sentence is false, in other words, to find the fiber of falsity in the
| + | the possession of k projection maps p_j : %B%^k -> %B%, for j = 1 to k, |
| − | indicted proposition, or to invert the indicator function that is being
| + | along with the property that any k maps f_j : W -> %B%, from an arbitrary |
| − | denoted by the sentence with respect to its possible value of falsity.
| + | set W to %B%, determine a unique map !f! : W -> %B%^k satisfying the system |
| | + | of equations p_j(!f!(w)) = f_j(w), for all j = 1 to k, and for all w in W. |
| | | | |
| − | According to this manner of definition, any sign that happens to denote
| + | Now suppose that the arbitrary set W in this construction is just |
| − | a proposition, any sign that is taken as denoting an indicator function,
| + | the relevant universe X. Given that the function !f! : X -> %B%^k |
| − | by that very fact alone successfully qualifies as a sentence. That is, | + | is uniquely determined by the imagination #f# : (X -> %B%)^k, or what |
| − | a sentence is any sign that actually succeeds in denoting a proposition,
| + | is the same thing, by the k-tuple of propositions #f# = <f_1, ..., f_k>, |
| − | any sign that one way or another brings to mind, as its actual object,
| + | it is safe to identify !f! and #f# as being a single function, and this |
| − | a function of the form f : X -> %B%. | + | makes it convenient on many occasions to refer to the identified function |
| | + | by means of its explicitly descriptive name "<f_1, ..., f_k>". This facility |
| | + | of address is especially appropriate whenever a concrete term or a constructive |
| | + | precision is demanded by the context of discussion. |
| | + | </pre> |
| | | | |
| − | There are many features of this definition that need to be understood.
| + | =====1.3.10.7. Stretching Operations===== |
| − | Indeed, there are problems involved in this whole style of definition
| |
| − | that need to be discussed, and doing this requires a slight excursion.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | + | The preceding discussion of stretch operations is slightly more general |
| | + | than is called for in the present context, and so it is probably a good |
| | + | idea to draw out the particular implications that are needed right away. |
| | | | |
| − | IDS. Note 130
| + | If F : %B%^k -> %B% is a boolean function on k variables, then it is possible |
| | + | to define a mapping F^$ : (X -> %B%)^k -> (X -> %B%), in effect, an operation |
| | + | that takes k propositions into a single proposition, where F^$ satisfies the |
| | + | following conditions: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | F^$ (f_1, ..., f_k) : X -> %B% |
| | | | |
| − | 1.3.10.4. Empirical Types and Rational Types
| + | such that: |
| | | | |
| − | In this Subsection, I want to examine the style of definition that I used
| + | F^$ (f_1, ..., f_k)(x) = F(#f#(x)) |
| − | to define a sentence as a type of sign, to adapt its application to other
| |
| − | problems of defining types, and to draw a lesson of general significance.
| |
| | | | |
| − | Notice that I am defining a sentence in terms of what it denotes, and not
| + | = F((f_1, ..., f_k)(x)) |
| − | in terms of its structure as a sign. In this way of reckoning, a sign is
| |
| − | not a sentence on account of any property that it has in itself, but only
| |
| − | due to the sign relation that actually works to interpret it. This makes
| |
| − | the property of being a sentence a question of actualities and contingent
| |
| − | relations, not merely a question of potentialities and absolute categories.
| |
| − | This does nothing to alter the level of interest that one is bound to have
| |
| − | in the structures of signs, it merely shifts the axis of the question from
| |
| − | the logical plane of definition to the pragmatic plane of effective action.
| |
| − | As a practical matter, of course, some signs are better for a given purpose
| |
| − | than others, more conducive to a particular result than others, and turn out
| |
| − | to be more effective in achieving an assigned objective than others, and the
| |
| − | reasons for this are at least partly explained by the relationships that can
| |
| − | be found to exist among a sign's structure, its object, and the sign relation
| |
| − | that fits the sign and its object to each other.
| |
| | | | |
| − | Notice the general character of this development. I start by
| + | = F(f_1(x), ..., f_k(x)). |
| − | defining a type of sign according to the type of object that it
| |
| − | happens to denote, ignoring at first the structural potential that
| |
| − | it brings to the task. According to this mode of definition, a type
| |
| − | of sign is singled out from other signs in terms of the type of object
| |
| − | that it actually denotes and not according to the type of object that it
| |
| − | is designed or destined to denote, nor in terms of the type of structure
| |
| − | that it possesses in itself. This puts the empirical categories, the
| |
| − | classes based on actualities, at odds with the rational categories,
| |
| − | the classes based on intentionalities. In hopes that this much
| |
| − | explanation is enough to rationalize the account of types that
| |
| − | I am using, I break off the digression at this point and
| |
| − | return to the main discussion.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | Thus, F^$ is just the sort of entity that a propositional connective denotes, |
| | + | a particular way of connecting the propositions that are denoted by a number |
| | + | of sentences into a proposition that is denoted by a single sentence. |
| | | | |
| − | IDS. Note 131
| + | Now "f_X" is sign that denotes the proposition f_X, |
| | + | and it certainly seems like a sufficient sign for it. |
| | + | Why would we need to recognize any other signs of it? |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | If one takes a sentence as a type of sign that denotes a proposition and |
| | + | a proposition as a type of function whose values serve to indicate a set, |
| | + | then one needs a way to grasp the overall relation between the sentence |
| | + | and the set as taking place within a higher order sign relation. |
| | | | |
| − | 1.3.10.5. Articulate Sentences
| + | The various relationships of denotation and indication that exist |
| | + | among sets, propositions, sentences, and values in this situation |
| | + | are illustrated very roughly by the array of materials in Table 10. |
| | | | |
| − | A sentence is called "articulate" if:
| + | Table 10. Levels of Indication |
| − | | + | o-------------------o-------------------o-------------------o |
| − | 1. It has a significant form, a compound construction,
| + | | Object | Sign | Higher Order Sign | |
| − | a multi-part constitution, a well-developed composition,
| + | o-------------------o-------------------o-------------------o |
| − | or a non-trivial structure as a sign.
| + | | | | | |
| | + | | Set | Proposition | Sentence | |
| | + | | | | | |
| | + | | f^(-1)(b) | f | "f" | |
| | + | | | | | |
| | + | o-------------------o-------------------o-------------------o |
| | + | | | | | |
| | + | | Q | %1% | "%1%" | |
| | + | | | | | |
| | + | | X - Q | %0% | "%0%" | |
| | + | | | | | |
| | + | o-------------------o-------------------o-------------------o |
| | | | |
| − | 2. There is an informative relationship that exists
| + | Strictly speaking, propositions are too abstract to be signs, hence the |
| − | between its structure as a sign and the content
| + | contents of Table 10 have to be taken with the indicated grains of salt. |
| − | of the proposition that it happens to denote.
| + | Propositions, as indicator functions, are abstract mathematical objects, |
| | + | not any kinds of syntactic elements, thus propositions cannot literally |
| | + | constitute the orders of concrete signs that remain of ultimate interest |
| | + | in the pragmatic theory of signs, or in any theory of effective meaning. |
| | | | |
| − | A sentence of the articulate kind is typically given in the form of
| + | Therefore, it needs to be understood that a proposition f can be said |
| − | a "description", an "expression", or a "formula", in other words, as
| + | to "indicate" the set Q only insofar as the values of %1% and %0% that |
| − | an articulated sign or a well-structured element of a formal language.
| + | it assigns to the elements of the universe X are positive and negative |
| − | As a general rule, the category of sentences that one will be willing to
| + | indications, respectively, of the elements in Q, and thus indications |
| − | contemplate is compiled from a particular selection of complex signs and
| + | of the set Q and of its complement ~X = X - Q, respectively. It is |
| − | syntactic strings, those that are assembled from the basic building blocks
| + | these logical values, when rendered by a concrete implementation of |
| − | of a formal language and held in especial esteem for the roles that they | + | the indicator function f, that are the actual signs of the objects |
| − | play within its grammar. Still, even if the typical sentence is a sign
| + | inside the set Q and the objects outside the set Q, respectively. |
| − | that is generated by a formal regimen, having its form, its meaning,
| |
| − | and its use governed by the principles of a comprehensive grammar,
| |
| − | the class of sentences that one has a mind to contemplate can also | |
| − | include among its number many other signs of an arbitrary nature.
| |
| | | | |
| − | Frequently this "formula" has a "variable" in it that "ranges over" the
| + | In order to deal with the higher order sign relations |
| − | universe X. A "variable" is an ambiguous or equivocal sign that can be
| + | that are involved in the present setting, I introduce |
| − | interpreted as denoting any element of the set that it "ranges over".
| + | a couple of new notations: |
| | | | |
| − | If a sentence denotes a proposition f : X -> %B%, then the "value" of the
| + | 1. To mark the relation of denotation between a sentence z and the |
| − | sentence with regard to x in X is the value f(x) of the proposition at x,
| + | proposition that it denotes, the "spiny bracket" notation "-[z]-" |
| − | where "%0%" is interpreted as "false" and "%1%" is interpreted as "true".
| + | will be used for "the indicator function denoted by the sentence z". |
| | | | |
| − | Since the value of a sentence or a proposition depends on the universe of discourse
| + | 2. To mark the relation of denotation between a proposition q and |
| − | to which it is "referred", and since it also depends on the element of the universe
| + | the set that it indicates, the "spiny brace" notation "-{Q}-" |
| − | with regard to which it is evaluated, it is conventional to say that a sentence or
| + | will be used for "the indicator function of the set Q". |
| − | a proposition "refers" to a universe of discourse and to its elements, though often
| |
| − | in a variety of different senses. Furthermore, a proposition, acting in the guise
| |
| − | of an indicator function, "refers" to the elements that it "indicates", namely, the
| |
| − | elements on which it takes a positive value. In order to sort out the potential
| |
| − | confusions that are capable of arising here, I need to examine how these various
| |
| − | notions of reference are related to the notion of denotation that is used in the
| |
| − | pragmatic theory of sign relations.
| |
| | | | |
| − | One way to resolve the various and sundry senses of "reference" that arise
| + | Notice that the spiny bracket operator "-[ ]-" takes one "downstream", |
| − | in this setting is to make the following brands of distinctions among them:
| + | confluent with the direction of denotation, from a sign to its object, |
| | + | whereas the spiny brace operator "-{ }-" takes one "upstream", against |
| | + | the usual direction of denotation, and thus from an object to its sign. |
| | | | |
| − | 1. Let the reference of a sentence or a proposition to a universe of discourse,
| + | In order to make these notations useful in practice, it is necessary to note |
| − | the one that it acquires by way of taking on any interpretation at all, be
| + | a couple of their finer points, points that might otherwise seem too fine to |
| − | taken as its "general reference", the kind of reference that one can safely
| + | take much trouble over. For the sake their ultimate utility, nevertheless, |
| − | ignore as irrelevant, at least, so long as one stays immersed in only one
| + | I will describe their usage a bit more carefully as follows: |
| − | context of discourse or only one moment of discussion.
| |
| | | | |
| − | 2. Let the references that an indicator function f has to the elements | + | 1. Let "spiny brackets", like "-[ ]-", be placed around a name |
| − | on which it evaluates to %0% be called its "negative references". | + | of a sentence z, as in the expression "-[z]-", or else around |
| | + | a token appearance of the sentence itself, to serve as a name |
| | + | for the proposition that z denotes. |
| | | | |
| − | 3. Let the references that an indicator function f has to the elements | + | 2. Let "spiny braces", like "-{ }-", be placed around a name of |
| − | on which it evaluates to %1% be called its "positive references" | + | a set Q, as in the expression "-{Q}-", to serve as a name for |
| − | or its "indications". | + | the indicator function f_Q. |
| | | | |
| − | Finally, unspecified references to the "references" of a sentence,
| + | In passing, let us recall the use of the "fiber bars" |
| − | a proposition, or an indicator function can all be taken by default | + | or the "ground marker" "[| ... |]" as an alternate |
| − | as references to their specific, positive references. | + | notation for the fiber of truth in a proposition q, |
| | + | as follows: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | [| q |] = q^(-1)(%1%). |
| | | | |
| − | IDS. Note 132
| + | Table 11 illustrates the use of this notation, listing in each Column |
| | + | several different but equivalent ways of referring to the same entity. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | + | Table 11. Illustrations of Notation |
| | + | o-------------------o-------------------o-------------------o |
| | + | | Object | Sign | Higher Order Sign | |
| | + | o-------------------o-------------------o-------------------o |
| | + | | | | | |
| | + | | Set | Proposition | Sentence | |
| | + | | | | | |
| | + | | Q | q | z | |
| | + | | | | | |
| | + | | [| -[z]- |] | -[z]- | z | |
| | + | | | | | |
| | + | | [| q |] | q | "q" | |
| | + | | | | | |
| | + | | [| f_Q |] | f_Q | "f_Q" | |
| | + | | | | | |
| | + | | Q | -{Q}- | "-{Q}-" | |
| | + | | | | | |
| | + | o-------------------o-------------------o-------------------o |
| | | | |
| − | 1.3.10.5. Articulate Sentences (concl.)
| + | In effect, one can observe the following relations |
| | + | and formulas, all of a purely notational character: |
| | | | |
| − | The universe of discourse for a sentence, the set whose elements the
| + | 1. If the sentence z denotes the proposition q : X -> %B%, |
| − | sentence is interpreted to be about, is not a property of the sentence | |
| − | by itself, but of the sentence in the presence of its interpretation.
| |
| − | Independently of how many explicit variables a sentence contains, its
| |
| − | value can always be interpreted as depending on any number of implicit
| |
| − | variables. For instance, even a sentence with no explicit variable,
| |
| − | a constant expression like "%0%" or "%1%", can be taken to denote
| |
| − | a constant proposition of the form c : X -> %B%. Whether or not it
| |
| − | has an explicit variable, I always take a sentence as referring to
| |
| − | a proposition, one whose values refer to elements of a universe X.
| |
| | | | |
| − | Notice that the letters "p" and "q", interpreted as signs that denote
| + | then -[z]- = q. |
| − | the indicator functions p, q : X -> %B%, have the character of sentences
| |
| − | in relation to propositions, at least, they have the same status in this
| |
| − | abstract discussion as genuine sentences have in concrete applications.
| |
| − | This illustrates the relation between sentences and propositions as
| |
| − | a special case of the relation between signs and objects.
| |
| | | | |
| − | To assist the reading of informal examples, I frequently use the letters
| + | 2. If the sentence z denotes the proposition q : X -> %B%, |
| − | "t", "u", "v", "z" to denote sentences. Thus, it is conceivable to have
| |
| − | a situation where z = "q" and where q : X -> %B%. Altogether, this means
| |
| − | that the sign "z" denotes the sentence z, that the sentence z is the same
| |
| − | thing as the sentence "q", and that the sentence "q" denotes the proposition,
| |
| − | characteristic function, or indicator function q : X -> %B%. In settings where
| |
| − | it is necessary to keep track of a large number of sentences, I use subscripted
| |
| − | letters like "e_1", ..., "e_n" to refer to the various expressions in question.
| |
| | | | |
| − | A "sentential connective" is a sign, a coordinated sequence of signs,
| + | hence [|q|] = q^(-1)(%1%) = Q c X, |
| − | a syntactic pattern of contextual arrangement, or any other syntactic
| |
| − | device that can be used to connect a number of sentences together in
| |
| − | order to form a single sentence. If k is the number of sentences that
| |
| − | are thereby connected, then the connective is said to be of "order k".
| |
| − | If the sentences acquire a logical relationship through this mechanism,
| |
| − | and are not just strung together by this device, then the connective
| |
| − | is called a "logical connective". If the value of the constructed
| |
| − | sentence depends on the values of the component sentences in such
| |
| − | a way that the value of the whole is a boolean function of the
| |
| − | values of the parts, then the connective earns the title of
| |
| − | a "propositional connective".
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | then -[z]- = q = f_Q = -{Q}-. |
| | | | |
| − | IDS. Note 133
| + | 3. Q = {x in X : x in Q} |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | = [| -{Q}- |] = -{Q}-^(-1)(%1%) |
| | | | |
| − | 1.3.10.6. Stretching Principles | + | = [| f_Q |] = (f_Q)^(-1)(%1%). |
| | | | |
| − | There is a principle of constant use in this work that needs
| + | 4. -{Q}- = -{ {x in X : x in Q} }- |
| − | to be made explicit at this point. In order to give it a name,
| |
| − | I will refer to it as the "stretching principle". Expressed in
| |
| − | a variety of different ways, it can be taken to say any one of
| |
| − | the following things:
| |
| | | | |
| − | 1. Any relation of values extends
| + | = -[x in Q]- |
| − | to a relation of what is valued.
| |
| | | | |
| − | 2. Any statement about values says something
| + | = f_Q. |
| − | about the things that are given these values.
| |
| | | | |
| − | 3. Any association among a range of values
| + | If a sentence z really denotes a proposition q, and if the notation "-[z]-" |
| − | establishes an association among the
| + | is merely meant to supply another name for the proposition that z already |
| − | domains of things that these values
| + | denotes, then why is there any need for all of this additional notation? |
| − | are the values of.
| + | It is because the interpretive mind habitually races from the sentence z, |
| | + | through the proposition q that it denotes, and on to the set Q = [|q|] |
| | + | that the proposition indicates, often jumping to the conclusion that |
| | + | the set Q is the only thing that the sentence z is meant to denote. |
| | + | The momentum of this type of higher order sign situation, together |
| | + | with the mind's inclination when placed within its setting, calls |
| | + | for a linguistic mechanism or a notational device that is capable |
| | + | of analyzing the compound action and controlling its articulate |
| | + | performance, and this requires a way to interrupt the flow of |
| | + | assertion that runs its unreflective course from z to q to Q. |
| | + | </pre> |
| | | | |
| − | 4. Any connection between two values can be stretched
| + | ====1.3.11. The Cactus Patch==== |
| − | to create a connection, of analogous form, between
| |
| − | the objects, persons, qualities, or relationships
| |
| − | that are valued in these connections.
| |
| | | | |
| − | 5. For every operation on values, there is a corresponding operation
| + | <pre> |
| − | on the actions, conducts, functions, procedures, or processes that
| + | | Thus, what looks to us like a sphere of scientific knowledge more accurately |
| − | lead to these values, as well as there being analogous operations
| + | | should be represented as the inside of a highly irregular and spiky object, |
| − | on the objects that instigate all of these various proceedings.
| + | | like a pincushion or porcupine, with very sharp extensions in certain |
| | + | | directions, and virtually no knowledge in immediately adjacent areas. |
| | + | | If our intellectual gaze could shift slightly, it would alter each |
| | + | | quill's direction, and suddenly our entire reality would change. |
| | + | | |
| | + | | Herbert J. Bernstein, "Idols", p. 38. |
| | + | | |
| | + | | Herbert J. Bernstein, |
| | + | |"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in: |
| | + | | |
| | + | | Marcus G. Raskin & Herbert J. Bernstein, |
| | + | |'New Ways of Knowing: The Sciences, Society, and Reconstructive Knowledge', |
| | + | | Rowman & Littlefield, Totowa, NJ, 1987. |
| | | | |
| − | Nothing about the application of the stretching principle guarantees that
| + | In this and the four Subsections that follow, I describe a calculus for |
| − | the analogues it generates will be as useful as the material it works on.
| + | representing propositions as sentences, in other words, as syntactically |
| − | It is another question entirely whether the links that are forged in this
| + | defined sequences of signs, and for manipulating these sentences chiefly |
| − | fashion are equal in their strength and apposite in their bearing to the
| + | in the light of their semantically defined contents, in other words, with |
| − | tried and true utilities of the original ties, but in principle they are
| + | respect to their logical values as propositions. In their computational |
| − | always there.
| + | representation, the expressions of this calculus parse into a class of |
| | + | tree-like data structures called "painted cacti". This is a family of |
| | + | graph-theoretic data structures that can be observed to have especially |
| | + | nice properties, turning out to be not only useful from a computational |
| | + | standpoint but also quite interesting from a theoretical point of view. |
| | + | The rest of this subsection serves to motivate the development of this |
| | + | calculus and treats a number of general issues that surround the topic. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | In order to facilitate the use of propositions as indicator functions |
| | + | it helps to acquire a flexible notation for referring to propositions |
| | + | in that light, for interpreting sentences in a corresponding role, and |
| | + | for negotiating the requirements of mutual sense between the two domains. |
| | + | If none of the formalisms that are readily available or in common use are |
| | + | able to meet all of the design requirements that come to mind, then it is |
| | + | necessary to contemplate the design of a new language that is especially |
| | + | tailored to the purpose. In the present application, there is a pressing |
| | + | need to devise a general calculus for composing propositions, computing |
| | + | their values on particular arguments, and inverting their indications to |
| | + | arrive at the sets of things in the universe that are indicated by them. |
| | | | |
| − | IDS. Note 134
| + | For computational purposes, it is convenient to have a middle ground or |
| | + | an intermediate language for negotiating between the koine of sentences |
| | + | regarded as strings of literal characters and the realm of propositions |
| | + | regarded as objects of logical value, even if this renders it necessary |
| | + | to introduce an artificial medium of exchange between these two domains. |
| | + | If one envisions these computations to be carried out in any organized |
| | + | fashion, and ultimately or partially by means of the familiar sorts of |
| | + | machines, then the strings that express these logical propositions are |
| | + | likely to find themselves parsed into tree-like data structures at some |
| | + | stage of the game. With regard to their abstract structures as graphs, |
| | + | there are several species of graph-theoretic data structures that can be |
| | + | used to accomplish this job in a reasonably effective and efficient way. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | Over the course of this project, I plan to use two species of graphs: |
| | | | |
| − | 1.3.10.6. Stretching Principles (cont.) | + | 1. "Painted And Rooted Cacti" (PARCAI). |
| | | | |
| − | The purpose of this exercise is to illuminate how a sentence,
| + | 2. "Painted And Rooted Conifers" (PARCOI). |
| − | a sign constituted as a string of characters, can be enfused
| |
| − | with a proposition, an object of no slight abstraction, in a
| |
| − | way that can speak about an external universe of discourse X.
| |
| | | | |
| − | To complete the general discussion of stretching principles,
| + | For now, it is enough to discuss the former class of data structures, |
| − | we will need to call back to mind the following definitions:
| + | leaving the consideration of the latter class to a part of the project |
| | + | where their distinctive features are key to developments at that stage. |
| | + | Accordingly, within the context of the current patch of discussion, or |
| | + | until it becomes necessary to attach further notice to the conceivable |
| | + | varieties of parse graphs, the acronym "PARC" is sufficient to indicate |
| | + | the pertinent genus of abstract graphs that are under consideration. |
| | | | |
| − | A "boolean connection" of degree k, also known as a "boolean function"
| + | By way of making these tasks feasible to carry out on a regular basis, |
| − | on k variables, is a map of the form F : %B%^k -> %B%. In other words,
| + | a prospective language designer is required not only to supply a fluent |
| − | a boolean connection of degree k is a proposition about things in the | + | medium for the expression of propositions, but further to accompany the |
| − | universe of discourse X = %B%^k.
| + | assertions of their sentences with a canonical mechanism for teasing out |
| | + | the fibers of their indicator functions. Accordingly, with regard to a |
| | + | body of conceivable propositions, one needs to furnish a standard array |
| | + | of techniques for following the threads of their indications from their |
| | + | objective universe to their values for the mind and back again, that is, |
| | + | for tracing the clues that sentences provide from the universe of their |
| | + | objects to the signs of their values, and, in turn, from signs to objects. |
| | + | Ultimately, one seeks to render propositions so functional as indicators |
| | + | of sets and so essential for examining the equality of sets that they can |
| | + | constitute a veritable criterion for the practical conceivability of sets. |
| | + | Tackling this task requires me to introduce a number of new definitions |
| | + | and a collection of additional notational devices, to which I now turn. |
| | | | |
| − | An "imagination" of degree k on X is a k-tuple of propositions about things
| + | Depending on whether a formal language is called by the type of sign |
| − | in the universe X. By way of displaying the various brands of notation that
| + | that makes it up or whether it is named after the type of object that |
| − | are used to express this idea, the imagination #f# = <f_1, ..., f_k> is given | + | its signs are intended to denote, one may refer to this cactus language |
| − | as a sequence of indicator functions f_j : X -> %B%, for j = 1 to k. All of | + | as a "sentential calculus" or as a "propositional calculus", respectively. |
| − | these features of the typical imagination #f# can be summed up in either one
| |
| − | of two ways: either in the form of a membership statement, to the effect that
| |
| − | #f# is in (X -> %B%)^k, or in the form of a type statement, to the effect that
| |
| − | #f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise
| |
| − | than the former.
| |
| | | | |
| − | The "play of images" that is determined by #f# and x, more specifically,
| + | When the syntactic definition of the language is well enough understood, |
| − | the play of the imagination #f# = <f_1, ..., f_k> that has to with the | + | then the language can begin to acquire a semantic function. In natural |
| − | element x in X, is the k-tuple #b# = <b_1, ..., b_k> of values in %B%
| + | circumstances, the syntax and the semantics are likely to be engaged in |
| − | that satisfies the equations b_j = f_j (x), for all j = 1 to k.
| + | a process of co-evolution, whether in ontogeny or in phylogeny, that is, |
| | + | the two developments probably form parallel sides of a single bootstrap. |
| | + | But this is not always the easiest way, at least, at first, to formally |
| | + | comprehend the nature of their action or the power of their interaction. |
| | | | |
| − | A "projection" of %B%^k, typically denoted by "p_j" or "pr_j",
| + | According to the customary mode of formal reconstruction, the language |
| − | is one of the maps p_j : %B%^k -> %B%, for j = 1 to k, that is | + | is first presented in terms of its syntax, in other words, as a formal |
| − | defined as follows:
| + | language of strings called "sentences", amounting to a particular subset |
| | + | of the possible strings that can be formed on a finite alphabet of signs. |
| | + | A syntactic definition of the "cactus language", one that proceeds along |
| | + | purely formal lines, is carried out in the next Subsection. After that, |
| | + | the development of the language's more concrete aspects can be seen as |
| | + | a matter of defining two functions: |
| | | | |
| − | If #b# = <b_1, ..., b_k> in %B%^k, | + | 1. The first is a function that takes each sentence of the language |
| | + | into a computational data structure, in this particular setting, |
| | + | a tree-like parse graph called a "painted cactus". |
| | | | |
| − | then p_j (#b#) = p_j (<b_1, ..., b_k>) = b_j in %B%. | + | 2. The second is a function that takes each sentence of the language, |
| | + | by way of its corresponding parse graph, into a logical proposition, |
| | + | in effect, ending up with an indicator function as the object denoted |
| | + | by the sentence. |
| | | | |
| − | The "projective imagination" of %B%^k is the imagination <p_1, ..., p_k>. | + | The discussion of syntax brings up a number of associated issues that |
| | + | have to be clarified before going on. These are questions of "style", |
| | + | that is, the sort of description, "grammar", or theory of the language |
| | + | that one finds available or chooses as preferable for a given language. |
| | + | These issues are discussed in Subsection 1.3.10.10. |
| | | | |
| − | As an application of the stretching principle, a connection F : %B%^k -> %B%
| + | There is an aspect of syntax that is so schematic in its basic character |
| − | can be understood to indicate a relation among boolean values, namely, the | + | that it can be conveyed by computational data structures, so algorithmic |
| − | k-ary relation L = F^(-1)(%1%) c %B%^k. If these k values are values of
| + | in its uses that it can be automated by routine mechanisms, and so fixed |
| − | things in a universe X, that is, if one imagines each value in a k-tuple
| + | in its nature that its practical exploitation can be served by the usual |
| − | of values to be the functional image that results from evaluating an
| + | devices of computation. Because it involves the transformation of signs |
| − | element of X under one of its possible aspects of value, then one
| + | it can be recognized as an aspect of semiotics. But given the fact that |
| − | has in mind the k propositions f_j : X -> %B%, for j = 1 to k,
| + | these transformations can be carried out in abstraction from meaning, it |
| − | in sum, one embodies the imagination #f# = <f_1, ..., f_k>.
| + | does not rise to the level of semantics, much less a complete pragmatics, |
| − | Together, the imagination #f# in (X -> %B%)^k and the
| + | although it does involve the "pragmatic" aspects of computation that are |
| − | connection F : %B%^k -> %B% stretch each other to
| + | auxiliary to, incidental to, or tangent to the use of language by humans. |
| − | cover the universe X, yielding a new proposition
| + | In light of these characteristics, I will refer to this aspect of formal |
| − | q : X -> %B%.
| + | language use as the "algorithmics" or "mechanics" of language processing. |
| | + | An algorithmic conversion of the cactus language into its corresponding |
| | + | data structures is discussed in Subsection 1.3.10.11. |
| | | | |
| − | To encapsulate the form of this general result, I define a scheme of composition
| + | In the usual way of proceeding on formal grounds, meaning is added by giving |
| − | that takes an imagination #f# = <f_1, ..., f_k> in (X -> %B%)^k and a boolean
| + | each "grammatical sentence", or each syntactically distinguished string, an |
| − | connection F : %B%^k -> %B% and gives a proposition q : X -> %B%. Depending
| + | interpretation as a logically meaningful sentence, in effect, equipping or |
| − | on the situation, specifically, according to whether many F and many #f#,
| + | providing each abstractly well-formed sentence with a logical proposition |
| − | a single F and many #f#, or many F and a single #f# are being considered,
| + | for it to denote. A semantic interpretation of the "cactus language", |
| − | I refer to the resultant q under one of three descriptions, respectively:
| + | just one of at least two classical interpretations, is carried out |
| | + | in Subsection 1.3.10.12. |
| | + | </pre> |
| | | | |
| − | 1. In a general setting, where the connection F and the imagination #f#
| + | =====1.3.11.1. The Cactus Language : Syntax===== |
| − | are both permitted to take up a variety of concrete possibilities,
| |
| − | call q the "stretch of F and #f# from X to %B%", and write it in
| |
| − | the style of a composition as "F $ #f#". This is meant to suggest
| |
| − | that the symbol "$", here read as "stretch", denotes an operator
| |
| − | of the form $ : (%B%^k -> %B%) x (X -> %B%)^k -> (X -> %B%).
| |
| | | | |
| − | 2. In a setting where the connection F is fixed but the imagination #f#
| + | <pre> |
| − | is allowed to vary over a wide range of possibilities, call q the
| + | | Picture two different configurations of such an irregular shape, superimposed |
| − | "stretch of F to #f# on X", and write it in the style "F^$ #f#",
| + | | on each other in space, like a double exposure photograph. Of the two images, |
| − | as if "F^$" denotes an operator F^$ : (X -> %B%)^k -> (X -> %B%)
| + | | the only part which coincides is the body. The two different sets of quills |
| − | that is derived from F and applied to #f#, ultimately yielding
| + | | stick out into very different regions of space. The objective reality we |
| − | a proposition F^$ #f# : X -> %B%.
| + | | see from within the first position, seemingly so full and spherical, |
| | + | | actually agrees with the shifted reality only in the body of common |
| | + | | knowledge. In every direction in which we look at all deeply, the |
| | + | | realm of discovered scientific truth could be quite different. |
| | + | | Yet in each of those two different situations, we would have |
| | + | | thought the world complete, firmly known, and rather round |
| | + | | in its penetration of the space of possible knowledge. |
| | + | | |
| | + | | Herbert J. Bernstein, "Idols", p. 38. |
| | + | | |
| | + | | Herbert J. Bernstein, |
| | + | |"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in: |
| | + | | |
| | + | | Marcus G. Raskin & Herbert J. Bernstein, |
| | + | |'New Ways of Knowing: The Sciences, Society, and Reconstructive Knowledge', |
| | + | | Rowman & Littlefield, Totowa, NJ, 1987. |
| | | | |
| − | 3. In a setting where the imagination #f# is fixed but the connection F
| + | In this Subsection, I describe the syntax of a family of formal languages |
| − | is allowed to range over a wide variety of possibilities, call q the
| + | that I intend to use as a sentential calculus, and thus to interpret for |
| − | "stretch of #f# by F to %B%", and write it in the fashion "#f#^$ F",
| + | the purpose of reasoning about propositions and their logical relations. |
| − | as if "#f#^$" denotes an operator #f#^$ : (%B%^k -> %B%) -> (X -> %B%)
| + | In order to carry out the discussion, I need a way of referring to signs |
| − | that is derived from #f# and applied to F, ultimately yielding
| + | as if they were objects like any others, in other words, as the sorts of |
| − | a proposition #f#^$ F : X -> %B%.
| + | things that are subject to being named, indicated, described, discussed, |
| | + | and renamed if necessary, that can be placed, arranged, and rearranged |
| | + | within a suitable medium of expression, or else manipulated in the mind, |
| | + | that can be articulated and decomposed into their elementary signs, and |
| | + | that can be strung together in sequences to form complex signs. Signs |
| | + | that have signs as their objects are called "higher order" (HO) signs, |
| | + | and this is a topic that demands an apt formalization, but in due time. |
| | + | The present discussion requires a quicker way to get into this subject, |
| | + | even if it takes informal means that cannot be made absolutely precise. |
| | + | |
| | + | As a temporary notation, let the relationship between a particular sign z |
| | + | and a particular object o, namely, the fact that z denotes o or the fact |
| | + | that o is denoted by z, be symbolized in one of the following two ways: |
| | | | |
| − | Because the stretch notation is used only in settings
| + | 1. z >-> o, |
| − | where the imagination #f# : (X -> %B%)^k and the
| |
| − | connection F : %B%^k -> %B% are distinguished
| |
| − | by their types, it does not really matter
| |
| − | whether one writes "F $ #f#" or "#f# $ F"
| |
| − | for the initial form of composition.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | + | z den o. |
| | | | |
| − | IDS. Note 135
| + | 2. o <-< z, |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | + | o ned z. |
| | | | |
| − | 1.3.10.6. Stretching Principles (concl.)
| + | Now consider the following paradigm: |
| | | | |
| − | Just as a sentence is a sign that denotes a proposition,
| + | 1. If "A" >-> Ann, |
| − | which thereby serves to indicate a set, a propositional
| |
| − | connective is a provision of syntax whose mediate effect
| |
| − | is to denote an operation on propositions, which thereby
| |
| − | manages to indicate the result of an operation on sets.
| |
| − | In order to see how these compound forms of indication
| |
| − | can be defined, it is useful to go through the steps
| |
| − | that are needed to construct them. In general terms,
| |
| − | the ingredients of the construction are as follows:
| |
| | | | |
| − | 1. An imagination of degree k on X, in other words, a k-tuple
| + | i.e. "A" den Ann, |
| − | of propositions f_j : X -> %B%, for j = 1 to k, or an object
| |
| − | of the form #f# = <f_1, ..., f_k> : (X -> %B%)^k. | |
| | | | |
| − | 2. A connection of degree k, in other words, a proposition | + | then A = Ann, |
| − | about things in %B%^k, or a boolean function of the form
| |
| − | F : %B%^k -> %B%.
| |
| | | | |
| − | From this 2-ply of material, it is required to construct a proposition
| + | thus "Ann" >-> A, |
| − | q : X -> %B% such that q(x) = F(f_1(x), ..., f_k(x)), for all x in X.
| |
| − | The desired construction can be developed in the following manner:
| |
| | | | |
| − | The cartesian power %B%^k, as a cartesian product, is characterized
| + | i.e. "Ann" den A. |
| − | by the possession of a projective imagination #p# = <p_1, ..., p_k>
| |
| − | of degree k on %B%^k, along with the property that any imagination
| |
| − | #f# = <f_1, ..., f_k> of degree k on an arbitrary set W determines
| |
| − | a unique map !f! : W -> %B%^k, the play of whose projective images
| |
| − | <p_1(!f!(w)), ..., p_k(!f!(w))> on the functional image !f!(w)
| |
| − | matches the play of images <f_1(w), ..., f_k(w)> under #f#,
| |
| − | term for term and at every element w in W.
| |
| | | | |
| − | Just to be on the safe side, I state this again in more standard terms.
| + | 2. If Bob <-< "B", |
| − | The cartesian power %B%^k, as a cartesian product, is characterized by
| |
| − | the possession of k projection maps p_j : %B%^k -> %B%, for j = 1 to k,
| |
| − | along with the property that any k maps f_j : W -> %B%, from an arbitrary
| |
| − | set W to %B%, determine a unique map !f! : W -> %B%^k satisfying the system
| |
| − | of equations p_j(!f!(w)) = f_j(w), for all j = 1 to k, and for all w in W.
| |
| | | | |
| − | Now suppose that the arbitrary set W in this construction is just
| + | i.e. Bob ned "B", |
| − | the relevant universe X. Given that the function !f! : X -> %B%^k
| |
| − | is uniquely determined by the imagination #f# : (X -> %B%)^k, or what
| |
| − | is the same thing, by the k-tuple of propositions #f# = <f_1, ..., f_k>,
| |
| − | it is safe to identify !f! and #f# as being a single function, and this
| |
| − | makes it convenient on many occasions to refer to the identified function
| |
| − | by means of its explicitly descriptive name "<f_1, ..., f_k>". This facility
| |
| − | of address is especially appropriate whenever a concrete term or a constructive
| |
| − | precision is demanded by the context of discussion.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | then Bob = B, |
| | | | |
| − | IDS. Note 136
| + | thus B <-< "Bob", |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | i.e. B ned "Bob". |
| | | | |
| − | 1.3.10.7. Stretching Operations
| + | When I say that the sign "blank" denotes the sign " ", |
| | + | it means that the string of characters inside the first |
| | + | pair of quotation marks can be used as another name for |
| | + | the string of characters inside the second pair of quotes. |
| | + | In other words, "blank" is a HO sign whose object is " ", |
| | + | and the string of five characters inside the first pair of |
| | + | quotation marks is a sign at a higher level of signification |
| | + | than the string of one character inside the second pair of |
| | + | quotation marks. This relationship can be abbreviated in |
| | + | either one of the following ways: |
| | | | |
| − | The preceding discussion of stretch operations is slightly more general
| + | " " <-< "blank" |
| − | than is called for in the present context, and so it is probably a good
| |
| − | idea to draw out the particular implications that are needed right away.
| |
| | | | |
| − | If F : %B%^k -> %B% is a boolean function on k variables, then it is possible
| + | "blank" >-> " " |
| − | to define a mapping F^$ : (X -> %B%)^k -> (X -> %B%), in effect, an operation
| |
| − | that takes k propositions into a single proposition, where F^$ satisfies the
| |
| − | following conditions:
| |
| | | | |
| − | F^$ (f_1, ..., f_k) : X -> %B%
| + | Using the raised dot "·" as a sign to mark the articulation of a |
| | + | quoted string into a sequence of possibly shorter quoted strings, |
| | + | and thus to mark the concatenation of a sequence of quoted strings |
| | + | into a possibly larger quoted string, one can write: |
| | | | |
| − | such that: | + | " " <-< "blank" = "b"·"l"·"a"·"n"·"k" |
| | | | |
| − | F^$ (f_1, ..., f_k)(x) = F(#f#(x))
| + | This usage allows us to refer to the blank as a type of character, and |
| | + | also to refer any blank we choose as a token of this type, referring to |
| | + | either of them in a marked way, but without the use of quotation marks, |
| | + | as I just did. Now, since a blank is just what the name "blank" names, |
| | + | it is possible to represent the denotation of the sign " " by the name |
| | + | "blank" in the form of an identity between the named objects, thus: |
| | | | |
| − | = F((f_1, ..., f_k)(x))
| + | " " = blank |
| | | | |
| − | = F(f_1(x), ..., f_k(x)).
| + | With these kinds of identity in mind, it is possible to extend the use of |
| | + | the "·" sign to mark the articulation of either named or quoted strings |
| | + | into both named and quoted strings. For example: |
| | | | |
| − | Thus, F^$ is just the sort of entity that a propositional connective denotes,
| + | " " = " "·" " = blank·blank |
| − | a particular way of connecting the propositions that are denoted by a number
| |
| − | of sentences into a proposition that is denoted by a single sentence.
| |
| | | | |
| − | Now "f_X" is sign that denotes the proposition f_X,
| + | " blank" = " "·"blank" = blank·"blank" |
| − | and it certainly seems like a sufficient sign for it.
| |
| − | Why would we need to recognize any other signs of it?
| |
| | | | |
| − | If one takes a sentence as a type of sign that denotes a proposition and
| + | "blank " = "blank"·" " = "blank"·blank |
| − | a proposition as a type of function whose values serve to indicate a set,
| |
| − | then one needs a way to grasp the overall relation between the sentence
| |
| − | and the set as taking place within a higher order sign relation.
| |
| | | | |
| − | The various relationships of denotation and indication that exist
| + | A few definitions from formal language theory are required at this point. |
| − | among sets, propositions, sentences, and values in this situation
| |
| − | are illustrated very roughly by the array of materials in Table 10.
| |
| | | | |
| − | Table 10. Levels of Indication
| + | An "alphabet" is a finite set of signs, typically, !A! = {a_1, ..., a_n}. |
| − | o-------------------o-------------------o-------------------o
| |
| − | | Object | Sign | Higher Order Sign |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | Set | Proposition | Sentence |
| |
| − | | | | |
| |
| − | | f^(-1)(b) | f | "f" |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | Q | %1% | "%1%" |
| |
| − | | | | |
| |
| − | | X - Q | %0% | "%0%" |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| | | | |
| − | Strictly speaking, propositions are too abstract to be signs, hence the
| + | A "string" over an alphabet !A! is a finite sequence of signs from !A!. |
| − | contents of Table 10 have to be taken with the indicated grains of salt.
| |
| − | Propositions, as indicator functions, are abstract mathematical objects,
| |
| − | not any kinds of syntactic elements, thus propositions cannot literally
| |
| − | constitute the orders of concrete signs that remain of ultimate interest
| |
| − | in the pragmatic theory of signs, or in any theory of effective meaning.
| |
| | | | |
| − | Therefore, it needs to be understood that a proposition f can be said
| + | The "length" of a string is just its length as a sequence of signs. |
| − | to "indicate" the set Q only insofar as the values of %1% and %0% that
| + | A sequence of length 0 yields the "empty string", here presented as "". |
| − | it assigns to the elements of the universe X are positive and negative
| + | A sequence of length k > 0 is typically presented in the concatenated forms: |
| − | indications, respectively, of the elements in Q, and thus indications
| |
| − | of the set Q and of its complement ~X = X - Q, respectively. It is | |
| − | these logical values, when rendered by a concrete implementation of
| |
| − | the indicator function f, that are the actual signs of the objects
| |
| − | inside the set Q and the objects outside the set Q, respectively.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | s_1 s_2 ... s_(k-1) s_k, |
| | | | |
| − | IDS. Note 137
| + | or: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | s_1 · s_2 · ... · s_(k-1) · s_k, |
| | | | |
| − | 1.3.10.7. Stretching Operations (concl.) | + | with s_j in !A!, for all j = 1 to k. |
| | | | |
| − | In order to deal with the higher order sign relations
| + | Two alternative notations are often useful: |
| − | that are involved in the present setting, I introduce
| |
| − | a couple of new notations:
| |
| | | | |
| − | 1. To mark the relation of denotation between a sentence z and the | + | 1. !e! = "" = the empty string. |
| − | proposition that it denotes, the "spiny bracket" notation "-[z]-"
| |
| − | will be used for "the indicator function denoted by the sentence z".
| |
| | | | |
| − | 2. To mark the relation of denotation between a proposition q and | + | 2. %e% = {!e!} = the language consisting of a single empty string. |
| − | the set that it indicates, the "spiny brace" notation "-{Q}-"
| |
| − | will be used for "the indicator function of the set Q".
| |
| | | | |
| − | Notice that the spiny bracket operator "-[ ]-" takes one "downstream",
| + | The "kleene star" !A!* of alphabet !A! is the set of all strings over !A!. |
| − | confluent with the direction of denotation, from a sign to its object,
| + | In particular, !A!* includes among its elements the empty string !e!. |
| − | whereas the spiny brace operator "-{ }-" takes one "upstream", against
| |
| − | the usual direction of denotation, and thus from an object to its sign.
| |
| | | | |
| − | In order to make these notations useful in practice, it is necessary to note
| + | The "surplus" !A!^+ of an alphabet !A! is the set of all positive length |
| − | a couple of their finer points, points that might otherwise seem too fine to
| + | strings over !A!, in other words, everything in !A!* but the empty string. |
| − | take much trouble over. For the sake their ultimate utility, nevertheless,
| |
| − | I will describe their usage a bit more carefully as follows:
| |
| | | | |
| − | 1. Let "spiny brackets", like "-[ ]-", be placed around a name
| + | A "formal language" !L! over an alphabet !A! is a subset !L! c !A!*. |
| − | of a sentence z, as in the expression "-[z]-", or else around
| + | If z is a string over !A! and if z is an element of !L!, then it is |
| − | a token appearance of the sentence itself, to serve as a name
| + | customary to call z a "sentence" of !L!. Thus, a formal language !L! |
| − | for the proposition that z denotes.
| + | is defined by specifying its elements, which amounts to saying what it |
| | + | means to be a sentence of !L!. |
| | | | |
| − | 2. Let "spiny braces", like "-{ }-", be placed around a name of
| + | One last device turns out to be useful in this connection. |
| − | a set Q, as in the expression "-{Q}-", to serve as a name for
| + | If z is a string that ends with a sign t, then z · t^-1 is |
| − | the indicator function f_Q.
| + | the string that results by "deleting" from z the terminal t. |
| | | | |
| − | In passing, let us recall the use of the "fiber bars" | + | In this context, I make the following distinction: |
| − | or the "ground marker" "[| ... |]" as an alternate
| |
| − | notation for the fiber of truth in a proposition q,
| |
| − | as follows:
| |
| | | | |
| − | [| q |] = q^(-1)(%1%). | + | 1. By "deleting" an appearance of a sign, |
| | + | I mean replacing it with an appearance |
| | + | of the empty string "". |
| | | | |
| − | Table 11 illustrates the use of this notation, listing in each Column
| + | 2. By "erasing" an appearance of a sign, |
| − | several different but equivalent ways of referring to the same entity.
| + | I mean replacing it with an appearance |
| | + | of the blank symbol " ". |
| | | | |
| − | Table 11. Illustrations of Notation
| + | A "token" is a particular appearance of a sign. |
| − | o-------------------o-------------------o-------------------o
| |
| − | | Object | Sign | Higher Order Sign |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | Set | Proposition | Sentence |
| |
| − | | | | |
| |
| − | | Q | q | z |
| |
| − | | | | |
| |
| − | | [| -[z]- |] | -[z]- | z |
| |
| − | | | | |
| |
| − | | [| q |] | q | "q" |
| |
| − | | | | |
| |
| − | | [| f_Q |] | f_Q | "f_Q" |
| |
| − | | | | |
| |
| − | | Q | -{Q}- | "-{Q}-" |
| |
| − | | | | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| | | | |
| − | In effect, one can observe the following relations
| + | The informal mechanisms that have been illustrated in the immediately preceding |
| − | and formulas, all of a purely notational character: | + | discussion are enough to equip the rest of this discussion with a moderately |
| | + | exact description of the so-called "cactus language" that I intend to use |
| | + | in both my conceptual and my computational representations of the minimal |
| | + | formal logical system that is variously known to sundry communities of |
| | + | interpretation as "propositional logic", "sentential calculus", or |
| | + | more inclusively, "zeroth order logic" (ZOL). |
| | | | |
| − | 1. If the sentence z denotes the proposition q : X -> %B%,
| + | The "painted cactus language" !C! is actually a parameterized |
| | + | family of languages, consisting of one language !C!(!P!) for |
| | + | each set !P! of "paints". |
| | | | |
| − | then -[z]- = q.
| + | The alphabet !A! = !M! |_| !P! is the disjoint union of two sets of symbols: |
| | | | |
| − | 2. If the sentence z denotes the proposition q : X -> %B%, | + | 1. !M! is the alphabet of "measures", the set of "punctuation marks", |
| | + | or the collection of "syntactic constants" that is common to all |
| | + | of the languages !C!(!P!). This set of signs is given as follows: |
| | | | |
| − | hence [|q|] = q^(-1)(%1%) = Q c X, | + | !M! = {m_1, m_2, m_3, m_4} |
| | | | |
| − | then -[z]- = q = f_Q = -{Q}-.
| + | = {" ", "-(", ",", ")-"} |
| | | | |
| − | 3. Q = {x in X : x in Q}
| + | = {blank, links, comma, right}. |
| | | | |
| − | = [| -{Q}- |] = -{Q}-^(-1)(%1%)
| + | 2. !P! is the "palette", the alphabet of "paints", or the collection |
| | + | of "syntactic variables" that is peculiar to the language !C!(!P!). |
| | + | This set of signs is given as follows: |
| | + | |
| | + | !P! = {p_j : j in J}. |
| | + | |
| | + | The easiest way to define the language !C!(!P!) is to indicate the general sorts |
| | + | of operations that suffice to construct the greater share of its sentences from |
| | + | the specified few of its sentences that require a special election. In accord |
| | + | with this manner of proceeding, I introduce a family of operations on strings |
| | + | of !A!* that are called "syntactic connectives". If the strings on which |
| | + | they operate are exclusively sentences of !C!(!P!), then these operations |
| | + | are tantamount to "sentential connectives", and if the syntactic sentences, |
| | + | considered as abstract strings of meaningless signs, are given a semantics |
| | + | in which they denote propositions, considered as indicator functions over |
| | + | some universe, then these operations amount to "propositional connectives". |
| | + | |
| | + | NB. In this transcription, the symbols "-(" and ")-" |
| | + | will serve for the logically significant parentheses. |
| | + | |
| | + | The discussion that follows is intended to serve a dual purpose, |
| | + | in its specific focus presenting the family of cactus languages |
| | + | with some degree of detail, but more generally and peripherally |
| | + | developing the subject material and demonstrating the technical |
| | + | methodology of formal languages and grammars. I will do this by |
| | + | taking up a particular method of "stepwise refinement" and using |
| | + | it to extract a rigorous formal grammar for the cactus language, |
| | + | starting with little more than a rough description of the target |
| | + | language and applying a systematic analysis to develop a series |
| | + | of increasingly more effective and more exact approximations to |
| | + | the desired form of grammar. |
| | | | |
| − | = [| f_Q |] = (f_Q)^(-1)(%1%).
| + | Rather than presenting the most concise description of these languages |
| | + | right from the beginning, it serves comprehension to develop a picture |
| | + | of their forms in gradual stages, starting from the most natural ways |
| | + | of viewing their elements, if somewhat at a distance, and working |
| | + | through the most easily grasped impressions of their structures, |
| | + | if not always the sharpest acquaintances with their details. |
| | | | |
| − | 4. -{Q}- = -{ {x in X : x in Q} }-
| + | The first step is to define two sets of basic operations on strings of !A!*. |
| | | | |
| − | = -[x in Q]-
| + | 1. The "concatenation" of one string z_1 is just the string z_1. |
| | | | |
| − | = f_Q.
| + | The "concatenation" of two strings z_1, z_2 is the string z_1 · z_2. |
| | | | |
| − | If a sentence z really denotes a proposition q, and if the notation "-[z]-"
| + | The "concatenation" of the k strings z_j, for j = 1 to k, |
| − | is merely meant to supply another name for the proposition that z already
| |
| − | denotes, then why is there any need for all of this additional notation?
| |
| − | It is because the interpretive mind habitually races from the sentence z,
| |
| − | through the proposition q that it denotes, and on to the set Q = [|q|]
| |
| − | that the proposition indicates, often jumping to the conclusion that
| |
| − | the set Q is the only thing that the sentence z is meant to denote.
| |
| − | The momentum of this type of higher order sign situation, together
| |
| − | with the mind's inclination when placed within its setting, calls
| |
| − | for a linguistic mechanism or a notational device that is capable
| |
| − | of analyzing the compound action and controlling its articulate
| |
| − | performance, and this requires a way to interrupt the flow of
| |
| − | assertion that runs its unreflective course from z to q to Q.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | is the string of the form z_1 · ... · z_k. |
| | | | |
| − | IDS. Note 138
| + | 2. The "surcatenation" of one string z_1 is the string "-(" · z_1 · ")-". |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | The "surcatenation" of two strings z_1, z_2 is "-(" · z_1 · "," · z_2 · ")-". |
| | | | |
| − | 1.3.10.8. The Cactus Patch | + | The "surcatenation" of k strings z_j, for j = 1 to k, |
| | | | |
| − | | Thus, what looks to us like a sphere of scientific knowledge more accurately
| + | is the string of the form "-(" · z_1 · "," · ... · "," · z_k · ")-". |
| − | | should be represented as the inside of a highly irregular and spiky object,
| + | |
| − | | like a pincushion or porcupine, with very sharp extensions in certain
| + | These definitions can be rendered a little more succinct by |
| − | | directions, and virtually no knowledge in immediately adjacent areas.
| + | defining the following set of generic operators on strings: |
| − | | If our intellectual gaze could shift slightly, it would alter each
| + | |
| − | | quill's direction, and suddenly our entire reality would change.
| + | 1. The "concatenation" Conc^k of the k strings z_j, |
| − | |
| + | for j = 1 to k, is defined recursively as follows: |
| − | | Herbert J. Bernstein, "Idols", p. 38.
| |
| − | |
| |
| − | | Herbert J. Bernstein,
| |
| − | |"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in:
| |
| − | |
| |
| − | | Marcus G. Raskin & Herbert J. Bernstein,
| |
| − | |'New Ways of Knowing: The Sciences, Society, and Reconstructive Knowledge',
| |
| − | | Rowman & Littlefield, Totowa, NJ, 1987.
| |
| | | | |
| − | In this and the four Subsections that follow, I describe a calculus for
| + | a. Conc^1_j z_j = z_1. |
| − | representing propositions as sentences, in other words, as syntactically
| |
| − | defined sequences of signs, and for manipulating these sentences chiefly
| |
| − | in the light of their semantically defined contents, in other words, with
| |
| − | respect to their logical values as propositions. In their computational
| |
| − | representation, the expressions of this calculus parse into a class of
| |
| − | tree-like data structures called "painted cacti". This is a family of
| |
| − | graph-theoretic data structures that can be observed to have especially
| |
| − | nice properties, turning out to be not only useful from a computational
| |
| − | standpoint but also quite interesting from a theoretical point of view.
| |
| − | The rest of this subsection serves to motivate the development of this
| |
| − | calculus and treats a number of general issues that surround the topic.
| |
| | | | |
| − | In order to facilitate the use of propositions as indicator functions
| + | b. For k > 1, |
| − | it helps to acquire a flexible notation for referring to propositions
| |
| − | in that light, for interpreting sentences in a corresponding role, and
| |
| − | for negotiating the requirements of mutual sense between the two domains.
| |
| − | If none of the formalisms that are readily available or in common use are
| |
| − | able to meet all of the design requirements that come to mind, then it is
| |
| − | necessary to contemplate the design of a new language that is especially
| |
| − | tailored to the purpose. In the present application, there is a pressing
| |
| − | need to devise a general calculus for composing propositions, computing
| |
| − | their values on particular arguments, and inverting their indications to
| |
| − | arrive at the sets of things in the universe that are indicated by them.
| |
| | | | |
| − | For computational purposes, it is convenient to have a middle ground or
| + | Conc^k_j z_j = (Conc^(k-1)_j z_j) · z_k. |
| − | an intermediate language for negotiating between the koine of sentences
| |
| − | regarded as strings of literal characters and the realm of propositions
| |
| − | regarded as objects of logical value, even if this renders it necessary
| |
| − | to introduce an artificial medium of exchange between these two domains.
| |
| − | If one envisions these computations to be carried out in any organized
| |
| − | fashion, and ultimately or partially by means of the familiar sorts of
| |
| − | machines, then the strings that express these logical propositions are
| |
| − | likely to find themselves parsed into tree-like data structures at some
| |
| − | stage of the game. With regard to their abstract structures as graphs,
| |
| − | there are several species of graph-theoretic data structures that can be
| |
| − | used to accomplish this job in a reasonably effective and efficient way.
| |
| | | | |
| − | Over the course of this project, I plan to use two species of graphs:
| + | 2. The "surcatenation" Surc^k of the k strings z_j, |
| | + | for j = 1 to k, is defined recursively as follows: |
| | | | |
| − | 1. "Painted And Rooted Cacti" (PARCAI).
| + | a. Surc^1_j z_j = "-(" · z_1 · ")-". |
| | | | |
| − | 2. "Painted And Rooted Conifers" (PARCOI).
| + | b. For k > 1, |
| | | | |
| − | For now, it is enough to discuss the former class of data structures,
| + | Surc^k_j z_j = (Surc^(k-1)_j z_j) · ")-"^(-1) · "," · z_k · ")-". |
| − | leaving the consideration of the latter class to a part of the project
| |
| − | where their distinctive features are key to developments at that stage.
| |
| − | Accordingly, within the context of the current patch of discussion, or
| |
| − | until it becomes necessary to attach further notice to the conceivable
| |
| − | varieties of parse graphs, the acronym "PARC" is sufficient to indicate
| |
| − | the pertinent genus of abstract graphs that are under consideration.
| |
| | | | |
| − | By way of making these tasks feasible to carry out on a regular basis,
| + | The definitions of the foregoing syntactic operations can now be organized in |
| − | a prospective language designer is required not only to supply a fluent
| + | a slightly better fashion, for the sake of both conceptual and computational |
| − | medium for the expression of propositions, but further to accompany the
| + | purposes, by making a few additional conventions and auxiliary definitions. |
| − | assertions of their sentences with a canonical mechanism for teasing out
| + | |
| − | the fibers of their indicator functions. Accordingly, with regard to a
| + | 1. The conception of the k-place concatenation operation |
| − | body of conceivable propositions, one needs to furnish a standard array
| + | can be extended to include its natural "prequel": |
| − | of techniques for following the threads of their indications from their
| |
| − | objective universe to their values for the mind and back again, that is,
| |
| − | for tracing the clues that sentences provide from the universe of their
| |
| − | objects to the signs of their values, and, in turn, from signs to objects.
| |
| − | Ultimately, one seeks to render propositions so functional as indicators
| |
| − | of sets and so essential for examining the equality of sets that they can | |
| − | constitute a veritable criterion for the practical conceivability of sets.
| |
| − | Tackling this task requires me to introduce a number of new definitions
| |
| − | and a collection of additional notational devices, to which I now turn.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | Conc^0 = "" = the empty string. |
| | | | |
| − | IDS. Note 139
| + | Next, the construction of the k-place concatenation can be |
| | + | broken into stages by means of the following conceptions: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | a. The "precatenation" Prec(z_1, z_2) of the two strings |
| | + | z_1, z_2 is the string that is defined as follows: |
| | | | |
| − | 1.3.10.8. The Cactus Patch (concl.)
| + | Prec(z_1, z_2) = z_1 · z_2. |
| | | | |
| − | Depending on whether a formal language is called by the type of sign
| + | b. The "concatenation" of the k strings z_1, ..., z_k can now be |
| − | that makes it up or whether it is named after the type of object that | + | defined as an iterated precatenation over the sequence of k+1 |
| − | its signs are intended to denote, one may refer to this cactus language
| + | strings that begins with the string z_0 = Conc^0 = "" and then |
| − | as a "sentential calculus" or as a "propositional calculus", respectively.
| + | continues on through the other k strings: |
| | | | |
| − | When the syntactic definition of the language is well enough understood,
| + | i. Conc^0_j z_j = Conc^0 = "". |
| − | then the language can begin to acquire a semantic function. In natural
| |
| − | circumstances, the syntax and the semantics are likely to be engaged in
| |
| − | a process of co-evolution, whether in ontogeny or in phylogeny, that is,
| |
| − | the two developments probably form parallel sides of a single bootstrap.
| |
| − | But this is not always the easiest way, at least, at first, to formally
| |
| − | comprehend the nature of their action or the power of their interaction.
| |
| | | | |
| − | According to the customary mode of formal reconstruction, the language
| + | ii. For k > 0, |
| − | is first presented in terms of its syntax, in other words, as a formal
| |
| − | language of strings called "sentences", amounting to a particular subset
| |
| − | of the possible strings that can be formed on a finite alphabet of signs.
| |
| − | A syntactic definition of the "cactus language", one that proceeds along
| |
| − | purely formal lines, is carried out in the next Subsection. After that,
| |
| − | the development of the language's more concrete aspects can be seen as
| |
| − | a matter of defining two functions:
| |
| | | | |
| − | 1. The first is a function that takes each sentence of the language
| + | Conc^k_j z_j = Prec(Conc^(k-1)_j z_j, z_k). |
| − | into a computational data structure, in this particular setting,
| |
| − | a tree-like parse graph called a "painted cactus".
| |
| | | | |
| − | 2. The second is a function that takes each sentence of the language, | + | 2. The conception of the k-place surcatenation operation |
| − | by way of its corresponding parse graph, into a logical proposition, | + | can be extended to include its natural "prequel": |
| − | in effect, ending up with an indicator function as the object denoted
| |
| − | by the sentence.
| |
| | | | |
| − | The discussion of syntax brings up a number of associated issues that
| + | Surc^0 = "-()-". |
| − | have to be clarified before going on. These are questions of "style",
| |
| − | that is, the sort of description, "grammar", or theory of the language
| |
| − | that one finds available or chooses as preferable for a given language.
| |
| − | These issues are discussed in Subsection 1.3.10.10.
| |
| | | | |
| − | There is an aspect of syntax that is so schematic in its basic character
| + | Finally, the construction of the k-place surcatenation can be |
| − | that it can be conveyed by computational data structures, so algorithmic
| + | broken into stages by means of the following conceptions: |
| − | in its uses that it can be automated by routine mechanisms, and so fixed
| |
| − | in its nature that its practical exploitation can be served by the usual
| |
| − | devices of computation. Because it involves the transformation of signs
| |
| − | it can be recognized as an aspect of semiotics. But given the fact that
| |
| − | these transformations can be carried out in abstraction from meaning, it
| |
| − | does not rise to the level of semantics, much less a complete pragmatics,
| |
| − | although it does involve the "pragmatic" aspects of computation that are
| |
| − | auxiliary to, incidental to, or tangent to the use of language by humans.
| |
| − | In light of these characteristics, I will refer to this aspect of formal
| |
| − | language use as the "algorithmics" or "mechanics" of language processing.
| |
| − | An algorithmic conversion of the cactus language into its corresponding
| |
| − | data structures is discussed in Subsection 1.3.10.11.
| |
| | | | |
| − | In the usual way of proceeding on formal grounds, meaning is added by giving
| + | a. A "subclause" in !A!* is a string that ends with a ")-". |
| − | each "grammatical sentence", or each syntactically distinguished string, an
| |
| − | interpretation as a logically meaningful sentence, in effect, equipping or
| |
| − | providing each abstractly well-formed sentence with a logical proposition
| |
| − | for it to denote. A semantic interpretation of the "cactus language",
| |
| − | just one of at least two classical interpretations, is carried out
| |
| − | in Subsection 1.3.10.12.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | b. The "subcatenation" Subc(z_1, z_2) |
| | + | of a subclause z_1 by a string z_2 is |
| | + | the string that is defined as follows: |
| | | | |
| − | IDS. Note 140
| + | Subc(z_1, z_2) = z_1 · ")-"^(-1) · "," · z_2 · ")-". |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | c. The "surcatenation" of the k strings z_1, ..., z_k can now be |
| | + | defined as an iterated subcatenation over the sequence of k+1 |
| | + | strings that starts with the string z_0 = Surc^0 = "-()-" and |
| | + | then continues on through the other k strings: |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax
| + | i. Surc^0_j z_j = Surc^0 = "-()-". |
| | | | |
| − | | Picture two different configurations of such an irregular shape, superimposed
| + | ii. For k > 0, |
| − | | on each other in space, like a double exposure photograph. Of the two images,
| |
| − | | the only part which coincides is the body. The two different sets of quills
| |
| − | | stick out into very different regions of space. The objective reality we
| |
| − | | see from within the first position, seemingly so full and spherical,
| |
| − | | actually agrees with the shifted reality only in the body of common
| |
| − | | knowledge. In every direction in which we look at all deeply, the
| |
| − | | realm of discovered scientific truth could be quite different.
| |
| − | | Yet in each of those two different situations, we would have
| |
| − | | thought the world complete, firmly known, and rather round
| |
| − | | in its penetration of the space of possible knowledge.
| |
| − | |
| |
| − | | Herbert J. Bernstein, "Idols", p. 38.
| |
| − | |
| |
| − | | Herbert J. Bernstein,
| |
| − | |"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in:
| |
| − | |
| |
| − | | Marcus G. Raskin & Herbert J. Bernstein,
| |
| − | |'New Ways of Knowing: The Sciences, Society, and Reconstructive Knowledge',
| |
| − | | Rowman & Littlefield, Totowa, NJ, 1987.
| |
| | | | |
| − | In this Subsection, I describe the syntax of a family of formal languages
| + | Surc^k_j z_j = Subc(Surc^(k-1)_j z_j, z_k). |
| − | that I intend to use as a sentential calculus, and thus to interpret for
| |
| − | the purpose of reasoning about propositions and their logical relations.
| |
| − | In order to carry out the discussion, I need a way of referring to signs
| |
| − | as if they were objects like any others, in other words, as the sorts of
| |
| − | things that are subject to being named, indicated, described, discussed,
| |
| − | and renamed if necessary, that can be placed, arranged, and rearranged
| |
| − | within a suitable medium of expression, or else manipulated in the mind,
| |
| − | that can be articulated and decomposed into their elementary signs, and
| |
| − | that can be strung together in sequences to form complex signs. Signs
| |
| − | that have signs as their objects are called "higher order" (HO) signs,
| |
| − | and this is a topic that demands an apt formalization, but in due time.
| |
| − | The present discussion requires a quicker way to get into this subject,
| |
| − | even if it takes informal means that cannot be made absolutely precise.
| |
| | | | |
| − | As a temporary notation, let the relationship between a particular sign z
| + | Notice that the expressions Conc^0_j z_j and Surc^0_j z_j |
| − | and a particular object o, namely, the fact that z denotes o or the fact | + | are defined in such a way that the respective operators |
| − | that o is denoted by z, be symbolized in one of the following two ways:
| + | Conc^0 and Surc^0 basically "ignore", in the manner of |
| | + | constant functions, whatever sequences of strings z_j |
| | + | may happen to be listed as their ostensible arguments. |
| | | | |
| − | 1. z >-> o,
| + | Having defined the basic operations of concatenation and surcatenation |
| | + | on arbitrary strings, in effect, giving them operational meaning for |
| | + | the all-inclusive language !L! = !A!*, it is time to adjoin the |
| | + | notion of a more discriminating grammaticality, in other words, |
| | + | a more properly restrictive concept of a sentence. |
| | | | |
| − | z den o.
| + | If !L! is an arbitrary formal language over an alphabet of the sort that |
| | + | we are talking about, that is, an alphabet of the form !A! = !M! |_| !P!, |
| | + | then there are a number of basic structural relations that can be defined |
| | + | on the strings of !L!. |
| | | | |
| − | 2. o <-< z, | + | 1. z is the "concatenation" of z_1 and z_2 in !L! if and only if |
| | | | |
| − | o ned z. | + | z_1 is a sentence of !L!, z_2 is a sentence of !L!, and |
| | | | |
| − | Now consider the following paradigm:
| + | z = z_1 · z_2. |
| | | | |
| − | 1. If "A" >-> Ann, | + | 2. z is the "concatenation" of the k strings z1, ..., z_k in !L!, |
| | | | |
| − | i.e. "A" den Ann, | + | if and only if z_j is a sentence of !L!, for all j = 1 to k, and |
| | | | |
| − | then A = Ann, | + | z = Conc^k_j z_j = z_1 · ... · z_k. |
| | | | |
| − | thus "Ann" >-> A,
| + | 3. z is the "discatenation" of z_1 by t if and only if |
| | | | |
| − | i.e. "Ann" den A. | + | z_1 is a sentence of !L!, t is an element of !A!, and |
| | | | |
| − | 2. If Bob <-< "B",
| + | z_1 = z · t. |
| | | | |
| − | i.e. Bob ned "B", | + | When this is the case, one more commonly writes: |
| | | | |
| − | then Bob = B, | + | z = z_1 · t^-1. |
| | | | |
| − | thus B <-< "Bob",
| + | 4. z is a "subclause" of !L! if and only if |
| | | | |
| − | i.e. B ned "Bob". | + | z is a sentence of !L! and z ends with a ")-". |
| | | | |
| − | When I say that the sign "blank" denotes the sign " ",
| + | 5. z is the "subcatenation" of z_1 by z_2 if and only if |
| − | it means that the string of characters inside the first
| |
| − | pair of quotation marks can be used as another name for
| |
| − | the string of characters inside the second pair of quotes.
| |
| − | In other words, "blank" is a HO sign whose object is " ",
| |
| − | and the string of five characters inside the first pair of | |
| − | quotation marks is a sign at a higher level of signification
| |
| − | than the string of one character inside the second pair of
| |
| − | quotation marks. This relationship can be abbreviated in
| |
| − | either one of the following ways:
| |
| | | | |
| − | " " <-< "blank"
| + | z_1 is a subclause of !L!, z_2 is a sentence of !L!, and |
| | | | |
| − | "blank" >-> " "
| + | z = z_1 · ")-"^(-1) · "," · z_2 · ")-". |
| | | | |
| − | Using the raised dot "·" as a sign to mark the articulation of a
| + | 6. z is the "surcatenation" of the k strings z_1, ..., z_k in !L!, |
| − | quoted string into a sequence of possibly shorter quoted strings,
| |
| − | and thus to mark the concatenation of a sequence of quoted strings
| |
| − | into a possibly larger quoted string, one can write:
| |
| | | | |
| − | " " <-< "blank" = "b"·"l"·"a"·"n"·"k"
| + | if and only if z_j is a sentence of !L!, for all j = 1 to k, and |
| | | | |
| − | This usage allows us to refer to the blank as a type of character, and
| + | z = Surc^k_j z_j = "-(" · z_1 · "," · ... · "," · z_k · ")-". |
| − | also to refer any blank we choose as a token of this type, referring to
| |
| − | either of them in a marked way, but without the use of quotation marks,
| |
| − | as I just did. Now, since a blank is just what the name "blank" names,
| |
| − | it is possible to represent the denotation of the sign " " by the name
| |
| − | "blank" in the form of an identity between the named objects, thus: | |
| | | | |
| − | " " = blank
| + | The converses of these decomposition relations are tantamount to the |
| | + | corresponding forms of composition operations, making it possible for |
| | + | these complementary forms of analysis and synthesis to articulate the |
| | + | structures of strings and sentences in two directions. |
| | | | |
| − | With these kinds of identity in mind, it is possible to extend the use of
| + | The "painted cactus language" with paints in the |
| − | the "·" sign to mark the articulation of either named or quoted strings
| + | set !P! = {p_j : j in J} is the formal language |
| − | into both named and quoted strings. For example:
| + | !L! = !C!(!P!) c !A!* = (!M! |_| !P!)* that is |
| | + | defined as follows: |
| | | | |
| − | " " = " "·" " = blank·blank | + | PC 1. The blank symbol m_1 is a sentence. |
| | | | |
| − | " blank" = " "·"blank" = blank·"blank" | + | PC 2. The paint p_j is a sentence, for each j in J. |
| | | | |
| − | "blank " = "blank"·" " = "blank"·blank | + | PC 3. Conc^0 and Surc^0 are sentences. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | PC 4. For each positive integer k, |
| | | | |
| − | IDS. Note 141
| + | if z_1, ..., z_k are sentences, |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | then Conc^k_j z_j is a sentence, |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | and Surc^k_j z_j is a sentence. |
| | | | |
| − | A few definitions from formal language theory are required at this point.
| + | As usual, saying that z is a sentence is just a conventional way of |
| | + | stating that the string z belongs to the relevant formal language !L!. |
| | + | An individual sentence of !C!(!P!), for any palette !P!, is referred to |
| | + | as a "painted and rooted cactus expression" (PARCE) on the palette !P!, |
| | + | or a "cactus expression", for short. Anticipating the forms that the |
| | + | parse graphs of these PARCE's will take, to be described in the next |
| | + | Subsection, the language !L! = !C!(!P!) is also described as the |
| | + | set PARCE(!P!) of PARCE's on the palette !P!, more generically, |
| | + | as the PARCE's that constitute the language PARCE. |
| | | | |
| − | An "alphabet" is a finite set of signs, typically, !A! = {a_1, ..., a_n}.
| + | A "bare" PARCE, a bit loosely referred to as a "bare cactus expression", |
| | + | is a PARCE on the empty palette !P! = {}. A bare PARCE is a sentence |
| | + | in the "bare cactus language", !C!^0 = !C!({}) = PARCE^0 = PARCE({}). |
| | + | This set of strings, regarded as a formal language in its own right, |
| | + | is a sublanguage of every cactus language !C!(!P!). A bare cactus |
| | + | expression is commonly encountered in practice when one has occasion |
| | + | to start with an arbitrary PARCE and then finds a reason to delete or |
| | + | to erase all of its paints. |
| | | | |
| − | A "string" over an alphabet !A! is a finite sequence of signs from !A!.
| + | Only one thing remains to cast this description of the cactus language |
| | + | into a form that is commonly found acceptable. As presently formulated, |
| | + | the principle PC 4 appears to be attempting to define an infinite number |
| | + | of new concepts all in a single step, at least, it appears to invoke the |
| | + | indefinitely long sequences of operators, Conc^k and Surc^k, for all k > 0. |
| | + | As a general rule, one prefers to have an effectively finite description of |
| | + | conceptual objects, and this means restricting the description to a finite |
| | + | number of schematic principles, each of which involves a finite number of |
| | + | schematic effects, that is, a finite number of schemata that explicitly |
| | + | relate conditions to results. |
| | | | |
| − | The "length" of a string is just its length as a sequence of signs.
| + | A start in this direction, taking steps toward an effective description |
| − | A sequence of length 0 yields the "empty string", here presented as "".
| + | of the cactus language, a finitary conception of its membership conditions, |
| − | A sequence of length k > 0 is typically presented in the concatenated forms:
| + | and a bounded characterization of a typical sentence in the language, can be |
| | + | made by recasting the present description of these expressions into the pattern |
| | + | of what is called, more or less roughly, a "formal grammar". |
| | | | |
| − | s_1 s_2 ... s_(k-1) s_k,
| + | A notation in the style of "S :> T" is now introduced, |
| | + | to be read among many others in this manifold of ways: |
| | | | |
| − | or: | + | S covers T |
| | | | |
| − | s_1 · s_2 · ... · s_(k-1) · s_k, | + | S governs T |
| | | | |
| − | with s_j in !A!, for all j = 1 to k.
| + | S rules T |
| | | | |
| − | Two alternative notations are often useful:
| + | S subsumes T |
| | | | |
| − | 1. !e! = "" = the empty string. | + | S types over T |
| | | | |
| − | 2. %e% = {!e!} = the language consisting of a single empty string.
| + | The form "S :> T" is here recruited for polymorphic |
| | + | employment in at least the following types of roles: |
| | | | |
| − | The "kleene star" !A!* of alphabet !A! is the set of all strings over !A!.
| + | 1. To signify that an individually named or quoted string T is |
| − | In particular, !A!* includes among its elements the empty string !e!.
| + | being typed as a sentence S of the language of interest !L!. |
| | | | |
| − | The "surplus" !A!^+ of an alphabet !A! is the set of all positive length
| + | 2. To express the fact or to make the assertion that each member |
| − | strings over !A!, in other words, everything in !A!* but the empty string. | + | of a specified set of strings T c !A!* also belongs to the |
| | + | syntactic category S, the one that qualifies a string as |
| | + | being a sentence in the relevant formal language !L!. |
| | | | |
| − | A "formal language" !L! over an alphabet !A! is a subset !L! c !A!*.
| + | 3. To specify the intension or to signify the intention that every |
| − | If z is a string over !A! and if z is an element of !L!, then it is
| + | string that fits the conditions of the abstract type T must also |
| − | customary to call z a "sentence" of !L!. Thus, a formal language !L!
| + | fall under the grammatical heading of a sentence, as indicated by |
| − | is defined by specifying its elements, which amounts to saying what it
| + | the type name "S", all within the target language !L!. |
| − | means to be a sentence of !L!.
| |
| | | | |
| − | One last device turns out to be useful in this connection.
| + | In these types of situation the letter "S", that signifies the type of |
| − | If z is a string that ends with a sign t, then z · t^-1 is
| + | a sentence in the language of interest, is called the "initial symbol" |
| − | the string that results by "deleting" from z the terminal t. | + | or the "sentence symbol" of a candidate formal grammar for the language, |
| | + | while any number of letters like "T", signifying other types of strings |
| | + | that are necessary to a reasonable account or a rational reconstruction |
| | + | of the sentences that belong to the language, are collectively referred |
| | + | to as "intermediate symbols". |
| | | | |
| − | In this context, I make the following distinction: | + | Combining the singleton set {"S"} whose sole member is the initial symbol |
| | + | with the set !Q! that assembles together all of the intermediate symbols |
| | + | results in the set {"S"} |_| !Q! of "non-terminal symbols". Completing |
| | + | the package, the alphabet !A! of the language is also known as the set |
| | + | of "terminal symbols". In this discussion, I will adopt the convention |
| | + | that !Q! is the set of intermediate symbols, but I will often use "q" |
| | + | as a typical variable that ranges over all of the non-terminal symbols, |
| | + | q in {"S"} |_| !Q!. Finally, it is convenient to refer to all of the |
| | + | symbols in {"S"} |_| !Q! |_| !A! as the "augmented alphabet" of the |
| | + | prospective grammar for the language, and accordingly to describe |
| | + | the strings in ({"S"} |_| !Q! |_| !A!)* as the "augmented strings", |
| | + | in effect, expressing the forms that are superimposed on a language |
| | + | by one of its conceivable grammars. In certain settings it becomes |
| | + | desirable to separate the augmented strings that contain the symbol |
| | + | "S" from all other sorts of augmented strings. In these situations, |
| | + | the strings in the disjoint union {"S"} |_| (!Q! |_| !A!)* are known |
| | + | as the "sentential forms" of the associated grammar. |
| | | | |
| − | 1. By "deleting" an appearance of a sign,
| + | In forming a grammar for a language, statements of the form W :> W', |
| − | I mean replacing it with an appearance
| + | where W and W' are augmented strings or sentential forms of specified |
| − | of the empty string "".
| + | types that depend on the style of the grammar that is being sought, are |
| | + | variously known as "characterizations", "covering rules", "productions", |
| | + | "rewrite rules", "subsumptions", "transformations", or "typing rules". |
| | + | These are collected together into a set !K! that serves to complete |
| | + | the definition of the formal grammar in question. |
| | | | |
| − | 2. By "erasing" an appearance of a sign,
| + | Correlative with the use of this notation, an expression of the |
| − | I mean replacing it with an appearance
| + | form "T <: S", read as "T is covered by S", can be interpreted |
| − | of the blank symbol " ".
| + | as saying that T is of the type S. Depending on the context, |
| | + | this can be taken in either one of two ways: |
| | | | |
| − | A "token" is a particular appearance of a sign.
| + | 1. Treating "T" as a string variable, it means |
| | + | that the individual string T is typed as S. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | 2. Treating "T" as a type name, it means that any |
| | + | instance of the type T also falls under the type S. |
| | | | |
| − | IDS. Note 142
| + | In accordance with these interpretations, an expression like "t <: T" can be |
| | + | read in all of the ways that one typically reads an expression like "t : T". |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | There are several abuses of notation that commonly tolerated in the use |
| | + | of covering relations. The worst offense is that of allowing symbols to |
| | + | stand equivocally either for individual strings or else for their types. |
| | + | There is a measure of consistency to this practice, considering the fact |
| | + | that perfectly individual entities are rarely if ever grasped by means of |
| | + | signs and finite expressions, which entails that every appearance of an |
| | + | apparent token is only a type of more particular tokens, and meaning in |
| | + | the end that there is never any recourse but to the sort of discerning |
| | + | interpretation that can decide just how each sign is intended. In view |
| | + | of all this, I continue to permit expressions like "t <: T" and "T <: S", |
| | + | where any of the symbols "t", "T", "S" can be taken to signify either the |
| | + | tokens or the subtypes of their covering types. |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | Employing the notion of a covering relation it becomes possible to |
| | + | redescribe the cactus language !L! = !C!(!P!) in the following way. |
| | | | |
| − | The informal mechanisms that have been illustrated in the immediately preceding
| + | Grammar 1 is something of a misnomer. It is nowhere near exemplifying |
| − | discussion are enough to equip the rest of this discussion with a moderately
| + | any kind of a standard form and it is only intended as a starting point |
| − | exact description of the so-called "cactus language" that I intend to use
| + | for the initiation of more respectable grammars. Such as it is, it uses |
| − | in both my conceptual and my computational representations of the minimal
| + | the terminal alphabet !A! = !M! |_| !P! that comes with the territory of |
| − | formal logical system that is variously known to sundry communities of
| + | the cactus language !C!(!P!), it specifies !Q! = {}, in other words, it |
| − | interpretation as "propositional logic", "sentential calculus", or
| + | employs no intermediate symbols, and it embodies the "covering set" !K! |
| − | more inclusively, "zeroth order logic" (ZOL).
| + | as listed in the following display. |
| | | | |
| − | The "painted cactus language" !C! is actually a parameterized
| + | o-------------------------------------------------o |
| − | family of languages, consisting of one language !C!(!P!) for
| + | | !C!(!P!). Grammar 1 !Q! = {} | |
| − | each set !P! of "paints".
| + | o-------------------------------------------------o |
| | + | | | |
| | + | | 1. S :> m_1 = " " | |
| | + | | | |
| | + | | 2. S :> p_j, for each j in J | |
| | + | | | |
| | + | | 3. S :> Conc^0 = "" | |
| | + | | | |
| | + | | 4. S :> Surc^0 = "-()-" | |
| | + | | | |
| | + | | 5. S :> S* | |
| | + | | | |
| | + | | 6. S :> "-(" · S · ("," · S)* · ")-" | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | | | |
| − | The alphabet !A! = !M! |_| !P! is the disjoint union of two sets of symbols:
| + | In this formulation, the last two lines specify that: |
| | | | |
| − | 1. !M! is the alphabet of "measures", the set of "punctuation marks", | + | 5. The concept of a sentence in !L! covers any |
| − | or the collection of "syntactic constants" that is common to all | + | concatenation of sentences in !L!, in effect, |
| − | of the languages !C!(!P!). This set of signs is given as follows: | + | any number of freely chosen sentences that are |
| | + | available to be concatenated one after another. |
| | | | |
| − | !M! = {m_1, m_2, m_3, m_4} | + | 6. The concept of a sentence in !L! covers any |
| − | | + | surcatenation of sentences in !L!, in effect, |
| − | = {" ", "-(", ",", ")-"}
| + | any string that opens with a "-(", continues |
| | + | with a sentence, possibly empty, follows with |
| | + | a finite number of phrases of the form "," · S, |
| | + | and closes with a ")-". |
| | | | |
| − | = {blank, links, comma, right}.
| + | This appears to be just about the most concise description |
| | + | of the cactus language !C!(!P!) that one can imagine, but |
| | + | there exist a couple of problems that are commonly felt |
| | + | to afflict this style of presentation and to make it |
| | + | less than completely acceptable. Briefly stated, |
| | + | these problems turn on the following properties |
| | + | of the presentation: |
| | | | |
| − | 2. !P! is the "palette", the alphabet of "paints", or the collection | + | a. The invocation of the kleene star operation |
| − | of "syntactic variables" that is peculiar to the language !C!(!P!). | + | is not reduced to a manifestly finitary form. |
| − | This set of signs is given as follows:
| |
| | | | |
| − | !P! = {p_j : j in J}. | + | b. The type of a sentence S is allowed to cover |
| | + | not only itself but also the empty string. |
| | | | |
| − | The easiest way to define the language !C!(!P!) is to indicate the general sorts
| + | I will discuss these issues at first in general, and especially in regard to |
| − | of operations that suffice to construct the greater share of its sentences from
| + | how the two features interact with one another, and then I return to address |
| − | the specified few of its sentences that require a special election. In accord | + | in further detail the questions that they engender on their individual bases. |
| − | with this manner of proceeding, I introduce a family of operations on strings | |
| − | of !A!* that are called "syntactic connectives". If the strings on which
| |
| − | they operate are exclusively sentences of !C!(!P!), then these operations
| |
| − | are tantamount to "sentential connectives", and if the syntactic sentences,
| |
| − | considered as abstract strings of meaningless signs, are given a semantics
| |
| − | in which they denote propositions, considered as indicator functions over | |
| − | some universe, then these operations amount to "propositional connectives".
| |
| | | | |
| − | NB. In this transcription, the symbols "-(" and ")-"
| + | In the process of developing a grammar for a language, it is possible |
| − | will serve for the logically significant parentheses.
| + | to notice a number of organizational, pragmatic, and stylistic questions, |
| | + | whose moment to moment answers appear to decide the ongoing direction of the |
| | + | grammar that develops and the impact of whose considerations work in tandem |
| | + | to determine, or at least to influence, the sort of grammar that turns out. |
| | + | The issues that I can see arising at this point I can give the following |
| | + | prospective names, putting off the discussion of their natures and the |
| | + | treatment of their details to the points in the development of the |
| | + | present example where they evolve their full import. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | 1. The "degree of intermediate organization" in a grammar. |
| | | | |
| − | IDS. Note 143
| + | 2. The "distinction between empty and significant strings", and thus |
| | + | the "distinction between empty and significant types of strings". |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | 3. The "principle of intermediate significance". This is a constraint |
| | + | on the grammar that arises from considering the interaction of the |
| | + | first two issues. |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | In responding to these issues, it is advisable at first to proceed in |
| | + | a stepwise fashion, all the better thereby to accommodate the chances |
| | + | of pursuing a series of parallel developments in the grammar, to allow |
| | + | for the possibility of reversing many steps in its development, indeed, |
| | + | to take into account the near certain necessity of having to revisit, |
| | + | to revise, and to reverse many decisions about how to proceed toward |
| | + | an optimal description or a satisfactory grammar for the language. |
| | + | Doing all this means exploring the effects of various alterations |
| | + | and innovations as independently from each other as possible. |
| | | | |
| − | The discussion that follows is intended to serve a dual purpose, | + | The degree of intermediate organization in a grammar is measured by how many |
| − | in its specific focus presenting the family of cactus languages
| + | intermediate symbols it has and by how they interact with each other by means |
| − | with some degree of detail, but more generally and peripherally
| + | of its productions. With respect to this issue, Grammar 1 has no intermediate |
| − | developing the subject material and demonstrating the technical
| + | symbols at all, !Q! = {}, and therefore remains at an ostensibly trivial degree |
| − | methodology of formal languages and grammars. I will do this by
| + | of intermediate organization. Some additions to the list of intermediate symbols |
| − | taking up a particular method of "stepwise refinement" and using
| + | are practically obligatory in order to arrive at any reasonable grammar at all, |
| − | it to extract a rigorous formal grammar for the cactus language,
| + | other inclusions appear to have a more optional character, though obviously |
| − | starting with little more than a rough description of the target
| + | useful from the standpoints of clarity and ease of comprehension. |
| − | language and applying a systematic analysis to develop a series
| |
| − | of increasingly more effective and more exact approximations to | |
| − | the desired form of grammar.
| |
| | | | |
| − | Rather than presenting the most concise description of these languages
| + | One of the troubles that is perceived to affect Grammar 1 is that it wastes |
| − | right from the beginning, it serves comprehension to develop a picture
| + | so much of the available potential for efficient description in recounting |
| − | of their forms in gradual stages, starting from the most natural ways | + | over and over again the simple fact that the empty string is present in |
| − | of viewing their elements, if somewhat at a distance, and working
| + | the language. This arises in part from the statement that S :> S*, |
| − | through the most easily grasped impressions of their structures,
| + | which implies that: |
| − | if not always the sharpest acquaintances with their details.
| |
| | | | |
| − | The first step is to define two sets of basic operations on strings of !A!*.
| + | S :> S* = %e% |_| S |_| S · S |_| S · S · S |_| ... |
| | | | |
| − | 1. The "concatenation" of one string z_1 is just the string z_1.
| + | There is nothing wrong with the more expansive pan of the covered equation, |
| | + | since it follows straightforwardly from the definition of the kleene star |
| | + | operation, but the covering statement, to the effect that S :> S*, is not |
| | + | necessarily a very productive piece of information, to the extent that it |
| | + | does always tell us very much about the language that is being supposed to |
| | + | fall under the type of a sentence S. In particular, since it implies that |
| | + | S :> %e%, and since !L! = %e%·!L! = !L!·%e%, for any formal language !L!, |
| | + | the empty string !e! = "" is counted over and over in every term of the union, |
| | + | and every non-empty sentence under S appears again and again in every term of |
| | + | the union that follows the initial appearance of S. As a result, this style |
| | + | of characterization has to be classified as "true but not very informative". |
| | + | If at all possible, one prefers to partition the language of interest into |
| | + | a disjoint union of subsets, thereby accounting for each sentence under |
| | + | its proper term, and one whose place under the sum serves as a useful |
| | + | parameter of its character or its complexity. In general, this form |
| | + | of description is not always possible to achieve, but it is usually |
| | + | worth the trouble to actualize it whenever it is. |
| | | | |
| − | The "concatenation" of two strings z_1, z_2 is the string z_1 · z_2.
| + | Suppose that one tries to deal with this problem by eliminating each use of |
| | + | the kleene star operation, by reducing it to a purely finitary set of steps, |
| | + | or by finding an alternative way to cover the sublanguage that it is used to |
| | + | generate. This amounts, in effect, to "recognizing a type", a complex process |
| | + | that involves the following steps: |
| | | | |
| − | The "concatenation" of the k strings z_j, for j = 1 to k,
| + | 1. Noticing a category of strings that |
| | + | is generated by iteration or recursion. |
| | | | |
| − | is the string of the form z_1 · ... · z_k. | + | 2. Acknowledging the circumstance that the noted category |
| | + | of strings needs to be covered by a non-terminal symbol. |
| | | | |
| − | 2. The "surcatenation" of one string z_1 is the string "-(" · z_1 · ")-". | + | 3. Making a note of it by declaring and instituting |
| | + | an explicitly and even expressively named category. |
| | | | |
| − | The "surcatenation" of two strings z_1, z_2 is "-(" · z_1 · "," · z_2 · ")-".
| + | In sum, one introduces a non-terminal symbol for each type of sentence and |
| | + | each "part of speech" or sentential component that is generated by means of |
| | + | iteration or recursion under the ruling constraints of the grammar. In order |
| | + | to do this one needs to analyze the iteration of each grammatical operation in |
| | + | a way that is analogous to a mathematically inductive definition, but further in |
| | + | a way that is not forced explicitly to recognize a distinct and separate type of |
| | + | expression merely to account for and to recount every increment in the parameter |
| | + | of iteration. |
| | | | |
| − | The "surcatenation" of k strings z_j, for j = 1 to k,
| + | Returning to the case of the cactus language, the process of recognizing an |
| | + | iterative type or a recursive type can be illustrated in the following way. |
| | + | The operative phrases in the simplest sort of recursive definition are its |
| | + | initial part and its generic part. For the cactus language !C!(!P!), one |
| | + | has the following definitions of concatenation as iterated precatenation |
| | + | and of surcatenation as iterated subcatenation, respectively: |
| | | | |
| − | is the string of the form "-(" · z_1 · "," · ... · "," · z_k · ")-". | + | 1. Conc^0 = "" |
| | | | |
| − | These definitions can be rendered a little more succinct by
| + | Conc^k_j S_j = Prec(Conc^(k-1)_j S_j, S_k) |
| − | defining the following set of generic operators on strings:
| |
| | | | |
| − | 1. The "concatenation" Conc^k of the k strings z_j, | + | 2. Surc^0 = "-()-" |
| − | for j = 1 to k, is defined recursively as follows: | |
| | | | |
| − | a. Conc^1_j z_j = z_1. | + | Surc^k_j S_j = Subc(Surc^(k-1)_j S_j, S_k) |
| | | | |
| − | b. For k > 1,
| + | In order to transform these recursive definitions into grammar rules, |
| | + | one introduces a new pair of intermediate symbols, "Conc" and "Surc", |
| | + | corresponding to the operations that share the same names but ignoring |
| | + | the inflexions of their individual parameters j and k. Recognizing the |
| | + | type of a sentence by means of the initial symbol "S", and interpreting |
| | + | "Conc" and "Surc" as names for the types of strings that are generated |
| | + | by concatenation and by surcatenation, respectively, one arrives at |
| | + | the following transformation of the ruling operator definitions |
| | + | into the form of covering grammar rules: |
| | | | |
| − | Conc^k_j z_j = (Conc^(k-1)_j z_j) · z_k.
| + | 1. Conc :> "" |
| | | | |
| − | 2. The "surcatenation" Surc^k of the k strings z_j,
| + | Conc :> Conc · S |
| − | for j = 1 to k, is defined recursively as follows:
| |
| | | | |
| − | a. Surc^1_j z_j = "-(" · z_1 · ")-".
| + | 2. Surc :> "-()-" |
| | | | |
| − | b. For k > 1, | + | Surc :> "-(" · S · ")-" |
| | | | |
| − | Surc^k_j z_j = (Surc^(k-1)_j z_j) · ")-"^(-1) · "," · z_k · ")-".
| + | Surc :> Surc ")-"^(-1) · "," · S · ")-" |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | As given, this particular fragment of the intended grammar |
| | + | contains a couple of features that are desirable to amend. |
| | | | |
| − | IDS. Note 144
| + | 1. Given the covering S :> Conc, the covering rule Conc :> Conc · S |
| | + | says no more than the covering rule Conc :> S · S. Consequently, |
| | + | all of the information contained in these two covering rules is |
| | + | already covered by the statement that S :> S · S. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | 2. A grammar rule that invokes a notion of decatenation, deletion, erasure, |
| | + | or any other sort of retrograde production, is frequently considered to |
| | + | be lacking in elegance, and a there is a style of critique for grammars |
| | + | that holds it preferable to avoid these types of operations if it is at |
| | + | all possible to do so. Accordingly, contingent on the prescriptions of |
| | + | the informal rule in question, and pursuing the stylistic dictates that |
| | + | are writ in the realm of its aesthetic regime, it becomes necessary for |
| | + | us to backtrack a little bit, to temporarily withdraw the suggestion of |
| | + | employing these elliptical types of operations, but without, of course, |
| | + | eliding the record of doing so. |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | One way to analyze the surcatenation of any number of sentences is to |
| | + | introduce an auxiliary type of string, not in general a sentence, but |
| | + | a proper component of any sentence that is formed by surcatenation. |
| | + | Doing this brings one to the following definition: |
| | | | |
| − | The definitions of the foregoing syntactic operations can now be organized in
| + | A "tract" is a concatenation of a finite sequence of sentences, with a |
| − | a slightly better fashion, for the sake of both conceptual and computational
| + | literal comma "," interpolated between each pair of adjacent sentences. |
| − | purposes, by making a few additional conventions and auxiliary definitions.
| + | Thus, a typical tract T takes the form: |
| | | | |
| − | 1. The conception of the k-place concatenation operation | + | T = S_1 · "," · ... · "," · S_k |
| − | can be extended to include its natural "prequel":
| |
| | | | |
| − | Conc^0 = "" = the empty string.
| + | A tract must be distinguished from the abstract sequence of sentences, |
| | + | S_1, ..., S_k, where the commas that appear to come to mind, as if being |
| | + | called up to separate the successive sentences of the sequence, remain as |
| | + | partially abstract conceptions, or as signs that retain a disengaged status |
| | + | on the borderline between the text and the mind. In effect, the types of |
| | + | commas that appear to follow in the abstract sequence continue to exist |
| | + | as conceptual abstractions and fail to be cognized in a wholly explicit |
| | + | fashion, whether as concrete tokens in the object language, or as marks |
| | + | in the strings of signs that are able to engage one's parsing attention. |
| | | | |
| − | Next, the construction of the k-place concatenation can be
| + | Returning to the case of the painted cactus language !L! = !C!(!P!), |
| − | broken into stages by means of the following conceptions:
| + | it is possible to put the currently assembled pieces of a grammar |
| | + | together in the light of the presently adopted canons of style, |
| | + | to arrive a more refined analysis of the fact that the concept |
| | + | of a sentence covers any concatenation of sentences and any |
| | + | surcatenation of sentences, and so to obtain the following |
| | + | form of a grammar: |
| | | | |
| − | a. The "precatenation" Prec(z_1, z_2) of the two strings
| + | o-------------------------------------------------o |
| − | z_1, z_2 is the string that is defined as follows:
| + | | !C!(!P!). Grammar 2 !Q! = {"T"} | |
| | + | o-------------------------------------------------o |
| | + | | | |
| | + | | 1. S :> !e! | |
| | + | | | |
| | + | | 2. S :> m_1 | |
| | + | | | |
| | + | | 3. S :> p_j, for each j in J | |
| | + | | | |
| | + | | 4. S :> S · S | |
| | + | | | |
| | + | | 5. S :> "-(" · T · ")-" | |
| | + | | | |
| | + | | 6. T :> S | |
| | + | | | |
| | + | | 7. T :> T · "," · S | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | | | |
| − | Prec(z_1, z_2) = z_1 · z_2.
| + | In this rendition, a string of type T is not in general |
| | + | a sentence itself but a proper "part of speech", that is, |
| | + | a strictly "lesser" component of a sentence in any suitable |
| | + | ordering of sentences and their components. In order to see |
| | + | how the grammatical category T gets off the ground, that is, |
| | + | to detect its minimal strings and to discover how its ensuing |
| | + | generations gets started from these, it is useful to observe |
| | + | that the covering rule T :> S means that T "inherits" all of |
| | + | the initial conditions of S, namely, T :> !e!, m_1, p_j. |
| | + | In accord with these simple beginnings it comes to parse |
| | + | that the rule T :> T · "," · S, with the substitutions |
| | + | T = !e! and S = !e! on the covered side of the rule, |
| | + | bears the germinal implication that T :> ",". |
| | | | |
| − | b. The "concatenation" of the k strings z_1, ..., z_k can now be
| + | Grammar 2 achieves a portion of its success through a higher degree of |
| − | defined as an iterated precatenation over the sequence of k+1
| + | intermediate organization. Roughly speaking, the level of organization |
| − | strings that begins with the string z_0 = Conc^0 = "" and then
| + | can be seen as reflected in the cardinality of the intermediate alphabet |
| − | continues on through the other k strings:
| + | !Q! = {"T"}, but it is clearly not explained by this simple circumstance |
| | + | alone, since it is taken for granted that the intermediate symbols serve |
| | + | a purpose, a purpose that is easily recognizable but that may not be so |
| | + | easy to pin down and to specify exactly. Nevertheless, it is worth the |
| | + | trouble of exploring this aspect of organization and this direction of |
| | + | development a little further. Although it is not strictly necessary |
| | + | to do so, it is possible to organize the materials of the present |
| | + | grammar in a slightly better fashion by recognizing two recurrent |
| | + | types of strings that appear in the typical cactus expression. |
| | + | In doing this, one arrives at the following two definitions: |
| | | | |
| − | i. Conc^0_j z_j = Conc^0 = "".
| + | A "rune" is a string of blanks and paints concatenated together. |
| | + | Thus, a typical rune R is a string over {m_1} |_| !P!, possibly |
| | + | the empty string. |
| | | | |
| − | ii. For k > 0,
| + | R in ({m_1} |_| !P!)*. |
| | | | |
| − | Conc^k_j z_j = Prec(Conc^(k-1)_j z_j, z_k).
| + | When there is no possibility of confusion, the letter "R" can be used |
| − | | + | either as a string variable that ranges over the set of runes or else |
| − | 2. The conception of the k-place surcatenation operation
| + | as a type name for the class of runes. The latter reading amounts to |
| − | can be extended to include its natural "prequel":
| + | the enlistment of a fresh intermediate symbol, "R" in !Q!, as a part |
| | + | of a new grammar for !C!(!P!). In effect, "R" affords a grammatical |
| | + | recognition for any rune that forms a part of a sentence in !C!(!P!). |
| | + | In situations where these variant usages are likely to be confused, |
| | + | the types of strings can be indicated by means of expressions like |
| | + | "r <: R" and "W <: R". |
| | | | |
| − | Surc^0 = "-()-".
| + | A "foil" is a string of the form "-(" · T · ")-", where T is a tract. |
| | + | Thus, a typical foil F has the form: |
| | | | |
| − | Finally, the construction of the k-place surcatenation can be
| + | F = "-(" · S_1 · "," · ... · "," · S_k · ")-". |
| − | broken into stages by means of the following conceptions:
| |
| | | | |
| − | a. A "subclause" in !A!* is a string that ends with a ")-".
| + | This is just the surcatenation of the sentences S_1, ..., S_k. |
| | + | Given the possibility that this sequence of sentences is empty, |
| | + | and thus that the tract T is the empty string, the minimum foil |
| | + | F is the expression "-()-". Explicitly marking each foil F that |
| | + | is embodied in a cactus expression is tantamount to recognizing |
| | + | another intermediate symbol, "F" in !Q!, further articulating the |
| | + | structures of sentences and expanding the grammar for the language |
| | + | !C!(!P!). All of the same remarks about the versatile uses of the |
| | + | intermediate symbols, as string variables and as type names, apply |
| | + | again to the letter "F". |
| | | | |
| − | b. The "subcatenation" Subc(z_1, z_2)
| + | o-------------------------------------------------o |
| − | of a subclause z_1 by a string z_2 is
| + | | !C!(!P!). Grammar 3 !Q! = {"F", "R", "T"} | |
| − | the string that is defined as follows:
| + | o-------------------------------------------------o |
| | + | | | |
| | + | | 1. S :> R | |
| | + | | | |
| | + | | 2. S :> F | |
| | + | | | |
| | + | | 3. S :> S · S | |
| | + | | | |
| | + | | 4. R :> !e! | |
| | + | | | |
| | + | | 5. R :> m_1 | |
| | + | | | |
| | + | | 6. R :> p_j, for each j in J | |
| | + | | | |
| | + | | 7. R :> R · R | |
| | + | | | |
| | + | | 8. F :> "-(" · T · ")-" | |
| | + | | | |
| | + | | 9. T :> S | |
| | + | | | |
| | + | | 10. T :> T · "," · S | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | | | |
| − | Subc(z_1, z_2) = z_1 · ")-"^(-1) · "," · z_2 · ")-".
| + | In Grammar 3, the first three Rules say that a sentence (a string of type S), |
| | + | is a rune (a string of type R), a foil (a string of type F), or an arbitrary |
| | + | concatenation of strings of these two types. Rules 4 through 7 specify that |
| | + | a rune R is an empty string !e! = "", a blank symbol m_1 = " ", a paint p_j, |
| | + | for j in J, or any concatenation of strings of these three types. Rule 8 |
| | + | characterizes a foil F as a string of the form "-(" · T · ")-", where T is |
| | + | a tract. The last two Rules say that a tract T is either a sentence S or |
| | + | else the concatenation of a tract, a comma, and a sentence, in that order. |
| | | | |
| − | c. The "surcatenation" of the k strings z_1, ..., z_k can now be
| + | At this point in the succession of grammars for !C!(!P!), the explicit |
| − | defined as an iterated subcatenation over the sequence of k+1
| + | uses of indefinite iterations, like the kleene star operator, are now |
| − | strings that starts with the string z_0 = Surc^0 = "-()-" and
| + | completely reduced to finite forms of concatenation, but the problems |
| − | then continues on through the other k strings:
| + | that some styles of analysis have with allowing non-terminal symbols |
| | + | to cover both themselves and the empty string are still present. |
| | | | |
| − | i. Surc^0_j z_j = Surc^0 = "-()-".
| + | Any degree of reflection on this difficulty raises the general question: |
| | + | What is a practical strategy for accounting for the empty string in the |
| | + | organization of any formal language that counts it among its sentences? |
| | + | One answer that presents itself is this: If the empty string belongs to |
| | + | a formal language, it suffices to count it once at the beginning of the |
| | + | formal account that enumerates its sentences and then to move on to more |
| | + | interesting materials. |
| | | | |
| − | ii. For k > 0,
| + | Returning to the case of the cactus language !C!(!P!), that is, |
| | + | the formal language of "painted and rooted cactus expressions", |
| | + | it serves the purpose of efficient accounting to partition the |
| | + | language PARCE into the following couple of sublanguages: |
| | | | |
| − | Surc^k_j z_j = Subc(Surc^(k-1)_j z_j, z_k).
| + | 1. The "emptily painted and rooted cactus expressions" |
| | + | make up the language EPARCE that consists of |
| | + | a single empty string as its only sentence. |
| | + | In short: |
| | | | |
| − | Notice that the expressions Conc^0_j z_j and Surc^0_j z_j
| + | EPARCE = {""}. |
| − | are defined in such a way that the respective operators
| |
| − | Conc^0 and Surc^0 basically "ignore", in the manner of
| |
| − | constant functions, whatever sequences of strings z_j
| |
| − | may happen to be listed as their ostensible arguments.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | 2. The "significantly painted and rooted cactus expressions" |
| | + | make up the language SPARCE that consists of everything else, |
| | + | namely, all of the non-empty strings in the language PARCE. |
| | + | In sum: |
| | | | |
| − | IDS. Note 145
| + | SPARCE = PARCE \ "". |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | As a result of marking the distinction between empty and significant sentences, |
| | + | that is, by categorizing each of these three classes of strings as an entity |
| | + | unto itself and by conceptualizing the whole of its membership as falling |
| | + | under a distinctive symbol, one obtains an equation of sets that connects |
| | + | the three languages being marked: |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | SPARCE = PARCE - EPARCE. |
| | | | |
| − | Having defined the basic operations of concatenation and surcatenation
| + | In sum, one has the disjoint union: |
| − | on arbitrary strings, in effect, giving them operational meaning for
| |
| − | the all-inclusive language !L! = !A!*, it is time to adjoin the
| |
| − | notion of a more discriminating grammaticality, in other words,
| |
| − | a more properly restrictive concept of a sentence.
| |
| | | | |
| − | If !L! is an arbitrary formal language over an alphabet of the sort that
| + | PARCE = EPARCE |_| SPARCE. |
| − | we are talking about, that is, an alphabet of the form !A! = !M! |_| !P!,
| |
| − | then there are a number of basic structural relations that can be defined
| |
| − | on the strings of !L!.
| |
| | | | |
| − | 1. z is the "concatenation" of z_1 and z_2 in !L! if and only if
| + | For brevity in the present case, and to serve as a generic device |
| | + | in any similar array of situations, let the symbol "S" be used to |
| | + | signify the type of an arbitrary sentence, possibly empty, whereas |
| | + | the symbol "S'" is reserved to designate the type of a specifically |
| | + | non-empty sentence. In addition, let the symbol "%e%" be employed |
| | + | to indicate the type of the empty sentence, in effect, the language |
| | + | %e% = {""} that contains a single empty string, and let a plus sign |
| | + | "+" signify a disjoint union of types. In the most general type of |
| | + | situation, where the type S is permitted to include the empty string, |
| | + | one notes the following relation among types: |
| | | | |
| − | z_1 is a sentence of !L!, z_2 is a sentence of !L!, and
| + | S = %e% + S'. |
| | | | |
| − | z = z_1 · z_2.
| + | Consequences of the distinction between empty expressions and |
| | + | significant expressions are taken up for discussion next time. |
| | | | |
| − | 2. z is the "concatenation" of the k strings z1, ..., z_k in !L!,
| + | With the distinction between empty and significant expressions in mind, |
| − | | + | I return to the grasp of the cactus language !L! = !C!(!P!) = PARCE(!P!) |
| − | if and only if z_j is a sentence of !L!, for all j = 1 to k, and
| + | that is afforded by Grammar 2, and, taking that as a point of departure, |
| | + | explore other avenues of possible improvement in the comprehension of |
| | + | these expressions. In order to observe the effects of this alteration |
| | + | as clearly as possible, in isolation from any other potential factors, |
| | + | it is useful to strip away the higher levels intermediate organization |
| | + | that are present in Grammar 3, and start again with a single intermediate |
| | + | symbol, as used in Grammar 2. One way of carrying out this strategy leads |
| | + | on to a grammar of the variety that will be articulated next. |
| | | | |
| − | z = Conc^k_j z_j = z_1 · ... · z_k.
| + | If one imposes the distinction between empty and significant types on |
| | + | each non-terminal symbol in Grammar 2, then the non-terminal symbols |
| | + | "S" and "T" give rise to the non-terminal symbols "S", "S'", "T", "T'", |
| | + | leaving the last three of these to form the new intermediate alphabet. |
| | + | Grammar 4 has the intermediate alphabet !Q! = {"S'", "T", "T'"}, with |
| | + | the set !K! of covering production rules as listed in the next display. |
| | | | |
| − | 3. z is the "discatenation" of z_1 by t if and only if
| + | o-------------------------------------------------o |
| | + | | !C!(!P!). Grammar 4 !Q! = {"S'", "T", "T'"} | |
| | + | o-------------------------------------------------o |
| | + | | | |
| | + | | 1. S :> !e! | |
| | + | | | |
| | + | | 2. S :> S' | |
| | + | | | |
| | + | | 3. S' :> m_1 | |
| | + | | | |
| | + | | 4. S' :> p_j, for each j in J | |
| | + | | | |
| | + | | 5. S' :> "-(" · T · ")-" | |
| | + | | | |
| | + | | 6. S' :> S' · S' | |
| | + | | | |
| | + | | 7. T :> !e! | |
| | + | | | |
| | + | | 8. T :> T' | |
| | + | | | |
| | + | | 9. T' :> T · "," · S | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | | | |
| − | z_1 is a sentence of !L!, t is an element of !A!, and
| + | In this version of a grammar for !L! = !C!(!P!), the intermediate type T |
| | + | is partitioned as T = %e% + T', thereby parsing the intermediate symbol T |
| | + | in parallel fashion with the division of its overlying type as S = %e% + S'. |
| | + | This is an option that I will choose to close off for now, but leave it open |
| | + | to consider at a later point. Thus, it suffices to give a brief discussion |
| | + | of what it involves, in the process of moving on to its chief alternative. |
| | | | |
| − | z_1 = z · t.
| + | There does not appear to be anything radically wrong with trying this |
| | + | approach to types. It is reasonable and consistent in its underlying |
| | + | principle, and it provides a rational and a homogeneous strategy toward |
| | + | all parts of speech, but it does require an extra amount of conceptual |
| | + | overhead, in that every non-trivial type has to be split into two parts |
| | + | and comprehended in two stages. Consequently, in view of the largely |
| | + | practical difficulties of making the requisite distinctions for every |
| | + | intermediate symbol, it is a common convention, whenever possible, to |
| | + | restrict intermediate types to covering exclusively non-empty strings. |
| | | | |
| − | When this is the case, one more commonly writes:
| + | For the sake of future reference, it is convenient to refer to this restriction |
| | + | on intermediate symbols as the "intermediate significance" constraint. It can |
| | + | be stated in a compact form as a condition on the relations between non-terminal |
| | + | symbols q in {"S"} |_| !Q! and sentential forms W in {"S"} |_| (!Q! |_| !A!)*. |
| | | | |
| − | z = z_1 · t^-1.
| + | o-------------------------------------------------o |
| − | | + | | Condition On Intermediate Significance | |
| − | 4. z is a "subclause" of !L! if and only if
| + | o-------------------------------------------------o |
| − | | + | | | |
| − | z is a sentence of !L! and z ends with a ")-".
| + | | If q :> W | |
| | + | | | |
| | + | | and W = !e! | |
| | + | | | |
| | + | | then q = "S" | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | | | |
| − | 5. z is the "subcatenation" of z_1 by z_2 if and only if
| + | If this is beginning to sound like a monotone condition, then it is |
| | + | not absurd to sharpen the resemblance and render the likeness more |
| | + | acute. This is done by declaring a couple of ordering relations, |
| | + | denoting them under variant interpretations by the same sign "<". |
| | | | |
| − | z_1 is a subclause of !L!, z_2 is a sentence of !L!, and | + | 1. The ordering "<" on the set of non-terminal symbols, |
| | + | q in {"S"} |_| !Q!, ordains the initial symbol "S" |
| | + | to be strictly prior to every intermediate symbol. |
| | + | This is tantamount to the axiom that "S" < q, |
| | + | for all q in !Q!. |
| | | | |
| − | z = z_1 · ")-"^(-1) · "," · z_2 · ")-".
| + | 2. The ordering "<" on the collection of sentential forms, |
| | + | W in {"S"} |_| (!Q! |_| !A!)*, ordains the empty string |
| | + | to be strictly minor to every other sentential form. |
| | + | This is stipulated in the axiom that !e! < W, |
| | + | for every non-empty sentential form W. |
| | | | |
| − | 6. z is the "surcatenation" of the k strings z_1, ..., z_k in !L!,
| + | Given these two orderings, the constraint in question |
| | + | on intermediate significance can be stated as follows: |
| | | | |
| − | if and only if z_j is a sentence of !L!, for all j = 1 to k, and
| + | o-------------------------------------------------o |
| | + | | Condition Of Intermediate Significance | |
| | + | o-------------------------------------------------o |
| | + | | | |
| | + | | If q :> W | |
| | + | | | |
| | + | | and q > "S" | |
| | + | | | |
| | + | | then W > !e! | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | | | |
| − | z = Surc^k_j z_j = "-(" · z_1 · "," · ... · "," · z_k · ")-".
| + | Achieving a grammar that respects this convention typically requires a more |
| | + | detailed account of the initial setting of a type, both with regard to the |
| | + | type of context that incites its appearance and also with respect to the |
| | + | minimal strings that arise under the type in question. In order to find |
| | + | covering productions that satisfy the intermediate significance condition, |
| | + | one must be prepared to consider a wider variety of calling contexts or |
| | + | inciting situations that can be noted to surround each recognized type, |
| | + | and also to enumerate a larger number of the smallest cases that can |
| | + | be observed to fall under each significant type. |
| | | | |
| − | The converses of these decomposition relations are tantamount to the
| + | With the array of foregoing considerations in mind, |
| − | corresponding forms of composition operations, making it possible for
| + | one is gradually led to a grammar for !L! = !C!(!P!) |
| − | these complementary forms of analysis and synthesis to articulate the
| + | in which all of the covering productions have either |
| − | structures of strings and sentences in two directions.
| + | one of the following two forms: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | 1. S :> !e! |
| | | | |
| − | IDS. Note 146
| + | 2. q :> W, with q in {"S"} |_| !Q!, and W in (!Q! |_| !A!)^+ |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | A grammar that fits into this mold is called a "context-free" grammar. |
| − | | + | The first type of rewrite rule is referred to as a "special production", |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | while the second type of rewrite rule is called an "ordinary production". |
| − | | + | An "ordinary derivation" is one that employs only ordinary productions. |
| − | The "painted cactus language" with paints in the
| + | In ordinary productions, those that have the form q :> W, the replacement |
| − | set !P! = {p_j : j in J} is the formal language
| + | string W is never the empty string, and so the lengths of the augmented |
| − | !L! = !C!(!P!) c !A!* = (!M! |_| !P!)* that is
| + | strings or the sentential forms that follow one another in an ordinary |
| − | defined as follows:
| + | derivation, on account of using the ordinary types of rewrite rules, |
| | + | never decrease at any stage of the process, up to and including the |
| | + | terminal string that is finally generated by the grammar. This type |
| | + | of feature is known as the "non-contracting property" of productions, |
| | + | derivations, and grammars. A grammar is said to have the property if |
| | + | all of its covering productions, with the possible exception of S :> e, |
| | + | are non-contracting. In particular, context-free grammars are special |
| | + | cases of non-contracting grammars. The presence of the non-contracting |
| | + | property within a formal grammar makes the length of the augmented string |
| | + | available as a parameter that can figure into mathematical inductions and |
| | + | motivate recursive proofs, and this handle on the generative process makes |
| | + | it possible to establish the kinds of results about the generated language |
| | + | that are not easy to achieve in more general cases, nor by any other means |
| | + | even in these brands of special cases. |
| | | | |
| − | PC 1. The blank symbol m_1 is a sentence.
| + | Grammar 5 is a context-free grammar for the painted cactus language |
| | + | that uses !Q! = {"S'", "T"}, with !K! as listed in the next display. |
| | | | |
| − | PC 2. The paint p_j is a sentence, for each j in J.
| + | o-------------------------------------------------o |
| | + | | !C!(!P!). Grammar 5 !Q! = {"S'", "T"} | |
| | + | o-------------------------------------------------o |
| | + | | | |
| | + | | 1. S :> !e! | |
| | + | | | |
| | + | | 2. S :> S' | |
| | + | | | |
| | + | | 3. S' :> m_1 | |
| | + | | | |
| | + | | 4. S' :> p_j, for each j in J | |
| | + | | | |
| | + | | 5. S' :> S' · S' | |
| | + | | | |
| | + | | 6. S' :> "-()-" | |
| | + | | | |
| | + | | 7. S' :> "-(" · T · ")-" | |
| | + | | | |
| | + | | 8. T :> "," | |
| | + | | | |
| | + | | 9. T :> S' | |
| | + | | | |
| | + | | 10. T :> T · "," | |
| | + | | | |
| | + | | 11. T :> T · "," · S' | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | | | |
| − | PC 3. Conc^0 and Surc^0 are sentences.
| + | Finally, it is worth trying to bring together the advantages of these |
| | + | diverse styles of grammar, to whatever extent that they are compatible. |
| | + | To do this, a prospective grammar must be capable of maintaining a high |
| | + | level of intermediate organization, like that arrived at in Grammar 2, |
| | + | while respecting the principle of intermediate significance, and thus |
| | + | accumulating all the benefits of the context-free format in Grammar 5. |
| | + | A plausible synthesis of most of these features is given in Grammar 6. |
| | | | |
| − | PC 4. For each positive integer k,
| + | o-----------------------------------------------------------o |
| − | | + | | !C!(!P!). Grammar 6 !Q! = {"S'", "R", "F", "T"} | |
| − | if z_1, ..., z_k are sentences,
| + | o-----------------------------------------------------------o |
| − | | + | | | |
| − | then Conc^k_j z_j is a sentence,
| + | | 1. S :> !e! | |
| − | | + | | | |
| − | and Surc^k_j z_j is a sentence.
| + | | 2. S :> S' | |
| − | | + | | | |
| − | As usual, saying that z is a sentence is just a conventional way of
| + | | 3. S' :> R | |
| − | stating that the string z belongs to the relevant formal language !L!.
| + | | | |
| − | An individual sentence of !C!(!P!), for any palette !P!, is referred to
| + | | 4. S' :> F | |
| − | as a "painted and rooted cactus expression" (PARCE) on the palette !P!,
| + | | | |
| − | or a "cactus expression", for short. Anticipating the forms that the
| + | | 5. S' :> S' · S' | |
| − | parse graphs of these PARCE's will take, to be described in the next
| + | | | |
| − | Subsection, the language !L! = !C!(!P!) is also described as the
| + | | 6. R :> m_1 | |
| − | set PARCE(!P!) of PARCE's on the palette !P!, more generically,
| + | | | |
| − | as the PARCE's that constitute the language PARCE.
| + | | 7. R :> p_j, for each j in J | |
| | + | | | |
| | + | | 8. R :> R · R | |
| | + | | | |
| | + | | 9. F :> "-()-" | |
| | + | | | |
| | + | | 10. F :> "-(" · T · ")-" | |
| | + | | | |
| | + | | 11. T :> "," | |
| | + | | | |
| | + | | 12. T :> S' | |
| | + | | | |
| | + | | 13. T :> T · "," | |
| | + | | | |
| | + | | 14. T :> T · "," · S' | |
| | + | | | |
| | + | o-----------------------------------------------------------o |
| | + | |
| | + | The preceding development provides a typical example of how an initially |
| | + | effective and conceptually succinct description of a formal language, but |
| | + | one that is terse to the point of allowing its prospective interpreter to |
| | + | waste exorbitant amounts of energy in trying to unravel its implications, |
| | + | can be converted into a form that is more efficient from the operational |
| | + | point of view, even if slightly more ungainly in regard to its elegance. |
| | | | |
| − | A "bare" PARCE, a bit loosely referred to as a "bare cactus expression",
| + | The basic idea behind all of this grammatical machinery remains the same: |
| − | is a PARCE on the empty palette !P! = {}. A bare PARCE is a sentence
| + | Aside from the select body of formulas introduced as boundary conditions, |
| − | in the "bare cactus language", !C!^0 = !C!({}) = PARCE^0 = PARCE({}).
| + | a grammar for the cactus language is nothing more or less than a device |
| − | This set of strings, regarded as a formal language in its own right,
| + | that institutes the following general rule: |
| − | is a sublanguage of every cactus language !C!(!P!). A bare cactus
| |
| − | expression is commonly encountered in practice when one has occasion
| |
| − | to start with an arbitrary PARCE and then finds a reason to delete or
| |
| − | to erase all of its paints.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | If the strings S_1, ..., S_k are sentences, |
| | | | |
| − | IDS. Note 147
| + | then their concatenation in the form |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | Conc^k_j S_j = S_1 · ... · S_k |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | is a sentence, |
| | | | |
| − | Only one thing remains to cast this description of the cactus language
| + | and their surcatenation in the form |
| − | into a form that is commonly found acceptable. As presently formulated,
| |
| − | the principle PC 4 appears to be attempting to define an infinite number
| |
| − | of new concepts all in a single step, at least, it appears to invoke the
| |
| − | indefinitely long sequences of operators, Conc^k and Surc^k, for all k > 0.
| |
| − | As a general rule, one prefers to have an effectively finite description of
| |
| − | conceptual objects, and this means restricting the description to a finite
| |
| − | number of schematic principles, each of which involves a finite number of
| |
| − | schematic effects, that is, a finite number of schemata that explicitly
| |
| − | relate conditions to results.
| |
| | | | |
| − | A start in this direction, taking steps toward an effective description
| + | Surc^k_j S_j = "-(" · S_1 · "," · ... · "," · S_k · ")-" |
| − | of the cactus language, a finitary conception of its membership conditions, | + | |
| − | and a bounded characterization of a typical sentence in the language, can be
| + | is a sentence. |
| − | made by recasting the present description of these expressions into the pattern
| + | |
| − | of what is called, more or less roughly, a "formal grammar". | + | It is fitting to wrap up the foregoing developments by summarizing the |
| | + | notion of a formal grammar that appeared to evolve in the present case. |
| | + | For the sake of future reference and the chance of a wider application, |
| | + | it is also useful to try to extract the scheme of a formalization that |
| | + | potentially holds for any formal language. The following presentation |
| | + | of the notion of a formal grammar is adapted, with minor modifications, |
| | + | from the treatment in (DDQ, 60-61). |
| | | | |
| − | A notation in the style of "S :> T" is now introduced, | + | A "formal grammar" !G! is given by a four-tuple !G! = ("S", !Q!, !A!, !K!) |
| − | to be read among many others in this manifold of ways:
| + | that takes the following form of description: |
| | | | |
| − | S covers T | + | 1. "S" is the "initial", "special", "start", or "sentence symbol". |
| | + | Since the letter "S" serves this function only in a special setting, |
| | + | its employment in this role need not create any confusion with its |
| | + | other typical uses as a string variable or as a sentence variable. |
| | | | |
| − | S governs T | + | 2. !Q! = {q_1, ..., q_m} is a finite set of "intermediate symbols", |
| | + | all distinct from "S". |
| | | | |
| − | S rules T | + | 3. !A! = {a_1, ..., a_n} is a finite set of "terminal symbols", |
| | + | also known as the "alphabet" of !G!, all distinct from "S" and |
| | + | disjoint from !Q!. Depending on the particular conception of the |
| | + | language !L! that is "covered", "generated", "governed", or "ruled" |
| | + | by the grammar !G!, that is, whether !L! is conceived to be a set of |
| | + | words, sentences, paragraphs, or more extended structures of discourse, |
| | + | it is usual to describe !A! as the "alphabet", "lexicon", "vocabulary", |
| | + | "liturgy", or "phrase book" of both the grammar !G! and the language !L! |
| | + | that it regulates. |
| | | | |
| − | S subsumes T | + | 4. !K! is a finite set of "characterizations". Depending on how they |
| | + | come into play, these are variously described as "covering rules", |
| | + | "formations", "productions", "rewrite rules", "subsumptions", |
| | + | "transformations", or "typing rules". |
| | | | |
| − | S types over T
| + | To describe the elements of !K! it helps to define some additional terms: |
| | | | |
| − | The form "S :> T" is here recruited for polymorphic | + | a. The symbols in {"S"} |_| !Q! |_| !A! form the "augmented alphabet" of !G!. |
| − | employment in at least the following types of roles:
| |
| | | | |
| − | 1. To signify that an individually named or quoted string T is | + | b. The symbols in {"S"} |_| !Q! are the "non-terminal symbols" of !G!. |
| − | being typed as a sentence S of the language of interest !L!.
| |
| | | | |
| − | 2. To express the fact or to make the assertion that each member | + | c. The symbols in !Q! |_| !A! are the "non-initial symbols" of !G!. |
| − | of a specified set of strings T c !A!* also belongs to the
| |
| − | syntactic category S, the one that qualifies a string as
| |
| − | being a sentence in the relevant formal language !L!.
| |
| | | | |
| − | 3. To specify the intension or to signify the intention that every | + | d. The strings in ({"S"} |_| !Q! |_| !A!)* are the "augmented strings" for G. |
| − | string that fits the conditions of the abstract type T must also
| |
| − | fall under the grammatical heading of a sentence, as indicated by
| |
| − | the type name "S", all within the target language !L!.
| |
| | | | |
| − | In these types of situation the letter "S", that signifies the type of
| + | e. The strings in {"S"} |_| (!Q! |_| !A!)* are the "sentential forms" for G. |
| − | a sentence in the language of interest, is called the "initial symbol"
| |
| − | or the "sentence symbol" of a candidate formal grammar for the language,
| |
| − | while any number of letters like "T", signifying other types of strings
| |
| − | that are necessary to a reasonable account or a rational reconstruction
| |
| − | of the sentences that belong to the language, are collectively referred
| |
| − | to as "intermediate symbols".
| |
| | | | |
| − | Combining the singleton set {"S"} whose sole member is the initial symbol
| + | Each characterization in !K! is an ordered pair of strings (S_1, S_2) |
| − | with the set !Q! that assembles together all of the intermediate symbols
| + | that takes the following form: |
| − | results in the set {"S"} |_| !Q! of "non-terminal symbols". Completing
| + | |
| − | the package, the alphabet !A! of the language is also known as the set
| + | S_1 = Q_1 · q · Q_2 |
| − | of "terminal symbols". In this discussion, I will adopt the convention | + | |
| − | that !Q! is the set of intermediate symbols, but I will often use "q" | + | S_2 = Q_1 · W · Q_2 |
| − | as a typical variable that ranges over all of the non-terminal symbols,
| |
| − | q in {"S"} |_| !Q!. Finally, it is convenient to refer to all of the | |
| − | symbols in {"S"} |_| !Q! |_| !A! as the "augmented alphabet" of the
| |
| − | prospective grammar for the language, and accordingly to describe
| |
| − | the strings in ({"S"} |_| !Q! |_| !A!)* as the "augmented strings",
| |
| − | in effect, expressing the forms that are superimposed on a language
| |
| − | by one of its conceivable grammars. In certain settings it becomes
| |
| − | desirable to separate the augmented strings that contain the symbol
| |
| − | "S" from all other sorts of augmented strings. In these situations,
| |
| − | the strings in the disjoint union {"S"} |_| (!Q! |_| !A!)* are known
| |
| − | as the "sentential forms" of the associated grammar.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | In this scheme, S_1 and S_2 are members of the augmented strings for !G!, |
| | + | more precisely, S_1 is a non-empty string and a sentential form over !G!, |
| | + | while S_2 is a possibly empty string and also a sentential form over !G!. |
| | | | |
| − | IDS. Note 148
| + | Here also, q is a non-terminal symbol, that is, q is in {"S"} |_| !Q!, |
| | + | while Q_1, Q_2, and W are possibly empty strings of non-initial symbols, |
| | + | a fact that can be expressed in the form: Q_1, Q_2, W in (!Q! |_| !A!)*. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | In practice, the ordered pairs of strings in !K! are used to "derive", |
| | + | to "generate", or to "produce" sentences of the language !L! = <!G!> |
| | + | that is then said to be "governed" or "regulated" by the grammar !G!. |
| | + | In order to facilitate this active employment of the grammar, it is |
| | + | conventional to write the characterization (S_1, S_2) in either one |
| | + | of the next two forms, where the more generic form is followed by |
| | + | the more specific form: |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | S_1 :> S_2 |
| | | | |
| − | In forming a grammar for a language, statements of the form W :> W', | + | Q_1 · q · Q_2 :> Q_1 · W · Q_2 |
| − | where W and W' are augmented strings or sentential forms of specified
| + | |
| − | types that depend on the style of the grammar that is being sought, are
| + | In this usage, the characterization S_1 :> S_2 is tantamount to a grammatical |
| − | variously known as "characterizations", "covering rules", "productions",
| + | license to transform a string of the form Q_1 · q · Q_2 into a string of the |
| − | "rewrite rules", "subsumptions", "transformations", or "typing rules". | + | form Q1 · W · Q2, in effect, replacing the non-terminal symbol q with the |
| − | These are collected together into a set !K! that serves to complete
| + | non-initial string W in any selected, preserved, and closely adjoining |
| − | the definition of the formal grammar in question.
| + | context of the form Q1 · ... · Q2. Accordingly, in this application |
| | + | the notation "S_1 :> S_2" can be read as "S_1 produces S_2" or as |
| | + | "S_1 transforms into S_2". |
| | | | |
| − | Correlative with the use of this notation, an expression of the
| + | An "immediate derivation" in !G! is an ordered pair (W, W') |
| − | form "T <: S", read as "T is covered by S", can be interpreted
| + | of sentential forms in !G! such that: |
| − | as saying that T is of the type S. Depending on the context,
| |
| − | this can be taken in either one of two ways:
| |
| | | | |
| − | 1. Treating "T" as a string variable, it means | + | W = Q_1 · X · Q_2 |
| − | that the individual string T is typed as S.
| |
| | | | |
| − | 2. Treating "T" as a type name, it means that any | + | W' = Q_1 · Y · Q_2 |
| − | instance of the type T also falls under the type S.
| |
| | | | |
| − | In accordance with these interpretations, an expression like "t <: T" can be
| + | and (X, Y) in !K! |
| − | read in all of the ways that one typically reads an expression like "t : T".
| |
| | | | |
| − | There are several abuses of notation that commonly tolerated in the use
| + | i.e. X :> Y in !G! |
| − | of covering relations. The worst offense is that of allowing symbols to
| |
| − | stand equivocally either for individual strings or else for their types.
| |
| − | There is a measure of consistency to this practice, considering the fact
| |
| − | that perfectly individual entities are rarely if ever grasped by means of
| |
| − | signs and finite expressions, which entails that every appearance of an
| |
| − | apparent token is only a type of more particular tokens, and meaning in
| |
| − | the end that there is never any recourse but to the sort of discerning
| |
| − | interpretation that can decide just how each sign is intended. In view
| |
| − | of all this, I continue to permit expressions like "t <: T" and "T <: S",
| |
| − | where any of the symbols "t", "T", "S" can be taken to signify either the
| |
| − | tokens or the subtypes of their covering types.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | This relation is indicated by saying that W "immediately derives" W', |
| | + | that W' is "immediately derived" from W in !G!, and also by writing: |
| | | | |
| − | IDS. Note 149
| + | W ::> W' |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | A "derivation" in !G! is a finite sequence (W_1, ..., W_k) |
| | + | of sentential forms over !G! such that each adjacent pair |
| | + | (W_j, W_(j+1)) of sentential forms in the sequence is an |
| | + | immediate derivation in !G!, in other words, such that: |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | W_j ::> W_(j+1), for all j = 1 to k-1 |
| | | | |
| − | Employing the notion of a covering relation it becomes possible to
| + | If there exists a derivation (W_1, ..., W_k) in !G!, |
| − | redescribe the cactus language !L! = !C!(!P!) in the following way.
| + | one says that W_1 "derives" W_k in !G!, conversely, |
| | + | that W_k is "derivable" from W_1 in !G!, and one |
| | + | typically summarizes the derivation by writing: |
| | | | |
| − | Grammar 1 is something of a misnomer. It is nowhere near exemplifying
| + | W_1 :*:> W_k |
| − | any kind of a standard form and it is only intended as a starting point
| |
| − | for the initiation of more respectable grammars. Such as it is, it uses
| |
| − | the terminal alphabet !A! = !M! |_| !P! that comes with the territory of
| |
| − | the cactus language !C!(!P!), it specifies !Q! = {}, in other words, it
| |
| − | employs no intermediate symbols, and it embodies the "covering set" !K!
| |
| − | as listed in the following display.
| |
| | | | |
| − | o-------------------------------------------------o
| + | The language !L! = !L!(!G!) = <!G!> that is "generated" |
| − | | !C!(!P!). Grammar 1 !Q! = {} |
| + | by the formal grammar !G! = ("S", !Q!, !A!, !K!) is the |
| − | o-------------------------------------------------o
| + | set of strings over the terminal alphabet !A! that are |
| − | | |
| + | derivable from the initial symbol "S" by way of the |
| − | | 1. S :> m_1 = " " |
| + | intermediate symbols in !Q! according to the |
| − | | |
| + | characterizations in K. In sum: |
| − | | 2. S :> p_j, for each j in J |
| |
| − | | |
| |
| − | | 3. S :> Conc^0 = "" |
| |
| − | | |
| |
| − | | 4. S :> Surc^0 = "-()-" |
| |
| − | | |
| |
| − | | 5. S :> S* |
| |
| − | | |
| |
| − | | 6. S :> "-(" · S · ("," · S)* · ")-" |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| | | | |
| − | In this formulation, the last two lines specify that:
| + | !L!(!G!) = <!G!> = {W in !A!* : "S" :*:> W} |
| | | | |
| − | 5. The concept of a sentence in !L! covers any
| + | Finally, a string W is called a "word", a "sentence", or so on, |
| − | concatenation of sentences in !L!, in effect,
| + | of the language generated by !G! if and only if W is in !L!(!G!). |
| − | any number of freely chosen sentences that are
| |
| − | available to be concatenated one after another.
| |
| | | | |
| − | 6. The concept of a sentence in !L! covers any
| + | Reference: |
| − | surcatenation of sentences in !L!, in effect,
| |
| − | any string that opens with a "-(", continues
| |
| − | with a sentence, possibly empty, follows with
| |
| − | a finite number of phrases of the form "," · S,
| |
| − | and closes with a ")-".
| |
| | | | |
| − | This appears to be just about the most concise description
| + | | Denning, P.J., Dennis, J.B., Qualitz, J.E., |
| − | of the cactus language !C!(!P!) that one can imagine, but
| + | |'Machines, Languages, and Computation', |
| − | there exist a couple of problems that are commonly felt
| + | | Prentice-Hall, Englewood Cliffs, NJ, 1978. |
| − | to afflict this style of presentation and to make it
| + | </pre> |
| − | less than completely acceptable. Briefly stated,
| |
| − | these problems turn on the following properties
| |
| − | of the presentation:
| |
| | | | |
| − | a. The invocation of the kleene star operation
| + | =====1.3.11.2. Generalities About Formal Grammars===== |
| − | is not reduced to a manifestly finitary form.
| |
| | | | |
| − | b. The type of a sentence S is allowed to cover
| + | =====1.3.11.3. The Cactus Language : Stylistics===== |
| − | not only itself but also the empty string.
| |
| | | | |
| − | I will discuss these issues at first in general, and especially in regard to
| + | <pre> |
| − | how the two features interact with one another, and then I return to address
| + | | As a result, we can hardly conceive of how many possibilities there are for what |
| − | in further detail the questions that they engender on their individual bases.
| + | | we call objective reality. Our sharp quills of knowledge are so narrow and so |
| | + | | concentrated in particular directions that with science there are myriads of |
| | + | | totally different real worlds, each one accessible from the next simply by |
| | + | | slight alterations -- shifts of gaze -- of every particular discipline |
| | + | | and subspecialty. |
| | + | | |
| | + | | Herbert J. Bernstein, "Idols", p. 38. |
| | + | | |
| | + | | Herbert J. Bernstein, |
| | + | |"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in: |
| | + | | |
| | + | | Marcus G. Raskin & Herbert J. Bernstein, |
| | + | |'New Ways of Knowing: The Sciences, Society, and Reconstructive Knowledge', |
| | + | | Rowman & Littlefield, Totowa, NJ, 1987. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | This Subsection highlights an issue of "style" that arises in describing |
| | + | a formal language. In broad terms, I use the word "style" to refer to a |
| | + | loosely specified class of formal systems, typically ones that have a set |
| | + | of distinctive features in common. For instance, a style of proof system |
| | + | usually dictates one or more rules of inference that are acknowledged as |
| | + | conforming to that style. In the present context, the word "style" is a |
| | + | natural choice to characterize the varieties of formal grammars, or any |
| | + | other sorts of formal systems that can be contemplated for deriving the |
| | + | sentences of a formal language. |
| | | | |
| − | IDS. Note 150
| + | In looking at what seems like an incidental issue, the discussion arrives |
| | + | at a critical point. The question is: What decides the issue of style? |
| | + | Taking a given language as the object of discussion, what factors enter |
| | + | into and determine the choice of a style for its presentation, that is, |
| | + | a particular way of arranging and selecting the materials that come to |
| | + | be involved in a description, a grammar, or a theory of the language? |
| | + | To what degree is the determination accidental, empirical, pragmatic, |
| | + | rhetorical, or stylistic, and to what extent is the choice essential, |
| | + | logical, and necessary? For that matter, what determines the order |
| | + | of signs in a word, a sentence, a text, or a discussion? All of |
| | + | the corresponding parallel questions about the character of this |
| | + | choice can be posed with regard to the constituent part as well |
| | + | as with regard to the main constitution of the formal language. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | In order to answer this sort of question, at any level of articulation, |
| | + | one has to inquire into the type of distinction that it invokes, between |
| | + | arrangements and orders that are essential, logical, and necessary and |
| | + | orders and arrangements that are accidental, rhetorical, and stylistic. |
| | + | As a rough guide to its comprehension, a "logical order", if it resides |
| | + | in the subject at all, can be approached by considering all of the ways |
| | + | of saying the same things, in all of the languages that are capable of |
| | + | saying roughly the same things about that subject. Of course, the "all" |
| | + | that appears in this rule of thumb has to be interpreted as a reasonably |
| | + | qualified type of universal. For all practical purposes, it simply means |
| | + | "all of the ways that a person can think of" and "all of the languages |
| | + | that a person can conceive of", with all things being relative to the |
| | + | particular moment of investigation. For all of these reasons, the rule |
| | + | must stand as little more than a rough idea of how to approach its object. |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | If it is demonstrated that a given formal language can be presented in |
| | + | any one of several styles of formal grammar, then the choice of a format |
| | + | is accidental, optional, and stylistic to the very extent that it is free. |
| | + | But if it can be shown that a particular language cannot be successfully |
| | + | presented in a particular style of grammar, then the issue of style is |
| | + | no longer free and rhetorical, but becomes to that very degree essential, |
| | + | necessary, and obligatory, in other words, a question of the objective |
| | + | logical order that can be found to reside in the object language. |
| | | | |
| − | In the process of developing a grammar for a language, it is possible
| + | As a rough illustration of the difference between logical and rhetorical |
| − | to notice a number of organizational, pragmatic, and stylistic questions,
| + | orders, consider the kinds of order that are expressed and exhibited in |
| − | whose moment to moment answers appear to decide the ongoing direction of the
| + | the following conjunction of implications: |
| − | grammar that develops and the impact of whose considerations work in tandem
| |
| − | to determine, or at least to influence, the sort of grammar that turns out.
| |
| − | The issues that I can see arising at this point I can give the following
| |
| − | prospective names, putting off the discussion of their natures and the
| |
| − | treatment of their details to the points in the development of the
| |
| − | present example where they evolve their full import.
| |
| | | | |
| − | 1. The "degree of intermediate organization" in a grammar. | + | X => Y and Y => Z |
| | | | |
| − | 2. The "distinction between empty and significant strings", and thus
| + | Here, there is a happy conformity between the logical content and the |
| − | the "distinction between empty and significant types of strings".
| + | rhetorical form, indeed, to such a degree that one hardly notices the |
| | + | difference between them. The rhetorical form is given by the order |
| | + | of sentences in the two implications and the order of implications |
| | + | in the conjunction. The logical content is given by the order of |
| | + | propositions in the extended implicational sequence: |
| | | | |
| − | 3. The "principle of intermediate significance". This is a constraint | + | X =< Y =< Z |
| − | on the grammar that arises from considering the interaction of the
| |
| − | first two issues.
| |
| | | | |
| − | In responding to these issues, it is advisable at first to proceed in
| + | To see the difference between form and content, or manner and matter, |
| − | a stepwise fashion, all the better thereby to accommodate the chances | + | it is enough to observe a few of the ways that the expression can be |
| − | of pursuing a series of parallel developments in the grammar, to allow
| + | varied without changing its meaning, for example: |
| − | for the possibility of reversing many steps in its development, indeed,
| |
| − | to take into account the near certain necessity of having to revisit,
| |
| − | to revise, and to reverse many decisions about how to proceed toward
| |
| − | an optimal description or a satisfactory grammar for the language.
| |
| − | Doing all this means exploring the effects of various alterations
| |
| − | and innovations as independently from each other as possible.
| |
| | | | |
| − | The degree of intermediate organization in a grammar is measured by how many
| + | Z <= Y and Y <= X |
| − | intermediate symbols it has and by how they interact with each other by means
| |
| − | of its productions. With respect to this issue, Grammar 1 has no intermediate
| |
| − | symbols at all, !Q! = {}, and therefore remains at an ostensibly trivial degree
| |
| − | of intermediate organization. Some additions to the list of intermediate symbols
| |
| − | are practically obligatory in order to arrive at any reasonable grammar at all,
| |
| − | other inclusions appear to have a more optional character, though obviously
| |
| − | useful from the standpoints of clarity and ease of comprehension.
| |
| | | | |
| − | One of the troubles that is perceived to affect Grammar 1 is that it wastes
| + | Any style of declarative programming, also called "logic programming", |
| − | so much of the available potential for efficient description in recounting
| + | depends on a capacity, as embodied in a programming language or other |
| − | over and over again the simple fact that the empty string is present in
| + | formal system, to describe the relation between problems and solutions |
| − | the language. This arises in part from the statement that S :> S*, | + | in logical terms. A recurring problem in building this capacity is in |
| − | which implies that:
| + | bridging the gap between ostensibly non-logical orders and the logical |
| | + | orders that are used to describe and to represent them. For instance, |
| | + | to mention just a couple of the most pressing cases, and the ones that |
| | + | are currently proving to be the most resistant to a complete analysis, |
| | + | one has the orders of dynamic evolution and rhetorical transition that |
| | + | manifest themselves in the process of inquiry and in the communication |
| | + | of its results. |
| | | | |
| − | S :> S* = %e% |_| S |_| S · S |_| S · S · S |_| ...
| + | This patch of the ongoing discussion is concerned with describing a |
| | + | particular variety of formal languages, whose typical representative |
| | + | is the painted cactus language !L! = !C!(!P!). It is the intention of |
| | + | this work to interpret this language for propositional logic, and thus |
| | + | to use it as a sentential calculus, an order of reasoning that forms an |
| | + | active ingredient and a significant component of all logical reasoning. |
| | + | To describe this language, the standard devices of formal grammars and |
| | + | formal language theory are more than adequate, but this only raises the |
| | + | next question: What sorts of devices are exactly adequate, and fit the |
| | + | task to a "T"? The ultimate desire is to turn the tables on the order |
| | + | of description, and so begins a process of eversion that evolves to the |
| | + | point of asking: To what extent can the language capture the essential |
| | + | features and laws of its own grammar and describe the active principles |
| | + | of its own generation? In other words: How well can the language be |
| | + | described by using the language itself to do so? |
| | | | |
| − | There is nothing wrong with the more expansive pan of the covered equation,
| + | In order to speak to these questions, I have to express what a grammar says |
| − | since it follows straightforwardly from the definition of the kleene star
| + | about a language in terms of what a language can say on its own. In effect, |
| − | operation, but the covering statement, to the effect that S :> S*, is not
| + | it is necessary to analyze the kinds of meaningful statements that grammars |
| − | necessarily a very productive piece of information, to the extent that it
| + | are capable of making about languages in general and to relate them to the |
| − | does always tell us very much about the language that is being supposed to
| + | kinds of meaningful statements that the syntactic "sentences" of the cactus |
| − | fall under the type of a sentence S. In particular, since it implies that
| + | language might be interpreted as making about the very same topics. So far |
| − | S :> %e%, and since !L! = %e%·!L! = !L!·%e%, for any formal language !L!,
| + | in the present discussion, the sentences of the cactus language do not make |
| − | the empty string !e! = "" is counted over and over in every term of the union, | + | any meaningful statements at all, much less any meaningful statements about |
| − | and every non-empty sentence under S appears again and again in every term of
| + | languages and their constitutions. As of yet, these sentences subsist in the |
| − | the union that follows the initial appearance of S. As a result, this style | + | form of purely abstract, formal, and uninterpreted combinatorial constructions. |
| − | of characterization has to be classified as "true but not very informative". | |
| − | If at all possible, one prefers to partition the language of interest into
| |
| − | a disjoint union of subsets, thereby accounting for each sentence under
| |
| − | its proper term, and one whose place under the sum serves as a useful
| |
| − | parameter of its character or its complexity. In general, this form
| |
| − | of description is not always possible to achieve, but it is usually
| |
| − | worth the trouble to actualize it whenever it is.
| |
| | | | |
| − | Suppose that one tries to deal with this problem by eliminating each use of
| + | Before the capacity of a language to describe itself can be evaluated, |
| − | the kleene star operation, by reducing it to a purely finitary set of steps,
| + | the missing link to meaning has to be supplied for each of its strings. |
| − | or by finding an alternative way to cover the sublanguage that it is used to
| + | This calls for a dimension of semantics and a notion of interpretation, |
| − | generate. This amounts, in effect, to "recognizing a type", a complex process
| + | topics that are taken up for the case of the cactus language !C!(!P!) |
| − | that involves the following steps:
| + | in Subsection 1.3.10.12. Once a plausible semantics is prescribed for |
| | + | this language it will be possible to return to these questions and to |
| | + | address them in a meaningful way. |
| | | | |
| − | 1. Noticing a category of strings that
| + | The prominent issue at this point is the distinct placements of formal |
| − | is generated by iteration or recursion.
| + | languages and formal grammars with respect to the question of meaning. |
| − | | + | The sentences of a formal language are merely the abstract strings of |
| − | 2. Acknowledging the circumstance that the noted category
| + | abstract signs that happen to belong to a certain set. They do not by |
| − | of strings needs to be covered by a non-terminal symbol.
| + | themselves make any meaningful statements at all, not without mounting |
| − | | + | a separate effort of interpretation, but the rules of a formal grammar |
| − | 3. Making a note of it by declaring and instituting
| + | make meaningful statements about a formal language, to the extent that |
| − | an explicitly and even expressively named category.
| + | they say what strings belong to it and what strings do not. Thus, the |
| | + | formal grammar, a formalism that appears to be even more skeletal than |
| | + | the formal language, still has bits and pieces of meaning attached to it. |
| | + | In a sense, the question of meaning is factored into two parts, structure |
| | + | and value, leaving the aspect of value reduced in complexity and subtlety |
| | + | to the simple question of belonging. Whether this single bit of meaningful |
| | + | value is enough to encompass all of the dimensions of meaning that we require, |
| | + | and whether it can be compounded to cover the complexity that actually exists |
| | + | in the realm of meaning -- these are questions for an extended future inquiry. |
| | | | |
| − | In sum, one introduces a non-terminal symbol for each type of sentence and
| + | Perhaps I ought to comment on the differences between the present and |
| − | each "part of speech" or sentential component that is generated by means of
| + | the standard definition of a formal grammar, since I am attempting to |
| − | iteration or recursion under the ruling constraints of the grammar. In order
| + | strike a compromise with several alternative conventions of usage, and |
| − | to do this one needs to analyze the iteration of each grammatical operation in
| + | thus to leave certain options open for future exploration. All of the |
| − | a way that is analogous to a mathematically inductive definition, but further in
| + | changes are minor, in the sense that they are not intended to alter the |
| − | a way that is not forced explicitly to recognize a distinct and separate type of
| + | classes of languages that are able to be generated, but only to clear up |
| − | expression merely to account for and to recount every increment in the parameter
| + | various ambiguities and sundry obscurities that affect their conception. |
| − | of iteration.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | Primarily, the conventional scope of non-terminal symbols was expanded |
| | + | to encompass the sentence symbol, mainly on account of all the contexts |
| | + | where the initial and the intermediate symbols are naturally invoked in |
| | + | the same breath. By way of compensating for the usual exclusion of the |
| | + | sentence symbol from the non-terminal class, an equivalent distinction |
| | + | was introduced in the fashion of a distinction between the initial and |
| | + | the intermediate symbols, and this serves its purpose in all of those |
| | + | contexts where the two kind of symbols need to be treated separately. |
| | | | |
| − | IDS. Note 151
| + | At the present point, I remain a bit worried about the motivations |
| | + | and the justifications for introducing this distinction, under any |
| | + | name, in the first place. It is purportedly designed to guarantee |
| | + | that the process of derivation at least gets started in a definite |
| | + | direction, while the real questions have to do with how it all ends. |
| | + | The excuses of efficiency and expediency that I offered as plausible |
| | + | and sufficient reasons for distinguishing between empty and significant |
| | + | sentences are likely to be ephemeral, if not entirely illusory, since |
| | + | intermediate symbols are still permitted to characterize or to cover |
| | + | themselves, not to mention being allowed to cover the empty string, |
| | + | and so the very types of traps that one exerts oneself to avoid at |
| | + | the outset are always there to afflict the process at all of the |
| | + | intervening times. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | If one reflects on the form of grammar that is being prescribed here, |
| | + | it looks as if one sought, rather futilely, to avoid the problems of |
| | + | recursion by proscribing the main program from calling itself, while |
| | + | allowing any subprogram to do so. But any trouble that is avoidable |
| | + | in the part is also avoidable in the main, while any trouble that is |
| | + | inevitable in the part is also inevitable in the main. Consequently, |
| | + | I am reserving the right to change my mind at a later stage, perhaps |
| | + | to permit the initial symbol to characterize, to cover, to regenerate, |
| | + | or to produce itself, if that turns out to be the best way in the end. |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | Before I leave this Subsection, I need to say a few things about |
| | + | the manner in which the abstract theory of formal languages and |
| | + | the pragmatic theory of sign relations interact with each other. |
| | | | |
| − | Returning to the case of the cactus language, the process of recognizing an
| + | Formal language theory can seem like an awfully picky subject at times, |
| − | iterative type or a recursive type can be illustrated in the following way.
| + | treating every symbol as a thing in itself the way it does, sorting out |
| − | The operative phrases in the simplest sort of recursive definition are its
| + | the nominal types of symbols as objects in themselves, and singling out |
| − | initial part and its generic part. For the cactus language !C!(!P!), one
| + | the passing tokens of symbols as distinct entities in their own rights. |
| − | has the following definitions of concatenation as iterated precatenation
| + | It has to continue doing this, if not for any better reason than to aid |
| − | and of surcatenation as iterated subcatenation, respectively: | + | in clarifying the kinds of languages that people are accustomed to use, |
| | + | to assist in writing computer programs that are capable of parsing real |
| | + | sentences, and to serve in designing programming languages that people |
| | + | would like to become accustomed to use. As a matter of fact, the only |
| | + | time that formal language theory becomes too picky, or a bit too myopic |
| | + | in its focus, is when it leads one to think that one is dealing with the |
| | + | thing itself and not just with the sign of it, in other words, when the |
| | + | people who use the tools of formal language theory forget that they are |
| | + | dealing with the mere signs of more interesting objects and not with the |
| | + | objects of ultimate interest in and of themselves. |
| | | | |
| − | 1. Conc^0 = ""
| + | As a result, there a number of deleterious effects that can arise from |
| | + | the extreme pickiness of formal language theory, arising, as is often the |
| | + | case, when formal theorists forget the practical context of theorization. |
| | + | It frequently happens that the exacting task of defining the membership |
| | + | of a formal language leads one to think that this object and this object |
| | + | alone is the justifiable end of the whole exercise. The distractions of |
| | + | this mediate objective render one liable to forget that one's penultimate |
| | + | interest lies always with various kinds of equivalence classes of signs, |
| | + | not entirely or exclusively with their more meticulous representatives. |
| | | | |
| − | Conc^k_j S_j = Prec(Conc^(k-1)_j S_j, S_k)
| + | When this happens, one typically goes on working oblivious to the fact |
| | + | that many details about what transpires in the meantime do not matter |
| | + | at all in the end, and one is likely to remain in blissful ignorance |
| | + | of the circumstance that many special details of language membership |
| | + | are bound, destined, and pre-determined to be glossed over with some |
| | + | measure of indifference, especially when it comes down to the final |
| | + | constitution of those equivalence classes of signs that are able to |
| | + | answer for the genuine objects of the whole enterprise of language. |
| | + | When any form of theory, against its initial and its best intentions, |
| | + | leads to this kind of absence of mind that is no longer beneficial in |
| | + | all of its main effects, the situation calls for an antidotal form of |
| | + | theory, one that can restore the presence of mind that all forms of |
| | + | theory are meant to augment. |
| | | | |
| − | 2. Surc^0 = "-()-"
| + | The pragmatic theory of sign relations is called for in settings where |
| | + | everything that can be named has many other names, that is to say, in |
| | + | the usual case. Of course, one would like to replace this superfluous |
| | + | multiplicity of signs with an organized system of canonical signs, one |
| | + | for each object that needs to be denoted, but reducing the redundancy |
| | + | too far, beyond what is necessary to eliminate the factor of "noise" in |
| | + | the language, that is, to clear up its effectively useless distractions, |
| | + | can destroy the very utility of a typical language, which is intended to |
| | + | provide a ready means to express a present situation, clear or not, and |
| | + | to describe an ongoing condition of experience in just the way that it |
| | + | seems to present itself. Within this fleshed out framework of language, |
| | + | moreover, the process of transforming the manifestations of a sign from |
| | + | its ordinary appearance to its canonical aspect is the whole problem of |
| | + | computation in a nutshell. |
| | | | |
| − | Surc^k_j S_j = Subc(Surc^(k-1)_j S_j, S_k)
| + | It is a well-known truth, but an often forgotten fact, that nobody |
| | + | computes with numbers, but solely with numerals in respect of numbers, |
| | + | and numerals themselves are symbols. Among other things, this renders |
| | + | all discussion of numeric versus symbolic computation a bit beside the |
| | + | point, since it is only a question of what kinds of symbols are best for |
| | + | one's immediate application or for one's selection of ongoing objectives. |
| | + | The numerals that everybody knows best are just the canonical symbols, |
| | + | the standard signs or the normal terms for numbers, and the process of |
| | + | computation is a matter of getting from the arbitrarily obscure signs |
| | + | that the data of a situation are capable of throwing one's way to the |
| | + | indications of its character that are clear enough to motivate action. |
| | | | |
| − | In order to transform these recursive definitions into grammar rules,
| + | Having broached the distinction between propositions and sentences, one |
| − | one introduces a new pair of intermediate symbols, "Conc" and "Surc", | + | can see its similarity to the distinction between numbers and numerals. |
| − | corresponding to the operations that share the same names but ignoring
| + | What are the implications of the foregoing considerations for reasoning |
| − | the inflexions of their individual parameters j and k. Recognizing the | + | about propositions and for the realm of reckonings in sentential logic? |
| − | type of a sentence by means of the initial symbol "S", and interpreting
| + | If the purpose of a sentence is just to denote a proposition, then the |
| − | "Conc" and "Surc" as names for the types of strings that are generated
| + | proposition is just the object of whatever sign is taken for a sentence. |
| − | by concatenation and by surcatenation, respectively, one arrives at | + | This means that the computational manifestation of a piece of reasoning |
| − | the following transformation of the ruling operator definitions | + | about propositions amounts to a process that takes place entirely within |
| − | into the form of covering grammar rules:
| + | a language of sentences, a procedure that can rationalize its account by |
| | + | referring to the denominations of these sentences among propositions. |
| | | | |
| − | 1. Conc :> ""
| + | The application of these considerations in the immediate setting is this: |
| | + | Do not worry too much about what roles the empty string "" and the blank |
| | + | symbol " " are supposed to play in a given species of formal languages. |
| | + | As it happens, it is far less important to wonder whether these types |
| | + | of formal tokens actually constitute genuine sentences than it is to |
| | + | decide what equivalence classes it makes sense to form over all of |
| | + | the sentences in the resulting language, and only then to bother |
| | + | about what equivalence classes these limiting cases of sentences |
| | + | are most conveniently taken to represent. |
| | | | |
| − | Conc :> Conc · S
| + | These concerns about boundary conditions betray a more general issue. |
| | + | Already by this point in discussion the limits of the purely syntactic |
| | + | approach to a language are beginning to be visible. It is not that one |
| | + | cannot go a whole lot further by this road in the analysis of a particular |
| | + | language and in the study of languages in general, but when it comes to the |
| | + | questions of understanding the purpose of a language, of extending its usage |
| | + | in a chosen direction, or of designing a language for a particular set of uses, |
| | + | what matters above all else are the "pragmatic equivalence classes" of signs that |
| | + | are demanded by the application and intended by the designer, and not so much the |
| | + | peculiar characters of the signs that represent these classes of practical meaning. |
| | | | |
| − | 2. Surc :> "-()-"
| + | Any description of a language is bound to have alternative descriptions. |
| | + | More precisely, a circumscribed description of a formal language, as any |
| | + | effectively finite description is bound to be, is certain to suggest the |
| | + | equally likely existence and the possible utility of other descriptions. |
| | + | A single formal grammar describes but a single formal language, but any |
| | + | formal language is described by many different formal grammars, not all |
| | + | of which afford the same grasp of its structure, provide an equivalent |
| | + | comprehension of its character, or yield an interchangeable view of its |
| | + | aspects. Consequently, even with respect to the same formal language, |
| | + | different formal grammars are typically better for different purposes. |
| | | | |
| − | Surc :> "-(" · S · ")-"
| + | With the distinctions that evolve among the different styles of grammar, |
| | + | and with the preferences that different observers display toward them, |
| | + | there naturally comes the question: What is the root of this evolution? |
| | | | |
| − | Surc :> Surc ")-"^(-1) · "," · S · ")-"
| + | One dimension of variation in the styles of formal grammars can be seen |
| | + | by treating the union of languages, and especially the disjoint union of |
| | + | languages, as a "sum", by treating the concatenation of languages as a |
| | + | "product", and then by distinguishing the styles of analysis that favor |
| | + | "sums of products" from those that favor "products of sums" as their |
| | + | canonical forms of description. If one examines the relation between |
| | + | languages and grammars carefully enough to see the presence and the |
| | + | influence of these different styles, and when one comes to appreciate |
| | + | the ways that different styles of grammars can be used with different |
| | + | degrees of success for different purposes, then one begins to see the |
| | + | possibility that alternative styles of description can be based on |
| | + | altogether different linguistic and logical operations. |
| | | | |
| − | As given, this particular fragment of the intended grammar
| + | It possible to trace this divergence of styles to an even more primitive |
| − | contains a couple of features that are desirable to amend.
| + | division, one that distinguishes the "additive" or the "parallel" styles |
| | + | from the "multiplicative" or the "serial" styles. The issue is somewhat |
| | + | confused by the fact that an "additive" analysis is typically expressed |
| | + | in the form of a "series", in other words, a disjoint union of sets or a |
| | + | linear sum of their independent effects. But it is easy enough to sort |
| | + | this out if one observes the more telling connection between "parallel" |
| | + | and "independent". Another way to keep the right associations straight |
| | + | is to employ the term "sequential" in preference to the more misleading |
| | + | term "serial". Whatever one calls this broad division of styles, the |
| | + | scope and sweep of their dimensions of variation can be delineated in |
| | + | the following way: |
| | | | |
| − | 1. Given the covering S :> Conc, the covering rule Conc :> Conc · S | + | 1. The "additive" or "parallel" styles favor "sums of products" as |
| − | says no more than the covering rule Conc :> S · S. Consequently, | + | canonical forms of expression, pulling sums, unions, co-products, |
| − | all of the information contained in these two covering rules is | + | and logical disjunctions to the outermost layers of analysis and |
| − | already covered by the statement that S :> S · S. | + | synthesis, while pushing products, intersections, concatenations, |
| | + | and logical conjunctions to the innermost levels of articulation |
| | + | and generation. In propositional logic, this style leads to the |
| | + | "disjunctive normal form" (DNF). |
| | + | |
| | + | 2. The "multiplicative" or "serial" styles favor "products of sums" |
| | + | as canonical forms of expression, pulling products, intersections, |
| | + | concatenations, and logical conjunctions to the outermost layers of |
| | + | analysis and synthesis, while pushing sums, unions, co-products, |
| | + | and logical disjunctions to the innermost levels of articulation |
| | + | and generation. In propositional logic, this style leads to the |
| | + | "conjunctive normal form" (CNF). |
| | | | |
| − | 2. A grammar rule that invokes a notion of decatenation, deletion, erasure,
| + | There is a curious sort of diagnostic clue, a veritable shibboleth, |
| − | or any other sort of retrograde production, is frequently considered to
| + | that often serves to reveal the dominance of one mode or the other |
| − | be lacking in elegance, and a there is a style of critique for grammars
| + | within an individual thinker's cognitive style. Examined on the |
| − | that holds it preferable to avoid these types of operations if it is at
| + | question of what constitutes the "natural numbers", an "additive" |
| − | all possible to do so. Accordingly, contingent on the prescriptions of
| + | thinker tends to start the sequence at 0, while a "multiplicative" |
| − | the informal rule in question, and pursuing the stylistic dictates that
| + | thinker tends to regard it as beginning at 1. |
| − | are writ in the realm of its aesthetic regime, it becomes necessary for
| |
| − | us to backtrack a little bit, to temporarily withdraw the suggestion of
| |
| − | employing these elliptical types of operations, but without, of course,
| |
| − | eliding the record of doing so.
| |
| | | | |
| − | One way to analyze the surcatenation of any number of sentences is to
| + | In any style of description, grammar, or theory of a language, it is |
| − | introduce an auxiliary type of string, not in general a sentence, but
| + | usually possible to tease out the influence of these contrasting traits, |
| − | a proper component of any sentence that is formed by surcatenation.
| + | namely, the "additive" attitude versus the "mutiplicative" tendency that |
| − | Doing this brings one to the following definition:
| + | go to make up the particular style in question, and even to determine the |
| | + | dominant inclination or point of view that establishes its perspective on |
| | + | the target domain. |
| | | | |
| − | A "tract" is a concatenation of a finite sequence of sentences, with a
| + | In each style of formal grammar, the "multiplicative" aspect is present |
| − | literal comma "," interpolated between each pair of adjacent sentences.
| + | in the sequential concatenation of signs, both in the augmented strings |
| − | Thus, a typical tract T takes the form:
| + | and in the terminal strings. In settings where the non-terminal symbols |
| | + | classify types of strings, the concatenation of the non-terminal symbols |
| | + | signifies the cartesian product over the corresponding sets of strings. |
| | | | |
| − | T = S_1 · "," · ... · "," · S_k
| + | In the context-free style of formal grammar, the "additive" aspect is |
| | + | easy enough to spot. It is signaled by the parallel covering of many |
| | + | augmented strings or sentential forms by the same non-terminal symbol. |
| | + | Expressed in active terms, this calls for the independent rewriting |
| | + | of that non-terminal symbol by a number of different successors, |
| | + | as in the following scheme: |
| | | | |
| − | A tract must be distinguished from the abstract sequence of sentences,
| + | q :> W_1 |
| − | S_1, ..., S_k, where the commas that appear to come to mind, as if being
| |
| − | called up to separate the successive sentences of the sequence, remain as
| |
| − | partially abstract conceptions, or as signs that retain a disengaged status
| |
| − | on the borderline between the text and the mind. In effect, the types of
| |
| − | commas that appear to follow in the abstract sequence continue to exist
| |
| − | as conceptual abstractions and fail to be cognized in a wholly explicit
| |
| − | fashion, whether as concrete tokens in the object language, or as marks
| |
| − | in the strings of signs that are able to engage one's parsing attention.
| |
| | | | |
| − | Returning to the case of the painted cactus language !L! = !C!(!P!),
| + | q :> W_2 |
| − | it is possible to put the currently assembled pieces of a grammar
| |
| − | together in the light of the presently adopted canons of style,
| |
| − | to arrive a more refined analysis of the fact that the concept
| |
| − | of a sentence covers any concatenation of sentences and any
| |
| − | surcatenation of sentences, and so to obtain the following
| |
| − | form of a grammar:
| |
| | | | |
| − | o-------------------------------------------------o
| + | ... ... ... |
| − | | !C!(!P!). Grammar 2 !Q! = {"T"} |
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | 1. S :> !e! |
| |
| − | | |
| |
| − | | 2. S :> m_1 |
| |
| − | | |
| |
| − | | 3. S :> p_j, for each j in J |
| |
| − | | |
| |
| − | | 4. S :> S · S |
| |
| − | | |
| |
| − | | 5. S :> "-(" · T · ")-" |
| |
| − | | |
| |
| − | | 6. T :> S |
| |
| − | | |
| |
| − | | 7. T :> T · "," · S |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| | | | |
| − | In this rendition, a string of type T is not in general
| + | q :> W_k |
| − | a sentence itself but a proper "part of speech", that is,
| |
| − | a strictly "lesser" component of a sentence in any suitable
| |
| − | ordering of sentences and their components. In order to see
| |
| − | how the grammatical category T gets off the ground, that is,
| |
| − | to detect its minimal strings and to discover how its ensuing
| |
| − | generations gets started from these, it is useful to observe
| |
| − | that the covering rule T :> S means that T "inherits" all of
| |
| − | the initial conditions of S, namely, T :> !e!, m_1, p_j.
| |
| − | In accord with these simple beginnings it comes to parse
| |
| − | that the rule T :> T · "," · S, with the substitutions
| |
| − | T = !e! and S = !e! on the covered side of the rule,
| |
| − | bears the germinal implication that T :> ",".
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | It is useful to examine the relationship between the grammatical covering |
| | + | or production relation ":>" and the logical relation of implication "=>", |
| | + | with one eye to what they have in common and one eye to how they differ. |
| | + | The production "q :> W" says that the appearance of the symbol "q" in |
| | + | a sentential form implies the possibility of exchanging it for "W". |
| | + | Although this sounds like a "possible implication", to the extent |
| | + | that "q implies a possible W" or that "q possibly implies W", the |
| | + | qualifiers "possible" and "possibly" are the critical elements in |
| | + | these statements, and they are crucial to the meaning of what is |
| | + | actually being implied. In effect, these qualifications reverse |
| | + | the direction of implication, yielding "q <= W" as the best |
| | + | analogue for the sense of the production. |
| | | | |
| − | IDS. Note 152
| + | One way to sum this up is to say that non-terminal symbols have the |
| | + | significance of hypotheses. The terminal strings form the empirical |
| | + | matter of a language, while the non-terminal symbols mark the patterns |
| | + | or the types of substrings that can be noticed in the profusion of data. |
| | + | If one observes a portion of a terminal string that falls into the pattern |
| | + | of the sentential form W, then it is an admissable hypothesis, according to |
| | + | the theory of the language that is constituted by the formal grammar, that |
| | + | this piece not only fits the type q but even comes to be generated under |
| | + | the auspices of the non-terminal symbol "q". |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | A moment's reflection on the issue of style, giving due consideration to the |
| − | | + | received array of stylistic choices, ought to inspire at least the question: |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | "Are these the only choices there are?" In the present setting, there are |
| − | | + | abundant indications that other options, more differentiated varieties of |
| − | Grammar 2 achieves a portion of its success through a higher degree of
| + | description and more integrated ways of approaching individual languages, |
| − | intermediate organization. Roughly speaking, the level of organization
| + | are likely to be conceivable, feasible, and even more ultimately viable. |
| − | can be seen as reflected in the cardinality of the intermediate alphabet
| + | If a suitably generic style, one that incorporates the full scope of |
| − | !Q! = {"T"}, but it is clearly not explained by this simple circumstance
| + | logical combinations and operations, is broadly available, then it |
| − | alone, since it is taken for granted that the intermediate symbols serve
| + | would no longer be necessary, or even apt, to argue in universal |
| − | a purpose, a purpose that is easily recognizable but that may not be so | + | terms about "which style is best", but more useful to investigate |
| − | easy to pin down and to specify exactly. Nevertheless, it is worth the
| + | how we might adapt the local styles to the local requirements. |
| − | trouble of exploring this aspect of organization and this direction of
| + | The medium of a generic style would yield a viable compromise |
| − | development a little further. Although it is not strictly necessary
| + | between "additive" and "multiplicative" canons, and render the |
| − | to do so, it is possible to organize the materials of the present
| + | choice between "parallel" and "serial" a false alternative, |
| − | grammar in a slightly better fashion by recognizing two recurrent
| + | at least, when expressed in the globally exclusive terms |
| − | types of strings that appear in the typical cactus expression.
| + | that are currently most commonly adopted for posing it. |
| − | In doing this, one arrives at the following two definitions:
| |
| | | | |
| − | A "rune" is a string of blanks and paints concatenated together.
| + | One set of indications comes from the study of machines, languages, and |
| − | Thus, a typical rune R is a string over {m_1} |_| !P!, possibly
| + | computation, especially the theories of their structures and relations. |
| − | the empty string. | + | The forms of composition and decomposition that are generally known as |
| | + | "parallel" and "serial" are merely the extreme special cases, in variant |
| | + | directions of specialization, of a more generic form, usually called the |
| | + | "cascade" form of combination. This is a well-known fact in the theories |
| | + | that deal with automata and their associated formal languages, but its |
| | + | implications do not seem to be widely appreciated outside these fields. |
| | + | In particular, it dispells the need to choose one extreme or the other, |
| | + | since most of the natural cases are likely to exist somewhere in between. |
| | | | |
| − | R in ({m_1} |_| !P!)*.
| + | Another set of indications appears in algebra and category theory, |
| | + | where forms of composition and decomposition related to the cascade |
| | + | combination, namely, the "semi-direct product" and its special case, |
| | + | the "wreath product", are encountered at higher levels of generality |
| | + | than the cartesian products of sets or the direct products of spaces. |
| | | | |
| − | When there is no possibility of confusion, the letter "R" can be used
| + | In these domains of operation, one finds it necessary to consider also |
| − | either as a string variable that ranges over the set of runes or else
| + | the "co-product" of sets and spaces, a construction that artificially |
| − | as a type name for the class of runes. The latter reading amounts to | + | creates a disjoint union of sets, that is, a union of spaces that are |
| − | the enlistment of a fresh intermediate symbol, "R" in !Q!, as a part
| + | being treated as independent. It does this, in effect, by "indexing", |
| − | of a new grammar for !C!(!P!). In effect, "R" affords a grammatical
| + | "coloring", or "preparing" the otherwise possibly overlapping domains |
| − | recognition for any rune that forms a part of a sentence in !C!(!P!).
| + | that are being combined. What renders this a "chimera" or a "hybrid" |
| − | In situations where these variant usages are likely to be confused,
| + | form of combination is the fact that this indexing is tantamount to a |
| − | the types of strings can be indicated by means of expressions like | + | cartesian product of a singleton set, namely, the conventional "index", |
| − | "r <: R" and "W <: R". | + | "color", or "affix" in question, with the individual domain that is |
| | + | entering as a factor, a term, or a participant in the final result. |
| | | | |
| − | A "foil" is a string of the form "-(" · T · ")-", where T is a tract.
| + | One of the insights that arises out of Peirce's logical work is that |
| − | Thus, a typical foil F has the form:
| + | the set operations of complementation, intersection, and union, along |
| − | | + | with the logical operations of negation, conjunction, and disjunction |
| − | F = "-(" · S_1 · "," · ... · "," · S_k · ")-".
| + | that operate in isomorphic tandem with them, are not as fundamental as |
| − | | + | they first appear. This is because all of them can be constructed from |
| − | This is just the surcatenation of the sentences S_1, ..., S_k.
| + | or derived from a smaller set of operations, in fact, taking the logical |
| − | Given the possibility that this sequence of sentences is empty,
| + | side of things, from either one of two "solely sufficient" operators, |
| − | and thus that the tract T is the empty string, the minimum foil
| + | called "amphecks" by Peirce, "strokes" by those who re-discovered them |
| − | F is the expression "-()-". Explicitly marking each foil F that
| + | later, and known in computer science as the NAND and the NNOR operators. |
| − | is embodied in a cactus expression is tantamount to recognizing
| + | For this reason, that is, by virtue of their precedence in the orders |
| − | another intermediate symbol, "F" in !Q!, further articulating the
| + | of construction and derivation, these operations have to be regarded |
| − | structures of sentences and expanding the grammar for the language
| + | as the simplest and the most primitive in principle, even if they are |
| − | !C!(!P!). All of the same remarks about the versatile uses of the
| + | scarcely recognized as lying among the more familiar elements of logic. |
| − | intermediate symbols, as string variables and as type names, apply
| |
| − | again to the letter "F".
| |
| | | | |
| − | o-------------------------------------------------o
| + | I am throwing together a wide variety of different operations into |
| − | | !C!(!P!). Grammar 3 !Q! = {"F", "R", "T"} |
| + | the bins labeled "additive" and "multiplicative", but it is easy to |
| − | o-------------------------------------------------o
| + | observe a natural organization and even some relations that approach |
| − | | |
| + | the level of isomorphisms among and between the members of each class. |
| − | | 1. S :> R |
| |
| − | | |
| |
| − | | 2. S :> F |
| |
| − | | |
| |
| − | | 3. S :> S · S |
| |
| − | | |
| |
| − | | 4. R :> !e! |
| |
| − | | |
| |
| − | | 5. R :> m_1 |
| |
| − | | |
| |
| − | | 6. R :> p_j, for each j in J |
| |
| − | | |
| |
| − | | 7. R :> R · R |
| |
| − | | |
| |
| − | | 8. F :> "-(" · T · ")-" |
| |
| − | | |
| |
| − | | 9. T :> S |
| |
| − | | |
| |
| − | | 10. T :> T · "," · S |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| | | | |
| − | In Grammar 3, the first three Rules say that a sentence (a string of type S),
| + | The relation between logical disjunction and set-theoretic union and |
| − | is a rune (a string of type R), a foil (a string of type F), or an arbitrary
| + | the relation between logical conjunction and set-theoretic intersection |
| − | concatenation of strings of these two types. Rules 4 through 7 specify that
| + | are most likely clear enough for the purposes of the immediately present |
| − | a rune R is an empty string !e! = "", a blank symbol m_1 = " ", a paint p_j,
| + | discussion. At any rate, all of these relations are scheduled to receive |
| − | for j in J, or any concatenation of strings of these three types. Rule 8
| + | a thorough examination in a subsequent discussion (Subsection 1.3.10.13). |
| − | characterizes a foil F as a string of the form "-(" · T · ")-", where T is
| + | But the relation of set-theoretic union to category-theoretic co-product |
| − | a tract. The last two Rules say that a tract T is either a sentence S or
| + | and the relation of set-theoretic intersection to syntactic concatenation |
| − | else the concatenation of a tract, a comma, and a sentence, in that order.
| + | deserve a closer look at this point. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | The effect of a co-product as a "disjointed union", in other words, that |
| − | | + | creates an object tantamount to a disjoint union of sets in the resulting |
| − | IDS. Note 153
| + | co-product even if some of these sets intersect non-trivially and even if |
| − | | + | some of them are identical "in reality", can be achieved in several ways. |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | The most usual conception is that of making a "separate copy", for each |
| | + | part of the intended co-product, of the set that is intended to go there. |
| | + | Often one thinks of the set that is assigned to a particular part of the |
| | + | co-product as being distinguished by a particular "color", in other words, |
| | + | by the attachment of a distinct "index", "label", or "tag", being a marker |
| | + | that is inherited by and passed on to every element of the set in that part. |
| | + | A concrete image of this construction can be achieved by imagining that each |
| | + | set and each element of each set is placed in an ordered pair with the sign |
| | + | of its color, index, label, or tag. One describes this as the "injection" |
| | + | of each set into the corresponding "part" of the co-product. |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.) | + | For example, given the sets P and Q, overlapping or not, one can define |
| | + | the "indexed" sets or the "marked" sets P_[1] and Q_[2], amounting to the |
| | + | copy of P into the first part of the co-product and the copy of Q into the |
| | + | second part of the co-product, in the following manner: |
| | | | |
| − | At this point in the succession of grammars for !C!(!P!), the explicit
| + | P_[1] = <P, 1> = {<x, 1> : x in P}, |
| − | uses of indefinite iterations, like the kleene star operator, are now
| + | |
| − | completely reduced to finite forms of concatenation, but the problems
| + | Q_[2] = <Q, 2> = {<x, 2> : x in Q}. |
| − | that some styles of analysis have with allowing non-terminal symbols
| |
| − | to cover both themselves and the empty string are still present.
| |
| | | | |
| − | Any degree of reflection on this difficulty raises the general question:
| + | Using the sign "]_[" for this construction, the "sum", the "co-product", |
| − | What is a practical strategy for accounting for the empty string in the
| + | or the "disjointed union" of P and Q in that order can be represented as |
| − | organization of any formal language that counts it among its sentences?
| + | the ordinary disjoint union of P_[1] and Q_[2], as follows: |
| − | One answer that presents itself is this: If the empty string belongs to
| |
| − | a formal language, it suffices to count it once at the beginning of the
| |
| − | formal account that enumerates its sentences and then to move on to more
| |
| − | interesting materials.
| |
| | | | |
| − | Returning to the case of the cactus language !C!(!P!), that is,
| + | P ]_[ Q = P_[1] |_| Q_[2]. |
| − | the formal language of "painted and rooted cactus expressions",
| |
| − | it serves the purpose of efficient accounting to partition the
| |
| − | language PARCE into the following couple of sublanguages:
| |
| | | | |
| − | 1. The "emptily painted and rooted cactus expressions"
| + | The concatenation L_1 · L_2 of the formal languages L_1 and L_2 is just |
| − | make up the language EPARCE that consists of
| + | the cartesian product of sets L_1 x L_2 without the extra x's, but the |
| − | a single empty string as its only sentence.
| + | relation of cartesian products to set-theoretic intersections and thus |
| − | In short:
| + | to logical conjunctions is far from being clear. |
| | | | |
| − | EPARCE = {""}.
| + | One way of seeing a type of relation in this setting is to focus on the |
| | + | information that is needed to specify each construction, and thereby to |
| | + | reflect on the signs that are used to carry this information. As a way |
| | + | of making a first approach to the topic of information, in accord with |
| | + | a strategy that seeks to be as elementary and as informal as possible, |
| | + | I introduce the following collection of ideas, intended to be taken |
| | + | in a very provisional way. |
| | | | |
| − | 2. The "significantly painted and rooted cactus expressions"
| + | A "stricture" is syntactic specification of a certain set in a certain place, |
| − | make up the language SPARCE that consists of everything else,
| + | relative to a number of other sets, yet to be specified. It is assumed that |
| − | namely, all of the non-empty strings in the language PARCE.
| + | one knows enough about the general form of the specifications in question to |
| − | In sum:
| + | tell if two strictures are equivalent as pieces of information, but any more |
| | + | determinate indications, like names for the places that are mentioned in the |
| | + | stricture, or bounds on the number of places that are involved, are regarded |
| | + | as being extraneous impositions, outside the chief concern of the definition, |
| | + | no matter how convenient they are found to be within a particular discussion. |
| | + | As a schematic form of illustration, a stricture can be pictured in this way: |
| | | | |
| − | SPARCE = PARCE \ "".
| + | "... x X x Q x X x ..." |
| | | | |
| − | As a result of marking the distinction between empty and significant sentences,
| + | A "strait" is the object that is specified by a stricture, in effect, |
| − | that is, by categorizing each of these three classes of strings as an entity | + | a certain set in a certain place of an otherwise yet to be specified |
| − | unto itself and by conceptualizing the whole of its membership as falling
| + | relation. Somewhat sketchily, the strait that corresponds to the |
| − | under a distinctive symbol, one obtains an equation of sets that connects
| + | stricture just given can be pictured in the following shape: |
| − | the three languages being marked:
| + | |
| | + | ... x X x Q x X x ... |
| | + | |
| | + | In this picture, Q is a certain set, and X is the universe of discourse that is |
| | + | pertinent to a given discussion. Since a stricture does not, by itself, contain |
| | + | a sufficient amount of information to specify the number of sets that it intends |
| | + | to set in place, or even to specify the absolute location of the set that it does |
| | + | set in place, it appears to place an unspecified number of unspecified sets in a |
| | + | vague and uncertain strait. Taken out of its interpretive context, the residual |
| | + | information that a stricture can convey makes all of the following potentially |
| | + | equivalent as strictures: |
| | | | |
| − | SPARCE = PARCE - EPARCE. | + | "Q", "X x Q x X", "X x X x Q x X x X", ... |
| | | | |
| − | In sum, one has the disjoint union:
| + | With respect to what these strictures specify, this |
| | + | leaves all of the following equivalent as straits: |
| | | | |
| − | PARCE = EPARCE |_| SPARCE.
| + | Q = X x Q x X = X x X x Q x X x X = ... |
| | | | |
| − | For brevity in the present case, and to serve as a generic device
| + | Within the framework of a particular discussion, it is customary to |
| − | in any similar array of situations, let the symbol "S" be used to
| + | set a bound on the number of places and to limit the variety of sets |
| − | signify the type of an arbitrary sentence, possibly empty, whereas
| + | that are regarded as being under active consideration, and it is also |
| − | the symbol "S'" is reserved to designate the type of a specifically
| + | convenient to index the places of the indicated relations, and of their |
| − | non-empty sentence. In addition, let the symbol "%e%" be employed
| + | encompassing cartesian products, in some fixed way. But the whole idea |
| − | to indicate the type of the empty sentence, in effect, the language | + | of a stricture is to specify a strait that is capable of extending through |
| − | %e% = {""} that contains a single empty string, and let a plus sign
| + | and beyond any fixed frame of discussion. In other words, a stricture is |
| − | "+" signify a disjoint union of types. In the most general type of
| + | conceived to constrain a strait at a certain point, and then to leave it |
| − | situation, where the type S is permitted to include the empty string,
| + | literally embedded, if tacitly expressed, in a yet to be fully specified |
| − | one notes the following relation among types: | + | relation, one that involves an unspecified number of unspecified domains. |
| | | | |
| − | S = %e% + S'.
| + | A quantity of information is a measure of constraint. In this respect, |
| | + | a set of comparable strictures is ordered on account of the information |
| | + | that each one conveys, and a system of comparable straits is ordered in |
| | + | accord with the amount of information that it takes to pin each one of |
| | + | them down. Strictures that are more constraining and straits that are |
| | + | more constrained are placed at higher levels of information than those |
| | + | that are less so. In other language that is often used, entities of |
| | + | either kind that involve more information are said to have a greater |
| | + | "complexity" in relation to comparable entities which involve less |
| | + | information, the latter being said to have a greater "simplicity". |
| | | | |
| − | Consequences of the distinction between empty expressions and
| + | In order to create a concrete example, let me now institute a frame of |
| − | significant expressions are taken up for discussion next time.
| + | discussion where the number of places in a relation is bounded at two, |
| | + | and where the variety of sets under active consideration is limited to |
| | + | the typical subsets P and Q of a universe X. Under these conditions, |
| | + | one can use the following sorts of expression as schematic strictures: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | "X" "P" "Q" |
| | | | |
| − | IDS. Note 154
| + | "X x X" "X x P" "X x Q" |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | "P x X" "P x P" "P x Q" |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | "Q x X" "Q x P" "Q x Q" |
| | | | |
| − | With the distinction between empty and significant expressions in mind,
| + | These strictures and their corresponding straits are stratified according |
| − | I return to the grasp of the cactus language !L! = !C!(!P!) = PARCE(!P!)
| + | to their amounts of information, or their levels of constraint, as follows: |
| − | that is afforded by Grammar 2, and, taking that as a point of departure,
| + | |
| − | explore other avenues of possible improvement in the comprehension of
| + | High: "P x P" "P x Q" "Q x P" "Q x Q" |
| − | these expressions. In order to observe the effects of this alteration
| |
| − | as clearly as possible, in isolation from any other potential factors,
| |
| − | it is useful to strip away the higher levels intermediate organization
| |
| − | that are present in Grammar 3, and start again with a single intermediate
| |
| − | symbol, as used in Grammar 2. One way of carrying out this strategy leads
| |
| − | on to a grammar of the variety that will be articulated next.
| |
| | | | |
| − | If one imposes the distinction between empty and significant types on
| + | Medium: "P" "X x P" "P x X" |
| − | each non-terminal symbol in Grammar 2, then the non-terminal symbols
| |
| − | "S" and "T" give rise to the non-terminal symbols "S", "S'", "T", "T'", | |
| − | leaving the last three of these to form the new intermediate alphabet.
| |
| − | Grammar 4 has the intermediate alphabet !Q! = {"S'", "T", "T'"}, with
| |
| − | the set !K! of covering production rules as listed in the next display.
| |
| | | | |
| − | o-------------------------------------------------o
| + | Medium: "Q" "X x Q" "Q x X" |
| − | | !C!(!P!). Grammar 4 !Q! = {"S'", "T", "T'"} |
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | 1. S :> !e! |
| |
| − | | |
| |
| − | | 2. S :> S' |
| |
| − | | |
| |
| − | | 3. S' :> m_1 |
| |
| − | | |
| |
| − | | 4. S' :> p_j, for each j in J |
| |
| − | | |
| |
| − | | 5. S' :> "-(" · T · ")-" |
| |
| − | | |
| |
| − | | 6. S' :> S' · S' |
| |
| − | | |
| |
| − | | 7. T :> !e! |
| |
| − | | |
| |
| − | | 8. T :> T' |
| |
| − | | |
| |
| − | | 9. T' :> T · "," · S |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| | | | |
| − | In this version of a grammar for !L! = !C!(!P!), the intermediate type T
| + | Low: "X" "X x X" |
| − | is partitioned as T = %e% + T', thereby parsing the intermediate symbol T
| |
| − | in parallel fashion with the division of its overlying type as S = %e% + S'.
| |
| − | This is an option that I will choose to close off for now, but leave it open
| |
| − | to consider at a later point. Thus, it suffices to give a brief discussion
| |
| − | of what it involves, in the process of moving on to its chief alternative.
| |
| | | | |
| − | There does not appear to be anything radically wrong with trying this
| + | Within this framework, the more complex strait P x Q can be expressed |
| − | approach to types. It is reasonable and consistent in its underlying
| + | in terms of the simpler straits, P x X and X x Q. More specifically, |
| − | principle, and it provides a rational and a homogeneous strategy toward
| + | it lends itself to being analyzed as their intersection, as follows: |
| − | all parts of speech, but it does require an extra amount of conceptual
| |
| − | overhead, in that every non-trivial type has to be split into two parts
| |
| − | and comprehended in two stages. Consequently, in view of the largely | |
| − | practical difficulties of making the requisite distinctions for every
| |
| − | intermediate symbol, it is a common convention, whenever possible, to
| |
| − | restrict intermediate types to covering exclusively non-empty strings.
| |
| | | | |
| − | For the sake of future reference, it is convenient to refer to this restriction
| + | P x Q = P x X |^| X x Q |
| − | on intermediate symbols as the "intermediate significance" constraint. It can
| |
| − | be stated in a compact form as a condition on the relations between non-terminal
| |
| − | symbols q in {"S"} |_| !Q! and sentential forms W in {"S"} |_| (!Q! |_| !A!)*.
| |
| | | | |
| − | o-------------------------------------------------o
| + | From here it is easy to see the relation of concatenation, by virtue of |
| − | | Condition On Intermediate Significance |
| + | these types of intersection, to the logical conjunction of propositions. |
| − | o-------------------------------------------------o
| + | A cartesian product P x Q is described by a conjunction of propositions, |
| − | | |
| + | namely, "P_<1> and Q_<2>", subject to the following interpretation: |
| − | | If q :> W |
| |
| − | | |
| |
| − | | and W = !e! |
| |
| − | | |
| |
| − | | then q = "S" |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| | | | |
| − | If this is beginning to sound like a monotone condition, then it is
| + | 1. "P_<1>" asserts that there is an element from |
| − | not absurd to sharpen the resemblance and render the likeness more
| + | the set P in the first place of the product. |
| − | acute. This is done by declaring a couple of ordering relations,
| |
| − | denoting them under variant interpretations by the same sign "<".
| |
| | | | |
| − | 1. The ordering "<" on the set of non-terminal symbols, | + | 2. "Q_<2>" asserts that there is an element from |
| − | q in {"S"} |_| !Q!, ordains the initial symbol "S"
| + | the set Q in the second place of the product. |
| − | to be strictly prior to every intermediate symbol.
| |
| − | This is tantamount to the axiom that "S" < q,
| |
| − | for all q in !Q!.
| |
| | | | |
| − | 2. The ordering "<" on the collection of sentential forms,
| + | The integration of these two pieces of information can be taken |
| − | W in {"S"} |_| (!Q! |_| !A!)*, ordains the empty string
| + | in that measure to specify a yet to be fully determined relation. |
| − | to be strictly minor to every other sentential form.
| |
| − | This is stipulated in the axiom that !e! < W,
| |
| − | for every non-empty sentential form W.
| |
| | | | |
| − | Given these two orderings, the constraint in question
| + | In a corresponding fashion at the level of the elements, |
| − | on intermediate significance can be stated as follows:
| + | the ordered pair <p, q> is described by a conjunction |
| | + | of propositions, namely, "p_<1> and q_<2>", subject |
| | + | to the following interpretation: |
| | | | |
| − | o-------------------------------------------------o
| + | 1. "p_<1>" says that p is in the first place |
| − | | Condition Of Intermediate Significance |
| + | of the product element under construction. |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | If q :> W |
| |
| − | | |
| |
| − | | and q > "S" |
| |
| − | | |
| |
| − | | then W > !e! |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| | | | |
| − | Achieving a grammar that respects this convention typically requires a more
| + | 2. "q_<2>" says that q is in the second place |
| − | detailed account of the initial setting of a type, both with regard to the
| + | of the product element under construction. |
| − | type of context that incites its appearance and also with respect to the
| |
| − | minimal strings that arise under the type in question. In order to find
| |
| − | covering productions that satisfy the intermediate significance condition,
| |
| − | one must be prepared to consider a wider variety of calling contexts or
| |
| − | inciting situations that can be noted to surround each recognized type,
| |
| − | and also to enumerate a larger number of the smallest cases that can
| |
| − | be observed to fall under each significant type.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | Notice that, in construing the cartesian product of the sets P and Q or the |
| − | | + | concatenation of the languages L_1 and L_2 in this way, one shifts the level |
| − | IDS. Note 155
| + | of the active construction from the tupling of the elements in P and Q or the |
| − | | + | concatenation of the strings that are internal to the languages L_1 and L_2 to |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | the concatenation of the external signs that it takes to indicate these sets or |
| − | | + | these languages, in other words, passing to a conjunction of indexed propositions, |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.) | + | "P_<1> and Q_<2>", or to a conjunction of assertions, "L_1_<1> and L_2_<2>", that |
| | + | marks the sets or the languages in question for insertion in the indicated places |
| | + | of a product set or a product language, respectively. In effect, the subscripting |
| | + | by the indices "<1>" and "<2>" can be recognized as a special case of concatenation, |
| | + | albeit through the posting of editorial remarks from an external "mark-up" language. |
| | + | |
| | + | In order to systematize the relationships that strictures and straits |
| | + | placed at higher levels of complexity, constraint, information, and |
| | + | organization bear toward strictures and straits that are placed at |
| | + | the corresponding lower levels of these measures, I introduce the |
| | + | following pair of definitions: |
| | | | |
| − | With the array of foregoing considerations in mind,
| + | The j^th "excerpt" of a stricture of the form "S_1 x ... x S_k", regarded |
| − | one is gradually led to a grammar for !L! = !C!(!P!)
| + | within a frame of discussion where the number of places is limited to k, |
| − | in which all of the covering productions have either
| + | is the stricture of the form "X x ... x S_j x ... x X". In the proper |
| − | one of the following two forms:
| + | context, this can be written more succinctly as the stricture "S_j_<j>", |
| | + | an assertion that places the j^th set in the j^th place of the product. |
| | + | |
| | + | The j^th "extract" of a strait of the form S_1 x ... x S_k, constrained |
| | + | to a frame of discussion where the number of places is restricted to k, |
| | + | is the strait of the form X x ... x S_j x ... x X. In the appropriate |
| | + | context, this can be denoted more succinctly by the stricture "S_j_<j>", |
| | + | an assertion that places the j^th set in the j^th place of the product. |
| | | | |
| − | 1. S :> !e!
| + | In these terms, a stricture of the form "S_1 x ... x S_k" |
| | + | can be expressed in terms of simpler strictures, namely, |
| | + | as a conjunction of its k excerpts: |
| | | | |
| − | 2. q :> W, with q in {"S"} |_| !Q!, and W in (!Q! |_| !A!)^+ | + | "S_1 x ... x S_k" = "S_1_<1>" & ... & "S_k_<k>". |
| | | | |
| − | A grammar that fits into this mold is called a "context-free" grammar.
| + | In a similar vein, a strait of the form S_1 x ... x S_k |
| − | The first type of rewrite rule is referred to as a "special production",
| + | can be expressed in terms of simpler straits, namely, |
| − | while the second type of rewrite rule is called an "ordinary production".
| + | as an intersection of its k extracts: |
| − | An "ordinary derivation" is one that employs only ordinary productions.
| |
| − | In ordinary productions, those that have the form q :> W, the replacement
| |
| − | string W is never the empty string, and so the lengths of the augmented
| |
| − | strings or the sentential forms that follow one another in an ordinary
| |
| − | derivation, on account of using the ordinary types of rewrite rules,
| |
| − | never decrease at any stage of the process, up to and including the
| |
| − | terminal string that is finally generated by the grammar. This type
| |
| − | of feature is known as the "non-contracting property" of productions,
| |
| − | derivations, and grammars. A grammar is said to have the property if
| |
| − | all of its covering productions, with the possible exception of S :> e,
| |
| − | are non-contracting. In particular, context-free grammars are special
| |
| − | cases of non-contracting grammars. The presence of the non-contracting
| |
| − | property within a formal grammar makes the length of the augmented string
| |
| − | available as a parameter that can figure into mathematical inductions and
| |
| − | motivate recursive proofs, and this handle on the generative process makes
| |
| − | it possible to establish the kinds of results about the generated language
| |
| − | that are not easy to achieve in more general cases, nor by any other means
| |
| − | even in these brands of special cases.
| |
| | | | |
| − | Grammar 5 is a context-free grammar for the painted cactus language
| + | S_1 x ... x S_k = S_1_<1> |^| ... |^| S_k_<k>. |
| − | that uses !Q! = {"S'", "T"}, with !K! as listed in the next display.
| |
| | | | |
| − | o-------------------------------------------------o
| + | There is a measure of ambiguity that remains in this formulation, |
| − | | !C!(!P!). Grammar 5 !Q! = {"S'", "T"} |
| + | but it is the best that I can do in the present informal context. |
| − | o-------------------------------------------------o
| + | </pre> |
| − | | |
| |
| − | | 1. S :> !e! |
| |
| − | | |
| |
| − | | 2. S :> S' |
| |
| − | | |
| |
| − | | 3. S' :> m_1 |
| |
| − | | |
| |
| − | | 4. S' :> p_j, for each j in J |
| |
| − | | |
| |
| − | | 5. S' :> S' · S' |
| |
| − | | |
| |
| − | | 6. S' :> "-()-" |
| |
| − | | |
| |
| − | | 7. S' :> "-(" · T · ")-" |
| |
| − | | |
| |
| − | | 8. T :> "," |
| |
| − | | |
| |
| − | | 9. T :> S' |
| |
| − | | |
| |
| − | | 10. T :> T · "," |
| |
| − | | |
| |
| − | | 11. T :> T · "," · S' |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | =====1.3.11.4. The Cactus Language : Mechanics===== |
| | | | |
| − | IDS. Note 156
| + | <pre> |
| | + | | We are only now beginning to see how this works. Clearly one of the |
| | + | | mechanisms for picking a reality is the sociohistorical sense of what |
| | + | | is important -- which research program, with all its particularity of |
| | + | | knowledge, seems most fundamental, most productive, most penetrating. |
| | + | | The very judgments which make us push narrowly forward simultaneously |
| | + | | make us forget how little we know. And when we look back at history, |
| | + | | where the lesson is plain to find, we often fail to imagine ourselves |
| | + | | in a parallel situation. We ascribe the differences in world view |
| | + | | to error, rather than to unexamined but consistent and internally |
| | + | | justified choice. |
| | + | | |
| | + | | Herbert J. Bernstein, "Idols", p. 38. |
| | + | | |
| | + | | Herbert J. Bernstein, |
| | + | |"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in: |
| | + | | |
| | + | | Marcus G. Raskin & Herbert J. Bernstein, |
| | + | |'New Ways of Knowing: The Sciences, Society, and Reconstructive Knowledge', |
| | + | | Rowman & Littlefield, Totowa, NJ, 1987. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | In this Subsection, I discuss the "mechanics" of parsing the |
| | + | cactus language into the corresponding class of computational |
| | + | data structures. This provides each sentence of the language |
| | + | with a translation into a computational form that articulates |
| | + | its syntactic structure and prepares it for automated modes of |
| | + | processing and evaluation. For this purpose, it is necessary |
| | + | to describe the target data structures at a fairly high level |
| | + | of abstraction only, ignoring the details of address pointers |
| | + | and record structures and leaving the more operational aspects |
| | + | of implementation to the imagination of prospective programmers. |
| | + | In this way, I can put off to another stage of elaboration and |
| | + | refinement the description of the program that constructs these |
| | + | pointers and operates on these graph-theoretic data structures. |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (cont.)
| + | The structure of a "painted cactus", insofar as it presents itself |
| | + | to the visual imagination, can be described as follows. The overall |
| | + | structure, as given by its underlying graph, falls within the species |
| | + | of graph that is commonly known as a "rooted cactus", and the only novel |
| | + | feature that it adds to this is that each of its nodes can be "painted" |
| | + | with a finite sequence of "paints", chosen from a "palette" that is given |
| | + | by the parametric set {" "} |_| !P! = {m_1} |_| {p_1, ..., p_k}. |
| | | | |
| − | Finally, it is worth trying to bring together the advantages of these
| + | It is conceivable, from a purely graph-theoretical point of view, to have |
| − | diverse styles of grammar, to whatever extent that they are compatible.
| + | a class of cacti that are painted but not rooted, and so it is frequently |
| − | To do this, a prospective grammar must be capable of maintaining a high
| + | necessary, for the sake of precision, to more exactly pinpoint the target |
| − | level of intermediate organization, like that arrived at in Grammar 2,
| + | species of graphical structure as a "painted and rooted cactus" (PARC). |
| − | while respecting the principle of intermediate significance, and thus
| |
| − | accumulating all the benefits of the context-free format in Grammar 5.
| |
| − | A plausible synthesis of most of these features is given in Grammar 6.
| |
| | | | |
| − | o-----------------------------------------------------------o
| + | A painted cactus, as a rooted graph, has a distinguished "node" that is |
| − | | !C!(!P!). Grammar 6 !Q! = {"S'", "R", "F", "T"} |
| + | called its "root". By starting from the root and working recursively, |
| − | o-----------------------------------------------------------o
| + | the rest of its structure can be described in the following fashion. |
| − | | |
| |
| − | | 1. S :> !e! |
| |
| − | | |
| |
| − | | 2. S :> S' |
| |
| − | | |
| |
| − | | 3. S' :> R |
| |
| − | | |
| |
| − | | 4. S' :> F |
| |
| − | | |
| |
| − | | 5. S' :> S' · S' |
| |
| − | | |
| |
| − | | 6. R :> m_1 |
| |
| − | | |
| |
| − | | 7. R :> p_j, for each j in J |
| |
| − | | |
| |
| − | | 8. R :> R · R |
| |
| − | | |
| |
| − | | 9. F :> "-()-" |
| |
| − | | |
| |
| − | | 10. F :> "-(" · T · ")-" |
| |
| − | | |
| |
| − | | 11. T :> "," |
| |
| − | | |
| |
| − | | 12. T :> S' |
| |
| − | | |
| |
| − | | 13. T :> T · "," |
| |
| − | | |
| |
| − | | 14. T :> T · "," · S' |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| | | | |
| − | The preceding development provides a typical example of how an initially
| + | Each "node" of a PARC consists of a graphical "point" or "vertex" plus |
| − | effective and conceptually succinct description of a formal language, but
| + | a finite sequence of "attachments", described in relative terms as the |
| − | one that is terse to the point of allowing its prospective interpreter to
| + | attachments "at" or "to" that node. An empty sequence of attachments |
| − | waste exorbitant amounts of energy in trying to unravel its implications,
| + | defines the "empty node". Otherwise, each attachment is one of three |
| − | can be converted into a form that is more efficient from the operational
| + | kinds: a blank, a paint, or a type of PARC that is called a "lobe". |
| − | point of view, even if slightly more ungainly in regard to its elegance.
| |
| | | | |
| − | The basic idea behind all of this grammatical machinery remains the same: | + | Each "lobe" of a PARC consists of a directed graphical "cycle" plus a |
| − | Aside from the select body of formulas introduced as boundary conditions,
| + | finite sequence of "accoutrements", described in relative terms as the |
| − | a grammar for the cactus language is nothing more or less than a device | + | accoutrements "of" or "on" that lobe. Recalling the circumstance that |
| − | that institutes the following general rule:
| + | every lobe that comes under consideration comes already attached to a |
| | + | particular node, exactly one vertex of the corresponding cycle is the |
| | + | vertex that comes from that very node. The remaining vertices of the |
| | + | cycle have their definitions filled out according to the accoutrements |
| | + | of the lobe in question. An empty sequence of accoutrements is taken |
| | + | to be tantamount to a sequence that contains a single empty node as its |
| | + | unique accoutrement, and either one of these ways of approaching it can |
| | + | be regarded as defining a graphical structure that is called a "needle" |
| | + | or a "terminal edge". Otherwise, each accoutrement of a lobe is itself |
| | + | an arbitrary PARC. |
| | | | |
| − | If the strings S_1, ..., S_k are sentences,
| + | Although this definition of a lobe in terms of its intrinsic structural |
| − | | + | components is logically sufficient, it is also useful to characterize the |
| − | then their concatenation in the form
| + | structure of a lobe in comparative terms, that is, to view the structure |
| | + | that typifies a lobe in relation to the structures of other PARC's and to |
| | + | mark the inclusion of this special type within the general run of PARC's. |
| | + | This approach to the question of types results in a form of description |
| | + | that appears to be a bit more analytic, at least, in mnemonic or prima |
| | + | facie terms, if not ultimately more revealing. Working in this vein, |
| | + | a "lobe" can be characterized as a special type of PARC that is called |
| | + | an "unpainted root plant" (UR-plant). |
| | | | |
| − | Conc^k_j S_j = S_1 · ... · S_k
| + | An "UR-plant" is a PARC of a simpler sort, at least, with respect to the |
| | + | recursive ordering of structures that is being followed here. As a type, |
| | + | it is defined by the presence of two properties, that of being "planted" |
| | + | and that of having an "unpainted root". These are defined as follows: |
| | | | |
| − | is a sentence,
| + | 1. A PARC is "planted" if its list of attachments has just one PARC. |
| | | | |
| − | and their surcatenation in the form | + | 2. A PARC is "UR" if its list of attachments has no blanks or paints. |
| | | | |
| − | Surc^k_j S_j = "-(" · S_1 · "," · ... · "," · S_k · ")-"
| + | In short, an UR-planted PARC has a single PARC as its only attachment, |
| | + | and since this attachment is prevented from being a blank or a paint, |
| | + | the single attachment at its root has to be another sort of structure, |
| | + | that which we call a "lobe". |
| | | | |
| − | is a sentence.
| + | To express the description of a PARC in terms of its nodes, each node |
| | + | can be specified in the fashion of a functional expression, letting a |
| | + | citation of the generic function name "Node" be followed by a list of |
| | + | arguments that enumerates the attachments of the node in question, and |
| | + | letting a citation of the generic function name "Lobe" be followed by a |
| | + | list of arguments that details the accoutrements of the lobe in question. |
| | + | Thus, one can write expressions of the following forms: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | 1. Node^0 = Node() |
| | | | |
| − | IDS. Note 157
| + | = a node with no attachments. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | Node^k_j C_j = Node(C_1, ..., C_k) |
| | | | |
| − | 1.3.10.9. The Cactus Language: Syntax (concl.)
| + | = a node with the attachments C_1, ..., C_k. |
| | | | |
| − | It is fitting to wrap up the foregoing developments by summarizing the
| + | 2. Lobe^0 = Lobe() |
| − | notion of a formal grammar that appeared to evolve in the present case.
| |
| − | For the sake of future reference and the chance of a wider application,
| |
| − | it is also useful to try to extract the scheme of a formalization that
| |
| − | potentially holds for any formal language. The following presentation
| |
| − | of the notion of a formal grammar is adapted, with minor modifications,
| |
| − | from the treatment in (DDQ, 60-61).
| |
| | | | |
| − | A "formal grammar" !G! is given by a four-tuple !G! = ("S", !Q!, !A!, !K!)
| + | = a lobe with no accoutrements. |
| − | that takes the following form of description:
| |
| | | | |
| − | 1. "S" is the "initial", "special", "start", or "sentence symbol".
| + | Lobe^k_j C_j = Lobe(C_1, ..., C_k) |
| − | Since the letter "S" serves this function only in a special setting,
| |
| − | its employment in this role need not create any confusion with its
| |
| − | other typical uses as a string variable or as a sentence variable.
| |
| | | | |
| − | 2. !Q! = {q_1, ..., q_m} is a finite set of "intermediate symbols",
| + | = a lobe with the accoutrements C_1, ..., C_k. |
| − | all distinct from "S".
| |
| | | | |
| − | 3. !A! = {a_1, ..., a_n} is a finite set of "terminal symbols",
| + | Working from a structural description of the cactus language, |
| − | also known as the "alphabet" of !G!, all distinct from "S" and
| + | or any suitable formal grammar for !C!(!P!), it is possible to |
| − | disjoint from !Q!. Depending on the particular conception of the
| + | give a recursive definition of the function called "Parse" that |
| − | language !L! that is "covered", "generated", "governed", or "ruled"
| + | maps each sentence in PARCE(!P!) to the corresponding graph in |
| − | by the grammar !G!, that is, whether !L! is conceived to be a set of
| + | PARC(!P!). One way to do this proceeds as follows: |
| − | words, sentences, paragraphs, or more extended structures of discourse,
| |
| − | it is usual to describe !A! as the "alphabet", "lexicon", "vocabulary",
| |
| − | "liturgy", or "phrase book" of both the grammar !G! and the language !L!
| |
| − | that it regulates.
| |
| | | | |
| − | 4. !K! is a finite set of "characterizations". Depending on how they | + | 1. The parse of the concatenation Conc^k of the k sentences S_j, |
| − | come into play, these are variously described as "covering rules", | + | for j = 1 to k, is defined recursively as follows: |
| − | "formations", "productions", "rewrite rules", "subsumptions",
| |
| − | "transformations", or "typing rules".
| |
| | | | |
| − | To describe the elements of !K! it helps to define some additional terms:
| + | a. Parse(Conc^0) = Node^0. |
| | | | |
| − | a. The symbols in {"S"} |_| !Q! |_| !A! form the "augmented alphabet" of !G!.
| + | b. For k > 0, |
| | | | |
| − | b. The symbols in {"S"} |_| !Q! are the "non-terminal symbols" of !G!.
| + | Parse(Conc^k_j S_j) = Node^k_j Parse(S_j). |
| | | | |
| − | c. The symbols in !Q! |_| !A! are the "non-initial symbols" of !G!. | + | 2. The parse of the surcatenation Surc^k of the k sentences S_j, |
| | + | for j = 1 to k, is defined recursively as follows: |
| | | | |
| − | d. The strings in ({"S"} |_| !Q! |_| !A!)* are the "augmented strings" for G.
| + | a. Parse(Surc^0) = Lobe^0. |
| | | | |
| − | e. The strings in {"S"} |_| (!Q! |_| !A!)* are the "sentential forms" for G.
| + | b. For k > 0, |
| | | | |
| − | Each characterization in !K! is an ordered pair of strings (S_1, S_2)
| + | Parse(Surc^k_j S_j) = Lobe^k_j Parse(S_j). |
| − | that takes the following form:
| |
| | | | |
| − | S_1 = Q_1 · q · Q_2
| + | For ease of reference, Table 12 summarizes the mechanics of these parsing rules. |
| | | | |
| − | S_2 = Q_1 · W · Q_2
| + | Table 12. Algorithmic Translation Rules |
| | + | o------------------------o---------o------------------------o |
| | + | | | Parse | | |
| | + | | Sentence in PARCE | --> | Graph in PARC | |
| | + | o------------------------o---------o------------------------o |
| | + | | | | | |
| | + | | Conc^0 | --> | Node^0 | |
| | + | | | | | |
| | + | | Conc^k_j S_j | --> | Node^k_j Parse(S_j) | |
| | + | | | | | |
| | + | | Surc^0 | --> | Lobe^0 | |
| | + | | | | | |
| | + | | Surc^k_j S_j | --> | Lobe^k_j Parse(S_j) | |
| | + | | | | | |
| | + | o------------------------o---------o------------------------o |
| | | | |
| − | In this scheme, S_1 and S_2 are members of the augmented strings for !G!,
| + | A "substructure" of a PARC is defined recursively as follows. Starting |
| − | more precisely, S_1 is a non-empty string and a sentential form over !G!,
| + | at the root node of the cactus C, any attachment is a substructure of C. |
| − | while S_2 is a possibly empty string and also a sentential form over !G!.
| + | If a substructure is a blank or a paint, then it constitutes a minimal |
| | + | substructure, meaning that no further substructures of C arise from it. |
| | + | If a substructure is a lobe, then each one of its accoutrements is also |
| | + | a substructure of C, and has to be examined for further substructures. |
| | | | |
| − | Here also, q is a non-terminal symbol, that is, q is in {"S"} |_| !Q!,
| + | The concept of substructure can be used to define varieties of deletion |
| − | while Q_1, Q_2, and W are possibly empty strings of non-initial symbols,
| + | and erasure operations that respect the structure of the abstract graph. |
| − | a fact that can be expressed in the form: Q_1, Q_2, W in (!Q! |_| !A!)*.
| + | For the purposes of this depiction, a blank symbol " " is treated as |
| | + | a "primer", in other words, as a "clear paint", a "neutral tint", or |
| | + | a "white wash". In effect, one is letting m_1 = p_0. In this frame |
| | + | of discussion, it is useful to make the following distinction: |
| | | | |
| − | In practice, the ordered pairs of strings in !K! are used to "derive",
| + | 1. To "delete" a substructure is to replace it with an empty node, |
| − | to "generate", or to "produce" sentences of the language !L! = <!G!>
| + | in effect, to reduce the whole structure to a trivial point. |
| − | that is then said to be "governed" or "regulated" by the grammar !G!.
| |
| − | In order to facilitate this active employment of the grammar, it is
| |
| − | conventional to write the characterization (S_1, S_2) in either one
| |
| − | of the next two forms, where the more generic form is followed by
| |
| − | the more specific form:
| |
| | | | |
| − | S_1 :> S_2 | + | 2. To "erase" a substructure is to replace it with a blank symbol, |
| | + | in effect, to paint it out of the picture or to overwrite it. |
| | | | |
| − | Q_1 · q · Q_2 :> Q_1 · W · Q_2
| + | A "bare" PARC, loosely referred to as a "bare cactus", is a PARC on the |
| | + | empty palette !P! = {}. In other veins, a bare cactus can be described |
| | + | in several different ways, depending on how the form arises in practice. |
| | | | |
| − | In this usage, the characterization S_1 :> S_2 is tantamount to a grammatical
| + | 1. Leaning on the definition of a bare PARCE, a bare PARC can be |
| − | license to transform a string of the form Q_1 · q · Q_2 into a string of the
| + | described as the kind of a parse graph that results from parsing |
| − | form Q1 · W · Q2, in effect, replacing the non-terminal symbol q with the
| + | a bare cactus expression, in other words, as the kind of a graph |
| − | non-initial string W in any selected, preserved, and closely adjoining
| + | that issues from the requirements of processing a sentence of |
| − | context of the form Q1 · ... · Q2. Accordingly, in this application
| + | the bare cactus language !C!^0 = PARCE^0. |
| − | the notation "S_1 :> S_2" can be read as "S_1 produces S_2" or as | |
| − | "S_1 transforms into S_2".
| |
| | | | |
| − | An "immediate derivation" in !G! is an ordered pair (W, W')
| + | 2. To express it more in its own terms, a bare PARC can be defined |
| − | of sentential forms in !G! such that: | + | by tracing the recursive definition of a generic PARC, but then |
| | + | by detaching an independent form of description from the source |
| | + | of that analogy. The method is sufficiently sketched as follows: |
| | | | |
| − | W = Q_1 · X · Q_2
| + | a. A "bare PARC" is a PARC whose attachments |
| | + | are limited to blanks and "bare lobes". |
| | | | |
| − | W' = Q_1 · Y · Q_2
| + | b. A "bare lobe" is a lobe whose accoutrements |
| | + | are limited to bare PARC's. |
| | | | |
| − | and (X, Y) in !K! | + | 3. In practice, a bare cactus is usually encountered in the process |
| | + | of analyzing or handling an arbitrary PARC, the circumstances of |
| | + | which frequently call for deleting or erasing all of its paints. |
| | + | In particular, this generally makes it easier to observe the |
| | + | various properties of its underlying graphical structure. |
| | + | </pre> |
| | | | |
| − | i.e. X :> Y in !G!
| + | =====1.3.11.5. The Cactus Language : Semantics===== |
| | | | |
| − | This relation is indicated by saying that W "immediately derives" W',
| + | <pre> |
| − | that W' is "immediately derived" from W in !G!, and also by writing:
| + | | Alas, and yet what 'are' you, my written and painted thoughts! |
| | + | | It is not long ago that you were still so many-coloured, |
| | + | | young and malicious, so full of thorns and hidden |
| | + | | spices you made me sneeze and laugh -- and now? |
| | + | | You have already taken off your novelty and |
| | + | | some of you, I fear, are on the point of |
| | + | | becoming truths: they already look so |
| | + | | immortal, so pathetically righteous, |
| | + | | so boring! |
| | + | | |
| | + | | Friedrich Nietzsche, 'Beyond Good and Evil', Paragraph 296. |
| | + | | |
| | + | | Friedrich Nietzsche, |
| | + | |'Beyond Good and Evil: Prelude to a Philosophy of the Future', |
| | + | | trans. by R.J. Hollingdale, intro. by Michael Tanner, |
| | + | | Penguin Books, London, UK, 1973, 1990. |
| | | | |
| − | W ::> W'
| + | In this Subsection, I describe a particular semantics for the |
| | + | painted cactus language, telling what meanings I aim to attach |
| | + | to its bare syntactic forms. This supplies an "interpretation" |
| | + | for this parametric family of formal languages, but it is good |
| | + | to remember that it forms just one of many such interpretations |
| | + | that are conceivable and even viable. In deed, the distinction |
| | + | between the object domain and the sign domain can be observed in |
| | + | the fact that many languages can be deployed to depict the same |
| | + | set of objects and that any language worth its salt is bound to |
| | + | to give rise to many different forms of interpretive saliency. |
| | | | |
| − | A "derivation" in !G! is a finite sequence (W_1, ..., W_k)
| + | In formal settings, it is common to speak of "interpretation" as if it |
| − | of sentential forms over !G! such that each adjacent pair | + | created a direct connection between the signs of a formal language and |
| − | (W_j, W_(j+1)) of sentential forms in the sequence is an
| + | the objects of the intended domain, in other words, as if it determined |
| − | immediate derivation in !G!, in other words, such that:
| + | the denotative component of a sign relation. But a closer attention to |
| − | | + | what goes on reveals that the process of interpretation is more indirect, |
| − | W_j ::> W_(j+1), for all j = 1 to k-1
| + | that what it does is to provide each sign of a prospectively meaningful |
| | + | source language with a translation into an already established target |
| | + | language, where "already established" means that its relationship to |
| | + | pragmatic objects is taken for granted at the moment in question. |
| | | | |
| − | If there exists a derivation (W_1, ..., W_k) in !G!,
| + | With this in mind, it is clear that interpretation is an affair of signs |
| − | one says that W_1 "derives" W_k in !G!, conversely,
| + | that at best respects the objects of all of the signs that enter into it, |
| − | that W_k is "derivable" from W_1 in !G!, and one | + | and so it is the connotative aspect of semiotics that is at stake here. |
| − | typically summarizes the derivation by writing:
| + | There is nothing wrong with my saying that I interpret a sentence of a |
| | + | formal language as a sign that refers to a function or to a proposition, |
| | + | so long as you understand that this reference is likely to be achieved |
| | + | by way of more familiar and perhaps less formal signs that you already |
| | + | take to denote those objects. |
| | | | |
| − | W_1 :*:> W_k
| + | On entering a context where a logical interpretation is intended for the |
| | + | sentences of a formal language there are a few conventions that make it |
| | + | easier to make the translation from abstract syntactic forms to their |
| | + | intended semantic senses. Although these conventions are expressed in |
| | + | unnecessarily colorful terms, from a purely abstract point of view, they |
| | + | do provide a useful array of connotations that help to negotiate what is |
| | + | otherwise a difficult transition. This terminology is introduced as the |
| | + | need for it arises in the process of interpreting the cactus language. |
| | | | |
| − | The language !L! = !L!(!G!) = <!G!> that is "generated" | + | The task of this Subsection is to specify a "semantic function" for |
| − | by the formal grammar !G! = ("S", !Q!, !A!, !K!) is the
| + | the sentences of the cactus language !L! = !C!(!P!), in other words, |
| − | set of strings over the terminal alphabet !A! that are
| + | to define a mapping that "interprets" each sentence of !C!(!P!) as |
| − | derivable from the initial symbol "S" by way of the
| + | a sentence that says something, as a sentence that bears a meaning, |
| − | intermediate symbols in !Q! according to the
| + | in short, as a sentence that denotes a proposition, and thus as a |
| − | characterizations in K. In sum:
| + | sign of an indicator function. When the syntactic sentences of a |
| | + | formal language are given a referent significance in logical terms, |
| | + | for example, as denoting propositions or indicator functions, then |
| | + | each form of syntactic combination takes on a corresponding form |
| | + | of logical significance. |
| | | | |
| − | !L!(!G!) = <!G!> = {W in !A!* : "S" :*:> W}
| + | By way of providing a logical interpretation for the cactus language, |
| | + | I introduce a family of operators on indicator functions that are |
| | + | called "propositional connectives", and I distinguish these from |
| | + | the associated family of syntactic combinations that are called |
| | + | "sentential connectives", where the relationship between these |
| | + | two realms of connection is exactly that between objects and |
| | + | their signs. A propositional connective, as an entity of a |
| | + | well-defined functional and operational type, can be treated |
| | + | in every way as a logical or a mathematical object, and thus |
| | + | as the type of object that can be denoted by the corresponding |
| | + | form of syntactic entity, namely, the sentential connective that |
| | + | is appropriate to the case in question. |
| | | | |
| − | Finally, a string W is called a "word", a "sentence", or so on,
| + | There are two basic types of connectives, called the "blank connectives" |
| − | of the language generated by !G! if and only if W is in !L!(!G!).
| + | and the "bound connectives", respectively, with one connective of each |
| | + | type for each natural number k = 0, 1, 2, 3, ... . |
| | | | |
| − | Reference:
| + | 1. The "blank connective" of k places is signified by the |
| | + | concatenation of the k sentences that fill those places. |
| | | | |
| − | | Denning, P.J., Dennis, J.B., Qualitz, J.E.,
| + | For the special case of k = 0, the "blank connective" is taken to |
| − | |'Machines, Languages, and Computation',
| + | be an empty string or a blank symbol -- it does not matter which, |
| − | | Prentice-Hall, Englewood Cliffs, NJ, 1978.
| + | since both are assigned the same denotation among propositions. |
| − | | + | For the generic case of k > 0, the "blank connective" takes |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | the form "S_1 · ... · S_k". In the type of data that is |
| | + | called a "text", the raised dots "·" are usually omitted, |
| | + | supplanted by whatever number of spaces and line breaks |
| | + | serve to improve the readability of the resulting text. |
| | | | |
| − | IDS. Note 158
| + | 2. The "bound connective" of k places is signified by the |
| | + | surcatenation of the k sentences that fill those places. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | For the special case of k = 0, the "bound connective" is taken to |
| | + | be an expression of the form "-()-", "-( )-", "-( )-", and so on, |
| | + | with any number of blank symbols between the parentheses, all of |
| | + | which are assigned the same logical denotation among propositions. |
| | + | For the generic case of k > 0, the "bound connective" takes the |
| | + | form "-(S_1, ..., S_k)-". |
| | | | |
| − | 1.3.10.10. The Cactus Language: Stylistics
| + | At this point, there are actually two different "dialects", "scripts", |
| | + | or "modes" of presentation for the cactus language that need to be |
| | + | interpreted, in other words, that need to have a semantic function |
| | + | defined on their domains. |
| | | | |
| − | | As a result, we can hardly conceive of how many possibilities there are for what
| + | a. There is the literal formal language of strings in PARCE(!P!), |
| − | | we call objective reality. Our sharp quills of knowledge are so narrow and so
| + | the "painted and rooted cactus expressions" that constitute |
| − | | concentrated in particular directions that with science there are myriads of
| + | the langauge !L! = !C!(!P!) c !A!* = (!M! |_| !P!)*. |
| − | | totally different real worlds, each one accessible from the next simply by
| |
| − | | slight alterations -- shifts of gaze -- of every particular discipline
| |
| − | | and subspecialty.
| |
| − | |
| |
| − | | Herbert J. Bernstein, "Idols", p. 38.
| |
| − | |
| |
| − | | Herbert J. Bernstein,
| |
| − | |"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in:
| |
| − | | | |
| − | | Marcus G. Raskin & Herbert J. Bernstein, | |
| − | |'New Ways of Knowing: The Sciences, Society, and Reconstructive Knowledge',
| |
| − | | Rowman & Littlefield, Totowa, NJ, 1987.
| |
| | | | |
| − | This Subsection highlights an issue of "style" that arises in describing
| + | b. There is the figurative formal language of graphs in PARC(!P!), |
| − | a formal language. In broad terms, I use the word "style" to refer to a
| + | the "painted and rooted cacti" themselves, a parametric family |
| − | loosely specified class of formal systems, typically ones that have a set
| + | of graphs or a species of computational data structures that |
| − | of distinctive features in common. For instance, a style of proof system | + | is graphically analogous to the language of literal strings. |
| − | usually dictates one or more rules of inference that are acknowledged as
| |
| − | conforming to that style. In the present context, the word "style" is a
| |
| − | natural choice to characterize the varieties of formal grammars, or any
| |
| − | other sorts of formal systems that can be contemplated for deriving the
| |
| − | sentences of a formal language.
| |
| | | | |
| − | In looking at what seems like an incidental issue, the discussion arrives
| + | Of course, these two modalities of formal language, like written and |
| − | at a critical point. The question is: What decides the issue of style?
| + | spoken natural languages, are meant to have compatible interpretations, |
| − | Taking a given language as the object of discussion, what factors enter
| + | and so it is usually sufficient to give just the meanings of either one. |
| − | into and determine the choice of a style for its presentation, that is,
| + | All that remains is to provide a "codomain" or a "target space" for the |
| − | a particular way of arranging and selecting the materials that come to
| + | intended semantic function, in other words, to supply a suitable range |
| − | be involved in a description, a grammar, or a theory of the language?
| + | of logical meanings for the memberships of these languages to map into. |
| − | To what degree is the determination accidental, empirical, pragmatic,
| + | Out of the many interpretations that are formally possible to arrange, |
| − | rhetorical, or stylistic, and to what extent is the choice essential,
| + | one way of doing this proceeds by making the following definitions: |
| − | logical, and necessary? For that matter, what determines the order | |
| − | of signs in a word, a sentence, a text, or a discussion? All of | |
| − | the corresponding parallel questions about the character of this
| |
| − | choice can be posed with regard to the constituent part as well
| |
| − | as with regard to the main constitution of the formal language.
| |
| | | | |
| − | In order to answer this sort of question, at any level of articulation,
| + | 1. The "conjunction" Conj^J_j Q_j of a set of propositions, {Q_j : j in J}, |
| − | one has to inquire into the type of distinction that it invokes, between
| + | is a proposition that is true if and only if each one of the Q_j is true. |
| − | arrangements and orders that are essential, logical, and necessary and
| |
| − | orders and arrangements that are accidental, rhetorical, and stylistic.
| |
| − | As a rough guide to its comprehension, a "logical order", if it resides
| |
| − | in the subject at all, can be approached by considering all of the ways
| |
| − | of saying the same things, in all of the languages that are capable of | |
| − | saying roughly the same things about that subject. Of course, the "all"
| |
| − | that appears in this rule of thumb has to be interpreted as a reasonably
| |
| − | qualified type of universal. For all practical purposes, it simply means
| |
| − | "all of the ways that a person can think of" and "all of the languages
| |
| − | that a person can conceive of", with all things being relative to the
| |
| − | particular moment of investigation. For all of these reasons, the rule
| |
| − | must stand as little more than a rough idea of how to approach its object.
| |
| | | | |
| − | If it is demonstrated that a given formal language can be presented in
| + | Conj^J_j Q_j is true <=> Q_j is true for every j in J. |
| − | any one of several styles of formal grammar, then the choice of a format
| |
| − | is accidental, optional, and stylistic to the very extent that it is free. | |
| − | But if it can be shown that a particular language cannot be successfully
| |
| − | presented in a particular style of grammar, then the issue of style is
| |
| − | no longer free and rhetorical, but becomes to that very degree essential,
| |
| − | necessary, and obligatory, in other words, a question of the objective
| |
| − | logical order that can be found to reside in the object language.
| |
| | | | |
| − | As a rough illustration of the difference between logical and rhetorical
| + | 2. The "surjunction" Surj^J_j Q_j of a set of propositions, {Q_j : j in J}, |
| − | orders, consider the kinds of order that are expressed and exhibited in
| + | is a proposition that is true if and only if just one of the Q_j is untrue. |
| − | the following conjunction of implications: | |
| | | | |
| − | X => Y and Y => Z
| + | Surj^J_j Q_j is true <=> Q_j is untrue for unique j in J. |
| | | | |
| − | Here, there is a happy conformity between the logical content and the
| + | If the number of propositions that are being joined together is finite, |
| − | rhetorical form, indeed, to such a degree that one hardly notices the
| + | then the conjunction and the surjunction can be represented by means of |
| − | difference between them. The rhetorical form is given by the order
| + | sentential connectives, incorporating the sentences that represent these |
| − | of sentences in the two implications and the order of implications
| + | propositions into finite strings of symbols. |
| − | in the conjunction. The logical content is given by the order of
| |
| − | propositions in the extended implicational sequence: | |
| | | | |
| − | X =< Y =< Z
| + | If J is finite, for instance, if J constitutes the interval j = 1 to k, |
| | + | and if each proposition Q_j is represented by a sentence S_j, then the |
| | + | following strategies of expression are open: |
| | | | |
| − | To see the difference between form and content, or manner and matter,
| + | 1. The conjunction Conj^J_j Q_j can be represented by a sentence that |
| − | it is enough to observe a few of the ways that the expression can be
| + | is constructed by concatenating the S_j in the following fashion: |
| − | varied without changing its meaning, for example:
| |
| | | | |
| − | Z <= Y and Y <= X
| + | Conj^J_j Q_j <-< S_1 S_2 ... S_k. |
| | | | |
| − | Any style of declarative programming, also called "logic programming",
| + | 2. The surjunction Surj^J_j Q_j can be represented by a sentence that |
| − | depends on a capacity, as embodied in a programming language or other
| + | is constructed by surcatenating the S_j in the following fashion: |
| − | formal system, to describe the relation between problems and solutions
| |
| − | in logical terms. A recurring problem in building this capacity is in
| |
| − | bridging the gap between ostensibly non-logical orders and the logical
| |
| − | orders that are used to describe and to represent them. For instance,
| |
| − | to mention just a couple of the most pressing cases, and the ones that
| |
| − | are currently proving to be the most resistant to a complete analysis,
| |
| − | one has the orders of dynamic evolution and rhetorical transition that
| |
| − | manifest themselves in the process of inquiry and in the communication
| |
| − | of its results.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | Surj^J_j Q_j <-< -(S_1, S_2, ..., S_k)-. |
| | | | |
| − | IDS. Note 159
| + | If one opts for a mode of interpretation that moves more directly from |
| | + | the parse graph of a sentence to the potential logical meaning of both |
| | + | the PARC and the PARCE, then the following specifications are in order: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | A cactus rooted at a particular node is taken to represent what that |
| | + | node denotes, its logical denotation or its logical interpretation. |
| | | | |
| − | 1.3.10.10. The Cactus Language: Stylistics (cont.) | + | 1. The logical denotation of a node is the logical conjunction of that node's |
| | + | "arguments", which are defined as the logical denotations of that node's |
| | + | attachments. The logical denotation of either a blank symbol or an empty |
| | + | node is the boolean value %1% = "true". The logical denotation of the |
| | + | paint p_j is the proposition P_j, a proposition that is regarded as |
| | + | "primitive", at least, with respect to the level of analysis that |
| | + | is represented in the current instance of !C!(!P!). |
| | | | |
| − | This patch of the ongoing discussion is concerned with describing a
| + | 2. The logical denotation of a lobe is the logical surjunction of that lobe's |
| − | particular variety of formal languages, whose typical representative
| + | "arguments", which are defined as the logical denotations of that lobe's |
| − | is the painted cactus language !L! = !C!(!P!). It is the intention of | + | accoutrements. As a corollary, the logical denotation of the parse graph |
| − | this work to interpret this language for propositional logic, and thus
| + | of "-()-", otherwise called a "needle", is the boolean value %0% = "false". |
| − | to use it as a sentential calculus, an order of reasoning that forms an
| |
| − | active ingredient and a significant component of all logical reasoning.
| |
| − | To describe this language, the standard devices of formal grammars and
| |
| − | formal language theory are more than adequate, but this only raises the
| |
| − | next question: What sorts of devices are exactly adequate, and fit the
| |
| − | task to a "T"? The ultimate desire is to turn the tables on the order
| |
| − | of description, and so begins a process of eversion that evolves to the
| |
| − | point of asking: To what extent can the language capture the essential
| |
| − | features and laws of its own grammar and describe the active principles
| |
| − | of its own generation? In other words: How well can the language be
| |
| − | described by using the language itself to do so?
| |
| | | | |
| − | In order to speak to these questions, I have to express what a grammar says
| + | If one takes the point of view that PARC's and PARCE's amount to a |
| − | about a language in terms of what a language can say on its own. In effect,
| + | pair of intertranslatable languages for the same domain of objects, |
| − | it is necessary to analyze the kinds of meaningful statements that grammars
| + | then the "spiny bracket" notation, as in "-[C_j]-" or "-[S_j]-", |
| − | are capable of making about languages in general and to relate them to the
| + | can be used on either domain of signs to indicate the logical |
| − | kinds of meaningful statements that the syntactic "sentences" of the cactus
| + | denotation of a cactus C_j or the logical denotation of |
| − | language might be interpreted as making about the very same topics. So far
| + | a sentence S_j, respectively. |
| − | in the present discussion, the sentences of the cactus language do not make | |
| − | any meaningful statements at all, much less any meaningful statements about
| |
| − | languages and their constitutions. As of yet, these sentences subsist in the
| |
| − | form of purely abstract, formal, and uninterpreted combinatorial constructions.
| |
| | | | |
| − | Before the capacity of a language to describe itself can be evaluated,
| + | Tables 13.1 and 13.2 summarize the relations that serve to connect the |
| − | the missing link to meaning has to be supplied for each of its strings. | + | formal language of sentences with the logical language of propositions. |
| − | This calls for a dimension of semantics and a notion of interpretation,
| + | Between these two realms of expression there is a family of graphical |
| − | topics that are taken up for the case of the cactus language !C!(!P!)
| + | data structures that arise in parsing the sentences and that serve to |
| − | in Subsection 1.3.10.12. Once a plausible semantics is prescribed for
| + | facilitate the performance of computations on the indicator functions. |
| − | this language it will be possible to return to these questions and to
| + | The graphical language supplies an intermediate form of representation |
| − | address them in a meaningful way.
| + | between the formal sentences and the indicator functions, and the form |
| | + | of mediation that it provides is very useful in rendering the possible |
| | + | connections between the other two languages conceivable in fact, not to |
| | + | mention in carrying out the necessary translations on a practical basis. |
| | + | These Tables include this intermediate domain in their Central Columns. |
| | + | Between their First and Middle Columns they illustrate the mechanics of |
| | + | parsing the abstract sentences of the cactus language into the graphical |
| | + | data structures of the corresponding species. Between their Middle and |
| | + | Final Columns they summarize the semantics of interpreting the graphical |
| | + | forms of representation for the purposes of reasoning with propositions. |
| | | | |
| − | The prominent issue at this point is the distinct placements of formal
| + | Table 13.1 Semantic Translations: Functional Form |
| − | languages and formal grammars with respect to the question of meaning.
| + | o-------------------o-----o-------------------o-----o-------------------o |
| − | The sentences of a formal language are merely the abstract strings of
| + | | | Par | | Den | | |
| − | abstract signs that happen to belong to a certain set. They do not by
| + | | Sentence | --> | Graph | --> | Proposition | |
| − | themselves make any meaningful statements at all, not without mounting
| + | o-------------------o-----o-------------------o-----o-------------------o |
| − | a separate effort of interpretation, but the rules of a formal grammar
| + | | | | | | | |
| − | make meaningful statements about a formal language, to the extent that
| + | | S_j | --> | C_j | --> | Q_j | |
| − | they say what strings belong to it and what strings do not. Thus, the
| + | | | | | | | |
| − | formal grammar, a formalism that appears to be even more skeletal than
| + | o-------------------o-----o-------------------o-----o-------------------o |
| − | the formal language, still has bits and pieces of meaning attached to it.
| + | | | | | | | |
| − | In a sense, the question of meaning is factored into two parts, structure
| + | | Conc^0 | --> | Node^0 | --> | %1% | |
| − | and value, leaving the aspect of value reduced in complexity and subtlety
| + | | | | | | | |
| − | to the simple question of belonging. Whether this single bit of meaningful
| + | | Conc^k_j S_j | --> | Node^k_j C_j | --> | Conj^k_j Q_j | |
| − | value is enough to encompass all of the dimensions of meaning that we require,
| + | | | | | | | |
| − | and whether it can be compounded to cover the complexity that actually exists
| + | o-------------------o-----o-------------------o-----o-------------------o |
| − | in the realm of meaning -- these are questions for an extended future inquiry.
| + | | | | | | | |
| | + | | Surc^0 | --> | Lobe^0 | --> | %0% | |
| | + | | | | | | | |
| | + | | Surc^k_j S_j | --> | Lobe^k_j C_j | --> | Surj^k_j Q_j | |
| | + | | | | | | | |
| | + | o-------------------o-----o-------------------o-----o-------------------o |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | + | Table 13.2 Semantic Translations: Equational Form |
| | + | o-------------------o-----o-------------------o-----o-------------------o |
| | + | | | Par | | Den | | |
| | + | | -[Sentence]- | = | -[Graph]- | = | Proposition | |
| | + | o-------------------o-----o-------------------o-----o-------------------o |
| | + | | | | | | | |
| | + | | -[S_j]- | = | -[C_j]- | = | Q_j | |
| | + | | | | | | | |
| | + | o-------------------o-----o-------------------o-----o-------------------o |
| | + | | | | | | | |
| | + | | -[Conc^0]- | = | -[Node^0]- | = | %1% | |
| | + | | | | | | | |
| | + | | -[Conc^k_j S_j]- | = | -[Node^k_j C_j]- | = | Conj^k_j Q_j | |
| | + | | | | | | | |
| | + | o-------------------o-----o-------------------o-----o-------------------o |
| | + | | | | | | | |
| | + | | -[Surc^0]- | = | -[Lobe^0]- | = | %0% | |
| | + | | | | | | | |
| | + | | -[Surc^k_j S_j]- | = | -[Lobe^k_j C_j]- | = | Surj^k_j Q_j | |
| | + | | | | | | | |
| | + | o-------------------o-----o-------------------o-----o-------------------o |
| | | | |
| − | IDS. Note 160
| + | Aside from their common topic, the two Tables present slightly different |
| | + | ways of conceptualizing the operations that go to establish their maps. |
| | + | Table 13.1 records the functional associations that connect each domain |
| | + | with the next, taking the triplings of a sentence S_j, a cactus C_j, and |
| | + | a proposition Q_j as basic data, and fixing the rest by recursion on these. |
| | + | Table 13.2 records these associations in the form of equations, treating |
| | + | sentences and graphs as alternative kinds of signs, and generalizing the |
| | + | spiny bracket operator to indicate the proposition that either denotes. |
| | + | It should be clear at this point that either scheme of translation puts |
| | + | the sentences, the graphs, and the propositions that it associates with |
| | + | each other roughly in the roles of the signs, the interpretants, and the |
| | + | objects, respectively, whose triples define an appropriate sign relation. |
| | + | Indeed, the "roughly" can be made "exactly" as soon as the domains of |
| | + | a suitable sign relation are specified precisely. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | A good way to illustrate the action of the conjunction and surjunction |
| | + | operators is to demonstate how they can be used to construct all of the |
| | + | boolean functions on k variables, just now, let us say, for k = 0, 1, 2. |
| | | | |
| − | 1.3.10.10. The Cactus Language: Stylistics (cont.) | + | A boolean function on 0 variables is just a boolean constant F^0 in the |
| | + | boolean domain %B% = {%0%, %1%}. Table 14 shows several different ways |
| | + | of referring to these elements, just for the sake of consistency using |
| | + | the same format that will be used in subsequent Tables, no matter how |
| | + | degenerate it tends to appears in the immediate case: |
| | | | |
| − | Perhaps I ought to comment on the differences between the present and
| + | Column 1 lists each boolean element or boolean function under its |
| − | the standard definition of a formal grammar, since I am attempting to
| + | ordinary constant name or under a succinct nickname, respectively. |
| − | strike a compromise with several alternative conventions of usage, and
| |
| − | thus to leave certain options open for future exploration. All of the
| |
| − | changes are minor, in the sense that they are not intended to alter the
| |
| − | classes of languages that are able to be generated, but only to clear up
| |
| − | various ambiguities and sundry obscurities that affect their conception.
| |
| | | | |
| − | Primarily, the conventional scope of non-terminal symbols was expanded
| + | Column 2 lists each boolean function in a style of function name "F^i_j" |
| − | to encompass the sentence symbol, mainly on account of all the contexts
| + | that is constructed as follows: The superscript "i" gives the dimension |
| − | where the initial and the intermediate symbols are naturally invoked in
| + | of the functional domain, that is, the number of its functional variables, |
| − | the same breath. By way of compensating for the usual exclusion of the
| + | and the subscript "j" is a binary string that recapitulates the functional |
| − | sentence symbol from the non-terminal class, an equivalent distinction
| + | values, using the obvious translation of boolean values into binary values. |
| − | was introduced in the fashion of a distinction between the initial and
| |
| − | the intermediate symbols, and this serves its purpose in all of those
| |
| − | contexts where the two kind of symbols need to be treated separately.
| |
| | | | |
| − | At the present point, I remain a bit worried about the motivations
| + | Column 3 lists the functional values for each boolean function, or possibly |
| − | and the justifications for introducing this distinction, under any
| + | a boolean element appearing in the guise of a function, for each combination |
| − | name, in the first place. It is purportedly designed to guarantee
| + | of its domain values. |
| − | that the process of derivation at least gets started in a definite
| |
| − | direction, while the real questions have to do with how it all ends.
| |
| − | The excuses of efficiency and expediency that I offered as plausible
| |
| − | and sufficient reasons for distinguishing between empty and significant
| |
| − | sentences are likely to be ephemeral, if not entirely illusory, since
| |
| − | intermediate symbols are still permitted to characterize or to cover
| |
| − | themselves, not to mention being allowed to cover the empty string,
| |
| − | and so the very types of traps that one exerts oneself to avoid at
| |
| − | the outset are always there to afflict the process at all of the
| |
| − | intervening times.
| |
| | | | |
| − | If one reflects on the form of grammar that is being prescribed here,
| + | Column 4 shows the usual expressions of these elements in the cactus language, |
| − | it looks as if one sought, rather futilely, to avoid the problems of
| + | conforming to the practice of omitting the strike-throughs in display formats. |
| − | recursion by proscribing the main program from calling itself, while
| + | Here I illustrate also the useful convention of sending the expression "(())" |
| − | allowing any subprogram to do so. But any trouble that is avoidable
| + | as a visible stand-in for the expression of a constantly "true" truth value, |
| − | in the part is also avoidable in the main, while any trouble that is
| + | one that would otherwise be represented by a blank expression, and tend to |
| − | inevitable in the part is also inevitable in the main. Consequently,
| + | elude our giving it much notice in the context of more demonstrative texts. |
| − | I am reserving the right to change my mind at a later stage, perhaps
| |
| − | to permit the initial symbol to characterize, to cover, to regenerate, | |
| − | or to produce itself, if that turns out to be the best way in the end.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | + | Table 14. Boolean Functions on Zero Variables |
| | + | o----------o----------o-------------------------------------------o----------o |
| | + | | Constant | Function | F() | Function | |
| | + | o----------o----------o-------------------------------------------o----------o |
| | + | | | | | | |
| | + | | %0% | F^0_0 | %0% | () | |
| | + | | | | | | |
| | + | | %1% | F^0_1 | %1% | (()) | |
| | + | | | | | | |
| | + | o----------o----------o-------------------------------------------o----------o |
| | | | |
| − | IDS. Note 161
| + | Table 15 presents the boolean functions on one variable, F^1 : %B% -> %B%, |
| | + | of which there are precisely four. Here, Column 1 codes the contents of |
| | + | Column 2 in a more concise form, compressing the lists of boolean values, |
| | + | recorded as bits in the subscript string, into their decimal equivalents. |
| | + | Naturally, the boolean constants reprise themselves in this new setting |
| | + | as constant functions on one variable. Thus, one has the synonymous |
| | + | expressions for constant functions that are expressed in the next |
| | + | two chains of equations: |
| | + | |
| | + | F^1_0 = F^1_00 = %0% : %B% -> %B% |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | F^1_3 = F^1_11 = %1% : %B% -> %B% |
| | + | |
| | + | As for the rest, the other two functions are easily recognized as corresponding |
| | + | to the one-place logical connectives, or the monadic operators on %B%. Thus, |
| | + | the function F^1_1 = F^1_01 is recognizable as the negation operation, and |
| | + | the function F^1_2 = F^1_10 is obviously the identity operation. |
| | | | |
| − | 1.3.10.10. The Cactus Language: Stylistics (cont.)
| + | Table 15. Boolean Functions on One Variable |
| | + | o----------o----------o-------------------------------------------o----------o |
| | + | | Function | Function | F(x) | Function | |
| | + | o----------o----------o---------------------o---------------------o----------o |
| | + | | | | F(%0%) | F(%1%) | | |
| | + | o----------o----------o---------------------o---------------------o----------o |
| | + | | | | | | | |
| | + | | F^1_0 | F^1_00 | %0% | %0% | ( ) | |
| | + | | | | | | | |
| | + | | F^1_1 | F^1_01 | %0% | %1% | (x) | |
| | + | | | | | | | |
| | + | | F^1_2 | F^1_10 | %1% | %0% | x | |
| | + | | | | | | | |
| | + | | F^1_3 | F^1_11 | %1% | %1% | (( )) | |
| | + | | | | | | | |
| | + | o----------o----------o---------------------o---------------------o----------o |
| | | | |
| − | Before I leave this Subsection, I need to say a few things about
| + | Table 16 presents the boolean functions on two variables, F^2 : %B%^2 -> %B%, |
| − | the manner in which the abstract theory of formal languages and | + | of which there are precisely sixteen in number. As before, all of the boolean |
| − | the pragmatic theory of sign relations interact with each other. | + | functions of fewer variables are subsumed in this Table, though under a set of |
| | + | alternative names and possibly different interpretations. Just to acknowledge |
| | + | a few of the more notable pseudonyms: |
| | | | |
| − | Formal language theory can seem like an awfully picky subject at times,
| + | The constant function %0% : %B%^2 -> %B% appears under the name of F^2_00. |
| − | treating every symbol as a thing in itself the way it does, sorting out
| |
| − | the nominal types of symbols as objects in themselves, and singling out
| |
| − | the passing tokens of symbols as distinct entities in their own rights.
| |
| − | It has to continue doing this, if not for any better reason than to aid
| |
| − | in clarifying the kinds of languages that people are accustomed to use,
| |
| − | to assist in writing computer programs that are capable of parsing real
| |
| − | sentences, and to serve in designing programming languages that people
| |
| − | would like to become accustomed to use. As a matter of fact, the only
| |
| − | time that formal language theory becomes too picky, or a bit too myopic
| |
| − | in its focus, is when it leads one to think that one is dealing with the
| |
| − | thing itself and not just with the sign of it, in other words, when the
| |
| − | people who use the tools of formal language theory forget that they are
| |
| − | dealing with the mere signs of more interesting objects and not with the
| |
| − | objects of ultimate interest in and of themselves.
| |
| | | | |
| − | As a result, there a number of deleterious effects that can arise from
| + | The constant function %1% : %B%^2 -> %B% appears under the name of F^2_15. |
| − | the extreme pickiness of formal language theory, arising, as is often the
| |
| − | case, when formal theorists forget the practical context of theorization.
| |
| − | It frequently happens that the exacting task of defining the membership
| |
| − | of a formal language leads one to think that this object and this object
| |
| − | alone is the justifiable end of the whole exercise. The distractions of
| |
| − | this mediate objective render one liable to forget that one's penultimate
| |
| − | interest lies always with various kinds of equivalence classes of signs,
| |
| − | not entirely or exclusively with their more meticulous representatives.
| |
| | | | |
| − | When this happens, one typically goes on working oblivious to the fact
| + | The negation and identity of the first variable are F^2_03 and F^2_12, resp. |
| − | that many details about what transpires in the meantime do not matter
| |
| − | at all in the end, and one is likely to remain in blissful ignorance
| |
| − | of the circumstance that many special details of language membership
| |
| − | are bound, destined, and pre-determined to be glossed over with some
| |
| − | measure of indifference, especially when it comes down to the final
| |
| − | constitution of those equivalence classes of signs that are able to
| |
| − | answer for the genuine objects of the whole enterprise of language.
| |
| − | When any form of theory, against its initial and its best intentions,
| |
| − | leads to this kind of absence of mind that is no longer beneficial in
| |
| − | all of its main effects, the situation calls for an antidotal form of
| |
| − | theory, one that can restore the presence of mind that all forms of
| |
| − | theory are meant to augment.
| |
| | | | |
| − | The pragmatic theory of sign relations is called for in settings where | + | The negation and identity of the other variable are F^2_05 and F^2_10, resp. |
| − | everything that can be named has many other names, that is to say, in
| |
| − | the usual case. Of course, one would like to replace this superfluous
| |
| − | multiplicity of signs with an organized system of canonical signs, one
| |
| − | for each object that needs to be denoted, but reducing the redundancy
| |
| − | too far, beyond what is necessary to eliminate the factor of "noise" in
| |
| − | the language, that is, to clear up its effectively useless distractions,
| |
| − | can destroy the very utility of a typical language, which is intended to
| |
| − | provide a ready means to express a present situation, clear or not, and
| |
| − | to describe an ongoing condition of experience in just the way that it
| |
| − | seems to present itself. Within this fleshed out framework of language,
| |
| − | moreover, the process of transforming the manifestations of a sign from
| |
| − | its ordinary appearance to its canonical aspect is the whole problem of
| |
| − | computation in a nutshell.
| |
| | | | |
| − | It is a well-known truth, but an often forgotten fact, that nobody
| + | The logical conjunction is given by the function F^2_08 (x, y) = x · y. |
| − | computes with numbers, but solely with numerals in respect of numbers,
| |
| − | and numerals themselves are symbols. Among other things, this renders
| |
| − | all discussion of numeric versus symbolic computation a bit beside the
| |
| − | point, since it is only a question of what kinds of symbols are best for
| |
| − | one's immediate application or for one's selection of ongoing objectives.
| |
| − | The numerals that everybody knows best are just the canonical symbols,
| |
| − | the standard signs or the normal terms for numbers, and the process of
| |
| − | computation is a matter of getting from the arbitrarily obscure signs
| |
| − | that the data of a situation are capable of throwing one's way to the
| |
| − | indications of its character that are clear enough to motivate action.
| |
| | | | |
| − | Having broached the distinction between propositions and sentences, one
| + | The logical disjunction is given by the function F^2_14 (x, y) = ((x)(y)). |
| − | can see its similarity to the distinction between numbers and numerals.
| |
| − | What are the implications of the foregoing considerations for reasoning
| |
| − | about propositions and for the realm of reckonings in sentential logic?
| |
| − | If the purpose of a sentence is just to denote a proposition, then the
| |
| − | proposition is just the object of whatever sign is taken for a sentence.
| |
| − | This means that the computational manifestation of a piece of reasoning
| |
| − | about propositions amounts to a process that takes place entirely within
| |
| − | a language of sentences, a procedure that can rationalize its account by
| |
| − | referring to the denominations of these sentences among propositions.
| |
| | | | |
| − | The application of these considerations in the immediate setting is this:
| + | Functions expressing the "conditionals", "implications", |
| − | Do not worry too much about what roles the empty string "" and the blank
| + | or "if-then" statements are given in the following ways: |
| − | symbol " " are supposed to play in a given species of formal languages.
| |
| − | As it happens, it is far less important to wonder whether these types
| |
| − | of formal tokens actually constitute genuine sentences than it is to
| |
| − | decide what equivalence classes it makes sense to form over all of
| |
| − | the sentences in the resulting language, and only then to bother
| |
| − | about what equivalence classes these limiting cases of sentences
| |
| − | are most conveniently taken to represent.
| |
| | | | |
| − | These concerns about boundary conditions betray a more general issue.
| + | [x => y] = F^2_11 (x, y) = (x (y)) = [not x without y]. |
| − | Already by this point in discussion the limits of the purely syntactic
| |
| − | approach to a language are beginning to be visible. It is not that one
| |
| − | cannot go a whole lot further by this road in the analysis of a particular
| |
| − | language and in the study of languages in general, but when it comes to the
| |
| − | questions of understanding the purpose of a language, of extending its usage
| |
| − | in a chosen direction, or of designing a language for a particular set of uses,
| |
| − | what matters above all else are the "pragmatic equivalence classes" of signs that
| |
| − | are demanded by the application and intended by the designer, and not so much the
| |
| − | peculiar characters of the signs that represent these classes of practical meaning.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | [x <= y] = F^2_13 (x, y) = ((x) y) = [not y without x]. |
| | | | |
| − | IDS. Note 162
| + | The function that corresponds to the "biconditional", |
| | + | the "equivalence", or the "if and only" statement is |
| | + | exhibited in the following fashion: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | [x <=> y] = [x = y] = F^2_09 (x, y) = ((x , y)). |
| | | | |
| − | 1.3.10.10. The Cactus Language: Stylistics (cont.)
| + | Finally, there is a boolean function that is logically associated with |
| | + | the "exclusive disjunction", "inequivalence", or "not equals" statement, |
| | + | algebraically associated with the "binary sum" or "bitsum" operation, |
| | + | and geometrically associated with the "symmetric difference" of sets. |
| | + | This function is given by: |
| | | | |
| − | Any description of a language is bound to have alternative descriptions.
| + | [x =/= y] = [x + y] = F^2_06 (x, y) = (x , y). |
| − | More precisely, a circumscribed description of a formal language, as any
| |
| − | effectively finite description is bound to be, is certain to suggest the
| |
| − | equally likely existence and the possible utility of other descriptions.
| |
| − | A single formal grammar describes but a single formal language, but any
| |
| − | formal language is described by many different formal grammars, not all
| |
| − | of which afford the same grasp of its structure, provide an equivalent
| |
| − | comprehension of its character, or yield an interchangeable view of its
| |
| − | aspects. Consequently, even with respect to the same formal language,
| |
| − | different formal grammars are typically better for different purposes.
| |
| | | | |
| − | With the distinctions that evolve among the different styles of grammar,
| + | Table 16. Boolean Functions on Two Variables |
| − | and with the preferences that different observers display toward them,
| + | o----------o----------o-------------------------------------------o----------o |
| − | there naturally comes the question: What is the root of this evolution?
| + | | Function | Function | F(x, y) | Function | |
| | + | o----------o----------o----------o----------o----------o----------o----------o |
| | + | | | | %1%, %1% | %1%, %0% | %0%, %1% | %0%, %0% | | |
| | + | o----------o----------o----------o----------o----------o----------o----------o |
| | + | | | | | | | | | |
| | + | | F^2_00 | F^2_0000 | %0% | %0% | %0% | %0% | () | |
| | + | | | | | | | | | |
| | + | | F^2_01 | F^2_0001 | %0% | %0% | %0% | %1% | (x)(y) | |
| | + | | | | | | | | | |
| | + | | F^2_02 | F^2_0010 | %0% | %0% | %1% | %0% | (x) y | |
| | + | | | | | | | | | |
| | + | | F^2_03 | F^2_0011 | %0% | %0% | %1% | %1% | (x) | |
| | + | | | | | | | | | |
| | + | | F^2_04 | F^2_0100 | %0% | %1% | %0% | %0% | x (y) | |
| | + | | | | | | | | | |
| | + | | F^2_05 | F^2_0101 | %0% | %1% | %0% | %1% | (y) | |
| | + | | | | | | | | | |
| | + | | F^2_06 | F^2_0110 | %0% | %1% | %1% | %0% | (x, y) | |
| | + | | | | | | | | | |
| | + | | F^2_07 | F^2_0111 | %0% | %1% | %1% | %1% | (x y) | |
| | + | | | | | | | | | |
| | + | | F^2_08 | F^2_1000 | %1% | %0% | %0% | %0% | x y | |
| | + | | | | | | | | | |
| | + | | F^2_09 | F^2_1001 | %1% | %0% | %0% | %1% | ((x, y)) | |
| | + | | | | | | | | | |
| | + | | F^2_10 | F^2_1010 | %1% | %0% | %1% | %0% | y | |
| | + | | | | | | | | | |
| | + | | F^2_11 | F^2_1011 | %1% | %0% | %1% | %1% | (x (y)) | |
| | + | | | | | | | | | |
| | + | | F^2_12 | F^2_1100 | %1% | %1% | %0% | %0% | x | |
| | + | | | | | | | | | |
| | + | | F^2_13 | F^2_1101 | %1% | %1% | %0% | %1% | ((x) y) | |
| | + | | | | | | | | | |
| | + | | F^2_14 | F^2_1110 | %1% | %1% | %1% | %0% | ((x)(y)) | |
| | + | | | | | | | | | |
| | + | | F^2_15 | F^2_1111 | %1% | %1% | %1% | %1% | (()) | |
| | + | | | | | | | | | |
| | + | o----------o----------o----------o----------o----------o----------o----------o |
| | | | |
| − | One dimension of variation in the styles of formal grammars can be seen
| + | Let me now address one last question that may have occurred to some. |
| − | by treating the union of languages, and especially the disjoint union of | + | What has happened, in this suggested scheme of functional reasoning, |
| − | languages, as a "sum", by treating the concatenation of languages as a
| + | to the distinction that is quite pointedly made by careful logicians |
| − | "product", and then by distinguishing the styles of analysis that favor | + | between (1) the connectives called "conditionals" and symbolized by |
| − | "sums of products" from those that favor "products of sums" as their | + | the signs "->" and "<-", and (2) the assertions called "implications" |
| − | canonical forms of description. If one examines the relation between
| + | and symbolized by the signs "=>" and "<=", and, in a related question: |
| − | languages and grammars carefully enough to see the presence and the
| + | What has happened to the distinction that is equally insistently made |
| − | influence of these different styles, and when one comes to appreciate
| + | between (3) the connective called the "biconditional" and signified by |
| − | the ways that different styles of grammars can be used with different | + | the sign "<->" and (4) the assertion that is called an "equivalence" |
| − | degrees of success for different purposes, then one begins to see the
| + | and signified by the sign "<=>"? My answer is this: For my part, |
| − | possibility that alternative styles of description can be based on
| + | I am deliberately avoiding making these distinctions at the level |
| − | altogether different linguistic and logical operations.
| + | of syntax, preferring to treat them instead as distinctions in |
| | + | the use of boolean functions, turning on whether the function |
| | + | is mentioned directly and used to compute values on arguments, |
| | + | or whether its inverse is being invoked to indicate the fibers |
| | + | of truth or untruth under the propositional function in question. |
| | + | </pre> |
| | | | |
| − | It possible to trace this divergence of styles to an even more primitive
| + | =====1.3.11.6. Stretching Exercises===== |
| − | division, one that distinguishes the "additive" or the "parallel" styles
| |
| − | from the "multiplicative" or the "serial" styles. The issue is somewhat
| |
| − | confused by the fact that an "additive" analysis is typically expressed
| |
| − | in the form of a "series", in other words, a disjoint union of sets or a
| |
| − | linear sum of their independent effects. But it is easy enough to sort
| |
| − | this out if one observes the more telling connection between "parallel"
| |
| − | and "independent". Another way to keep the right associations straight
| |
| − | is to employ the term "sequential" in preference to the more misleading
| |
| − | term "serial". Whatever one calls this broad division of styles, the
| |
| − | scope and sweep of their dimensions of variation can be delineated in
| |
| − | the following way:
| |
| | | | |
| − | 1. The "additive" or "parallel" styles favor "sums of products" as
| + | <pre> |
| − | canonical forms of expression, pulling sums, unions, co-products,
| + | For ease of reference, I repeat here a couple of the |
| − | and logical disjunctions to the outermost layers of analysis and
| + | definitions that are needed again in this discussion. |
| − | synthesis, while pushing products, intersections, concatenations,
| |
| − | and logical conjunctions to the innermost levels of articulation
| |
| − | and generation. In propositional logic, this style leads to the
| |
| − | "disjunctive normal form" (DNF).
| |
| | | | |
| − | 2. The "multiplicative" or "serial" styles favor "products of sums" | + | | A "boolean connection" of degree k, also known as a "boolean function" |
| − | as canonical forms of expression, pulling products, intersections,
| + | | on k variables, is a map of the form F : %B%^k -> %B%. In other words, |
| − | concatenations, and logical conjunctions to the outermost layers of
| + | | a boolean connection of degree k is a proposition about things in the |
| − | analysis and synthesis, while pushing sums, unions, co-products,
| + | | universe of discourse X = %B%^k. |
| − | and logical disjunctions to the innermost levels of articulation
| + | | |
| − | and generation. In propositional logic, this style leads to the
| + | | An "imagination" of degree k on X is a k-tuple of propositions |
| − | "conjunctive normal form" (CNF).
| + | | about things in the universe X. By way of displaying the kinds |
| | + | | of notation that are used to express this idea, the imagination |
| | + | | #f# = <f_1, ..., f_k> is can be given as a sequence of indicator |
| | + | | functions f_j : X -> %B%, for j = 1 to k. All of these features |
| | + | | of the typical imagination #f# can be summed up in either one of |
| | + | | two ways: either in the form of a membership statement, stating |
| | + | | words to the effect that #f# belongs to the space (X -> %B%)^k, |
| | + | | or in the form of the type declaration that #f# : (X -> %B%)^k, |
| | + | | though perhaps the latter specification is slightly more precise |
| | + | | than the former. |
| | | | |
| − | There is a curious sort of diagnostic clue, a veritable shibboleth,
| + | The definition of the "stretch" operation and the uses of the |
| − | that often serves to reveal the dominance of one mode or the other
| + | various brands of denotational operators can be reviewed here: |
| − | within an individual thinker's cognitive style. Examined on the
| |
| − | question of what constitutes the "natural numbers", an "additive"
| |
| − | thinker tends to start the sequence at 0, while a "multiplicative"
| |
| − | thinker tends to regard it as beginning at 1.
| |
| | | | |
| − | In any style of description, grammar, or theory of a language, it is
| + | IDS 133. http://stderr.org/pipermail/inquiry/2004-June/001578.html |
| − | usually possible to tease out the influence of these contrasting traits,
| + | IDS 134. http://stderr.org/pipermail/inquiry/2004-June/001579.html |
| − | namely, the "additive" attitude versus the "mutiplicative" tendency that
| + | IDS 136. http://stderr.org/pipermail/inquiry/2004-June/001581.html |
| − | go to make up the particular style in question, and even to determine the
| + | IDS 137. http://stderr.org/pipermail/inquiry/2004-June/001582.html |
| − | dominant inclination or point of view that establishes its perspective on
| |
| − | the target domain.
| |
| | | | |
| − | In each style of formal grammar, the "multiplicative" aspect is present
| + | Taking up the preceding arrays of particular connections, namely, |
| − | in the sequential concatenation of signs, both in the augmented strings
| + | the boolean functions on two or less variables, it possible to |
| − | and in the terminal strings. In settings where the non-terminal symbols
| + | illustrate the use of the stretch operation in a variety of |
| − | classify types of strings, the concatenation of the non-terminal symbols
| + | concrete cases. |
| − | signifies the cartesian product over the corresponding sets of strings.
| |
| | | | |
| − | In the context-free style of formal grammar, the "additive" aspect is
| + | For example, suppose that F is a connection of the form F : %B%^2 -> %B%, |
| − | easy enough to spot. It is signaled by the parallel covering of many
| + | that is, any one of the sixteen possibilities in Table 16, while p and q |
| − | augmented strings or sentential forms by the same non-terminal symbol.
| + | are propositions of the form p, q : X -> %B%, that is, propositions about |
| − | Expressed in active terms, this calls for the independent rewriting
| + | things in the universe X, or else the indicators of sets contained in X. |
| − | of that non-terminal symbol by a number of different successors, | |
| − | as in the following scheme:
| |
| | | | |
| − | q :> W_1
| + | Then one has the imagination #f# = <f_1, f_2> = <p, q> : (X -> %B%)^2, |
| | + | and the stretch of the connection F to #f# on X amounts to a proposition |
| | + | F^$ <p, q> : X -> %B%, usually written as "F^$ (p, q)" and vocalized as |
| | + | the "stretch of F to p and q". If one is concerned with many different |
| | + | propositions about things in X, or if one is abstractly indifferent to |
| | + | the particular choices for p and q, then one can detach the operator |
| | + | F^$ : (X -> %B%)^2 -> (X -> %B%), called the "stretch of F over X", |
| | + | and consider it in isolation from any concrete application. |
| | | | |
| − | q :> W_2
| + | When the "cactus notation" is used to represent boolean functions, |
| | + | a single "$" sign at the end of the expression is enough to remind |
| | + | a reader that the connections are meant to be stretched to several |
| | + | propositions on a universe X. |
| | | | |
| − | ... ... ...
| + | For instance, take the connection F : %B%^2 -> %B% such that: |
| | | | |
| − | q :> W_k | + | F(x, y) = F^2_06 (x, y) = -(x, y)-. |
| | | | |
| − | It is useful to examine the relationship between the grammatical covering
| + | This connection is the boolean function on a couple of variables x, y |
| − | or production relation ":>" and the logical relation of implication "=>",
| + | that yields a value of %1% if and only if just one of x, y is not %1%, |
| − | with one eye to what they have in common and one eye to how they differ.
| + | that is, if and only if just one of x, y is %1%. There is clearly an |
| − | The production "q :> W" says that the appearance of the symbol "q" in
| + | isomorphism between this connection, viewed as an operation on the |
| − | a sentential form implies the possibility of exchanging it for "W".
| + | boolean domain %B% = {%0%, %1%}, and the dyadic operation on binary |
| − | Although this sounds like a "possible implication", to the extent
| + | values x, y in !B! = GF(2) that is otherwise known as "x + y". |
| − | that "q implies a possible W" or that "q possibly implies W", the
| |
| − | qualifiers "possible" and "possibly" are the critical elements in
| |
| − | these statements, and they are crucial to the meaning of what is
| |
| − | actually being implied. In effect, these qualifications reverse
| |
| − | the direction of implication, yielding "q <= W" as the best
| |
| − | analogue for the sense of the production.
| |
| | | | |
| − | One way to sum this up is to say that non-terminal symbols have the
| + | The same connection F : %B%^2 -> %B% can also be read as a proposition |
| − | significance of hypotheses. The terminal strings form the empirical
| + | about things in the universe X = %B%^2. If S is a sentence that denotes |
| − | matter of a language, while the non-terminal symbols mark the patterns
| + | the proposition F, then the corresponding assertion says exactly what one |
| − | or the types of substrings that can be noticed in the profusion of data.
| + | otherwise states by uttering "x is not equal to y". In such a case, one |
| − | If one observes a portion of a terminal string that falls into the pattern | + | has -[S]- = F, and all of the following expressions are ordinarily taken |
| − | of the sentential form W, then it is an admissable hypothesis, according to
| + | as equivalent descriptions of the same set: |
| − | the theory of the language that is constituted by the formal grammar, that | + | |
| − | this piece not only fits the type q but even comes to be generated under
| + | [| -[S]- |] = [| F |] |
| − | the auspices of the non-terminal symbol "q".
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | = F^(-1)(%1%) |
| | | | |
| − | IDS. Note 163
| + | = {<x, y> in %B%^2 : S} |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | = {<x, y> in %B%^2 : F(x, y) = %1%} |
| | | | |
| − | 1.3.10.10. The Cactus Language: Stylistics (cont.)
| + | = {<x, y> in %B%^2 : F(x, y)} |
| | | | |
| − | A moment's reflection on the issue of style, giving due consideration to the
| + | = {<x, y> in %B%^2 : -(x, y)- = %1%} |
| − | received array of stylistic choices, ought to inspire at least the question:
| |
| − | "Are these the only choices there are?" In the present setting, there are
| |
| − | abundant indications that other options, more differentiated varieties of
| |
| − | description and more integrated ways of approaching individual languages,
| |
| − | are likely to be conceivable, feasible, and even more ultimately viable.
| |
| − | If a suitably generic style, one that incorporates the full scope of
| |
| − | logical combinations and operations, is broadly available, then it
| |
| − | would no longer be necessary, or even apt, to argue in universal
| |
| − | terms about "which style is best", but more useful to investigate
| |
| − | how we might adapt the local styles to the local requirements.
| |
| − | The medium of a generic style would yield a viable compromise
| |
| − | between "additive" and "multiplicative" canons, and render the
| |
| − | choice between "parallel" and "serial" a false alternative,
| |
| − | at least, when expressed in the globally exclusive terms
| |
| − | that are currently most commonly adopted for posing it.
| |
| | | | |
| − | One set of indications comes from the study of machines, languages, and
| + | = {<x, y> in %B%^2 : -(x, y)- } |
| − | computation, especially the theories of their structures and relations.
| |
| − | The forms of composition and decomposition that are generally known as
| |
| − | "parallel" and "serial" are merely the extreme special cases, in variant
| |
| − | directions of specialization, of a more generic form, usually called the
| |
| − | "cascade" form of combination. This is a well-known fact in the theories
| |
| − | that deal with automata and their associated formal languages, but its
| |
| − | implications do not seem to be widely appreciated outside these fields.
| |
| − | In particular, it dispells the need to choose one extreme or the other,
| |
| − | since most of the natural cases are likely to exist somewhere in between.
| |
| | | | |
| − | Another set of indications appears in algebra and category theory,
| + | = {<x, y> in %B%^2 : x exclusive-or y} |
| − | where forms of composition and decomposition related to the cascade
| |
| − | combination, namely, the "semi-direct product" and its special case,
| |
| − | the "wreath product", are encountered at higher levels of generality
| |
| − | than the cartesian products of sets or the direct products of spaces.
| |
| | | | |
| − | In these domains of operation, one finds it necessary to consider also
| + | = {<x, y> in %B%^2 : just one true of x, y} |
| − | the "co-product" of sets and spaces, a construction that artificially
| |
| − | creates a disjoint union of sets, that is, a union of spaces that are
| |
| − | being treated as independent. It does this, in effect, by "indexing",
| |
| − | "coloring", or "preparing" the otherwise possibly overlapping domains
| |
| − | that are being combined. What renders this a "chimera" or a "hybrid"
| |
| − | form of combination is the fact that this indexing is tantamount to a
| |
| − | cartesian product of a singleton set, namely, the conventional "index",
| |
| − | "color", or "affix" in question, with the individual domain that is
| |
| − | entering as a factor, a term, or a participant in the final result.
| |
| | | | |
| − | One of the insights that arises out of Peirce's logical work is that
| + | = {<x, y> in %B%^2 : x not equal to y} |
| − | the set operations of complementation, intersection, and union, along
| |
| − | with the logical operations of negation, conjunction, and disjunction
| |
| − | that operate in isomorphic tandem with them, are not as fundamental as
| |
| − | they first appear. This is because all of them can be constructed from
| |
| − | or derived from a smaller set of operations, in fact, taking the logical
| |
| − | side of things, from either one of two "solely sufficient" operators,
| |
| − | called "amphecks" by Peirce, "strokes" by those who re-discovered them
| |
| − | later, and known in computer science as the NAND and the NNOR operators.
| |
| − | For this reason, that is, by virtue of their precedence in the orders
| |
| − | of construction and derivation, these operations have to be regarded
| |
| − | as the simplest and the most primitive in principle, even if they are
| |
| − | scarcely recognized as lying among the more familiar elements of logic.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | = {<x, y> in %B%^2 : x <=/=> y} |
| | | | |
| − | IDS. Note 164
| + | = {<x, y> in %B%^2 : x =/= y} |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | = {<x, y> in %B%^2 : x + y} |
| | | | |
| − | 1.3.10.10. The Cactus Language: Stylistics (cont.) | + | Notice the slight distinction, that I continue to maintain at this point, |
| | + | between the logical values {false, true} and the algebraic values {0, 1}. |
| | + | This makes it legitimate to write a sentence directly into the right side |
| | + | of the set-builder expression, for instance, weaving the sentence S or the |
| | + | sentence "x is not equal to y" into the context "{<x, y> in %B%^2 : ... }", |
| | + | thereby obtaining the corresponding expressions listed above, while the |
| | + | proposition F(x, y) can also be asserted more directly without equating |
| | + | it to %1%, since it already has a value in {false, true}, and thus can |
| | + | be taken as tantamount to an actual sentence. |
| | | | |
| − | I am throwing together a wide variety of different operations into
| + | If the appropriate safeguards can be kept in mind, avoiding all danger of |
| − | the bins labeled "additive" and "multiplicative", but it is easy to
| + | confusing propositions with sentences and sentences with assertions, then |
| − | observe a natural organization and even some relations that approach
| + | the marks of these distinctions need not be forced to clutter the account |
| − | the level of isomorphisms among and between the members of each class.
| + | of the more substantive indications, that is, the ones that really matter. |
| | + | If this level of understanding can be achieved, then it may be possible |
| | + | to relax these restrictions, along with the absolute dichotomy between |
| | + | algebraic and logical values, which tends to inhibit the flexibility |
| | + | of interpretation. |
| | | | |
| − | The relation between logical disjunction and set-theoretic union and
| + | This covers the properties of the connection F(x, y) = -(x, y)-, |
| − | the relation between logical conjunction and set-theoretic intersection
| + | treated as a proposition about things in the universe X = %B%^2. |
| − | are most likely clear enough for the purposes of the immediately present
| + | Staying with this same connection, it is time to demonstrate how |
| − | discussion. At any rate, all of these relations are scheduled to receive
| + | it can be "stretched" into an operator on arbitrary propositions. |
| − | a thorough examination in a subsequent discussion (Subsection 1.3.10.13). | |
| − | But the relation of set-theoretic union to category-theoretic co-product
| |
| − | and the relation of set-theoretic intersection to syntactic concatenation
| |
| − | deserve a closer look at this point.
| |
| | | | |
| − | The effect of a co-product as a "disjointed union", in other words, that
| + | To continue the exercise, let p and q be arbitrary propositions about |
| − | creates an object tantamount to a disjoint union of sets in the resulting
| + | things in the universe X, that is, maps of the form p, q : X -> %B%, |
| − | co-product even if some of these sets intersect non-trivially and even if
| + | and suppose that p, q are indicator functions of the sets P, Q c X, |
| − | some of them are identical "in reality", can be achieved in several ways.
| + | respectively. In other words, one has the following set of data: |
| − | The most usual conception is that of making a "separate copy", for each
| |
| − | part of the intended co-product, of the set that is intended to go there.
| |
| − | Often one thinks of the set that is assigned to a particular part of the
| |
| − | co-product as being distinguished by a particular "color", in other words,
| |
| − | by the attachment of a distinct "index", "label", or "tag", being a marker
| |
| − | that is inherited by and passed on to every element of the set in that part.
| |
| − | A concrete image of this construction can be achieved by imagining that each
| |
| − | set and each element of each set is placed in an ordered pair with the sign
| |
| − | of its color, index, label, or tag. One describes this as the "injection"
| |
| − | of each set into the corresponding "part" of the co-product.
| |
| | | | |
| − | For example, given the sets P and Q, overlapping or not, one can define
| + | p = -{P}- : X -> %B% |
| − | the "indexed" sets or the "marked" sets P_[1] and Q_[2], amounting to the
| |
| − | copy of P into the first part of the co-product and the copy of Q into the
| |
| − | second part of the co-product, in the following manner:
| |
| | | | |
| − | P_[1] = <P, 1> = {<x, 1> : x in P},
| + | q = -{Q}- : X -> %B% |
| | | | |
| − | Q_[2] = <Q, 2> = {<x, 2> : x in Q}. | + | <p, q> = < -{P}- , -{Q}- > : (X -> %B%)^2 |
| | | | |
| − | Using the sign "]_[" for this construction, the "sum", the "co-product",
| + | Then one has an operator F^$, the stretch of the connection F over X, |
| − | or the "disjointed union" of P and Q in that order can be represented as
| + | and a proposition F^$ (p, q), the stretch of F to <p, q> on X, with |
| − | the ordinary disjoint union of P_[1] and Q_[2], as follows: | + | the following properties: |
| | | | |
| − | P ]_[ Q = P_[1] |_| Q_[2]. | + | F^$ = -( , )-^$ : (X -> %B%)^2 -> (X -> %B%) |
| | | | |
| − | The concatenation L_1 · L_2 of the formal languages L_1 and L_2 is just
| + | F^$ (p, q) = -(p, q)-^$ : X -> %B% |
| − | the cartesian product of sets L_1 x L_2 without the extra x's, but the
| |
| − | relation of cartesian products to set-theoretic intersections and thus
| |
| − | to logical conjunctions is far from being clear.
| |
| | | | |
| − | One way of seeing a type of relation in this setting is to focus on the
| + | As a result, the application of the proposition F^$ (p, q) to each x in X |
| − | information that is needed to specify each construction, and thereby to
| + | yields a logical value in %B%, all in accord with the following equations: |
| − | reflect on the signs that are used to carry this information. As a way
| |
| − | of making a first approach to the topic of information, in accord with
| |
| − | a strategy that seeks to be as elementary and as informal as possible,
| |
| − | I introduce the following collection of ideas, intended to be taken
| |
| − | in a very provisional way.
| |
| | | | |
| − | A "stricture" is syntactic specification of a certain set in a certain place,
| + | F^$ (p, q)(x) = -(p, q)-^$ (x) in %B% |
| − | relative to a number of other sets, yet to be specified. It is assumed that
| |
| − | one knows enough about the general form of the specifications in question to
| |
| − | tell if two strictures are equivalent as pieces of information, but any more
| |
| − | determinate indications, like names for the places that are mentioned in the
| |
| − | stricture, or bounds on the number of places that are involved, are regarded
| |
| − | as being extraneous impositions, outside the chief concern of the definition,
| |
| − | no matter how convenient they are found to be within a particular discussion.
| |
| − | As a schematic form of illustration, a stricture can be pictured in this way:
| |
| | | | |
| − | "... x X x Q x X x ..."
| + | ^ ^ |
| | + | | | |
| | + | = = |
| | + | | | |
| | + | v v |
| | | | |
| − | A "strait" is the object that is specified by a stricture, in effect,
| + | F(p(x), q(x)) = -(p(x), q(x))- in %B% |
| − | a certain set in a certain place of an otherwise yet to be specified
| |
| − | relation. Somewhat sketchily, the strait that corresponds to the
| |
| − | stricture just given can be pictured in the following shape:
| |
| | | | |
| − | ... x X x Q x X x ...
| + | For each choice of propositions p and q about things in X, the stretch of |
| | + | F to p and q on X is just another proposition about things in X, a simple |
| | + | proposition in its own right, no matter how complex its current expression |
| | + | or its present construction as F^$ (p, q) = -(p, q)^$ makes it appear in |
| | + | relation to p and q. Like any other proposition about things in X, it |
| | + | indicates a subset of X, namely, the fiber that is variously described |
| | + | in the following ways: |
| | | | |
| − | In this picture, Q is a certain set, and X is the universe of discourse that is
| + | [| F^$ (p, q) |] = [| -(p, q)-^$ |] |
| − | pertinent to a given discussion. Since a stricture does not, by itself, contain
| |
| − | a sufficient amount of information to specify the number of sets that it intends
| |
| − | to set in place, or even to specify the absolute location of the set that it does
| |
| − | set in place, it appears to place an unspecified number of unspecified sets in a
| |
| − | vague and uncertain strait. Taken out of its interpretive context, the residual
| |
| − | information that a stricture can convey makes all of the following potentially
| |
| − | equivalent as strictures:
| |
| | | | |
| − | "Q", "X x Q x X", "X x X x Q x X x X", ...
| + | = (F^$ (p, q))^(-1)(%1%) |
| | | | |
| − | With respect to what these strictures specify, this
| + | = {x in X : F^$ (p, q)(x)} |
| − | leaves all of the following equivalent as straits:
| |
| | | | |
| − | Q = X x Q x X = X x X x Q x X x X = ...
| + | = {x in X : -(p, q)-^$ (x)} |
| | | | |
| − | Within the framework of a particular discussion, it is customary to
| + | = {x in X : -(p(x), q(x))- } |
| − | set a bound on the number of places and to limit the variety of sets
| |
| − | that are regarded as being under active consideration, and it is also
| |
| − | convenient to index the places of the indicated relations, and of their
| |
| − | encompassing cartesian products, in some fixed way. But the whole idea
| |
| − | of a stricture is to specify a strait that is capable of extending through
| |
| − | and beyond any fixed frame of discussion. In other words, a stricture is
| |
| − | conceived to constrain a strait at a certain point, and then to leave it
| |
| − | literally embedded, if tacitly expressed, in a yet to be fully specified
| |
| − | relation, one that involves an unspecified number of unspecified domains.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | = {x in X : p(x) + q(x)} |
| | | | |
| − | IDS. Note 165
| + | = {x in X : p(x) =/= q(x)} |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | = {x in X : -{P}- (x) =/= -{Q}- (x)} |
| | | | |
| − | 1.3.10.10. The Cactus Language: Stylistics (cont.)
| + | = {x in X : x in P <=/=> x in Q} |
| | | | |
| − | A quantity of information is a measure of constraint. In this respect,
| + | = {x in X : x in P-Q or x in Q-P} |
| − | a set of comparable strictures is ordered on account of the information
| |
| − | that each one conveys, and a system of comparable straits is ordered in
| |
| − | accord with the amount of information that it takes to pin each one of
| |
| − | them down. Strictures that are more constraining and straits that are
| |
| − | more constrained are placed at higher levels of information than those
| |
| − | that are less so. In other language that is often used, entities of
| |
| − | either kind that involve more information are said to have a greater
| |
| − | "complexity" in relation to comparable entities which involve less
| |
| − | information, the latter being said to have a greater "simplicity".
| |
| | | | |
| − | In order to create a concrete example, let me now institute a frame of
| + | = {x in X : x in P-Q |_| Q-P} |
| − | discussion where the number of places in a relation is bounded at two,
| |
| − | and where the variety of sets under active consideration is limited to
| |
| − | the typical subsets P and Q of a universe X. Under these conditions,
| |
| − | one can use the following sorts of expression as schematic strictures:
| |
| | | | |
| − | "X" "P" "Q"
| + | = {x in X : x in P + Q} |
| | | | |
| − | "X x X" "X x P" "X x Q"
| + | = P + Q c X |
| | | | |
| − | "P x X" "P x P" "P x Q"
| + | = [|p|] + [|q|] c X |
| | | | |
| − | "Q x X" "Q x P" "Q x Q"
| + | Which was to be shown. |
| | + | </pre> |
| | | | |
| − | These strictures and their corresponding straits are stratified according
| + | ====1.3.12. Syntactic Transformations==== |
| − | to their amounts of information, or their levels of constraint, as follows:
| |
| | | | |
| − | High: "P x P" "P x Q" "Q x P" "Q x Q"
| + | We have been examining several distinct but closely related notions of ''indication''. To discuss the import of these ideas in greater depth, it serves to establish a number of logical relations and set-theoretic identities that can be found to hold among their roughly parallel arrays of conceptions and constructions. Facilitating this task requires in turn a number of auxiliary concepts and notations. |
| | | | |
| − | Medium: "P" "X x P" "P x X"
| + | The diverse notions of indication presently under discussion are expressed in a variety of different notations, enumerated as follows: |
| | | | |
| − | Medium: "Q" "X x Q" "Q x X"
| + | # The functional language of propositions |
| | + | # The logical language of sentences |
| | + | # The geometric language of sets |
| | | | |
| − | Low: "X" "X x X"
| + | Correspondingly, one way to explain the relationships that exist among the various notions of indication is to describe the translations that they induce among the associated families of notation. |
| | | | |
| − | Within this framework, the more complex strait P x Q can be expressed
| + | =====1.3.12.1. Syntactic Transformation Rules===== |
| − | in terms of the simpler straits, P x X and X x Q. More specifically,
| |
| − | it lends itself to being analyzed as their intersection, as follows:
| |
| | | | |
| − | P x Q = P x X |^| X x Q
| + | A good way to summarize the necessary translations between different styles of indication, and along the way to organize their use in practice, is by means of the ''rules of syntactic transformation'' (ROSTs) that partially formalize the translations in question. |
| | | | |
| − | From here it is easy to see the relation of concatenation, by virtue of
| + | Rudimentary examples of ROSTs are readily mined from the raw materials that are already available in this area of discussion. To begin as near the beginning as possible, let the definition of an indicator function be recorded in the following form: |
| − | these types of intersection, to the logical conjunction of propositions.
| |
| − | A cartesian product P x Q is described by a conjunction of propositions,
| |
| − | namely, "P_<1> and Q_<2>", subject to the following interpretation:
| |
| | | | |
| − | 1. "P_<1>" asserts that there is an element from
| + | <pre> |
| − | the set P in the first place of the product.
| + | o-------------------------------------------------o |
| − | | + | | Definition 1. Indicator Function | |
| − | 2. "Q_<2>" asserts that there is an element from
| + | o-------------------------------------------------o |
| − | the set Q in the second place of the product.
| + | | | |
| − | | + | | If Q c X, | |
| − | The integration of these two pieces of information can be taken
| + | | | |
| − | in that measure to specify a yet to be fully determined relation.
| + | | then -{Q}- : X -> %B% | |
| | + | | | |
| | + | | such that, for all x in X: | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | + | | | |
| | + | | D1a. -{Q}-(x) <=> x in Q. | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | + | </pre> |
| | | | |
| − | In a corresponding fashion at the level of the elements, | + | In practice, a definition like this is commonly used to substitute one of two logically equivalent expressions or sentences for the other in a context where the conditions of using the definition in this way are satisfied and where the change is perceived as potentially advancing a proof. The employment of a definition in this way can be expressed in the form of a ROST that allows one to exchange two expressions of logically equivalent forms for one another in every context where their logical values are the only consideration. To be specific, the ''logical value'' of an expression is the value in the boolean domain %B% = {%0%, %1%} that the expression represents to its context or that it stands for in its context. |
| − | the ordered pair <p, q> is described by a conjunction | |
| − | of propositions, namely, "p_<1> and q_<2>", subject | |
| − | to the following interpretation: | |
| | | | |
| − | 1. "p_<1>" says that p is in the first place
| + | In the case of Definition 1, the corresponding ROST permits one to exchange a sentence of the form "x in Q" with an expression of the form "-{Q}-(x)" in any context that satisfies the conditions of its use, namely, the conditions of the definition that lead up to the stated equivalence. The relevant ROST is recorded in Rule 1. By way of convention, I list the items that fall under a rule in rough order of their ascending conceptual subtlety or their increasing syntactic complexity, without regard for the normal or the typical orders of their exchange, since this can vary from widely from case to case. |
| − | of the product element under construction.
| |
| | | | |
| − | 2. "q_<2>" says that q is in the second place
| + | <pre> |
| − | of the product element under construction.
| + | o-------------------------------------------------o |
| | + | | Rule 1 | |
| | + | o-------------------------------------------------o |
| | + | | | |
| | + | | If Q c X, | |
| | + | | | |
| | + | | then -{Q}- : X -> %B%, | |
| | + | | | |
| | + | | and if x in X, | |
| | + | | | |
| | + | | then the following are equivalent: | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | + | | | |
| | + | | R1a. x in Q. | |
| | + | | | |
| | + | | R1b. -{Q}-(x). | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | + | </pre> |
| | | | |
| − | Notice that, in construing the cartesian product of the sets P and Q or the
| + | Conversely, any rule of this sort, properly qualified by the conditions under which it applies, can be turned back into a summary statement of the logical equivalence that is involved in its application. This mode of conversion between a static principle and a transformational rule, in other words, between a statement of equivalence and an equivalence of statements, is so automatic that it is usually not necessary to make a separate note of the "horizontal" versus the "vertical" versions of what amounts to the same abstract principle. |
| − | concatenation of the languages L_1 and L_2 in this way, one shifts the level
| |
| − | of the active construction from the tupling of the elements in P and Q or the | |
| − | concatenation of the strings that are internal to the languages L_1 and L_2 to
| |
| − | the concatenation of the external signs that it takes to indicate these sets or
| |
| − | these languages, in other words, passing to a conjunction of indexed propositions,
| |
| − | "P_<1> and Q_<2>", or to a conjunction of assertions, "L_1_<1> and L_2_<2>", that
| |
| − | marks the sets or the languages in question for insertion in the indicated places
| |
| − | of a product set or a product language, respectively. In effect, the subscripting
| |
| − | by the indices "<1>" and "<2>" can be recognized as a special case of concatenation,
| |
| − | albeit through the posting of editorial remarks from an external "mark-up" language.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | As another example of a ROST, consider the following logical equivalence, that holds for any <math>X \subseteq U\!</math> and for all <math>u \in U.</math> |
| | | | |
| − | IDS. Note 166
| + | : -{X}-(u) <=> -{X}-(u) = 1. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | In practice, this logical equivalence is used to exchange an expression of the form "-{X}-(u)" with a sentence of the form "-{X}-(u) = 1" in any context where one has a relatively fixed X c U in mind and where one is conceiving u in U to vary over its whole domain, namely, the universe U. This leads to the ROST that is given in Rule 2. |
| | | | |
| − | 1.3.10.10. The Cactus Language: Stylistics (concl.)
| + | <pre> |
| − | | + | o-------------------------------------------------o |
| − | In order to systematize the relationships that strictures and straits
| + | | Rule 2 | |
| − | placed at higher levels of complexity, constraint, information, and
| + | o-------------------------------------------------o |
| − | organization bear toward strictures and straits that are placed at
| + | | | |
| − | the corresponding lower levels of these measures, I introduce the
| + | | If f : U -> %B% | |
| − | following pair of definitions:
| + | | | |
| − | | + | | and u in U, | |
| − | The j^th "excerpt" of a stricture of the form "S_1 x ... x S_k", regarded
| + | | | |
| − | within a frame of discussion where the number of places is limited to k,
| + | | then the following are equivalent: | |
| − | is the stricture of the form "X x ... x S_j x ... x X". In the proper
| + | | | |
| − | context, this can be written more succinctly as the stricture "S_j_<j>",
| + | o-------------------------------------------------o |
| − | an assertion that places the j^th set in the j^th place of the product.
| + | | | |
| | + | | R2a. f(u). | |
| | + | | | |
| | + | | R2b. f(u) = 1. | |
| | + | | | |
| | + | o-------------------------------------------------o |
| | + | </pre> |
| | + | |
| | + | Rules like these can be chained together to establish extended rules, just so long as their antecedent conditions are compatible. For example, Rules 1 and 2 combine to give the equivalents that are listed in Rule 3. This follows from a recognition that the function -{X}- : U -> %B% that is introduced in Rule 1 is an instance of the function f : U -> %B% that is mentioned in Rule 2. By the time one arrives in the "consequence box" of either Rule, then, one has in mind a comparatively fixed X c U, a proposition f or -{X}- about things in U, and a variable argument u in U. |
| | + | |
| | + | <pre> |
| | + | o-------------------------------------------------o---------o |
| | + | | Rule 3 | | |
| | + | o-------------------------------------------------o---------o |
| | + | | | | |
| | + | | If X c U | | |
| | + | | | | |
| | + | | and u in U, | | |
| | + | | | | |
| | + | | then the following are equivalent: | | |
| | + | | | | |
| | + | o-------------------------------------------------o---------o |
| | + | | | | |
| | + | | R3a. u in X. | : R1a | |
| | + | | | :: | |
| | + | | R3b. -{X}-(u). | : R1b | |
| | + | | | : R2a | |
| | + | | | :: | |
| | + | | R3c. -{X}-(u) = 1. | : R2b | |
| | + | | | | |
| | + | o-------------------------------------------------o---------o |
| | + | </pre> |
| | + | |
| | + | A large stock of rules can be derived in this way, by chaining together segments that are selected from a stock of previous rules, with perhaps the whole process of derivation leading back to an axial body or a core stock of rules, with all recurring to and relying on an axiomatic basis. In order to keep track of their derivations, as their pedigrees help to remember the reasons for trusting their use in the first place, derived rules can be annotated by citing the rules from which they are derived. |
| | + | |
| | + | In the present discussion, I am using a particular style of annotation for rule derivations, one that is called "proof by grammatical paradigm" or "proof by syntactic analogy". The annotations in the right margin of the Rule box can be read as the "denominators" of the paradigm that is being employed, in other words, as the alternating terms of comparison in a sequence of analogies. This can be illustrated by considering the derivation Rule 3 in detail. Taking the steps marked in the box one at a time, one can interweave the applications in the central body of the box with the annotations in the right margin of the box, reading "is to" for the ":" sign and "as" for the "::" sign, in the following fashion: |
| | | | |
| − | The j^th "extract" of a strait of the form S_1 x ... x S_k, constrained
| + | <pre> |
| − | to a frame of discussion where the number of places is restricted to k,
| + | R3a. "u in X" is to R1a, namely, "u in X", |
| − | is the strait of the form X x ... x S_j x ... x X. In the appropriate
| |
| − | context, this can be denoted more succinctly by the stricture "S_j_<j>",
| |
| − | an assertion that places the j^th set in the j^th place of the product.
| |
| | | | |
| − | In these terms, a stricture of the form "S_1 x ... x S_k"
| + | as |
| − | can be expressed in terms of simpler strictures, namely,
| |
| − | as a conjunction of its k excerpts: | |
| | | | |
| − | "S_1 x ... x S_k" = "S_1_<1>" & ... & "S_k_<k>".
| + | R3b. "{X}(u)" is to R1b, namely, "{X}(u)", |
| | | | |
| − | In a similar vein, a strait of the form S_1 x ... x S_k
| + | and |
| − | can be expressed in terms of simpler straits, namely,
| |
| − | as an intersection of its k extracts:
| |
| | | | |
| − | S_1 x ... x S_k = S_1_<1> |^| ... |^| S_k_<k>.
| + | "{X}(u)" is to R2a, namely, "f(u)", |
| | | | |
| − | There is a measure of ambiguity that remains in this formulation,
| + | as |
| − | but it is the best that I can do in the present informal context.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | R3c. "{X}(u) = 1" is to R2b, namely, "f(u) = 1". |
| | + | </pre> |
| | | | |
| − | IDS. Note 167
| + | Notice how the sequence of analogies pivots on the item R3b, viewing it first under the aegis of R1b, as the second term of the first analogy, and then turning to view it again under the guise of R2a, as the first term of the second analogy. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | By way of convention, rules that are tailored to a particular application, case, or subject, and rules that are adapted to a particular goal, object, or purpose, I frequently refer to as "Facts". |
| | | | |
| − | 1.3.10.11. The Cactus Language: Mechanics
| + | Besides linking rules together into extended sequences of equivalents, there is one other way that is commonly used to get new rules from old. Novel starting points for rules can be obtained by extracting pairs of equivalent expressions from a sequence that falls under an established rule, and then by stating their equality in the proper form of equation. For example, by extracting the equivalent expressions that are annotated as "R3a" and "R3c" in Rule 3 and by explictly stating their equivalence, one obtains the specialized result that is recorded in Corollary 1. |
| | | | |
| − | | We are only now beginning to see how this works. Clearly one of the
| + | <pre> |
| − | | mechanisms for picking a reality is the sociohistorical sense of what
| + | Corollary 1 |
| − | | is important -- which research program, with all its particularity of
| + | |
| − | | knowledge, seems most fundamental, most productive, most penetrating.
| + | If X c U |
| − | | The very judgments which make us push narrowly forward simultaneously
| + | |
| − | | make us forget how little we know. And when we look back at history,
| + | and u C U, |
| − | | where the lesson is plain to find, we often fail to imagine ourselves
| |
| − | | in a parallel situation. We ascribe the differences in world view
| |
| − | | to error, rather than to unexamined but consistent and internally
| |
| − | | justified choice.
| |
| − | |
| |
| − | | Herbert J. Bernstein, "Idols", p. 38.
| |
| − | |
| |
| − | | Herbert J. Bernstein,
| |
| − | |"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in:
| |
| − | |
| |
| − | | Marcus G. Raskin & Herbert J. Bernstein,
| |
| − | |'New Ways of Knowing: The Sciences, Society, and Reconstructive Knowledge',
| |
| − | | Rowman & Littlefield, Totowa, NJ, 1987.
| |
| | | | |
| − | In this Subsection, I discuss the "mechanics" of parsing the
| + | then the following statement is true: |
| − | cactus language into the corresponding class of computational
| |
| − | data structures. This provides each sentence of the language
| |
| − | with a translation into a computational form that articulates
| |
| − | its syntactic structure and prepares it for automated modes of
| |
| − | processing and evaluation. For this purpose, it is necessary
| |
| − | to describe the target data structures at a fairly high level
| |
| − | of abstraction only, ignoring the details of address pointers
| |
| − | and record structures and leaving the more operational aspects
| |
| − | of implementation to the imagination of prospective programmers.
| |
| − | In this way, I can put off to another stage of elaboration and
| |
| − | refinement the description of the program that constructs these
| |
| − | pointers and operates on these graph-theoretic data structures.
| |
| | | | |
| − | The structure of a "painted cactus", insofar as it presents itself
| + | C1a. u C X <=> {X}(u) = 1. R3a=R3c |
| − | to the visual imagination, can be described as follows. The overall
| + | </pre> |
| − | structure, as given by its underlying graph, falls within the species
| |
| − | of graph that is commonly known as a "rooted cactus", and the only novel
| |
| − | feature that it adds to this is that each of its nodes can be "painted"
| |
| − | with a finite sequence of "paints", chosen from a "palette" that is given
| |
| − | by the parametric set {" "} |_| !P! = {m_1} |_| {p_1, ..., p_k}.
| |
| | | | |
| − | It is conceivable, from a purely graph-theoretical point of view, to have
| + | There are a number of issues, that arise especially in establishing the proper use of ROSTs, that are appropriate to discuss at this juncture. The notation "[S]" is intended to represent "the proposition denoted by the sentence S". There is only one problem with the use of this form. There is, in general, no such thing as "the" proposition denoted by S. Generally speaking, if a sentence is taken out of context and considered across a variety of different contexts, there is no unique proposition that it can be said to denote. But one is seldom ever speaking at the maximum level of generality, or even found to be thinking of it, and so this notation is usually meaningful and readily understandable whenever it is read in the proper frame of mind. Still, once the issue is raised, the question of how these meanings and understandings are possible has to be addressed, especially if one desires to express the regulations of their syntax in a partially computational form. This requires a closer examination of the very notion of "context", and it involves engaging in enough reflection on the "contextual evaluation" of sentences that the relevant principles of its successful operation can be discerned and rationalized in explicit terms. |
| − | a class of cacti that are painted but not rooted, and so it is frequently
| |
| − | necessary, for the sake of precision, to more exactly pinpoint the target
| |
| − | species of graphical structure as a "painted and rooted cactus" (PARC).
| |
| | | | |
| − | A painted cactus, as a rooted graph, has a distinguished "node" that is | + | A sentence that is written in a context where it represents a value of 1 or 0 as a function of things in the universe U, where it stands for a value of "true" or "false", depending on how the signs that constitute its proper syntactic arguments are interpreted as denoting objects in U, in other words, where it is bound to lead its interpreter to view its own truth or falsity as determined by a choice of objects in U, is a sentence that might as well be written in the context "[ ... ]", whether or not this frame is explicitly marked around it. |
| − | called its "root". By starting from the root and working recursively,
| |
| − | the rest of its structure can be described in the following fashion.
| |
| | | | |
| − | Each "node" of a PARC consists of a graphical "point" or "vertex" plus
| + | More often than not, the context of interpretation fixes the denotations of most of the signs that make up a sentence, and so it is safe to adopt the convention that only those signs whose objects are not already fixed are free to vary in their denotations. Thus, only the signs that remain in default of prior specification are subject to treatment as variables, with a decree of functional abstraction hanging over all of their heads. |
| − | a finite sequence of "attachments", described in relative terms as the
| |
| − | attachments "at" or "to" that node. An empty sequence of attachments
| |
| − | defines the "empty node". Otherwise, each attachment is one of three
| |
| − | kinds: a blank, a paint, or a type of PARC that is called a "lobe".
| |
| | | | |
| − | Each "lobe" of a PARC consists of a directed graphical "cycle" plus a
| + | : [u C X] = Lambda (u, C, X).(u C X). |
| − | finite sequence of "accoutrements", described in relative terms as the
| |
| − | accoutrements "of" or "on" that lobe. Recalling the circumstance that
| |
| − | every lobe that comes under consideration comes already attached to a
| |
| − | particular node, exactly one vertex of the corresponding cycle is the
| |
| − | vertex that comes from that very node. The remaining vertices of the
| |
| − | cycle have their definitions filled out according to the accoutrements
| |
| − | of the lobe in question. An empty sequence of accoutrements is taken
| |
| − | to be tantamount to a sequence that contains a single empty node as its
| |
| − | unique accoutrement, and either one of these ways of approaching it can
| |
| − | be regarded as defining a graphical structure that is called a "needle"
| |
| − | or a "terminal edge". Otherwise, each accoutrement of a lobe is itself
| |
| − | an arbitrary PARC.
| |
| | | | |
| − | Although this definition of a lobe in terms of its intrinsic structural
| + | As it is presently stated, Rule 1 lists a couple of manifest sentences, and it authorizes one to make exchanges in either direction between the syntactic items that have these two forms. But a sentence is any sign that denotes a proposition, and thus there are a number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged. Consider the sense of equivalence among sentences that is recorded in Rule 4. |
| − | components is logically sufficient, it is also useful to characterize the
| |
| − | structure of a lobe in comparative terms, that is, to view the structure
| |
| − | that typifies a lobe in relation to the structures of other PARC's and to
| |
| − | mark the inclusion of this special type within the general run of PARC's.
| |
| − | This approach to the question of types results in a form of description
| |
| − | that appears to be a bit more analytic, at least, in mnemonic or prima | |
| − | facie terms, if not ultimately more revealing. Working in this vein,
| |
| − | a "lobe" can be characterized as a special type of PARC that is called
| |
| − | an "unpainted root plant" (UR-plant).
| |
| | | | |
| − | An "UR-plant" is a PARC of a simpler sort, at least, with respect to the
| + | <pre> |
| − | recursive ordering of structures that is being followed here. As a type,
| + | Rule 4 |
| − | it is defined by the presence of two properties, that of being "planted"
| |
| − | and that of having an "unpainted root". These are defined as follows:
| |
| | | | |
| − | 1. A PARC is "planted" if its list of attachments has just one PARC.
| + | If X c U is fixed |
| | | | |
| − | 2. A PARC is "UR" if its list of attachments has no blanks or paints.
| + | and u C U is varied, |
| | | | |
| − | In short, an UR-planted PARC has a single PARC as its only attachment,
| + | then the following are equivalent: |
| − | and since this attachment is prevented from being a blank or a paint,
| |
| − | the single attachment at its root has to be another sort of structure, | |
| − | that which we call a "lobe".
| |
| | | | |
| − | To express the description of a PARC in terms of its nodes, each node
| + | R4a. u C X. |
| − | can be specified in the fashion of a functional expression, letting a
| |
| − | citation of the generic function name "Node" be followed by a list of
| |
| − | arguments that enumerates the attachments of the node in question, and
| |
| − | letting a citation of the generic function name "Lobe" be followed by a
| |
| − | list of arguments that details the accoutrements of the lobe in question.
| |
| − | Thus, one can write expressions of the following forms:
| |
| | | | |
| − | 1. Node^0 = Node()
| + | R4b. [u C X]. |
| | | | |
| − | = a node with no attachments.
| + | R4c. [u C X](u). |
| | | | |
| − | Node^k_j C_j = Node(C_1, ..., C_k)
| + | R4d. {X}(u). |
| | | | |
| − | = a node with the attachments C_1, ..., C_k.
| + | R4e. {X}(u) = 1. |
| | + | </pre> |
| | | | |
| − | 2. Lobe^0 = Lobe()
| + | The first and last items on this list, namely, the sentences "u C X" and "{X}(u) = 1" that are annotated as "R4a" and "R4e", respectively, are just the pair of sentences from Rule 3 whose equivalence for all u C U is usually taken to define the idea of an indicator function {X} : U -> B. At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their own ostensible types and the ruling type of a sentence. On reflection, and taken in context, these problems are not as serious as they initially seem. For instance, the expression "[u C X]" ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence. As a general rule, if one can see it on the page, then it cannot be a proposition, but can be, at best, a sign of one. |
| | | | |
| − | = a lobe with no accoutrements.
| + | The use of the basic connectives can be expressed in the form of a ROST as follows: |
| | | | |
| − | Lobe^k_j C_j = Lobe(C_1, ..., C_k)
| + | <pre> |
| | + | Logical Translation Rule 0 |
| | | | |
| − | = a lobe with the accoutrements C_1, ..., C_k.
| + | If Sj is a sentence |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | about things in the universe U |
| | | | |
| − | IDS. Note 168
| + | and Pj is a proposition |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | about things in the universe U |
| | | | |
| − | 1.3.10.11. The Cactus Language: Mechanics (concl.)
| + | such that: |
| | | | |
| − | Working from a structural description of the cactus language,
| + | L0a. [Sj] = Pj, for all j C J, |
| − | or any suitable formal grammar for !C!(!P!), it is possible to
| |
| − | give a recursive definition of the function called "Parse" that
| |
| − | maps each sentence in PARCE(!P!) to the corresponding graph in
| |
| − | PARC(!P!). One way to do this proceeds as follows:
| |
| | | | |
| − | 1. The parse of the concatenation Conc^k of the k sentences S_j,
| + | then the following equations are true: |
| − | for j = 1 to k, is defined recursively as follows:
| |
| | | | |
| − | a. Parse(Conc^0) = Node^0.
| + | L0b. [ConcJj Sj] = ConjJj [Sj] = ConjJj Pj. |
| | | | |
| − | b. For k > 0,
| + | L0c. [SurcJj Sj] = SurjJj [Sj] = SurjJj Pj. |
| | + | </pre> |
| | | | |
| − | Parse(Conc^k_j S_j) = Node^k_j Parse(S_j).
| + | As a general rule, the application of a ROST involves the recognition of an antecedent condition and the facilitation of a consequent condition. The antecedent condition is a state whose initial expression presents a match, in a formal sense, to one of the sentences that are listed in the STR, and the consequent condition is achieved by taking its suggestions seriously, in other words, by following its sequence of equivalents and implicants to some other link in its chain. |
| | | | |
| − | 2. The parse of the surcatenation Surc^k of the k sentences S_j,
| + | Generally speaking, the application of a rule involves the recognition of an antecedent condition as a case that falls under a clause of the rule. This means that the antecedent condition is able to be captured in the form, conceived in the guise, expressed in the manner, grasped in the pattern, or recognized in the shape of one of the sentences in a list of equivalents or a chain of implicants. |
| − | for j = 1 to k, is defined recursively as follows:
| |
| | | | |
| − | a. Parse(Surc^0) = Lobe^0.
| + | A condition is "amenable" to a rule if any of its conceivable expressions formally match any of the expressions that are enumerated by the rule. Further, it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that needs to be checked on input for whether it fits the antecedent condition and there are several types of output that are generated as a consequence, only a few of which are usually needed at any given time. |
| | | | |
| − | b. For k > 0,
| + | <pre> |
| | + | Logical Translation Rule 1 |
| | | | |
| − | Parse(Surc^k_j S_j) = Lobe^k_j Parse(S_j).
| + | If S is a sentence |
| | | | |
| − | For ease of reference, Table 12 summarizes the mechanics of these parsing rules.
| + | about things in the universe U |
| | | | |
| − | Table 12. Algorithmic Translation Rules
| + | and P is a proposition : U -> B, such that: |
| − | o------------------------o---------o------------------------o
| |
| − | | | Parse | |
| |
| − | | Sentence in PARCE | --> | Graph in PARC |
| |
| − | o------------------------o---------o------------------------o
| |
| − | | | | |
| |
| − | | Conc^0 | --> | Node^0 |
| |
| − | | | | |
| |
| − | | Conc^k_j S_j | --> | Node^k_j Parse(S_j) |
| |
| − | | | | |
| |
| − | | Surc^0 | --> | Lobe^0 |
| |
| − | | | | |
| |
| − | | Surc^k_j S_j | --> | Lobe^k_j Parse(S_j) |
| |
| − | | | | |
| |
| − | o------------------------o---------o------------------------o
| |
| | | | |
| − | A "substructure" of a PARC is defined recursively as follows. Starting
| + | L1a. [S] = P, |
| − | at the root node of the cactus C, any attachment is a substructure of C.
| |
| − | If a substructure is a blank or a paint, then it constitutes a minimal
| |
| − | substructure, meaning that no further substructures of C arise from it.
| |
| − | If a substructure is a lobe, then each one of its accoutrements is also
| |
| − | a substructure of C, and has to be examined for further substructures.
| |
| | | | |
| − | The concept of substructure can be used to define varieties of deletion
| + | then the following equations hold: |
| − | and erasure operations that respect the structure of the abstract graph.
| |
| − | For the purposes of this depiction, a blank symbol " " is treated as
| |
| − | a "primer", in other words, as a "clear paint", a "neutral tint", or
| |
| − | a "white wash". In effect, one is letting m_1 = p_0. In this frame
| |
| − | of discussion, it is useful to make the following distinction:
| |
| | | | |
| − | 1. To "delete" a substructure is to replace it with an empty node,
| + | L1b00. [False] = () = 0 : U->B. |
| − | in effect, to reduce the whole structure to a trivial point.
| |
| | | | |
| − | 2. To "erase" a substructure is to replace it with a blank symbol,
| + | L1b01. [Not S] = ([S]) = (P) : U->B. |
| − | in effect, to paint it out of the picture or to overwrite it.
| |
| | | | |
| − | A "bare" PARC, loosely referred to as a "bare cactus", is a PARC on the
| + | L1b10. [S] = [S] = P : U->B. |
| − | empty palette !P! = {}. In other veins, a bare cactus can be described
| |
| − | in several different ways, depending on how the form arises in practice.
| |
| | | | |
| − | 1. Leaning on the definition of a bare PARCE, a bare PARC can be
| + | L1b11. [True] = (()) = 1 : U->B. |
| − | described as the kind of a parse graph that results from parsing
| + | </pre> |
| − | a bare cactus expression, in other words, as the kind of a graph
| |
| − | that issues from the requirements of processing a sentence of
| |
| − | the bare cactus language !C!^0 = PARCE^0.
| |
| | | | |
| − | 2. To express it more in its own terms, a bare PARC can be defined
| + | <pre> |
| − | by tracing the recursive definition of a generic PARC, but then
| + | Geometric Translation Rule 1 |
| − | by detaching an independent form of description from the source
| |
| − | of that analogy. The method is sufficiently sketched as follows:
| |
| | | | |
| − | a. A "bare PARC" is a PARC whose attachments
| + | If X c U |
| − | are limited to blanks and "bare lobes".
| |
| | | | |
| − | b. A "bare lobe" is a lobe whose accoutrements
| + | and P : U -> B, such that: |
| − | are limited to bare PARC's.
| |
| | | | |
| − | 3. In practice, a bare cactus is usually encountered in the process
| + | G1a. {X} = P, |
| − | of analyzing or handling an arbitrary PARC, the circumstances of
| |
| − | which frequently call for deleting or erasing all of its paints.
| |
| − | In particular, this generally makes it easier to observe the
| |
| − | various properties of its underlying graphical structure.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | then the following equations hold: |
| | | | |
| − | IDS. Note 169
| + | G1b00. {{}} = () = 0 : U->B. |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | G1b10. {~X} = ({X}) = (P) : U->B. |
| | | | |
| − | 1.3.10.12. The Cactus Language: Semantics
| + | G1b01. {X} = {X} = P : U->B. |
| | | | |
| − | | Alas, and yet what 'are' you, my written and painted thoughts!
| + | G1b11. {U} = (()) = 1 : U->B. |
| − | | It is not long ago that you were still so many-coloured,
| + | </pre> |
| − | | young and malicious, so full of thorns and hidden
| |
| − | | spices you made me sneeze and laugh -- and now?
| |
| − | | You have already taken off your novelty and
| |
| − | | some of you, I fear, are on the point of
| |
| − | | becoming truths: they already look so
| |
| − | | immortal, so pathetically righteous,
| |
| − | | so boring!
| |
| − | |
| |
| − | | Friedrich Nietzsche, 'Beyond Good and Evil', Paragraph 296.
| |
| − | |
| |
| − | | Friedrich Nietzsche,
| |
| − | |'Beyond Good and Evil: Prelude to a Philosophy of the Future',
| |
| − | | trans. by R.J. Hollingdale, intro. by Michael Tanner,
| |
| − | | Penguin Books, London, UK, 1973, 1990.
| |
| | | | |
| − | In this Subsection, I describe a particular semantics for the
| + | <pre> |
| − | painted cactus language, telling what meanings I aim to attach
| + | Logical Translation Rule 2 |
| − | to its bare syntactic forms. This supplies an "interpretation"
| |
| − | for this parametric family of formal languages, but it is good
| |
| − | to remember that it forms just one of many such interpretations
| |
| − | that are conceivable and even viable. In deed, the distinction
| |
| − | between the object domain and the sign domain can be observed in
| |
| − | the fact that many languages can be deployed to depict the same
| |
| − | set of objects and that any language worth its salt is bound to
| |
| − | to give rise to many different forms of interpretive saliency.
| |
| | | | |
| − | In formal settings, it is common to speak of "interpretation" as if it
| + | If S, T are sentences |
| − | created a direct connection between the signs of a formal language and
| |
| − | the objects of the intended domain, in other words, as if it determined
| |
| − | the denotative component of a sign relation. But a closer attention to
| |
| − | what goes on reveals that the process of interpretation is more indirect,
| |
| − | that what it does is to provide each sign of a prospectively meaningful
| |
| − | source language with a translation into an already established target
| |
| − | language, where "already established" means that its relationship to
| |
| − | pragmatic objects is taken for granted at the moment in question.
| |
| | | | |
| − | With this in mind, it is clear that interpretation is an affair of signs
| + | about things in the universe U |
| − | that at best respects the objects of all of the signs that enter into it,
| |
| − | and so it is the connotative aspect of semiotics that is at stake here.
| |
| − | There is nothing wrong with my saying that I interpret a sentence of a
| |
| − | formal language as a sign that refers to a function or to a proposition,
| |
| − | so long as you understand that this reference is likely to be achieved
| |
| − | by way of more familiar and perhaps less formal signs that you already
| |
| − | take to denote those objects.
| |
| | | | |
| − | On entering a context where a logical interpretation is intended for the
| + | and P, Q are propositions: U -> B, such that: |
| − | sentences of a formal language there are a few conventions that make it
| |
| − | easier to make the translation from abstract syntactic forms to their
| |
| − | intended semantic senses. Although these conventions are expressed in
| |
| − | unnecessarily colorful terms, from a purely abstract point of view, they
| |
| − | do provide a useful array of connotations that help to negotiate what is
| |
| − | otherwise a difficult transition. This terminology is introduced as the
| |
| − | need for it arises in the process of interpreting the cactus language.
| |
| | | | |
| − | The task of this Subsection is to specify a "semantic function" for
| + | L2a. [S] = P and [T] = Q, |
| − | the sentences of the cactus language !L! = !C!(!P!), in other words,
| + | |
| − | to define a mapping that "interprets" each sentence of !C!(!P!) as
| + | then the following equations hold: |
| − | a sentence that says something, as a sentence that bears a meaning,
| |
| − | in short, as a sentence that denotes a proposition, and thus as a
| |
| − | sign of an indicator function. When the syntactic sentences of a
| |
| − | formal language are given a referent significance in logical terms,
| |
| − | for example, as denoting propositions or indicator functions, then
| |
| − | each form of syntactic combination takes on a corresponding form
| |
| − | of logical significance.
| |
| | | | |
| − | By way of providing a logical interpretation for the cactus language,
| + | L2b00. [False] = () = 0 : U->B. |
| − | I introduce a family of operators on indicator functions that are
| |
| − | called "propositional connectives", and I distinguish these from
| |
| − | the associated family of syntactic combinations that are called
| |
| − | "sentential connectives", where the relationship between these
| |
| − | two realms of connection is exactly that between objects and
| |
| − | their signs. A propositional connective, as an entity of a
| |
| − | well-defined functional and operational type, can be treated
| |
| − | in every way as a logical or a mathematical object, and thus
| |
| − | as the type of object that can be denoted by the corresponding
| |
| − | form of syntactic entity, namely, the sentential connective that
| |
| − | is appropriate to the case in question.
| |
| | | | |
| − | There are two basic types of connectives, called the "blank connectives"
| + | L2b01. [Neither S nor T] = ([S])([T]) = (P)(Q). |
| − | and the "bound connectives", respectively, with one connective of each
| |
| − | type for each natural number k = 0, 1, 2, 3, ... .
| |
| | | | |
| − | 1. The "blank connective" of k places is signified by the
| + | L2b02. [Not S, but T] = ([S])[T] = (P) Q. |
| − | concatenation of the k sentences that fill those places.
| |
| | | | |
| − | For the special case of k = 0, the "blank connective" is taken to
| + | L2b03. [Not S] = ([S]) = (P). |
| − | be an empty string or a blank symbol -- it does not matter which,
| |
| − | since both are assigned the same denotation among propositions.
| |
| − | For the generic case of k > 0, the "blank connective" takes
| |
| − | the form "S_1 · ... · S_k". In the type of data that is
| |
| − | called a "text", the raised dots "·" are usually omitted,
| |
| − | supplanted by whatever number of spaces and line breaks
| |
| − | serve to improve the readability of the resulting text.
| |
| | | | |
| − | 2. The "bound connective" of k places is signified by the
| + | L2b04. [S and not T] = [S]([T]) = P (Q). |
| − | surcatenation of the k sentences that fill those places.
| |
| | | | |
| − | For the special case of k = 0, the "bound connective" is taken to
| + | L2b05. [Not T] = ([T]) = (Q). |
| − | be an expression of the form "-()-", "-( )-", "-( )-", and so on,
| |
| − | with any number of blank symbols between the parentheses, all of
| |
| − | which are assigned the same logical denotation among propositions.
| |
| − | For the generic case of k > 0, the "bound connective" takes the
| |
| − | form "-(S_1, ..., S_k)-".
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | L2b06. [S or T, not both] = ([S], [T]) = (P, Q). |
| | | | |
| − | IDS. Note 170
| + | L2b07. [Not both S and T] = ([S].[T]) = (P Q). |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | L2b08. [S and T] = [S].[T] = P.Q. |
| | | | |
| − | 1.3.10.12. The Cactus Language: Semantics (cont.)
| + | L2b09. [S <=> T] = (([S], [T])) = ((P, Q)). |
| | | | |
| − | At this point, there are actually two different "dialects", "scripts",
| + | L2b10. [T] = [T] = Q. |
| − | or "modes" of presentation for the cactus language that need to be
| |
| − | interpreted, in other words, that need to have a semantic function
| |
| − | defined on their domains.
| |
| | | | |
| − | a. There is the literal formal language of strings in PARCE(!P!),
| + | L2b11. [S => T] = ([S]([T])) = (P (Q)). |
| − | the "painted and rooted cactus expressions" that constitute
| |
| − | the langauge !L! = !C!(!P!) c !A!* = (!M! |_| !P!)*.
| |
| | | | |
| − | b. There is the figurative formal language of graphs in PARC(!P!),
| + | L2b12. [S] = [S] = P. |
| − | the "painted and rooted cacti" themselves, a parametric family
| |
| − | of graphs or a species of computational data structures that
| |
| − | is graphically analogous to the language of literal strings.
| |
| | | | |
| − | Of course, these two modalities of formal language, like written and
| + | L2b13. [S <= T] = (([S]) [T]) = ((P) Q). |
| − | spoken natural languages, are meant to have compatible interpretations,
| + | |
| − | and so it is usually sufficient to give just the meanings of either one.
| + | L2b14. [S or T] = (([S])([T])) = ((P)(Q)). |
| − | All that remains is to provide a "codomain" or a "target space" for the
| |
| − | intended semantic function, in other words, to supply a suitable range
| |
| − | of logical meanings for the memberships of these languages to map into.
| |
| − | Out of the many interpretations that are formally possible to arrange,
| |
| − | one way of doing this proceeds by making the following definitions:
| |
| | | | |
| − | 1. The "conjunction" Conj^J_j Q_j of a set of propositions, {Q_j : j in J},
| + | L2b15. [True] = (()) = 1 : U->B. |
| − | is a proposition that is true if and only if each one of the Q_j is true.
| + | </pre> |
| | | | |
| − | Conj^J_j Q_j is true <=> Q_j is true for every j in J.
| + | <pre> |
| | + | Geometric Translation Rule 2 |
| | | | |
| − | 2. The "surjunction" Surj^J_j Q_j of a set of propositions, {Q_j : j in J},
| + | If X, Y c U |
| − | is a proposition that is true if and only if just one of the Q_j is untrue.
| |
| | | | |
| − | Surj^J_j Q_j is true <=> Q_j is untrue for unique j in J.
| + | and P, Q U -> B, such that: |
| | | | |
| − | If the number of propositions that are being joined together is finite,
| + | G2a. {X} = P and {Y} = Q, |
| − | then the conjunction and the surjunction can be represented by means of
| |
| − | sentential connectives, incorporating the sentences that represent these
| |
| − | propositions into finite strings of symbols.
| |
| | | | |
| − | If J is finite, for instance, if J constitutes the interval j = 1 to k,
| + | then the following equations hold: |
| − | and if each proposition Q_j is represented by a sentence S_j, then the
| |
| − | following strategies of expression are open: | |
| | | | |
| − | 1. The conjunction Conj^J_j Q_j can be represented by a sentence that
| + | G2b00. {{}} = () = 0 : U->B. |
| − | is constructed by concatenating the S_j in the following fashion:
| |
| | | | |
| − | Conj^J_j Q_j <-< S_1 S_2 ... S_k.
| + | G2b01. {~X n ~Y} = ({X})({Y}) = (P)(Q). |
| | | | |
| − | 2. The surjunction Surj^J_j Q_j can be represented by a sentence that
| + | G2b02. {~X n Y} = ({X}){Y} = (P) Q. |
| − | is constructed by surcatenating the S_j in the following fashion:
| |
| | | | |
| − | Surj^J_j Q_j <-< -(S_1, S_2, ..., S_k)-.
| + | G2b03. {~X} = ({X}) = (P). |
| | | | |
| − | If one opts for a mode of interpretation that moves more directly from
| + | G2b04. {X n ~Y} = {X}({Y}) = P (Q). |
| − | the parse graph of a sentence to the potential logical meaning of both
| |
| − | the PARC and the PARCE, then the following specifications are in order:
| |
| | | | |
| − | A cactus rooted at a particular node is taken to represent what that
| + | G2b05. {~Y} = ({Y}) = (Q). |
| − | node denotes, its logical denotation or its logical interpretation.
| |
| | | | |
| − | 1. The logical denotation of a node is the logical conjunction of that node's
| + | G2b06. {X + Y} = ({X}, {Y}) = (P, Q). |
| − | "arguments", which are defined as the logical denotations of that node's
| |
| − | attachments. The logical denotation of either a blank symbol or an empty
| |
| − | node is the boolean value %1% = "true". The logical denotation of the
| |
| − | paint p_j is the proposition P_j, a proposition that is regarded as
| |
| − | "primitive", at least, with respect to the level of analysis that
| |
| − | is represented in the current instance of !C!(!P!).
| |
| | | | |
| − | 2. The logical denotation of a lobe is the logical surjunction of that lobe's
| + | G2b07. {~(X n Y)} = ({X}.{Y}) = (P Q). |
| − | "arguments", which are defined as the logical denotations of that lobe's
| |
| − | accoutrements. As a corollary, the logical denotation of the parse graph
| |
| − | of "-()-", otherwise called a "needle", is the boolean value %0% = "false".
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | G2b08. {X n Y} = {X}.{Y} = P.Q. |
| | | | |
| − | IDS. Note 171
| + | G2b09. {~(X + Y)} = (({X}, {Y})) = ((P, Q)). |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | G2b10. {Y} = {Y} = Q. |
| | | | |
| − | 1.3.10.12. The Cactus Language: Semantics (cont.)
| + | G2b11. {~(X n ~Y)} = ({X}({Y})) = (P (Q)). |
| | | | |
| − | If one takes the point of view that PARC's and PARCE's amount to a
| + | G2b12. {X} = {X} = P. |
| − | pair of intertranslatable languages for the same domain of objects,
| + | |
| − | then the "spiny bracket" notation, as in "-[C_j]-" or "-[S_j]-",
| + | G2b13. {~(~X n Y)} = (({X}) {Y}) = ((P) Q). |
| − | can be used on either domain of signs to indicate the logical
| |
| − | denotation of a cactus C_j or the logical denotation of
| |
| − | a sentence S_j, respectively.
| |
| | | | |
| − | Tables 13.1 and 13.2 summarize the relations that serve to connect the
| + | G2b14. {X u Y} = (({X})({Y})) = ((P)(Q)). |
| − | formal language of sentences with the logical language of propositions.
| |
| − | Between these two realms of expression there is a family of graphical
| |
| − | data structures that arise in parsing the sentences and that serve to
| |
| − | facilitate the performance of computations on the indicator functions.
| |
| − | The graphical language supplies an intermediate form of representation
| |
| − | between the formal sentences and the indicator functions, and the form
| |
| − | of mediation that it provides is very useful in rendering the possible
| |
| − | connections between the other two languages conceivable in fact, not to
| |
| − | mention in carrying out the necessary translations on a practical basis.
| |
| − | These Tables include this intermediate domain in their Central Columns.
| |
| − | Between their First and Middle Columns they illustrate the mechanics of
| |
| − | parsing the abstract sentences of the cactus language into the graphical
| |
| − | data structures of the corresponding species. Between their Middle and
| |
| − | Final Columns they summarize the semantics of interpreting the graphical
| |
| − | forms of representation for the purposes of reasoning with propositions.
| |
| | | | |
| − | Table 13.1 Semantic Translations: Functional Form
| + | G2b15. {U} = (()) = 1 : U->B. |
| − | o-------------------o-----o-------------------o-----o-------------------o
| + | </pre> |
| − | | | Par | | Den | |
| |
| − | | Sentence | --> | Graph | --> | Proposition |
| |
| − | o-------------------o-----o-------------------o-----o-------------------o
| |
| − | | | | | | |
| |
| − | | S_j | --> | C_j | --> | Q_j |
| |
| − | | | | | | |
| |
| − | o-------------------o-----o-------------------o-----o-------------------o
| |
| − | | | | | | |
| |
| − | | Conc^0 | --> | Node^0 | --> | %1% |
| |
| − | | | | | | |
| |
| − | | Conc^k_j S_j | --> | Node^k_j C_j | --> | Conj^k_j Q_j |
| |
| − | | | | | | |
| |
| − | o-------------------o-----o-------------------o-----o-------------------o
| |
| − | | | | | | |
| |
| − | | Surc^0 | --> | Lobe^0 | --> | %0% |
| |
| − | | | | | | |
| |
| − | | Surc^k_j S_j | --> | Lobe^k_j C_j | --> | Surj^k_j Q_j |
| |
| − | | | | | | |
| |
| − | o-------------------o-----o-------------------o-----o-------------------o
| |
| | | | |
| − | Table 13.2 Semantic Translations: Equational Form
| + | <pre> |
| − | o-------------------o-----o-------------------o-----o-------------------o
| + | Value Rule 1 |
| − | | | Par | | Den | |
| |
| − | | -[Sentence]- | = | -[Graph]- | = | Proposition |
| |
| − | o-------------------o-----o-------------------o-----o-------------------o
| |
| − | | | | | | |
| |
| − | | -[S_j]- | = | -[C_j]- | = | Q_j |
| |
| − | | | | | | |
| |
| − | o-------------------o-----o-------------------o-----o-------------------o
| |
| − | | | | | | |
| |
| − | | -[Conc^0]- | = | -[Node^0]- | = | %1% |
| |
| − | | | | | | |
| |
| − | | -[Conc^k_j S_j]- | = | -[Node^k_j C_j]- | = | Conj^k_j Q_j |
| |
| − | | | | | | |
| |
| − | o-------------------o-----o-------------------o-----o-------------------o
| |
| − | | | | | | |
| |
| − | | -[Surc^0]- | = | -[Lobe^0]- | = | %0% |
| |
| − | | | | | | |
| |
| − | | -[Surc^k_j S_j]- | = | -[Lobe^k_j C_j]- | = | Surj^k_j Q_j |
| |
| − | | | | | | |
| |
| − | o-------------------o-----o-------------------o-----o-------------------o
| |
| | | | |
| − | Aside from their common topic, the two Tables present slightly different
| + | If v, w C B |
| − | ways of conceptualizing the operations that go to establish their maps.
| |
| − | Table 13.1 records the functional associations that connect each domain
| |
| − | with the next, taking the triplings of a sentence S_j, a cactus C_j, and
| |
| − | a proposition Q_j as basic data, and fixing the rest by recursion on these.
| |
| − | Table 13.2 records these associations in the form of equations, treating
| |
| − | sentences and graphs as alternative kinds of signs, and generalizing the
| |
| − | spiny bracket operator to indicate the proposition that either denotes.
| |
| − | It should be clear at this point that either scheme of translation puts
| |
| − | the sentences, the graphs, and the propositions that it associates with
| |
| − | each other roughly in the roles of the signs, the interpretants, and the
| |
| − | objects, respectively, whose triples define an appropriate sign relation.
| |
| − | Indeed, the "roughly" can be made "exactly" as soon as the domains of
| |
| − | a suitable sign relation are specified precisely.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | then "v = w" is a sentence about <v, w> C B2, |
| | | | |
| − | IDS. Note 172
| + | [v = w] is a proposition : B2 -> B, |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | and the following are identical values in B: |
| | | | |
| − | 1.3.10.12. The Cactus Language: Semantics (concl.)
| + | V1a. [ v = w ](v, w) |
| | | | |
| − | A good way to illustrate the action of the conjunction and surjunction
| + | V1b. [ v <=> w ](v, w) |
| − | operators is to demonstate how they can be used to construct all of the
| |
| − | boolean functions on k variables, just now, let us say, for k = 0, 1, 2.
| |
| | | | |
| − | A boolean function on 0 variables is just a boolean constant F^0 in the
| + | V1c. ((v , w)) |
| − | boolean domain %B% = {%0%, %1%}. Table 14 shows several different ways
| + | </pre> |
| − | of referring to these elements, just for the sake of consistency using
| |
| − | the same format that will be used in subsequent Tables, no matter how
| |
| − | degenerate it tends to appears in the immediate case:
| |
| | | | |
| − | Column 1 lists each boolean element or boolean function under its
| + | <pre> |
| − | ordinary constant name or under a succinct nickname, respectively.
| + | Value Rule 1 |
| | | | |
| − | Column 2 lists each boolean function in a style of function name "F^i_j"
| + | If v, w C B, |
| − | that is constructed as follows: The superscript "i" gives the dimension
| |
| − | of the functional domain, that is, the number of its functional variables,
| |
| − | and the subscript "j" is a binary string that recapitulates the functional
| |
| − | values, using the obvious translation of boolean values into binary values.
| |
| | | | |
| − | Column 3 lists the functional values for each boolean function, or possibly
| + | then the following are equivalent: |
| − | a boolean element appearing in the guise of a function, for each combination
| |
| − | of its domain values.
| |
| | | | |
| − | Column 4 shows the usual expressions of these elements in the cactus language,
| + | V1a. v = w. |
| − | conforming to the practice of omitting the strike-throughs in display formats.
| |
| − | Here I illustrate also the useful convention of sending the expression "(())"
| |
| − | as a visible stand-in for the expression of a constantly "true" truth value,
| |
| − | one that would otherwise be represented by a blank expression, and tend to
| |
| − | elude our giving it much notice in the context of more demonstrative texts.
| |
| | | | |
| − | Table 14. Boolean Functions on Zero Variables
| + | V1b. v <=> w. |
| − | o----------o----------o-------------------------------------------o----------o
| |
| − | | Constant | Function | F() | Function |
| |
| − | o----------o----------o-------------------------------------------o----------o
| |
| − | | | | | |
| |
| − | | %0% | F^0_0 | %0% | () |
| |
| − | | | | | |
| |
| − | | %1% | F^0_1 | %1% | (()) |
| |
| − | | | | | |
| |
| − | o----------o----------o-------------------------------------------o----------o
| |
| | | | |
| − | Table 15 presents the boolean functions on one variable, F^1 : %B% -> %B%,
| + | V1c. (( v , w )). |
| − | of which there are precisely four. Here, Column 1 codes the contents of
| + | </pre> |
| − | Column 2 in a more concise form, compressing the lists of boolean values,
| |
| − | recorded as bits in the subscript string, into their decimal equivalents.
| |
| − | Naturally, the boolean constants reprise themselves in this new setting
| |
| − | as constant functions on one variable. Thus, one has the synonymous
| |
| − | expressions for constant functions that are expressed in the next
| |
| − | two chains of equations:
| |
| | | | |
| − | F^1_0 = F^1_00 = %0% : %B% -> %B%
| + | A rule that allows one to turn equivalent sentences into identical propositions: |
| | | | |
| − | F^1_3 = F^1_11 = %1% : %B% -> %B%
| + | : (S <=> T) <=> ([S] = [T]) |
| | | | |
| − | As for the rest, the other two functions are easily recognized as corresponding
| + | Consider [ v = w ](v, w) and [ v(u) = w(u) ](u) |
| − | to the one-place logical connectives, or the monadic operators on %B%. Thus,
| |
| − | the function F^1_1 = F^1_01 is recognizable as the negation operation, and
| |
| − | the function F^1_2 = F^1_10 is obviously the identity operation.
| |
| | | | |
| − | Table 15. Boolean Functions on One Variable
| + | <pre> |
| − | o----------o----------o-------------------------------------------o----------o
| + | Value Rule 1 |
| − | | Function | Function | F(x) | Function |
| |
| − | o----------o----------o---------------------o---------------------o----------o
| |
| − | | | | F(%0%) | F(%1%) | |
| |
| − | o----------o----------o---------------------o---------------------o----------o
| |
| − | | | | | | |
| |
| − | | F^1_0 | F^1_00 | %0% | %0% | ( ) |
| |
| − | | | | | | |
| |
| − | | F^1_1 | F^1_01 | %0% | %1% | (x) |
| |
| − | | | | | | |
| |
| − | | F^1_2 | F^1_10 | %1% | %0% | x |
| |
| − | | | | | | |
| |
| − | | F^1_3 | F^1_11 | %1% | %1% | (( )) |
| |
| − | | | | | | |
| |
| − | o----------o----------o---------------------o---------------------o----------o
| |
| | | | |
| − | Table 16 presents the boolean functions on two variables, F^2 : %B%^2 -> %B%,
| + | If v, w C B, |
| − | of which there are precisely sixteen in number. As before, all of the boolean
| |
| − | functions of fewer variables are subsumed in this Table, though under a set of
| |
| − | alternative names and possibly different interpretations. Just to acknowledge
| |
| − | a few of the more notable pseudonyms:
| |
| | | | |
| − | The constant function %0% : %B%^2 -> %B% appears under the name of F^2_00.
| + | then the following are identical values in B: |
| | | | |
| − | The constant function %1% : %B%^2 -> %B% appears under the name of F^2_15.
| + | V1a. [ v = w ] |
| | | | |
| − | The negation and identity of the first variable are F^2_03 and F^2_12, resp.
| + | V1b. [ v <=> w ] |
| | | | |
| − | The negation and identity of the other variable are F^2_05 and F^2_10, resp.
| + | V1c. (( v , w )) |
| | + | </pre> |
| | | | |
| − | The logical conjunction is given by the function F^2_08 (x, y) = x · y.
| + | <pre> |
| | + | Value Rule 1 |
| | | | |
| − | The logical disjunction is given by the function F^2_14 (x, y) = ((x)(y)).
| + | If f, g : U -> B, |
| | | | |
| − | Functions expressing the "conditionals", "implications",
| + | and u C U |
| − | or "if-then" statements are given in the following ways:
| |
| | | | |
| − | [x => y] = F^2_11 (x, y) = (x (y)) = [not x without y].
| + | then the following are identical values in B: |
| | | | |
| − | [x <= y] = F^2_13 (x, y) = ((x) y) = [not y without x].
| + | V1a. [ f(u) = g(u) ] |
| | | | |
| − | The function that corresponds to the "biconditional",
| + | V1b. [ f(u) <=> g(u) ] |
| − | the "equivalence", or the "if and only" statement is
| |
| − | exhibited in the following fashion:
| |
| | | | |
| − | [x <=> y] = [x = y] = F^2_09 (x, y) = ((x , y)).
| + | V1c. (( f(u) , g(u) )) |
| | + | </pre> |
| | | | |
| − | Finally, there is a boolean function that is logically associated with
| + | <pre> |
| − | the "exclusive disjunction", "inequivalence", or "not equals" statement,
| + | Value Rule 1 |
| − | algebraically associated with the "binary sum" or "bitsum" operation,
| |
| − | and geometrically associated with the "symmetric difference" of sets.
| |
| − | This function is given by:
| |
| | | | |
| − | [x =/= y] = [x + y] = F^2_06 (x, y) = (x , y).
| + | If f, g : U -> B, |
| | | | |
| − | Table 16. Boolean Functions on Two Variables
| + | then the following are identical propositions on U: |
| − | o----------o----------o-------------------------------------------o----------o
| + | |
| − | | Function | Function | F(x, y) | Function |
| + | V1a. [ f = g ] |
| − | o----------o----------o----------o----------o----------o----------o----------o
| |
| − | | | | %1%, %1% | %1%, %0% | %0%, %1% | %0%, %0% | |
| |
| − | o----------o----------o----------o----------o----------o----------o----------o
| |
| − | | | | | | | | |
| |
| − | | F^2_00 | F^2_0000 | %0% | %0% | %0% | %0% | () |
| |
| − | | | | | | | | |
| |
| − | | F^2_01 | F^2_0001 | %0% | %0% | %0% | %1% | (x)(y) |
| |
| − | | | | | | | | |
| |
| − | | F^2_02 | F^2_0010 | %0% | %0% | %1% | %0% | (x) y |
| |
| − | | | | | | | | |
| |
| − | | F^2_03 | F^2_0011 | %0% | %0% | %1% | %1% | (x) |
| |
| − | | | | | | | | |
| |
| − | | F^2_04 | F^2_0100 | %0% | %1% | %0% | %0% | x (y) |
| |
| − | | | | | | | | |
| |
| − | | F^2_05 | F^2_0101 | %0% | %1% | %0% | %1% | (y) |
| |
| − | | | | | | | | |
| |
| − | | F^2_06 | F^2_0110 | %0% | %1% | %1% | %0% | (x, y) |
| |
| − | | | | | | | | |
| |
| − | | F^2_07 | F^2_0111 | %0% | %1% | %1% | %1% | (x y) |
| |
| − | | | | | | | | |
| |
| − | | F^2_08 | F^2_1000 | %1% | %0% | %0% | %0% | x y |
| |
| − | | | | | | | | |
| |
| − | | F^2_09 | F^2_1001 | %1% | %0% | %0% | %1% | ((x, y)) |
| |
| − | | | | | | | | |
| |
| − | | F^2_10 | F^2_1010 | %1% | %0% | %1% | %0% | y |
| |
| − | | | | | | | | |
| |
| − | | F^2_11 | F^2_1011 | %1% | %0% | %1% | %1% | (x (y)) |
| |
| − | | | | | | | | |
| |
| − | | F^2_12 | F^2_1100 | %1% | %1% | %0% | %0% | x |
| |
| − | | | | | | | | |
| |
| − | | F^2_13 | F^2_1101 | %1% | %1% | %0% | %1% | ((x) y) |
| |
| − | | | | | | | | |
| |
| − | | F^2_14 | F^2_1110 | %1% | %1% | %1% | %0% | ((x)(y)) |
| |
| − | | | | | | | | |
| |
| − | | F^2_15 | F^2_1111 | %1% | %1% | %1% | %1% | (()) |
| |
| − | | | | | | | | |
| |
| − | o----------o----------o----------o----------o----------o----------o----------o
| |
| | | | |
| − | Let me now address one last question that may have occurred to some.
| + | V1b. [ f <=> g ] |
| − | What has happened, in this suggested scheme of functional reasoning,
| |
| − | to the distinction that is quite pointedly made by careful logicians
| |
| − | between (1) the connectives called "conditionals" and symbolized by
| |
| − | the signs "->" and "<-", and (2) the assertions called "implications"
| |
| − | and symbolized by the signs "=>" and "<=", and, in a related question:
| |
| − | What has happened to the distinction that is equally insistently made
| |
| − | between (3) the connective called the "biconditional" and signified by
| |
| − | the sign "<->" and (4) the assertion that is called an "equivalence"
| |
| − | and signified by the sign "<=>"? My answer is this: For my part,
| |
| − | I am deliberately avoiding making these distinctions at the level
| |
| − | of syntax, preferring to treat them instead as distinctions in
| |
| − | the use of boolean functions, turning on whether the function
| |
| − | is mentioned directly and used to compute values on arguments,
| |
| − | or whether its inverse is being invoked to indicate the fibers
| |
| − | of truth or untruth under the propositional function in question.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | V1c. (( f , g ))$ |
| | + | </pre> |
| | | | |
| − | IDS. Note 173
| + | <pre> |
| | + | Evaluation Rule 1 |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | If f, g : U -> B |
| | | | |
| − | 1.3.10.13. Stretching Exercises
| + | and u C U, |
| | | | |
| − | For ease of reference, I repeat here a couple of the
| + | then the following are equivalent: |
| − | definitions that are needed again in this discussion.
| |
| | | | |
| − | | A "boolean connection" of degree k, also known as a "boolean function"
| + | E1a. f(u) = g(u). :V1a |
| − | | on k variables, is a map of the form F : %B%^k -> %B%. In other words,
| |
| − | | a boolean connection of degree k is a proposition about things in the
| |
| − | | universe of discourse X = %B%^k.
| |
| − | |
| |
| − | | An "imagination" of degree k on X is a k-tuple of propositions
| |
| − | | about things in the universe X. By way of displaying the kinds
| |
| − | | of notation that are used to express this idea, the imagination
| |
| − | | #f# = <f_1, ..., f_k> is can be given as a sequence of indicator
| |
| − | | functions f_j : X -> %B%, for j = 1 to k. All of these features
| |
| − | | of the typical imagination #f# can be summed up in either one of
| |
| − | | two ways: either in the form of a membership statement, stating
| |
| − | | words to the effect that #f# belongs to the space (X -> %B%)^k,
| |
| − | | or in the form of the type declaration that #f# : (X -> %B%)^k,
| |
| − | | though perhaps the latter specification is slightly more precise
| |
| − | | than the former.
| |
| | | | |
| − | The definition of the "stretch" operation and the uses of the
| + | :: |
| − | various brands of denotational operators can be reviewed here:
| |
| | | | |
| − | IDS 133. http://stderr.org/pipermail/inquiry/2004-June/001578.html
| + | E1b. f(u) <=> g(u). :V1b |
| − | IDS 134. http://stderr.org/pipermail/inquiry/2004-June/001579.html
| |
| − | IDS 136. http://stderr.org/pipermail/inquiry/2004-June/001581.html
| |
| − | IDS 137. http://stderr.org/pipermail/inquiry/2004-June/001582.html
| |
| | | | |
| − | Taking up the preceding arrays of particular connections, namely,
| + | :: |
| − | the boolean functions on two or less variables, it possible to
| |
| − | illustrate the use of the stretch operation in a variety of
| |
| − | concrete cases.
| |
| | | | |
| − | For example, suppose that F is a connection of the form F : %B%^2 -> %B%,
| + | E1c. (( f(u) , g(u) )). :V1c |
| − | that is, any one of the sixteen possibilities in Table 16, while p and q
| |
| − | are propositions of the form p, q : X -> %B%, that is, propositions about
| |
| − | things in the universe X, or else the indicators of sets contained in X.
| |
| | | | |
| − | Then one has the imagination #f# = <f_1, f_2> = <p, q> : (X -> %B%)^2,
| + | :$1a |
| − | and the stretch of the connection F to #f# on X amounts to a proposition
| |
| − | F^$ <p, q> : X -> %B%, usually written as "F^$ (p, q)" and vocalized as
| |
| − | the "stretch of F to p and q". If one is concerned with many different
| |
| − | propositions about things in X, or if one is abstractly indifferent to
| |
| − | the particular choices for p and q, then one can detach the operator
| |
| − | F^$ : (X -> %B%)^2 -> (X -> %B%), called the "stretch of F over X",
| |
| − | and consider it in isolation from any concrete application.
| |
| | | | |
| − | When the "cactus notation" is used to represent boolean functions,
| + | :: |
| − | a single "$" sign at the end of the expression is enough to remind
| |
| − | a reader that the connections are meant to be stretched to several
| |
| − | propositions on a universe X.
| |
| | | | |
| − | For instance, take the connection F : %B%^2 -> %B% such that:
| + | E1d. (( f , g ))$(u). :$1b |
| | + | </pre> |
| | | | |
| − | F(x, y) = F^2_06 (x, y) = -(x, y)-.
| + | <pre> |
| | + | Evaluation Rule 1 |
| | | | |
| − | This connection is the boolean function on a couple of variables x, y
| + | If S, T are sentences |
| − | that yields a value of %1% if and only if just one of x, y is not %1%,
| |
| − | that is, if and only if just one of x, y is %1%. There is clearly an
| |
| − | isomorphism between this connection, viewed as an operation on the
| |
| − | boolean domain %B% = {%0%, %1%}, and the dyadic operation on binary
| |
| − | values x, y in !B! = GF(2) that is otherwise known as "x + y".
| |
| | | | |
| − | The same connection F : %B%^2 -> %B% can also be read as a proposition
| + | about things in the universe U, |
| − | about things in the universe X = %B%^2. If S is a sentence that denotes | |
| − | the proposition F, then the corresponding assertion says exactly what one
| |
| − | otherwise states by uttering "x is not equal to y". In such a case, one
| |
| − | has -[S]- = F, and all of the following expressions are ordinarily taken
| |
| − | as equivalent descriptions of the same set:
| |
| | | | |
| − | [| -[S]- |] = [| F |]
| + | f, g are propositions: U -> B, |
| | | | |
| − | = F^(-1)(%1%)
| + | and u C U, |
| | | | |
| − | = {<x, y> in %B%^2 : S}
| + | then the following are equivalent: |
| | | | |
| − | = {<x, y> in %B%^2 : F(x, y) = %1%}
| + | E1a. f(u) = g(u). :V1a |
| | | | |
| − | = {<x, y> in %B%^2 : F(x, y)}
| + | :: |
| | | | |
| − | = {<x, y> in %B%^2 : -(x, y)- = %1%}
| + | E1b. f(u) <=> g(u). :V1b |
| | | | |
| − | = {<x, y> in %B%^2 : -(x, y)- }
| + | :: |
| | | | |
| − | = {<x, y> in %B%^2 : x exclusive-or y}
| + | E1c. (( f(u) , g(u) )). :V1c |
| | | | |
| − | = {<x, y> in %B%^2 : just one true of x, y}
| + | :$1a |
| | | | |
| − | = {<x, y> in %B%^2 : x not equal to y}
| + | :: |
| | | | |
| − | = {<x, y> in %B%^2 : x <=/=> y}
| + | E1d. (( f , g ))$(u). :$1b |
| | + | </pre> |
| | | | |
| − | = {<x, y> in %B%^2 : x =/= y}
| + | <pre> |
| | + | Definition 2 |
| | | | |
| − | = {<x, y> in %B%^2 : x + y}
| + | If X, Y c U, |
| | | | |
| − | Notice the slight distinction, that I continue to maintain at this point,
| + | then the following are equivalent: |
| − | between the logical values {false, true} and the algebraic values {0, 1}.
| |
| − | This makes it legitimate to write a sentence directly into the right side
| |
| − | of the set-builder expression, for instance, weaving the sentence S or the
| |
| − | sentence "x is not equal to y" into the context "{<x, y> in %B%^2 : ... }",
| |
| − | thereby obtaining the corresponding expressions listed above, while the
| |
| − | proposition F(x, y) can also be asserted more directly without equating
| |
| − | it to %1%, since it already has a value in {false, true}, and thus can
| |
| − | be taken as tantamount to an actual sentence.
| |
| | | | |
| − | If the appropriate safeguards can be kept in mind, avoiding all danger of
| + | D2a. X = Y. |
| − | confusing propositions with sentences and sentences with assertions, then
| |
| − | the marks of these distinctions need not be forced to clutter the account
| |
| − | of the more substantive indications, that is, the ones that really matter.
| |
| − | If this level of understanding can be achieved, then it may be possible
| |
| − | to relax these restrictions, along with the absolute dichotomy between
| |
| − | algebraic and logical values, which tends to inhibit the flexibility
| |
| − | of interpretation.
| |
| | | | |
| − | This covers the properties of the connection F(x, y) = -(x, y)-,
| + | D2b. u C X <=> u C Y, for all u C U. |
| − | treated as a proposition about things in the universe X = %B%^2.
| + | </pre> |
| − | Staying with this same connection, it is time to demonstrate how
| |
| − | it can be "stretched" into an operator on arbitrary propositions.
| |
| | | | |
| − | To continue the exercise, let p and q be arbitrary propositions about
| + | <pre> |
| − | things in the universe X, that is, maps of the form p, q : X -> %B%,
| + | Definition 3 |
| − | and suppose that p, q are indicator functions of the sets P, Q c X,
| |
| − | respectively. In other words, one has the following set of data:
| |
| | | | |
| − | p = -{P}- : X -> %B%
| + | If f, g : U -> V, |
| | | | |
| − | q = -{Q}- : X -> %B%
| + | then the following are equivalent: |
| | | | |
| − | <p, q> = < -{P}- , -{Q}- > : (X -> %B%)^2
| + | D3a. f = g. |
| | | | |
| − | Then one has an operator F^$, the stretch of the connection F over X,
| + | D3b. f(u) = g(u), for all u C U. |
| − | and a proposition F^$ (p, q), the stretch of F to <p, q> on X, with
| + | </pre> |
| − | the following properties:
| |
| | | | |
| − | F^$ = -( , )-^$ : (X -> %B%)^2 -> (X -> %B%)
| + | <pre> |
| | + | Definition 4 |
| | | | |
| − | F^$ (p, q) = -(p, q)-^$ : X -> %B%
| + | If X c U, |
| | | | |
| − | As a result, the application of the proposition F^$ (p, q) to each x in X
| + | then the following are identical subsets of UxB: |
| − | yields a logical value in %B%, all in accord with the following equations:
| |
| | | | |
| − | F^$ (p, q)(x) = -(p, q)-^$ (x) in %B%
| + | D4a. {X} |
| | | | |
| − | ^ ^
| + | D4b. {< u, v> C UxB : v = [u C X]} |
| − | | |
| + | </pre> |
| − | = =
| |
| − | | |
| |
| − | v v
| |
| | | | |
| − | F(p(x), q(x)) = -(p(x), q(x))- in %B%
| + | <pre> |
| | + | Definition 5 |
| | | | |
| − | For each choice of propositions p and q about things in X, the stretch of
| + | If X c U, |
| − | F to p and q on X is just another proposition about things in X, a simple
| |
| − | proposition in its own right, no matter how complex its current expression
| |
| − | or its present construction as F^$ (p, q) = -(p, q)^$ makes it appear in
| |
| − | relation to p and q. Like any other proposition about things in X, it
| |
| − | indicates a subset of X, namely, the fiber that is variously described
| |
| − | in the following ways:
| |
| | | | |
| − | [| F^$ (p, q) |] = [| -(p, q)-^$ |]
| + | then the following are identical propositions: |
| | | | |
| − | = (F^$ (p, q))^(-1)(%1%)
| + | D5a. {X}. |
| | | | |
| − | = {x in X : F^$ (p, q)(x)}
| + | D5b. f : U -> B |
| | | | |
| − | = {x in X : -(p, q)-^$ (x)}
| + | : f(u) = [u C X], for all u C U. |
| | + | </pre> |
| | | | |
| − | = {x in X : -(p(x), q(x))- }
| + | Given an indexed set of sentences, Sj for j C J, it is possible to consider the logical conjunction of the corresponding propositions. Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6. |
| | | | |
| − | = {x in X : p(x) + q(x)}
| + | <pre> |
| | + | Definition 6 |
| | | | |
| − | = {x in X : p(x) =/= q(x)}
| + | If Sj is a sentence |
| | | | |
| − | = {x in X : -{P}- (x) =/= -{Q}- (x)}
| + | about things in the universe U, |
| | | | |
| − | = {x in X : x in P <=/=> x in Q}
| + | for all j C J, |
| | | | |
| − | = {x in X : x in P-Q or x in Q-P}
| + | then the following are equivalent: |
| | | | |
| − | = {x in X : x in P-Q |_| Q-P}
| + | D6a. Sj, for all j C J. |
| | | | |
| − | = {x in X : x in P + Q}
| + | D6b. For all j C J, Sj. |
| | | | |
| − | = P + Q c X
| + | D6c. Conj(j C J) Sj. |
| | | | |
| − | = [|p|] + [|q|] c X
| + | D6d. ConjJ,j Sj. |
| | | | |
| − | Which was to be shown.
| + | D6e. ConjJj Sj. |
| | + | </pre> |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | + | Definition 7 |
| | | | |
| − | IDS. Note 174
| + | If S, T are sentences |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | about things in the universe U, |
| | | | |
| − | 1.3.10.14. Syntactic Transformations
| + | then the following are equivalent: |
| | | | |
| − | We have been examining several distinct but closely related
| + | D7a. S <=> T. |
| − | notions of indication. To discuss the import of these ideas
| |
| − | in greater depth, it serves to establish a number of logical
| |
| − | relations and set-theoretic identities that can be found to
| |
| − | hold among their roughly parallel arrays of conceptions and
| |
| − | constructions. Facilitating this task, in turn, requires
| |
| − | a number of auxiliary concepts and notations.
| |
| | | | |
| − | The diverse notions of "indication" presently under discussion
| + | D7b. [S] = [T]. |
| − | are expressed in a variety of different notations, for example,
| + | </pre> |
| − | the functional language of propositions, the geometric language
| |
| − | of sets, and the logical language of sentences. Correspondingly,
| |
| − | one way to explain the relationships that exist among the various
| |
| − | notions of indication is to describe the "translations" that they
| |
| − | induce among the asssociated families of notation. A good way to
| |
| − | summarize the necessary translations between different styles of
| |
| − | indication, and along the way to organize their use in practice,
| |
| − | is by means of the "rules of syntactic transformation" (ROST's)
| |
| − | that partially formalize the translations in question.
| |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | <pre> |
| | + | Rule 5 |
| | | | |
| − | IDS. Note 175
| + | If X, Y c U, |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | then the following are equivalent: |
| | + | |
| | + | R5a. X = Y. :D2a |
| | + | |
| | + | :: |
| | + | |
| | + | R5b. u C X <=> u C Y, for all u C U. :D2b |
| | + | |
| | + | :D7a |
| | + | |
| | + | :: |
| | + | |
| | + | R5c. [u C X] = [u C Y], for all u C U. :D7b |
| | + | |
| | + | :??? |
| | | | |
| − | 1.3.10.14. Syntactic Transformations (cont.)
| + | :: |
| | | | |
| − | Rudimentary examples of ROST's are readily mined from the
| + | R5d. {< u, v> C UxB : v = [u C X]} |
| − | raw materials that are already available in this area of
| |
| − | discussion. To begin as near the beginning as possible,
| |
| − | let the definition of an indicator function be recorded
| |
| − | in the following form:
| |
| | | | |
| − | o-------------------------------------------------o
| + | = |
| − | | Definition 1. Indicator Function |
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | If Q c X, |
| |
| − | | |
| |
| − | | then -{Q}- : X -> %B% |
| |
| − | | |
| |
| − | | such that, for all x in X: |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | D1a. -{Q}-(x) <=> x in Q. |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| | | | |
| − | In practice, a definition like this is commonly used to substitute
| + | {< u, v> C UxB : v = [u C Y]}. :??? |
| − | one of two logically equivalent expressions or sentences for the
| |
| − | other in a context where the conditions of using the definition
| |
| − | in this way are satisfied and where the change is perceived as
| |
| − | potentially advancing a proof. The employment of a definition
| |
| − | in this way can be expressed in the form of a ROST that allows
| |
| − | one to exchange two expressions of logically equivalent forms
| |
| − | for one another in every context where their logical values are
| |
| − | the only consideration. To be specific, the "logical value" of
| |
| − | an expression is the value in the boolean domain %B% = {%0%, %1%}
| |
| − | that the expression represents to its context or that it stands for
| |
| − | in its context.
| |
| | | | |
| − | In the case of Definition 1, the corresponding ROST permits one
| + | :D5b |
| − | to exchange a sentence of the form "x in Q" with an expression of
| |
| − | the form "-{Q}-(x)" in any context that satisfies the conditions of
| |
| − | its use, namely, the conditions of the definition that lead up to the
| |
| − | stated equivalence. The relevant ROST is recorded in Rule 1. By way
| |
| − | of convention, I list the items that fall under a rule in rough order
| |
| − | of their ascending conceptual subtlety or their increasing syntactic
| |
| − | complexity, without regard for the normal or the typical orders of
| |
| − | their exchange, since this can vary from widely from case to case.
| |
| | | | |
| − | o-------------------------------------------------o
| + | :: |
| − | | Rule 1 |
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | If Q c X, |
| |
| − | | |
| |
| − | | then -{Q}- : X -> %B%, |
| |
| − | | |
| |
| − | | and if x in X, |
| |
| − | | |
| |
| − | | then the following are equivalent: |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | R1a. x in Q. |
| |
| − | | |
| |
| − | | R1b. -{Q}-(x). |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| | | | |
| − | Conversely, any rule of this sort, properly qualified by the
| + | R5e. {X} = {Y}. :D5a |
| − | conditions under which it applies, can be turned back into a
| |
| − | summary statement of the logical equivalence that is involved
| |
| − | in its application. This mode of conversion between a static
| |
| − | principle and a transformational rule, in other words, between
| |
| − | a statement of equivalence and an equivalence of statements, is
| |
| − | so automatic that it is usually not necessary to make a separate
| |
| − | note of the "horizontal" versus the "vertical" versions of what
| |
| − | amounts to the same abstract principle.
| |
| | </pre> | | </pre> |
| | | | |
| − | ==Where I Left Off In June 2004==
| + | <pre> |
| | + | Rule 6 |
| | | | |
| − | <pre>
| + | If f, g : U -> V, |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| | | | |
| − | IDS. Note 176
| + | then the following are equivalent: |
| | | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| + | R6a. f = g. :D3a |
| | | | |
| − | 1.3.10.14. Syntactic Transformations (cont.)
| + | :: |
| | | | |
| − | As another example of a ROST, consider the
| + | R6b. f(u) = g(u), for all u C U. :D3b |
| − | following logical equivalence, that holds
| |
| − | for any X c U and for all u in U:
| |
| | | | |
| − | -{X}-(u) <=> -{X}-(u) = 1.
| + | :D6a |
| | | | |
| − | In practice, this logical equivalence is used to exchange
| + | :: |
| − | an expression of the form "-{X}-(u)" with a sentence of the
| |
| − | form "-{X}-(u) = 1" in any context where one has a relatively
| |
| − | fixed X c U in mind and where one is conceiving u in U to vary
| |
| − | over its whole domain, namely, the universe U. This leads to
| |
| − | the ROST that is given in Rule 2.
| |
| | | | |
| − | o-------------------------------------------------o
| + | R6c. ConjUu (f(u) = g(u)). :D6e |
| − | | Rule 2 |
| + | </pre> |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | If f : U -> %B% |
| |
| − | | |
| |
| − | | and u in U, |
| |
| − | | |
| |
| − | | then the following are equivalent: |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | R2a. f(u). |
| |
| − | | |
| |
| − | | R2b. f(u) = 1. |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| | | | |
| − | Rules like these can be chained together to establish extended
| + | <pre> |
| − | rules, just so long as their antecedent conditions are compatible.
| + | Rule 7 |
| − | For example, Rules 1 and 2 combine to give the equivalents that are
| |
| − | listed in Rule 3. This follows from a recognition that the function
| |
| − | -{X}- : U -> %B% that is introduced in Rule 1 is an instance of the
| |
| − | function f : U -> %B% that is mentioned in Rule 2. By the time one
| |
| − | arrives in the "consequence box" of either Rule, then, one has in
| |
| − | mind a comparatively fixed X c U, a proposition f or -{X}- about
| |
| − | things in U, and a variable argument u in U.
| |
| | | | |
| − | o-------------------------------------------------o---------o
| + | If P, Q : U -> B, |
| − | | Rule 3 | |
| |
| − | o-------------------------------------------------o---------o
| |
| − | | | |
| |
| − | | If X c U | |
| |
| − | | | |
| |
| − | | and u in U, | |
| |
| − | | | |
| |
| − | | then the following are equivalent: | |
| |
| − | | | |
| |
| − | o-------------------------------------------------o---------o
| |
| − | | | |
| |
| − | | R3a. u in X. | : R1a |
| |
| − | | | :: |
| |
| − | | R3b. -{X}-(u). | : R1b |
| |
| − | | | : R2a |
| |
| − | | | :: |
| |
| − | | R3c. -{X}-(u) = 1. | : R2b |
| |
| − | | | |
| |
| − | o-------------------------------------------------o---------o
| |
| | | | |
| − | A large stock of rules can be derived in this way, by chaining together
| + | then the following are equivalent: |
| − | segments that are selected from a stock of previous rules, with perhaps
| |
| − | the whole process of derivation leading back to an axial body or a core
| |
| − | stock of rules, with all recurring to and relying on an axiomatic basis.
| |
| − | In order to keep track of their derivations, as their pedigrees help to
| |
| − | remember the reasons for trusting their use in the first place, derived
| |
| − | rules can be annotated by citing the rules from which they are derived.
| |
| | | | |
| − | In the present discussion, I am using a particular style of annotation
| + | R7a. P = Q. :R6a |
| − | for rule derivations, one that is called "proof by grammatical paradigm"
| |
| − | or "proof by syntactic analogy". The annotations in the right margin of
| |
| − | the Rule box can be read as the "denominators" of the paradigm that is
| |
| − | being employed, in other words, as the alternating terms of comparison
| |
| − | in a sequence of analogies. This can be illustrated by considering the
| |
| − | derivation Rule 3 in detail. Taking the steps marked in the box one at
| |
| − | a time, one can interweave the applications in the central body of the
| |
| − | box with the annotations in the right margin of the box, reading "is to"
| |
| − | for the ":" sign and "as" for the "::" sign, in the following fashion:
| |
| | | | |
| − | R3a. "u in X" is to R1a, namely, "u in X",
| + | :: |
| | | | |
| − | as
| + | R7b. P(u) = Q(u), for all u C U. :R6b |
| | | | |
| − | R3b. "{X}(u)" is to R1b, namely, "{X}(u)",
| + | :: |
| | | | |
| − | and
| + | R7c. ConjUu (P(u) = Q(u)). :R6c |
| | | | |
| − | "{X}(u)" is to R2a, namely, "f(u)",
| + | :P1a |
| | | | |
| − | as
| + | :: |
| | | | |
| − | R3c. "{X}(u) = 1" is to R2b, namely, "f(u) = 1".
| + | R7d. ConjUu (P(u) <=> Q(u)). :P1b |
| | | | |
| − | Notice how the sequence of analogies pivots on the item R3b,
| + | :: |
| − | viewing it first under the aegis of R1b, as the second term of
| |
| − | the first analogy, and then turning to view it again under the
| |
| − | guise of R2a, as the first term of the second analogy.
| |
| | | | |
| − | By way of convention, rules that are tailored to a particular
| + | R7e. ConjUu (( P(u) , Q(u) )). :P1c |
| − | application, case, or subject, and rules that are adapted to
| |
| − | a particular goal, object, or purpose, I frequently refer to
| |
| − | as "Facts".
| |
| − | </pre>
| |
| | | | |
| − | =====1.3.10.14. Syntactic Transformations (cont.)=====
| + | :$1a |
| | | | |
| − | Besides linking rules together into extended sequences of equivalents,
| + | :: |
| − | there is one other way that is commonly used to get new rules from old.
| |
| − | Novel starting points for rules can be obtained by extracting pairs of
| |
| − | equivalent expressions from a sequence that falls under an established
| |
| − | rule, and then by stating their equality in the proper form of equation.
| |
| − | For example, by extracting the equivalent expressions that are annotated
| |
| − | as "R3a" and "R3c" in Rule 3 and by explictly stating their equivalence,
| |
| − | one obtains the specialized result that is recorded in Corollary 1.
| |
| | | | |
| − | Corollary 1
| + | R7f. ConjUu (( P , Q ))$(u). :$1b |
| | + | </pre> |
| | | | |
| − | If X c U
| + | <pre> |
| | + | Rule 8 |
| | | | |
| − | and u C U,
| + | If S, T are sentences |
| | | | |
| − | then the following statement is true:
| + | about things in the universe U, |
| | | | |
| − | C1a. u C X <=> {X}(u) = 1. R3a=R3c
| + | then the following are equivalent: |
| | | | |
| − | There are a number of issues, that arise especially in establishing the proper use of STR's, that are appropriate to discuss at this juncture. The notation "[S]" is intended to represent "the proposition denoted by the sentence S". There is only one problem with the use of this form. There is, in general, no such thing as "the" proposition denoted by S. Generally speaking, if a sentence is taken out of context and considered across a variety of different contexts, there is no unique proposition that it can be said to denote. But one is seldom ever speaking at the maximum level of generality, or even found to be thinking of it, and so this notation is usually meaningful and readily understandable whenever it is read in the proper frame of mind. Still, once the issue is raised, the question of how these meanings and understandings are possible has to be addressed, especially if one desires to express the regulations of their syntax in a partially computational form. This requires a closer examination of the very notion of "context", and it involves engaging in enough reflection on the "contextual evaluation" of sentences that the relevant principles of its successful operation can be discerned and rationalized in explicit terms.
| + | R8a. S <=> T. :D7a |
| | | | |
| − | A sentence that is written in a context where it represents a value of 1 or 0 as a function of things in the universe U, where it stands for a value of "true" or "false", depending on how the signs that constitute its proper syntactic arguments are interpreted as denoting objects in U, in other words, where it is bound to lead its interpreter to view its own truth or falsity as determined by a choice of objects in U, is a sentence that might as well be written in the context "[ ... ]", whether or not this frame is explicitly marked around it.
| + | :: |
| | | | |
| − | More often than not, the context of interpretation fixes the denotations of most of the signs that make up a sentence, and so it is safe to adopt the convention that only those signs whose objects are not already fixed are free to vary in their denotations. Thus, only the signs that remain in default of prior specification are subject to treatment as variables, with a decree of functional abstraction hanging over all of their heads.
| + | R8b. [S] = [T]. :D7b |
| | | | |
| − | [u C X] = Lambda (u, C, X).(u C X).
| + | :R7a |
| | | | |
| − | As it is presently stated, Rule 1 lists a couple of manifest sentences, and it authorizes one to make exchanges in either direction between the syntactic items that have these two forms. But a sentence is any sign that denotes a proposition, and thus there are a number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged. Consider the sense of equivalence among sentences that is recorded in Rule 4.
| + | :: |
| | | | |
| − | Rule 4
| + | R8c. [S](u) = [T](u), for all u C U. :R7b |
| | | | |
| − | If X c U is fixed
| + | :: |
| | | | |
| − | and u C U is varied,
| + | R8d. ConjUu ( [S](u) = [T](u) ). :R7c |
| | | | |
| − | then the following are equivalent:
| + | :: |
| | | | |
| − | R4a. u C X.
| + | R8e. ConjUu ( [S](u) <=> [T](u) ). :R7d |
| | | | |
| − | R4b. [u C X].
| + | :: |
| | | | |
| − | R4c. [u C X](u).
| + | R8f. ConjUu (( [S](u) , [T](u) )). :R7e |
| | | | |
| − | R4d. {X}(u).
| + | :: |
| | | | |
| − | R4e. {X}(u) = 1.
| + | R8g. ConjUu (( [S] , [T] ))$(u). :R7f |
| | + | </pre> |
| | | | |
| − | The first and last items on this list, namely, the sentences "u C X" and "{X}(u) = 1" that are annotated as "R4a" and "R4e", respectively, are just the pair of sentences from Rule 3 whose equivalence for all u C U is usually taken to define the idea of an indicator function {X} : U -> B. At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their own ostensible types and the ruling type of a sentence. On reflection, and taken in context, these problems are not as serious as they initially seem. For instance, the expression "[u C X]" ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence. As a general rule, if one can see it on the page, then it cannot be a proposition, but can be, at best, a sign of one.
| + | For instance, the observation that expresses the equality of sets in terms of their indicator functions can be formalized according to the pattern in Rule 9, namely, at lines (a, b, c), and these components of Rule 9 can be cited in future uses as "R9a", "R9b", "R9c", respectively. Using Rule 7, annotated as "R7", to adduce a few properties of indicator functions to the account, it is possible to extend Rule 9 by another few steps, referenced as "R9d", "R9e", "R9f", "R9g". |
| | | | |
| − | The use of the basic connectives can be expressed in the form of a STR as follows:
| + | <pre> |
| | + | Rule 9 |
| | | | |
| − | Logical Translation Rule 0
| + | If X, Y c U, |
| | | | |
| − | If Sj is a sentence
| + | then the following are equivalent: |
| | | | |
| − | about things in the universe U
| + | R9a. X = Y. :R5a |
| | | | |
| − | and Pj is a proposition
| + | :: |
| | | | |
| − | about things in the universe U
| + | R9b. {X} = {Y}. :R5e |
| | | | |
| − | such that:
| + | :R7a |
| | | | |
| − | L0a. [Sj] = Pj, for all j C J,
| + | :: |
| | | | |
| − | then the following equations are true:
| + | R9c. {X}(u) = {Y}(u), for all u C U. :R7b |
| | | | |
| − | L0b. [ConcJj Sj] = ConjJj [Sj] = ConjJj Pj.
| + | :: |
| | | | |
| − | L0c. [SurcJj Sj] = SurjJj [Sj] = SurjJj Pj.
| + | R9d. ConjUu ( {X}(u) = {Y}(u) ). :R7c |
| | | | |
| − | As a general rule, the application of a STR involves the recognition of an antecedent condition and the facilitation of a consequent condition. The antecedent condition is a state whose initial expression presents a match, in a formal sense, to one of the sentences that are listed in the STR, and the consequent condition is achieved by taking its suggestions seriously, in other words, by following its sequence of equivalents and implicants to some other link in its chain.
| + | :: |
| | | | |
| − | Generally speaking, the application of a rule involves the recognition of an antecedent condition as a case that falls under a clause of the rule. This means that the antecedent condition is able to be captured in the form, conceived in the guise, expressed in the manner, grasped in the pattern, or recognized in the shape of one of the sentences in a list of equivalents or a chain of implicants.
| + | R9e. ConjUu ( {X}(u) <=> {Y}(u) ). :R7d |
| | | | |
| − | A condition is "amenable" to a rule if any of its conceivable expressions formally match any of the expressions that are enumerated by the rule. Further, it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that needs to be checked on input for whether it fits the antecedent condition and there are several types of output that are generated as a consequence, only a few of which are usually needed at any given time.
| + | :: |
| | | | |
| − | Logical Translation Rule 1
| + | R9f. ConjUu (( {X}(u) , {Y}(u) )). :R7e |
| | | | |
| − | If S is a sentence
| + | :: |
| | | | |
| − | about things in the universe U
| + | R9g. ConjUu (( {X} , {Y} ))$(u). :R7f |
| | + | </pre> |
| | | | |
| − | and P is a proposition : U -> B, such that:
| + | <pre> |
| | + | Rule 10 |
| | | | |
| − | L1a. [S] = P,
| + | If X, Y c U, |
| | | | |
| − | then the following equations hold: | + | then the following are equivalent: |
| | | | |
| − | L1b00. [False] = () = 0 : U->B.
| + | R10a. X = Y. :D2a |
| | | | |
| − | L1b01. [Not S] = ([S]) = (P) : U->B.
| + | :: |
| | | | |
| − | L1b10. [S] = [S] = P : U->B.
| + | R10b. u C X <=> u C Y, for all u C U. :D2b |
| | | | |
| − | L1b11. [True] = (()) = 1 : U->B.
| + | :R8a |
| | | | |
| − | Geometric Translation Rule 1
| + | :: |
| | | | |
| − | If X c U
| + | R10c. [u C X] = [u C Y]. :R8b |
| | | | |
| − | and P : U -> B, such that:
| + | :: |
| | | | |
| − | G1a. {X} = P,
| + | R10d. For all u C U, |
| | | | |
| − | then the following equations hold:
| + | [u C X](u) = [u C Y](u). :R8c |
| | | | |
| − | G1b00. {{}} = () = 0 : U->B.
| + | :: |
| | | | |
| − | G1b10. {~X} = ({X}) = (P) : U->B.
| + | R10e. ConjUu ( [u C X](u) = [u C Y](u) ). :R8d |
| | | | |
| − | G1b01. {X} = {X} = P : U->B.
| + | :: |
| | | | |
| − | G1b11. {U} = (()) = 1 : U->B.
| + | R10f. ConjUu ( [u C X](u) <=> [u C Y](u) ). :R8e |
| | | | |
| − | Logical Translation Rule 2
| + | :: |
| | | | |
| − | If S, T are sentences
| + | R10g. ConjUu (( [u C X](u) , [u C Y](u) )). :R8f |
| | | | |
| − | about things in the universe U
| + | :: |
| | | | |
| − | and P, Q are propositions: U -> B, such that:
| + | R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g |
| | + | </pre> |
| | | | |
| − | L2a. [S] = P and [T] = Q,
| + | <pre> |
| | + | Rule 11 |
| | | | |
| − | then the following equations hold:
| + | If X c U |
| | | | |
| − | L2b00. [False] = () = 0 : U->B.
| + | then the following are equivalent: |
| | | | |
| − | L2b01. [Neither S nor T] = ([S])([T]) = (P)(Q).
| + | R11a. X = {u C U : S}. :R5a |
| | | | |
| − | L2b02. [Not S, but T] = ([S])[T] = (P) Q.
| + | :: |
| | | | |
| − | L2b03. [Not S] = ([S]) = (P).
| + | R11b. {X} = { {u C U : S} }. :R5e |
| | | | |
| − | L2b04. [S and not T] = [S]([T]) = P (Q).
| + | :: |
| | | | |
| − | L2b05. [Not T] = ([T]) = (Q).
| + | R11c. {X} c UxB |
| | | | |
| − | L2b06. [S or T, not both] = ([S], [T]) = (P, Q).
| + | : {X} = {< u, v> C UxB : v = [S](u)}. :R |
| | | | |
| − | L2b07. [Not both S and T] = ([S].[T]) = (P Q).
| + | :: |
| | | | |
| − | L2b08. [S and T] = [S].[T] = P.Q.
| + | R11d. {X} : U -> B |
| | | | |
| − | L2b09. [S <=> T] = (([S], [T])) = ((P, Q)).
| + | : {X}(u) = [S](u), for all u C U. :R |
| | | | |
| − | L2b10. [T] = [T] = Q.
| + | :: |
| | | | |
| − | L2b11. [S => T] = ([S]([T])) = (P (Q)).
| + | R11e. {X} = [S]. :R |
| | + | </pre> |
| | | | |
| − | L2b12. [S] = [S] = P.
| + | An application of Rule 11 involves the recognition of an antecedent condition as a case under the Rule, that is, as a condition that matches one of the sentences in the Rule's chain of equivalents, and it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that has to be checked on input for whether it fits the antecedent condition, and there is the choice of three types of output that are generated as a consequence, only one of which is generally needed at any given time. More often than not, though, a rule is applied in only a few of its possible ways. The usual antecedent and the usual consequents for Rule 11 can be distinguished in form and specialized in practice as follows: |
| | | | |
| − | L2b13. [S <= T] = (([S]) [T]) = ((P) Q).
| + | a. R11a marks the usual starting place for an application of the Rule, that is, the standard form of antecedent condition that is likely to lead to an invocation of the Rule. |
| | | | |
| − | L2b14. [S or T] = (([S])([T])) = ((P)(Q)).
| + | b. R11b records the trivial consequence of applying the spiny braces to both sides of the initial equation. |
| | | | |
| − | L2b15. [True] = (()) = 1 : U->B.
| + | c. R11c gives a version of the indicator function with {X} c UxB, called its "extensional form". |
| | | | |
| | + | d. R11d gives a version of the indicator function with {X} : U->B, called its "functional form". |
| | | | |
| − | Geometric Translation Rule 2
| + | Applying Rule 9, Rule 8, and the Logical Rules to the special case where S <=> (X = Y), one obtains the following general fact. |
| | | | |
| − | If X, Y c U
| + | <pre> |
| | + | Fact 1 |
| | | | |
| | + | If X,Y c U, |
| | | | |
| | + | then the following are equivalent: |
| | | | |
| − | and P, Q U -> B, such that:
| + | F1a. S <=> X = Y. :R9a |
| | | | |
| | + | :: |
| | | | |
| | + | F1b. S <=> {X} = {Y}. :R9b |
| | | | |
| − | G2a. {X} = P and {Y} = Q,
| + | :: |
| | | | |
| | + | F1c. S <=> {X}(u) = {Y}(u), for all u C U. :R9c |
| | | | |
| | + | :: |
| | | | |
| − | then the following equations hold:
| + | F1d. S <=> ConjUu ( {X}(u) = {Y}(u) ). :R9d |
| | | | |
| | + | :R8a |
| | | | |
| | + | :: |
| | | | |
| − | G2b00. {{}} = () = 0 : U->B.
| + | F1e. [S] = [ ConjUu ( {X}(u) = {Y}(u) ) ]. :R8b |
| | | | |
| | + | :??? |
| | | | |
| | + | :: |
| | | | |
| − | G2b01. {~X n ~Y} = ({X})({Y}) = (P)(Q).
| + | F1f. [S] = ConjUu [ {X}(u) = {Y}(u) ]. :??? |
| | | | |
| | + | :: |
| | | | |
| | + | F1g. [S] = ConjUu (( {X}(u) , {Y}(u) )). :$1a |
| | | | |
| − | G2b02. {~X n Y} = ({X}){Y} = (P) Q.
| + | :: |
| | | | |
| | + | F1h. [S] = ConjUu (( {X} , {Y} ))$(u). :$1b |
| | | | |
| | + | /// |
| | | | |
| − | G2b03. {~X} = ({X}) = (P).
| + | {u C U : (f, g)$(u)} |
| | | | |
| | + | = {u C U : (f(u), g(u))} |
| | | | |
| | + | = {u C |
| | | | |
| − | G2b04. {X n ~Y} = {X}({Y}) = P (Q).
| + | /// |
| | + | </pre> |
| | | | |
| | + | =====1.3.12.2. Derived Equivalence Relations===== |
| | | | |
| | + | One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote. |
| | | | |
| − | G2b05. {~Y} = ({Y}) = (Q).
| + | A classic way of showing that two sets are equal is to show that every element of the first belongs to the second and that every element of the second belongs to the first. The problem with this strategy is that one can exhaust a considerable amount of time trying to prove that two sets are equal before it occurs to one to look for a counterexample, that is, an element of the first that does not belong to the second or an element of the second that does not belong to the first, in cases where that is precisely what one ought to be seeking. It would be nice if there were a more balanced, impartial, neutral, or nonchalant way to go about this task, one that did not require such an undue commitment to either side, a technique that helps to pinpoint the counterexamples when they exist, and a method that keeps in mind the original relation of "proving that" and "showing that" to probing, testing, and seeing "whether". |
| | | | |
| | + | A different way of seeing that two sets are equal, or of seeing whether two sets are equal, is based on the following observation: |
| | | | |
| | + | <pre> |
| | + | Two sets are equal as sets |
| | | | |
| − | G2b06. {X + Y} = ({X}, {Y}) = (P, Q).
| + | <=> the indicator functions of these sets are equal as functions |
| | | | |
| | + | <=> the values of these functions are equal on all domain elements. |
| | + | </pre> |
| | | | |
| | + | It is important to notice the hidden quantifier, of a universal kind, that lurks in all three equivalent statements but is only revealed in the last. |
| | | | |
| − | G2b07. {~(X n Y)} = ({X}.{Y}) = (P Q).
| + | In making the next set of definitions and in using the corresponding terminology it is taken for granted that all of the references of signs are relative to a particular sign relation R c OxSxI that either remains to be specified or is already understood. Further, I continue to assume that S = I, in which case this set is called the "syntactic domain" of R. |
| | | | |
| | + | In the following definitions let R c OxSxI, let S = I, and let x, y C S. |
| | | | |
| | + | Recall the definition of Con(R), the connotative component of R, in the following form: |
| | | | |
| − | G2b08. {X n Y} = {X}.{Y} = P.Q.
| + | : Con(R) = RSI = {< s, i> C SxI : <o, s, i> C R for some o C O}. |
| | | | |
| | + | Equivalent expressions for this concept are recorded in Definition 8. |
| | | | |
| | + | <pre> |
| | + | Definition 8 |
| | | | |
| − | G2b09. {~(X + Y)} = (({X}, {Y})) = ((P, Q)).
| + | If R c OxSxI, |
| | | | |
| | + | then the following are identical subsets of SxI: |
| | | | |
| | + | D8a. RSI |
| | | | |
| − | G2b10. {Y} = {Y} = Q.
| + | D8b. ConR |
| | | | |
| | + | D8c. Con(R) |
| | | | |
| | + | D8d. PrSI(R) |
| | | | |
| − | G2b11. {~(X n ~Y)} = ({X}({Y})) = (P (Q)).
| + | D8e. {< s, i> C SxI : <o, s, i> C R for some o C O} |
| | + | </pre> |
| | | | |
| | + | The dyadic relation RIS that constitutes the converse of the connotative relation RSI can be defined directly in the following fashion: |
| | | | |
| | + | : Con(R)^ = RIS = {< i, s> C IxS : <o, s, i> C R for some o C O}. |
| | | | |
| − | G2b12. {X} = {X} = P.
| + | A few of the many different expressions for this concept are recorded in Definition 9. |
| | | | |
| | + | <pre> |
| | + | Definition 9 |
| | | | |
| | + | If R c OxSxI, |
| | | | |
| − | G2b13. {~(~X n Y)} = (({X}) {Y}) = ((P) Q).
| + | then the following are identical subsets of IxS: |
| | | | |
| | + | D9a. RIS |
| | | | |
| | + | D9b. RSI^ |
| | | | |
| − | G2b14. {X u Y} = (({X})({Y})) = ((P)(Q)).
| + | D9c. ConR^ |
| | | | |
| | + | D9d. Con(R)^ |
| | | | |
| | + | D9e. PrIS(R) |
| | | | |
| − | G2b15. {U} = (()) = 1 : U->B.
| + | D9f. Conv(Con(R)) |
| | | | |
| | + | D9g. {< i, s> C IxS : <o, s, i> C R for some o C O} |
| | + | </pre> |
| | | | |
| | + | Recall the definition of Den(R), the denotative component of R, in the following form: |
| | | | |
| | + | : Den(R) = ROS = {<o, s> C OxS : <o, s, i> C R for some i C I}. |
| | | | |
| | + | Equivalent expressions for this concept are recorded in Definition 10. |
| | | | |
| − | Value Rule 1
| + | <pre> |
| | + | Definition 10 |
| | | | |
| − | If v, w C B | + | If R c OxSxI, |
| | | | |
| | + | then the following are identical subsets of OxS: |
| | | | |
| | + | D10a. ROS |
| | | | |
| − | then "v = w" is a sentence about <v, w> C B2,
| + | D10b. DenR |
| | | | |
| | + | D10c. Den(R) |
| | | | |
| | + | D10d. PrOS(R) |
| | | | |
| − | [v = w] is a proposition : B2 -> B,
| + | D10e. {<o, s> C OxS : <o, s, i> C R for some i C I} |
| | + | </pre> |
| | | | |
| | + | The dyadic relation RSO that constitutes the converse of the denotative relation ROS can be defined directly in the following fashion: |
| | | | |
| | + | : Den(R)^ = RSO = {< s, o> C SxO : <o, s, i> C R for some i C I}. |
| | | | |
| − | and the following are identical values in B:
| + | A few of the many different expressions for this concept are recorded in Definition 11. |
| | | | |
| | + | <pre> |
| | + | Definition 11 |
| | | | |
| | + | If R c OxSxI, |
| | | | |
| − | V1a. [ v = w ](v, w)
| + | then the following are identical subsets of SxO: |
| | | | |
| | + | D11a. RSO |
| | | | |
| | + | D11b. ROS^ |
| | | | |
| − | V1b. [ v <=> w ](v, w)
| + | D11c. DenR^ |
| | | | |
| | + | D11d. Den(R)^ |
| | | | |
| | + | D11e. PrSO(R) |
| | | | |
| − | V1c. ((v , w))
| + | D11f. Conv(Den(R)) |
| | | | |
| | + | D11g. {< s, o> C SxO : <o, s, i> C R for some i C I} |
| | + | </pre> |
| | | | |
| | + | The "denotation of x in R", written "Den(R, x)", is defined as follows: |
| | | | |
| − | Value Rule 1
| + | : Den(R, x) = {o C O : <o, x> C Den(R)}. |
| | | | |
| − | If v, w C B,
| + | In other words: |
| | | | |
| | + | : Den(R, x) = {o C O : <o, x, i> C R for some i C I}. |
| | | | |
| | + | Equivalent expressions for this concept are recorded in Definition 12. |
| | | | |
| − | then the following are equivalent:
| + | <pre> |
| | + | Definition 12 |
| | | | |
| | + | If R c OxSxI, |
| | | | |
| | + | and x C S, |
| | | | |
| − | V1a. v = w.
| + | then the following are identical subsets of O: |
| | | | |
| | + | D12a. ROS.x |
| | | | |
| | + | D12b. DenR.x |
| | | | |
| − | V1b. v <=> w.
| + | D12c. DenR|x |
| | | | |
| | + | D12d. DenR(, x) |
| | | | |
| | + | D12e. Den(R, x) |
| | | | |
| − | V1c. (( v , w )).
| + | D12f. Den(R).x |
| | | | |
| − | A rule that allows one to turn equivalent sentences into identical propositions:
| + | D12g. {o C O : <o, x> C Den(R)} |
| | | | |
| − | (S <=> T) <=> ([S] = [T])
| + | D12h. {o C O : <o, x, i> C R for some i C I} |
| | + | </pre> |
| | | | |
| − | Consider [ v = w ](v, w) and [ v(u) = w(u) ](u)
| + | Signs are "equiferent" if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be "denotatively equivalent" or "referentially equivalent", but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumping to the conclusions that are implied by these latter terms. |
| | | | |
| − | Value Rule 1
| + | To define the "equiference" of signs in terms of their denotations, one says that "x is equiferent to y under R", and writes "x =R y", to mean that Den(R, x) = Den(R, y). Taken in extension, this notion of a relation between signs induces an "equiference relation" on the syntactic domain. |
| | | | |
| − | If v, w C B,
| + | For each sign relation R, this yields a binary relation Der(R) c SxI that is defined as follows: |
| | | | |
| | + | : Der(R) = DerR = {<x, y> C SxI : Den(R, x) = Den(R, y)}. |
| | | | |
| | + | These definitions and notations are recorded in the following display. |
| | | | |
| − | then the following are identical values in B:
| + | <pre> |
| | + | Definition 13 |
| | | | |
| | + | If R c OxSxI, |
| | | | |
| | + | then the following are identical subsets of SxI: |
| | | | |
| − | V1a. [ v = w ]
| + | D13a. DerR |
| | | | |
| | + | D13b. Der(R) |
| | | | |
| | + | D13c. {<x,y> C SxI : DenR|x = DenR|y} |
| | | | |
| − | V1b. [ v <=> w ]
| + | D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)} |
| | + | </pre> |
| | | | |
| | + | The relation Der(R) is defined and the notation "x =R y" is meaningful in every situation where Den(-,-) makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation. |
| | | | |
| | + | # Reflexive property. Is it true that x =R x for every x C S = I? By definition, x =R x if and only if Den(R, x) = Den(R, x). Thus, the reflexive property holds in any setting where the denotations Den(R, x) are defined for all signs x in the syntactic domain of R. |
| | + | # Symmetric property. Does x =R y => y =R x for all x, y C S? In effect, does Den(R, x) = Den(R, y) imply Den(R, y) = Den(R, x) for all signs x and y in the syntactic domain S? Yes, so long as the sets Den(R, x) and Den(R, y) are well-defined, a fact which is already being assumed. |
| | + | # Transitive property. Does x =R y & y =R z => x =R z for all x, y, z C S? To belabor the point, does Den(R, x) = Den(R, y) and Den(R, y) = Den(R, z) imply Den(R, x) = Den(R, z) for all x, y, z in S? Yes, again, under the stated conditions. |
| | | | |
| − | V1c. (( v , w ))
| + | It should be clear at this point that any question about the equiference of signs reduces to a question about the equality of sets, specifically, the sets that are indexed by these signs. As a result, so long as these sets are well-defined, the issue of whether equiference relations induce equivalence relations on their syntactic domains is almost as trivial as it initially appears. |
| | | | |
| | + | Taken in its set-theoretic extension, a relation of equiference induces a "denotative equivalence relation" (DER) on its syntactic domain S = I. This leads to the formation of "denotative equivalence classes" (DEC's), "denotative partitions" (DEP's), and "denotative equations" (DEQ's) on the syntactic domain. But what does it mean for signs to be equiferent? |
| | | | |
| | + | Notice that this is not the same thing as being "semiotically equivalent", in the sense of belonging to a single "semiotic equivalence class" (SEC), falling into the same part of a "semiotic partition" (SEP), or having a "semiotic equation" (SEQ) between them. It is only when very felicitous conditions obtain, establishing a concord between the denotative and the connotative components of a sign relation, that these two ideas coalesce. |
| | | | |
| − | Value Rule 1
| + | In general, there is no necessity that the equiference of signs, that is, their denotational equivalence or their referential equivalence, induces the same equivalence relation on the syntactic domain as that defined by their semiotic equivalence, even though this state of accord seems like an especially desirable situation. This makes it necessary to find a distinctive nomenclature for these structures, for which I adopt the term "denotative equivalence relations" (DER's). In their train they bring the allied structures of "denotative equivalence classes" (DEC's) and "denotative partitions" (DEP's), while the corresponding statements of "denotative equations" (DEQ's) are expressible in the form "x =R y". |
| | | | |
| − | If f, g : U -> B,
| + | The uses of the equal sign for denoting equations or equivalences are recalled and extended in the following ways: |
| | | | |
| | + | 1. If E is an arbitrary equivalence relation, |
| | | | |
| | + | then the equation "x =E y" means that <x, y> C E. |
| | | | |
| − | and u C U
| + | 2. If R is a sign relation such that RSI is a SER on S = I, |
| | | | |
| | + | then the semiotic equation "x =R y" means that <x, y> C RSI. |
| | | | |
| | + | 3. If R is a sign relation such that F is its DER on S = I, |
| | | | |
| − | then the following are identical values in B: | + | then the denotative equation "x =R y" means that <x, y> C F, |
| | | | |
| | + | in other words, that Den(R, x) = Den(R, y). |
| | | | |
| | + | The uses of square brackets for denoting equivalence classes are recalled and extended in the following ways: |
| | | | |
| − | V1a. [ f(u) = g(u) ]
| + | 1. If E is an arbitrary equivalence relation, |
| | | | |
| | + | then "[x]E" denotes the equivalence class of x under E. |
| | | | |
| | + | 2. If R is a sign relation such that Con(R) is a SER on S = I, |
| | | | |
| − | V1b. [ f(u) <=> g(u) ]
| + | then "[x]R" denotes the SEC of x under Con(R). |
| | | | |
| | + | 3. If R is a sign relation such that Der(R) is a DER on S = I, |
| | | | |
| | + | then "[x]R" denotes the DEC of x under Der(R). |
| | | | |
| − | V1c. (( f(u) , g(u) ))
| + | By applying the form of Fact 1 to the special case where X = Den(R, x) and Y = Den(R, y), one obtains the following facts. |
| | | | |
| | + | <pre> |
| | + | Fact 2.1 |
| | | | |
| | + | If R c OxSxI, |
| | | | |
| − | Value Rule 1
| + | then the following are identical subsets of SxI: |
| | | | |
| − | If f, g : U -> B,
| + | F2.1a. DerR :D13a |
| | | | |
| | + | :: |
| | | | |
| | + | F2.1b. Der(R) :D13b |
| | | | |
| − | then the following are identical propositions on U:
| + | :: |
| | | | |
| | + | F2.1c. {<x, y> C SxI : |
| | | | |
| | + | Den(R, x) = Den(R, y) |
| | | | |
| − | V1a. [ f = g ]
| + | } :D13c |
| | | | |
| | + | :R9a |
| | | | |
| | + | :: |
| | | | |
| − | V1b. [ f <=> g ]
| + | F2.1d. {<x, y> C SxI : |
| | | | |
| | + | {Den(R, x)} = {Den(R, y)} |
| | | | |
| | + | } :R9b |
| | | | |
| − | V1c. (( f , g ))$
| + | :: |
| | | | |
| | + | F2.1e. {<x, y> C SxI : |
| | | | |
| | + | for all o C O |
| | | | |
| − | Evaluation Rule 1
| + | {Den(R, x)}(o) = {Den(R, y)}(o) |
| | | | |
| − | If f, g : U -> B
| + | } :R9c |
| | | | |
| | + | :: |
| | | | |
| | + | F2.1f. {<x, y> C SxI : |
| | | | |
| − | and u C U,
| + | Conj(o C O) |
| | | | |
| | + | {Den(R, x)}(o) = {Den(R, y)}(o) |
| | | | |
| | + | } :R9d |
| | | | |
| − | then the following are equivalent:
| + | :: |
| | | | |
| | + | F2.1g. {<x, y> C SxI : |
| | | | |
| | + | Conj(o C O) |
| | | | |
| − | E1a. f(u) = g(u). :V1a
| + | (( {Den(R, x)}(o) , {Den(R, y)}(o) )) |
| | | | |
| − | ::
| + | } :R9e |
| | | | |
| − | E1b. f(u) <=> g(u). :V1b
| + | :: |
| | | | |
| − | ::
| + | F2.1h. {<x, y> C SxI : |
| | | | |
| − | E1c. (( f(u) , g(u) )). :V1c
| + | Conj(o C O) |
| | | | |
| − | :$1a
| + | (( {Den(R, x)} , {Den(R, y)} ))$(o) |
| | | | |
| − | ::
| + | } :R9f |
| | | | |
| − | E1d. (( f , g ))$(u). :$1b
| + | :D12e |
| | | | |
| | + | :: |
| | | | |
| | + | F2.1i. {<x, y> C SxI : |
| | | | |
| − | Evaluation Rule 1
| + | Conj(o C O) |
| | | | |
| − | If S, T are sentences
| + | (( {ROS.x} , {ROS.y} ))$(o) |
| | | | |
| − | about things in the universe U,
| + | } :D12a |
| | + | </pre> |
| | | | |
| | + | <pre> |
| | + | Fact 2.2 |
| | | | |
| | + | If R c OxSxI, |
| | | | |
| − | f, g are propositions: U -> B,
| + | then the following are equivalent: |
| | | | |
| | + | F2.2a. DerR = {<x, y> C SxI : |
| | | | |
| | + | Conj(o C O) |
| | | | |
| − | and u C U,
| + | {Den(R, x)}(o) = |
| | | | |
| | + | {Den(R, y)}(o) |
| | | | |
| | + | } :R11a |
| | + | :: |
| | | | |
| − | then the following are equivalent:
| + | F2.2b. {DerR} = { {<x, y> C SxI : |
| | | | |
| | + | Conj(o C O) |
| | | | |
| | + | {Den(R, x)}(o) = |
| | | | |
| − | E1a. f(u) = g(u). :V1a
| + | {Den(R, y)}(o) |
| | | | |
| − | :: | + | } |
| | | | |
| − | E1b. f(u) <=> g(u). :V1b
| + | } :R11b |
| | | | |
| − | ::
| + | :: |
| | | | |
| − | E1c. (( f(u) , g(u) )). :V1c
| + | F2.2c. {DerR} c SxIxB |
| | | | |
| − | :$1a
| + | : |
| | | | |
| − | ::
| + | {DerR} = {<x, y, v> C SxIxB : |
| | | | |
| − | E1d. (( f , g ))$(u). :$1b
| + | v = |
| | | | |
| | + | [ Conj(o C O) |
| | | | |
| | + | {Den(R, x)}(o) = |
| | | | |
| | + | {Den(R, y)}(o) |
| | | | |
| | + | ] |
| | | | |
| − | Definition 2
| + | } :R11c |
| | | | |
| − | If X, Y c U,
| + | :: |
| | | | |
| | + | F2.2d. {DerR} = {<x, y, v> C SxIxB : |
| | | | |
| | + | v = |
| | | | |
| − | then the following are equivalent:
| + | Conj(o C O) |
| | | | |
| | + | [ {Den(R, x)}(o) = |
| | | | |
| | + | {Den(R, y)}(o) |
| | | | |
| − | D2a. X = Y.
| + | ] |
| | | | |
| | + | } :Log |
| | | | |
| | + | F2.2e. {DerR} = {<x, y, v> C SxIxB : |
| | | | |
| − | D2b. u C X <=> u C Y, for all u C U.
| + | v = |
| | | | |
| | + | Conj(o C O) |
| | | | |
| | + | (( {Den(R, x)}(o), |
| | | | |
| − | Definition 3
| + | {Den(R, y)}(o) |
| | | | |
| − | If f, g : U -> V,
| + | )) |
| | | | |
| | + | } :Log |
| | | | |
| | + | F2.2f. {DerR} = {<x, y, v> C SxIxB : |
| | | | |
| − | then the following are equivalent:
| + | v = |
| | | | |
| | + | Conj(o C O) |
| | | | |
| | + | (( {Den(R, x)}, |
| | | | |
| − | D3a. f = g.
| + | {Den(R, y)} |
| | | | |
| | + | ))$(o) |
| | | | |
| | + | } :$ |
| | + | </pre> |
| | | | |
| − | D3b. f(u) = g(u), for all u C U.
| + | <pre> |
| | + | Fact 2.3 |
| | | | |
| | + | If R c OxSxI, |
| | | | |
| | + | then the following are equivalent: |
| | | | |
| − | Definition 4
| + | F2.3a. DerR = {<x, y> C SxI : |
| | | | |
| − | If X c U,
| + | Conj(o C O) |
| | | | |
| | + | {Den(R, x)}(o) = |
| | | | |
| | + | {Den(R, y)}(o) |
| | | | |
| − | then the following are identical subsets of UxB:
| + | } :R11a |
| | | | |
| | + | :: |
| | | | |
| | + | F2.3b. {DerR} : SxI -> B |
| | | | |
| − | D4a. {X}
| + | : |
| | | | |
| | + | {DerR}(x, y) = [ Conj(o C O) |
| | | | |
| | + | {Den(R, x)}(o) = |
| | | | |
| − | D4b. {< u, v> C UxB : v = [u C X]}
| + | {Den(R, y)}(o) |
| | | | |
| | + | ] :R11d |
| | | | |
| | + | :: |
| | | | |
| − | Definition 5
| + | F2.3c. {DerR}(x, y) = Conj(o C O) |
| | | | |
| − | If X c U,
| + | [ {Den(R, x)}(o) = |
| | | | |
| | + | {Den(R, y)}(o) |
| | | | |
| | + | ] :Log |
| | | | |
| − | then the following are identical propositions:
| + | :: |
| | | | |
| | + | F2.3d. {DerR}(x, y) = Conj(o C O) |
| | | | |
| | + | [ {DenR}(o, x) = |
| | | | |
| − | D5a. {X}.
| + | {DenR}(o, y) |
| | | | |
| | + | ] :Def |
| | | | |
| | + | :: |
| | | | |
| − | D5b. f : U -> B
| + | F2.3e. {DerR}(x, y) = Conj(o C O) |
| | | | |
| | + | (( {DenR}(o, x), |
| | | | |
| | + | {DenR}(o, y) |
| | | | |
| − | : f(u) = [u C X], for all u C U. | + | )) :Log |
| | | | |
| − | Given an indexed set of sentences, Sj for j C J, it is possible to consider the logical conjunction of the corresponding propositions. Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6.
| + | :D10b |
| | | | |
| − | Definition 6
| + | :: |
| | | | |
| − | If Sj is a sentence
| + | F2.3f. {DerR}(x, y) = Conj(o C O) |
| | | | |
| − | about things in the universe U,
| + | (( {ROS}(o, x), |
| | | | |
| − | for all j C J,
| + | {ROS}(o, y) |
| | | | |
| | + | )) :D10a |
| | + | </pre> |
| | | | |
| | + | =====1.3.12.3. Digression on Derived Relations===== |
| | | | |
| − | then the following are equivalent:
| + | A better understanding of derived equivalence relations (DER's) can be achieved by placing their constructions within a more general context, and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation R into a dyadic relation Der(R), with other types of operations on triadic relations. The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations. |
| | | | |
| | + | To that end, let the derivation Der(R) be expressed in the following way: |
| | | | |
| | + | : {DerR}(x, y) = Conj(o C O) (( {RSO}(x, o) , {ROS}(o, y) )). |
| | | | |
| − | D6a. Sj, for all j C J.
| + | From this abstract a form of composition, temporarily notated as "P#Q", where P c XxM and Q c MxY are otherwise arbitrary dyadic relations, and where P#Q c XxY is defined as follows: |
| | | | |
| | + | : {P#Q}(x, y) = Conj(m C M) (( {P}(x, m) , {Q}(m, y) )). |
| | | | |
| | + | Compare this with the usual form of composition, typically notated as "P.Q" and defined as follows: |
| | | | |
| − | D6b. For all j C J, Sj.
| + | : {P.Q}(x, y) = Disj(m C M) ( {P}(x, m) . {Q}(m, y) ). |
| | | | |
| | + | ===1.4. Outlook of the Project : All Ways Lead to Inquiry=== |
| | | | |
| | + | I am using the word ''inquiry'' in a way that is roughly synonymous with the term ''scientific method''. Use of ''inquiry'' is more convenient, aside from being the shorter term, because of the following advantages: |
| | | | |
| − | D6c. Conj(j C J) Sj.
| + | # It allows one to broaden the scope of investigation to include any form of proceeding toward knowledge that merely aims at such a method. |
| | + | # It allows one to finesse the issue, for the time being, of how much "method" there is in science. |
| | | | |
| | + | This Subdivision and the next deal with opposite aspects of inquiry. In many ways it might have been better to interlace the opposing points of comparison, taking them up in a parallel fashion, but this plan was judged to be too distracting for a first approach. In other ways, the negative sides of each topic are prior in point of time to the positive sides of the issue, but sensible people like to see the light at the end of the tunnel before they trouble themselves with the obscurities of the intervening journey. Thus, this Subdivision of the text emphasizes the positive features of inquiry and the positive qualities of its objective, while the next Subdivision is reserved to examine the negative aspects of each question. |
| | | | |
| | + | In the order of nature, the absence of a feature naturally precedes the full development of its presence. In the order of discussion, however, positive terms must be proposed if it is desired to say anything at all. |
| | | | |
| − | D6d. ConjJ,j Sj.
| + | The discussion in this Subdivision is placed to serve a primer, declaring at least the names of enough positive concepts to propose addressing the negative conditions of knowledge in which inquiry necessarily starts. |
| | | | |
| | + | In this Subdivision I stand back once again from the problem of inquiry and allow myself take a more distant view of the subject, settling into what I think is a comfortable and a natural account of inquiry, the best that I have at my command, and attending to the task of describing its positive features in a positive light. I present my personal view of inquiry as I currently understand it, without stopping to justify every concept in detail or to examine every objection that might be made to this view. In the next Subdivision I discuss a few of the more obvious problems that stand in the way of this view and I try to remove a few of the more tractable obscurities that appear ready to be cleared up. The fact that I treat them as my "personal insights" does not mean that all of these ideas about inquiry originate with me, but only that I have come to adopt them for my personal use. There will be many occasions, the next time that I go over this ground, to point out the sources of these ideas, so far as I know them. |
| | | | |
| | + | The reader may take my apology for this style of presentation to be implicit in its dogmatic character. It is done this way in a first approach for the sake of avoiding an immense number of distractions, each of which is not being slighted but demands to be addressed in its own good time. I want to convey the general drift of my current model, however conjectural, naive, uncritical, and unreflective it may seem. |
| | | | |
| − | D6e. ConjJj Sj.
| + | ====1.4.1. The Matrix of Inquiry==== |
| | | | |
| | + | <blockquote> |
| | + | <p>Thus when mothers have children suffering from sleeplessness, and want to lull them to rest, the treatment they apply is to give them, not quiet, but motion, for they rock them constantly in their arms; and instead of silence, they use a kind of crooning noise; and thus they literally cast a spell upon the children (like the victims of a Bacchic frenzy) by employing the combined movements of dance and song as a remedy.</p> |
| | | | |
| | + | <p>(Plato, ''Laws'', VII, 790D).</p> |
| | + | </blockquote> |
| | | | |
| − | Definition 7
| + | Try as I might, I do not see a way to develop a theory of inquiry from nothing: To take for granted nothing more than is already given, to set out from nothing but absolutely certain beginnings, or to move forward with nothing but absolutely certain means of proceeding. In particular, the present inquiry into inquiry, <math>y_0 = y \cdot y,</math> ought not to be misconstrued as a device for magically generating a theory of inquiry from nothing. Like any other inquiry, it requires an agent to invest in a conjecture, to make a guess about the relevant features of the subject of interest, and to choose the actions, the aspects, and the attitudes with regard to the subject that are critical to achieving the objectives of the study. |
| | | | |
| − | If S, T are sentences
| + | I can sum all this up by saying that an inquiry requires an inquirer to suggest a hypothesis about the subject of interest and then to put that particular model of the subject to the test. This in turn requires one to devote a modicum of personal effort to the task of testing the chosen hypothesis, to put a quantum of personal interest at stake for the sake of finding out whether the model fits the subject, and, overall, to take the risk of being wrong. Any model that is feasible is also defeasible, at least, where it concerns a contingent subject of inquiry. |
| | | | |
| − | about things in the universe U,
| + | The first step, then, of an inquiry into inquiry, is to put forth a tentative model of inquiry, to make a hypothesis about the features of inquiry that are essential to explaining its experienced characteristics, and thus, in a sense, to make a guess at the very definition of inquiry. This requirement seems both obvious and outrageous at the same time. One is perfectly justified in objecting that there is much that precedes this so-called "first step", namely, the body of experience that prepares one to see it and the mass of observation that prompts one to take it. I can deal with this objection by making a distinction between mundane experience and olympian theory, and then by saying that the making of a conjecture is really the first "theoretical" step, but this is a hedge that covers the tracks of theory in a deceptive way, hiding how early in the empirical process the "cloven hoof" of theory actually enters. |
| | | | |
| | + | Leaving behind the mythical conditions of pure experience and naive observation, and at least by the time that one comes to give a name to the subject of investigation, one's trek through the data is already half-shod, half-fettered by the connotations of the name, and in turn by all of the concepts that it invokes in its train. The name, the concepts that it suggests, and the tacit but vague definition of the subject that this complex of associations is already beginning to constellate, attract certain experiences to the complex and filter out other observations from having any bearing on the subject matter. By this point, one is already busy translating one's empirical acquaintance with the subject into an arrangement of concepts that is intended to define its essential nature. |
| | | | |
| | + | An array of concepts that is set up to capture the essence of a subject is a provisional definition of it, an implicit model of the subject that contains the makings of an explicit theory. It amounts to a selection from the phenomenal aspects of the subject, expresses a guess about its relevant features, and constitutes a hypothesis in explanation of its experienced characteristics. This incipient order of model or theory is tantamount to a definition because it sets bounds on the "stretches" and the "holds" of a term — its extension, intension, and intention — but this is not the kind of definition that has to be taken on faith, or that constitutes the first and the last word on the subject. In other words, it is an empirical definition, one that is subject to being falsified in reference to its intended subject, by failing to indicate the necessary, the pertinent, or the relevant features that account for the presence of its phenomena or the persistence of its process. |
| | | | |
| − | then the following are equivalent:
| + | If I reflect on the conduct of inquiry, seeking to fix it in a fitting image and trying to cast it in a positive light, the best I can do is this: |
| | | | |
| | + | : Inquiry is a process that aims at achieving belief or knowledge. |
| | | | |
| | + | But even this simple a description already plunges the discussion deep into a number of obscurities. Most prominently, there is the disjunction between belief and knowledge that cries out to be explained or resolved. Stirring beneath the surface, and not quite fading into the background, many of the other terms that are invoked in the description are capable of hiding the entire contents of the original ignorance that the image as a whole is aimed to dispel. And yet, there is nothing that I can do in this avowedly positive context but to mark these points down as topics for future discussion. |
| | | | |
| − | D7a. S <=> T.
| + | There is already a model of inquiry that is implicit, at least partially, in the text of the above description. Let me see if I can tease out a few of its tacit assumptions. |
| | | | |
| | + | =====1.4.1.1. Inquiry as Conduct===== |
| | | | |
| | + | First of all, inquiry is conceived to be a form of conduct. This invokes the technical term ''conduct'', referring to the species of prototypically human action that is both dynamic and deliberate, or conceived to fall under a form of purposeful control, usually conscious but possibly not. For the sake of clarity, it helps to seek a more formal definition of conduct, one that expresses the concept in terms of abstract features rather than trying to suggest it by means of typical examples. |
| | | | |
| − | D7b. [S] = [T].
| + | Conduct is action with respect to an object. The distinction between action and conduct, reduced to the level of the most abstract formal relations that are involved, can be described in the following manner. |
| | | | |
| | + | Action is a matter of going from A to B, whereas conduct is matter of going from A to B in relation to C. In describing particular cases and types of conduct, the phrase "in relation to" can be filled out in more detail as "on account of", "in the cause of", "in order to bring about", "for the sake of", "in the interests of", or in many other ways. Thus, action by itself has a dyadic character, involving transitions through pairs of states, while conduct has a triadic character, involving the kinds of transactions between states that relate throughout to an object. |
| | | | |
| | + | With regard to this distinction, notice that "action" is used inclusively, to name the genus of which "conduct" names a species, and thus depicts whatever has the aspect of action, even if it is actually more complex. |
| | | | |
| − | Rule 5
| + | This creates the difficulty that the reputed "genus" is less than fully "generative", "generic", "genetic", or even "genuine" -- and so it is necessary to remain on guard against this source of misunderstanding. |
| | | | |
| − | If X, Y c U,
| + | What does this definition of conduct say about the temporal ordering of the object with respect to the states? The states are conceived to be ordered in time, but so far nothing has been said to pin down where in relation to these states the object must be conceived to fall in time. Nor does the definition make any particular specification necessary. This makes the question of relative time a secular parameter of the definition, allowing the consideration of the following options: |
| | | | |
| | + | # If the object is thought to precede the action of the conduct, then it tends to be regarded as a creative act, an initial intention, an original stimulus, a principal cause, or a prime mover. |
| | + | # If the object is thought to succeed the action of the conduct, then it tends to be regarded as an end, a goal, or a purpose, in other words, a state envisioned to be fulfilled. |
| | + | # If the object is thought to be concurrent, immanent, or transcendent throughout the action of the conduct, then it tends to be regarded as falling under one of the following possibilities: a prevailing value, a controlling parameter, a universal system of effective forces, a pervasive field of potentials, a ruling law, or a governing principle. |
| | | | |
| | + | A prevailing value or a controlling parameter, which guides the temporal development of a system, is a term that fits into a law or a principle, which governs the system at a higher level. The existence of a value or a law that rules a system, and the information that an agent of the system has about its parameters and its principles, are two different matters. Indeed, a major task of development for an inquiring agent is to inform itself about the values and the laws that form its own system. Thus, one of the objects of the conduct of inquiry is a description in terms of laws and values of the rules that govern and guide inquiry. |
| | | | |
| − | then the following are equivalent:
| + | The elaboration of an object in terms of this rich vocabulary — as a cause, end, field, force, goal, intention, law, parameter, principle, purpose, system, or value — adds colorful detail and concrete sensation to the account, and it helps to establish connections with the arrays of terminology that are widely used to discuss these issues. From a formal and relational point of view, however, all of these concepts are simply different ways of describing, at possibly different levels of generality, the object of a form of conduct. With that in mind, I find it useful to return to the simpler form of description as often as possible. |
| | | | |
| | + | This account of conduct brings to the fore a number of issues, some of them new and some of them familiar, but each of them allowing itself to be approached from a fresh direction by treating it as an implication of a critical thesis just laid down. I next examine these issues in accord with the tenets from which they stem. |
| | | | |
| | + | 1. Inquiry is a form of conduct. |
| | | | |
| − | R5a. X = Y. :D2a
| + | This makes inquiry into inquiry a special case of inquiry into conduct. |
| | | | |
| − | ::
| + | Certainly, it must be possible to reason about conduct in general, especially if forms of conduct need to be learned, examined, modified, and improved. |
| | | | |
| − | R5b. u C X <=> u C Y, for all u C U. :D2b
| + | Placing the subject of inquiry within the subject of conduct and making the inquiry into inquiry a subordinate part of the inquiry into conduct does not automatically further the investigation, especially if it turns out that the general subject of conduct is more difficult to understand than the specialized subject of inquiry. But in those realms of inquiry where it is feasible to proceed hypothetically and recursively, stretching the appropriate sort of hypothesis over a wider subject area can act to prime the pump of mathematical induction all the more generously, and actually increase the power of the recursion. Of course, the use of a recursive strategy comes at the expense of having to establish a more extended result at the base. |
| | | | |
| − | :D7a
| + | 2. The existence of an object that rules a form of conduct and the information that an agent of the conduct has about the object are two different matters. |
| | | | |
| − | ::
| + | This means that the exact specification of the object can demand an order of information that the agent does not have available, at least, not for use in reflective action, or even require an amount of information that the agent lacks the capacity to store. No matter how true it is that the actual course of the agent's conduct exactly reflects the influence of the object, and thus, in a sense, represents the object exactly, the question is whether the agent possesses the equivalent of this information in any kind of accessible, exploitable, reflective, surveyable, or usable form of representation, in effect, in any mode of information that the agent can use to forsee, to modify, or to temper its own temporal course. |
| | | | |
| − | R5c. [u C X] = [u C Y], for all u C U. :D7b
| + | This issue may seem familiar as a repetition of the "meta" question. |
| | | | |
| − | :???
| + | Once again, there is a distinction between (a) the properties of an action, agent, conduct, or system, as expressible by the agent that is engaged in the conduct, or as representable within the system that is undergoing the action, and (b) the properties of the same entities, as evident from an "external viewpoint", or as statable by the equivalent of an "outside observer". |
| | | | |
| − | ::
| + | 3. Reflection is a part of inquiry. Reflection is a form of conduct. |
| | | | |
| − | R5d. {< u, v> C UxB : v = [u C X]}
| + | The task of reflection on conduct is to pass from a purely interior view of one's own conduct to an outlook that is, effectively, an exterior view. |
| | | | |
| − | =
| + | What is sought is a wider perspective, one that is able to incorporate the sort of information that might be available to an outside observer, that ought to be evident from an external vantage point, or that one reasonably imagines might be obvious from an independent viewpoint. I am tempted to refer to such a view as a "quasi-objective perspective", but only so long as it possible to keep in mind that there is no such thing as a "completely outside perspective", at least, not one that a finite and mortal agent can hope to achieve, nor one that a reasonably socialized member of a community can wish to take up as a permanent station in life. |
| | | | |
| − | {< u, v> C UxB : v = [u C Y]}. :???
| + | With these qualifications, reflection is a form of conduct that can serve inquiry into conduct. Inquiry and its component reflection, applied to a form of conduct, are intended to provide information that can be used to develop the conduct in question. The "reflective development" that occurs depends on the nature of the case. It can be the continuation, the correction, or the complete cessation of the conduct in question. |
| | | | |
| − | :D5b
| + | If it is to have the properties that it is commonly thought to have, then reflection must be capable of running in parallel, and not interfering too severely, with the conduct on which it reflects. If this turns out to be an illusion of reflection that is not really possible in actuality, then reflection must be capable, at the very least, of reviewing the memory record of the conduct in question, in ways that appear concurrent with a replay of its action. But these are the abilities that reflection is "pre-reflectively" thought to have, that is, before the reflection on reflection can get under way. If reflection is truly a form of conduct, then it becomes conceivable as a project to reflect on reflection itself, and this reflection can even lead to the conclusion that reflection does not have all of the powers that it is commonly portrayed to have. |
| | | | |
| − | ::
| + | First of all, inquiry is conceived to be a form of conduct. This invokes the technical term "conduct", referring to the species of prototypically human action that is both dynamic and deliberate, or conceived to fall under a form of purposeful control, usually conscious but possibly not. For the sake of clarity, it helps to seek a more formal definition of conduct, one that expresses the concept in terms of abstract features rather than trying to suggest it by means of typical examples. |
| | | | |
| − | R5e. {X} = {Y}. :D5a
| + | Conduct is action with respect to an object. The distinction between action and conduct, reduced to the level of the most abstract formal relations that are involved, can be described in the following manner. Action is a matter of going from A to B, whereas conduct is matter of going from A to B in relation to C. In describing particular cases and types of conduct, the phrase "in relation to" can be filled out in more detail as "on account of", "in the cause of", "in order to bring about", "for the sake of", "in the interests of", or in many other ways. Thus, action by itself has a dyadic character, involving transitions through pairs of states, while conduct has a triadic character, involving the kinds of transactions between states that relate throughout to an object. |
| | | | |
| | + | With regard to this distinction, notice that "action" is used inclusively, to name the genus of which "conduct" names a species, and thus depicts whatever has the aspect of action, even if it is actually more complex. This creates the difficulty that the reputed "genus" is less than fully "generative", "generic", "genetic", or even "genuine" - and so it is necessary to remain on guard against this source of misunderstanding. |
| | | | |
| | + | What does this definition of conduct say about the temporal ordering of the object with respect to the states? The states are conceived to be ordered in time, but so far nothing has been said to pin down where in relation to these states the object must be conceived to fall in time. Nor does the definition make any particular specification necessary. This makes the question of relative time a secular parameter of the definition, allowing the consideration of the following options: |
| | | | |
| − | Rule 6
| + | # If the object is thought to precede the action of the conduct, then it tends to be regarded as a creative act, an initial intention, an original stimulus, a principal cause, or a prime mover. |
| | + | # If the object is thought to succeed the action of the conduct, then it tends to be regarded as an end, a goal, or a purpose, in other words, a state envisioned to be fulfilled. |
| | + | # If the object is thought to be concurrent, immanent, or transcendent throughout the action of the conduct, then it tends to be regarded as falling under one of the following possibilities: a prevailing value, a controlling parameter, a universal system of effective forces, a pervasive field of potentials, a ruling law, or a governing principle. |
| | | | |
| − | If f, g : U -> V,
| + | A prevailing value or a controlling parameter, which guides the temporal development of a system, is a term that fits into a law or a principle, which governs the system at a higher level. The existence of a value or a law that rules a system, and the information that an agent of the system has about its parameters and its principles, are two different matters. Indeed, a major task of development for an inquiring agent is to inform itself about the values and the laws that form its own system. Thus, one of the objects of the conduct of inquiry is a description in terms of laws and values of the rules that govern and guide inquiry. |
| | | | |
| | + | The elaboration of an object in terms of this rich vocabulary — as a cause, end, field, force, goal, intention, law, parameter, principle, purpose, system, or value — adds colorful detail and concrete sensation to the account, and it helps to establish connections with the arrays of terminology that are widely used to discuss these issues. From a formal and relational point of view, however, all of these concepts are simply different ways of describing, at possibly different levels of generality, the object of a form of conduct. With that in mind, I find it useful to return to the simpler form of description as often as possible. |
| | | | |
| | + | This account of conduct brings to the fore a number of issues, some of them new and some of them familiar, but each of them allowing itself to be approached from a fresh direction by treating it as an implication of a critical thesis just laid down. I next examine these issues in accord with the tenets from which they stem. |
| | | | |
| − | then the following are equivalent:
| + | 1. Inquiry is a form of conduct. |
| | | | |
| | + | This makes inquiry into inquiry a special case of inquiry into conduct. Certainly, it must be possible to reason about conduct in general, especially if forms of conduct need to be learned, examined, modified, and improved. |
| | | | |
| | + | Placing the subject of inquiry within the subject of conduct and making the inquiry into inquiry a subordinate part of the inquiry into conduct does not automatically further the investigation, especially if it turns out that the general subject of conduct is more difficult to understand than the specialized subject of inquiry. But in those realms of inquiry where it is feasible to proceed hypothetically and recursively, stretching the appropriate sort of hypothesis over a wider subject area can act to prime the pump of mathematical induction all the more generously, and actually increase the power of the recursion. Of course, the use of a recursive strategy comes at the expense of having to establish a more extended result at the base. |
| | | | |
| − | R6a. f = g. :D3a
| + | 2. The existence of an object that rules a form of conduct and the information that an agent of the conduct has about the object are two different matters. |
| | | | |
| − | ::
| + | This means that the exact specification of the object can require an order of information that the agent does not have available, at least, not for use in reflective action, or even an amount of information that the agent lacks the capacity to store. No matter how true it is that the actual course of the agent's conduct exactly reflects the influence of the object, and thus, in a sense, represents the object exactly, the question is whether the agent possesses the equivalent of this information in any kind of accessible, exploitable, reflective, surveyable, or usable form of representation, in effect, any mode of information that the agent can use to forsee, to modify, or to temper its own temporal course. |
| | | | |
| − | R6b. f(u) = g(u), for all u C U. :D3b
| + | This issue may seem familiar as a repetition of the "meta" question. Once again, there is a distinction between (a) the properties of an action, agent, conduct, or system, as expressible by the agent that is engaged in the conduct, or as representable within the system that is undergoing the action, and (b) the properties of the same entities, as evident from an "external viewpoint", or as statable by the equivalent of an "outside observer". |
| | | | |
| − | :D6a
| + | 3. Reflection is a part of inquiry. Reflection is a form of conduct. |
| | | | |
| − | ::
| + | The task of reflection on conduct is to pass from a purely interior view of one's own conduct to an outlook that is, effectively, an exterior view. What is sought is a wider perspective, one that is able to incorporate the sort of information that might be available to an outside observer, that ought to be evident from an external vantage point, or that one reasonably imagines might be obvious from an independent viewpoint. I am tempted to refer to such a view as a "quasi-objective perspective", but only so long as it possible to keep in mind that there is no such thing as a "completely outside perspective", at least, not one that a finite and mortal agent can hope to achieve, nor one that a reasonably socialized member of a community can wish to take up as a permanent station in life. |
| | | | |
| − | R6c. ConjUu (f(u) = g(u)). :D6e
| + | With these qualifications, reflection is a form of conduct that can serve inquiry into conduct. Inquiry and its component reflection, applied to a form of conduct, are intended to provide information that can be used to develop the conduct in question. The "reflective development" that occurs depends on the nature of the case. It can be the continuation, the correction, or the complete cessation of the conduct in question. |
| | | | |
| | + | If it is to have the properties that it is commonly thought to have, then reflection must be capable of running in parallel, and not interfering too severely, with the conduct on which it reflects. If this turns out to be an illusion of reflection that is not really possible in actuality, then reflection must be capable, at the very least, of reviewing the memory record of the conduct in question, in ways that appear concurrent with a replay of its action. But these are the abilities that reflection is "pre-reflectively" thought to have, that is, before the reflection on reflection can get under way. If reflection is truly a form of conduct, then it becomes conceivable as a project to reflect on reflection itself, and this reflection can even lead to the conclusion that reflection does not have all of the powers that it is commonly portrayed to have. |
| | | | |
| | + | =====1.4.1.2. Types of Conduct===== |
| | | | |
| − | Rule 7
| + | The chief distinction that applies to different forms of conduct is whether the object is the same sort of thing as the states or whether it is something entirely different, a thing apart, of a wholly other order. Although I am using different words for objects and states, it is always possible that these words are indicative of different roles in a formal relation and not indicative of substantially different types of things. If objects and states are but formal points and naturally belong to the same domain, then it is conceivable that a temporal sequence of states can include the object in its succession, in other words, that a path through a state space can reach or pass through an object of conduct. But if a form of conduct has an object that is completely different from any one of its temporal states, then the role of the object in regard to the action cannot be like the end or goal of a temporal development. |
| | | | |
| − | If P, Q : U -> B,
| + | What names can be given to these two orders of conduct? |
| | | | |
| | + | =====1.4.1.3. Perils of Inquiry===== |
| | | | |
| | + | Now suppose that making a hypothesis is a kind of action, no matter how covert, or that testing a hypothesis takes an action that is more overt. If entertaining a hypothesis in any serious way requires action, and if action is capable of altering the situation in which it acts, then what prevents this action from interfering with the subject of inquiry in a way that undermines, with positive or negative intentions, the very aim of inquiry, namely, to understand the situation as it is in itself? |
| | | | |
| − | then the following are equivalent:
| + | That making a hypothesis is a type of action may seem like a hypothesis that is too far-fetched, but it appears to follow without exception from thinking that thinking is a form of conduct, in other words, an activity with a purpose or an action that wants an end. The justification of a hypothesis is not to be found in a rational pedigree, by searching back through a deductive genealogy, or determined by that which precedes it in the logical order, since a perfectly trivial tautology caps them all. Since a logical tautology, that conveys no empirical information, finds every proposition appearing to implicate it, in other words, since it is an ultimate implication of every proposition and a conceivable conclusion that is implicit in every piece of reasoning, it is obvious that seeking logical precedents is the wrong way to go for empirical content. |
| | | | |
| | + | In making a hypothesis or choosing a model, one appears to select from a vaster number of conceivable possibilities than a finite agent could ever enumerate in complete detail or consider as an articulate totality. As the very nature of a contingent description and the very character of a discriminate action is to apply in some cases but not in others, there is no escaping the making of a risky hypothesis or a speculative interpretation, even in the realm of a purely mental action. Thus, all significant thought, even thinking to any purpose about thought itself, demands a guess at the subject or a grasp of the situation that is contingent, dubious, fallible, and uncertain. |
| | | | |
| | + | If all this is true — if inquiry begins with doubt, if every significant hypothesis is itself a dubious proposition, if the making and the testing of a hypothesis are instances of equally doubtful actions, and if every action has the potential to alter the very situation and the very subject matter that are being addressed — then it leads to the critical question: How is the conduct of inquiry, that begins by making a hypothesis and that continues by testing this description in action, supposed to help with the situation of uncertainty that incites it in the first place and that is supposed to maintain its motivation until the end is reached? The danger is that the posing of a hypothesis may literally introduce an irreversible change in the situation or the subject matter in question. The fear is that this change might be one that too conveniently fulfills or too perversely subverts the very hypothesis that engenders it, that it may obstruct the hypothesis from ever being viewed with equanimity again, and thus prevent the order of reflection that is needed to amend or discard the hypothesis when the occasion to do so arises. |
| | | | |
| − | R7a. P = Q. :R6a
| + | If one fears that merely contemplating a special hypothesis is enough to admit a spurious demonstration into the foundations of one's reasoning, even to allow a specious demon to subvert all one's hopes of a future rationality and to destroy all one's chances of a reasonable share of knowledge, then one is hardly in a state of mind that can tolerate the tensions of a full-fledged, genuine inquiry. If one is beset with such radical doubts, then all inquiry is no more comfort than pure enchoiry. Sometimes it seems like the best you can do is sing yourself a song that soothes your doubts. Perhaps it is even quite literally true that all inquiry comes back at last to a form of "enchoiry", the invocation of a nomos, a way of life, or a song and a dance. But even if this is the ultimate case, it does no harm and it does not seem like a bad idea to store up in this song one or two bits of useful lore, and to weave into its lyric a few suggestions of a practical character. |
| | | | |
| − | ::
| + | Let us now put aside these more radical doubts. This putting aside of doubts is itself a form of inquiry, that is, a way of allaying doubts. The fact that I appear to do this by fiat, and to beg for tacit assent, tends to make me suspect the validity of this particular tactic. Still, it is not too inanely dismissive, as its appeal is based on an argument, the argument that continuing to entertain this type of doubt leads to a paralysis of the reason, and that paralyzing the ability to think is not in the interests of the agent concerned. Thus, I adopt the hypothesis that the relationship between the world and the mind is not so perverse that merely making a hypothesis is enough to alter the nature of either. If, in future, I or anyone sees the need to reconsider this hypothesis, then I see nothing about making it that prevents anyone from doing so. Indeed, making it explicit only renders it more subject to reflection. |
| | | | |
| − | R7b. P(u) = Q(u), for all u C U. :R6b
| + | Of course, a finite person can only take up so many causes in a single lifetime, and so there is always the excuse of time for not chasing down every conceivable hypothesis that comes to mind. |
| | | | |
| − | ::
| + | =====1.4.1.4. Forms of Relations===== |
| | | | |
| − | R7c. ConjUu (P(u) = Q(u)). :R6c
| + | The next distinguishing trait that I can draw out of this incipient treatise is its emphasis on the forms of relations. From a sufficiently formal and relational point of view, many of the complexities that arise from throwing intentions, objectives, and purposes into the mix of discussion are conceivably due to the greater arity of triadic relations over dyadic relations, and do not necessarily implicate any differences of essence inhering in the entities and the states invoked. As far as this question goes, whether a dynamic object is essentially different from a deliberate object, I intend to remain as neutral as possible, at least, until forced by some good reason to do otherwise. In the meantime, the factors that are traceable to formal differences among relations are ready to be investigated and useful to examine. With this in mind, it it useful to make the following definition: |
| | | | |
| − | :P1a
| + | A ''conduct relation'' is a triadic relation involving a domain of objects and two domains of states. When a shorter term is desired, I refer to a conduct relation as a ''conduit''. A conduit is given in terms of its extension as a subset C c XxYxZ, where X is the ''object domain'' and where Y and Z are the ''state domains''. Typically, Y = Z. |
| | | | |
| − | ::
| + | In general, a conduct relation serves as a ''model of conduct'' (MOC), not always the kind of model that is meant to be emulated, but the type of model that captures an aspect of structure in a form of conduct. |
| | | | |
| − | R7d. ConjUu (P(u) <=> Q(u)). :P1b
| + | The question arises: What is the relationship between signs and states? On the assumption that signs and states are comparable in their levels of generality, consider the following possibilities: |
| | | | |
| − | ::
| + | # Signs are special cases of states. |
| | + | # Signs and states are the same sorts of things. |
| | + | # States are special cases of signs. |
| | | | |
| − | R7e. ConjUu (( P(u) , Q(u) )). :P1c
| + | Depending on how one answers this question, one is also choosing among the following options: |
| | | | |
| − | :$1a
| + | # Sign relations are special cases of conduct relations. |
| | + | # Sign relations and conduct relations are the same sorts of things. |
| | + | # Conduct relations are special cases of sign relations. |
| | | | |
| − | ::
| + | I doubt if there is any hard and fast answer to this question, but think that it depends on particular interpreters and particular observers, to what extent each one interprets a state as a sign, and to what degree each one recognizes a sign as a component of a state. |
| | | | |
| − | R7f. ConjUu (( P , Q ))$(u). :$1b
| + | =====1.4.1.5. Models of Inquiry===== |
| | | | |
| | + | The value of a hypothesis, or the worth of a model, is not to be given a prior justification, as by a deductive proof, but has to be examined in practice, as by an empirical probation. It is not intended to be taken for granted or to go untested, but its meaning in practice has to be articulated before its usefulness can be judged. This means that the conceivable practical import of the hypothesis or the model has to be developed in terms of its predicted and its promised consequences, after which it is judged by the comparison of these speculative consequences with the actual results. But this is not the end of the matter, for it can be a useful piece of information to discover that a particular kind of conception fails a particular kind of comparison. Thus, the final justification for a hypothesis or a model is contained in the order of work that it leads one to do, and the value of this work is often the same whether or not its premiss is true. Indeed, the fruitfulness of a suggestion can lie in the work that proves it untrue. |
| | | | |
| | + | My plan then has to be, rather than trying to derive a model of inquiry in a deductive fashion from a number of conditions like <math>y_0 = y \cdot y,</math> only to propose a plausible model, and then to test it under such conditions. Each of these tests is a two-edged sword, and the result of applying a particular test to a proposed model can have either one of two effects. If one believes that a particular test is a hard and fast rule of inquiry, or a condition that any inquiry is required to satisfy, then the failure of a model to live up to its standard tends only to rule out that model. If one has reason to believe that a particular model of inquiry covers a significant number of genuine examples, then the failure of these models to follow the prescribed rule can reflect badly on the test itself. |
| | | | |
| | + | In order to prime the pump, therefore, let me offer the following account of inquiry in general, the whole of which can be taken as a plausible hypothesis about the nature of inquiry in general. |
| | | | |
| | + | My observations of inquiry in general, together with a few suggestions that seem apt to me, have led me to believe that inquiry begins with a "surprise" or a "problem". The way I understand these words, they refer to departures, differences, or discrepancies among various modalities of experience, in particular, among "observations", "expectations", and "intentions". |
| | | | |
| − | Rule 8
| + | # A ''surprise'' is a departure of an observation from an expectation, and thus it invokes a comparison between present experience and past experience, since expectations are based on the remembered disposition of past experience. |
| | + | # A ''problem'' is a departure of an observation from an intention, and thus it invokes a comparison between present experience and future experience, since intentions choose from the envisioned disposition of future experience. |
| | | | |
| − | If S, T are sentences
| + | With respect to these |
| | | | |
| − | about things in the universe U,
| + | With respect to this hypothetical |
| | | | |
| | + | I now test this model of inquiry under the conditions of an inquiry into inquiry, asking whether it is consistent in its application to itself. This leaves others to test the models they like best under the same conditions, should they ever see the need to do so. |
| | | | |
| | + | Does the inquiry into inquiry begin with a surprise or a problem concerning the process or the conduct of inquiry? In other words, does the inquiry into inquiry start with one of the following forms of departure: (1) a surprising difference between what is expected of inquiry and what is observed about it, or (2) a problematic difference between what is observed about inquiry and what is intended for it? |
| | | | |
| − | then the following are equivalent:
| + | ====1.4.2. The Moment of Inquiry==== |
| | | | |
| | + | <blockquote> |
| | + | <p>Every young man — not to speak of old men — on hearing or seeing anything unusual and strange, is likely to avoid jumping to a hasty and impulsive solution of his doubts about it, and to stand still; just as a man who has come to a crossroads and is not quite sure of his way, if he be travelling alone, will question himself, or if travelling with others, will question them too about the matter in doubt, and refuse to proceed until he has made sure by investigation of the direction of his path.</p> |
| | | | |
| | + | <p>(Plato, ''Laws'', VII, 799C).</p> |
| | + | </blockquote> |
| | | | |
| − | R8a. S <=> T. :D7a
| + | Observe the paradox of this precise ambiguity: That both the occasion and the impulse of inquiry are instances of a negative moment. But the immediate discussion is aimed at the positive aspects of inquiry, and so I convert this issue into its corresponding positive form. |
| | | | |
| − | ::
| + | The positive aim of inquiry is a state of belief, certainty, or knowledge. There are distinctions that can be made in the use of these words, but the question remains as to what kind of distinctions these are. In my opinion, the differences that arise in practice have more to do with the purely grammatical distinctions of "case", "mood", "number", "person", and "voice", and thus raise the issues of plurality and point of view, as opposed to indicating substantial differences in the relevant features of state, as actually experienced by the agent concerned. |
| | | | |
| − | R8b. [S] = [T]. :D7b
| + | It is often claimed that there are signficant differences between the conditions of belief and knowledge, but the way that I understand the distinction is as follows. One says that a person "knows" something when that person believes exactly the same thing that one believes. When one is none other than the person in question, then one says that one "knows" exactly what one believes. Differences arise between the invocations of "belief" and "knowledge" only when more than one person is involved in the issue. Thus, there is no occasion for a difference between belief and knowledge unless there is more than one person that is being consulted about the matter in question, or else a single person in a divided state of opinion, in any case, when there is more than one impulse, moment, or occasion that currently falls under consideration. |
| | | | |
| − | :R7a
| + | In any case, belief or knowledge is the feature of state that an agent of inquiry lacks at the moment of setting out. Inquiry begins in a state of impoverishment, need, or privation, a state that is absent the quality of certainty. It is due to this feature that the agent is motivated, and it is on account of its continuing absence that the agent keeps on striving to achieve it, at least, with respect to the subject in question, and, at any rate, in sufficient measure to make action possible. |
| | | | |
| − | ::
| + | ====1.4.3. The Modes of Inquiry==== |
| | | | |
| − | R8c. [S](u) = [T](u), for all u C U. :R7b
| + | <blockquote> |
| | + | <p>Let the strange fact be granted, we say, that our hymns are now made into "nomes" (laws), just as the men of old, it would seem, gave this name to harp-tunes, — so that they, too, perhaps, would not wholly disagree with our present suggestion, but one of them may have divined it vaguely, as in a dream by night or a waking vision: anyhow, let this be the decree on the matter: — In violation of public tunes and sacred songs and the whole choristry of the young, just as in violation of any other "nome" (law), no person shall utter a note or move a limb in the dance.</p> |
| | | | |
| − | ::
| + | <p>(Plato, ''Laws'', VII, 799E–800A).</p> |
| | + | </blockquote> |
| | | | |
| − | R8d. ConjUu ( [S](u) = [T](u) ). :R7c
| + | In the present section, I am concerned with the kinds of reasoning that might be involved in the choice of a method, that is, in discovering a way to go about inquiry, in constructing a way to carry it through, and in justifying the way that one chooses. If the choice of a method can be established on the basis of reasoning, if it can be rationalized or reconstructed on grounds that are commonly thought to be sensible, or if it is likely to be affected or influenced in any way by a rational argument, then there is reason to examine the kinds of reasoning that go into this choice. All of this requires a minimal discussion of different modes of reasoning. |
| | | | |
| − | ::
| + | In this work as a whole, each instance of inquiry is analyzed in accord with various modes of reasoning, the prospective "elements of inquiry", and its structure as an object of inquiry is articulated, rationalized, and reconstructed with respect to the corresponding "form of analysis", "form of synthesis", or "objective genre" (OG). |
| | | | |
| − | R8e. ConjUu ( [S](u) <=> [T](u) ). :R7d
| + | According to my current understanding, the elements of inquiry can be found to rest on three types of steps, called "abductive", "deductive", and "inductive" modes of inference. As a result of this opinion, I do not believe that I can do any better at present than to articulate the structure of each instance of learning or reasoning according to these three types of motions of the mind. But since this work as a whole is nowhere near complete, I cannot dictate these steps in a dogmatic style, nor will it do for me to to call the tune of this form of analysis in a purely ritual or a wholly routine fashion. |
| | | | |
| − | ::
| + | Since the complexity of reasoning about different modes of reasoning is enough of a complication to occupy my attention at the present stage of development in this work, it is proably best to restrain this discussion along the majority of its other dimensions. A convenient way to do this is to limit its scope to simple examples and concrete situations, just enough to illustrate the selected modes of reasoning. |
| | | | |
| − | R8f. ConjUu (( [S](u) , [T](u) )). :R7e
| + | With all of these considerations in mind, the best plan that I can find for addressing the tasks of the present section is to proceed as follows: I make it my primary aim to examine only a few of the simplest settings in which these different modes of reasoning are able to appear, and I try to plot my path through this domain by way of concrete examples. Along the way, I discuss a few of the problems that are associated with reasoning about different modes of reasoning. Given the present stage of development, the majority of these issues have to be put aside almost as quickly as they are taken up. If they are ever going to be subject to resolution, it is not within reach of the present moment of discussion. In the body of this section, I therefore return to the initial strategy: to examine a few of the simplest cases and situations that can serve to illustrate the distinctions among the chosen modes of reasoning. |
| | | | |
| − | ::
| + | In trying to initiate a general discussion of the different modes of reasoning that might be available, and thus to motivate a model of this subject matter that makes an initial kind of sense to me, I meet once again with all the old "difficulties at the beginning", the kinds of obstructions that always seem to arise on trying to open up any new subject for discussion or in trying to introduce any new model of an old subject area. Much of this gratuitous bedevilment is probably due to the inherent conservatism of the human mind. Everything familiar is taken for granted, but each new picture of the situation is immediately subjected to the severest suspicions. |
| | | | |
| − | R8g. ConjUu (( [S] , [T] ))$(u). :R7f
| + | Now, I cannot reason with necessary force that the mind must use these particular modes of reasoning, any more than I can say that it must use a given language in order to express itself. But I can argue, relative to a particular model of thinking that must be proposed hypothetically, that certain modes of reasoning are available to the mind and are likely to be evident in its operation, if one only takes the trouble to look. |
| | | | |
| − | For instance, the observation that expresses the equality of sets in terms of their indicator functions can be formalized according to the pattern in Rule 9, namely, at lines (a, b, c), and these components of Rule 9 can be cited in future uses as "R9a", "R9b", "R9c", respectively. Using Rule 7, annotated as "R7", to adduce a few properties of indicator functions to the account, it is possible to extend Rule 9 by another few steps, referenced as "R9d", "R9e", "R9f", "R9g".
| + | Ultimately, the model of thinking that I plan to propose makes use of the proposition that all thinking takes place in signs, and thus that inquiry is the transformation of a sign relation. Relative to this hypothesis, it would be possible to discharge the current assumptions about the basic modes of reasoning, that is, to derive the elementary modes of inquiry from a sign relational model of inquiry, and then to compare them with the current suggestions. Until this work is done, however, the assumption that these really are the most basic modes of reasoning has to be treated as a still more tentative hypothesis. |
| | | | |
| − | Rule 9
| + | When a subject matter is so familiar that the logical connections between its parts are known both forwards and backwards, then it is reasonable and convenient to organize its presentation in an axiomatic fashion. This would not be such a bad idea, if it did not make it so easy to forget the nature of the reorganization that goes into a representation, and it would not constitute such a deceptive conception of the subject, if it did not mean that the exposition of the subject matter is just as often the falsification of its actual development and the covering up of its real excavation. Indeed, the logical order of axioms and theorems may have little to do with the original order of discovery and invention. In practice, the deepest axioms are often the last to come to light. |
| | | | |
| − | If X, Y c U,
| + | Once again, the structure of a reflective context means that each mode of reasoning is able to appear in a double role, once as an object and once as an instrument of the same extended discussion. And once again, the discussion runs into an array of obstructions, whose structures are becoming, if not more clear, at least, more familiar with each encounter. In particular, a description of different modes of reasoning involves a classification, and a classification presupposes a basis of distinctive features that cannot be treated as categorical, or objectively neutral, but has to be regarded as hypothetical, or potentially biased. In other words, the language that I use to describe different modes of reasoning may already have a particular model of reasoning built into it, and this disposition to a particular conception of logic may be lodged in such a way that it makes it nearly impossible to reflect on the operations and the limitations of this model. |
| | | | |
| | + | Inquiry begins when a law is violated. It marks a time when a certain peace of mind is breached, it reigns all the while that a common accord is broken, disturbed, forgotten, or lost, and it rules right up until the time when a former condition of harmony is restored or until the moment when a new state of accord is established. Of course, the word "law" is a highly equivocal choice, especially to convey the sense of a founding principle. It renders not just its own meaning irrevocably subject to interpretation, but delivers into a similar subjection all the forms of understanding that depend on it. But the letter must release its hold on the spirit, if the word "law" is meant to evoke the requisite variety of connotations, and yet to maintain a sensible degree of order among their concrete meanings. Only in this way can it rise above the many different kinds of law that come into play. |
| | | | |
| | + | There are descriptive laws, that organize experiences into expectations. There are prescriptive laws, that organize performances into intentions. |
| | | | |
| − | then the following are equivalent:
| + | Other names for descriptive laws are "declarative" or "empirical" laws. Other names for prescriptive laws are "procedural" or "normative" laws. |
| | | | |
| | + | Implicit in a descriptive law is the connection to be found or made, discovered or created, between past experience and present expectation. What one knows about these connections is kept in a descriptive model. |
| | | | |
| | + | Implicit in a prescriptive law is the connection to be found or made, discovered or created, between current conduct and future experience. What one knows about these connections is kept in a prescriptive model. |
| | | | |
| − | R9a. X = Y. :R5a
| + | A violation of an expectation, the contravention of a descriptive law, occurs when a present experience departs from a predicted experience, which is what a past expectation or description projected to be present. This is a "surprise", a state of affairs that calls for an explanation. An explanation points to other descriptions that better predict the actual experience, and suggests an alteration to the descriptive model that generated the expectation from a past experience. |
| | | | |
| − | ::
| + | A violation of an intention, the contravention of a prescriptive law, occurs when a present experience departs from a desired experience, which is what a past intention or prescription projected to be present. This is a "problem", a state of affairs that calls for a plan of action. , A plan of action points to other actions that better achieve the desired experience, and suggests an alteration to the prescriptive model that generated the conduct toward a prospective experience. |
| | | | |
| − | R9b. {X} = {Y}. :R5e
| + | In the rest of this section, I treat the different modes of reasoning according to the forms that Aristotle gave them, collectively referred to as the "syllogistic" model. The discussion is kept within the bounds of propositional reasoning by considering only those "figures of syllogism" that are "purely universal", that is, the forms of argument all of whose premisses, and therefore all of whose conclusions, involve nothing but universal quantifications. |
| | | | |
| − | :R7a
| + | If it were only a matter of doing propositional reasoning as efficiently as possible, I would simply use the cactus language and be done with it, but there are several other reasons for revisiting the syllogistic model. Treating the discipline that is commonly called "logic" as a cultural subject with a rich and varied history of development, and attending to the thread of tradition in which I currently find myself, I observe what looks like a critical transition that occurs between the classical and the modern ages. Aside from supplying the barest essentials of a historical approach to the subject, a consideration of this elder standard makes it easier to appreciate the nature and the character of this transformation. In addition, and surprisingly enough to warrant further attention, there appear to be a number of cryptic relationships that exist between the syllogistic patterns of reasoning and the ostensibly more advanced forms of analysis and synthesis that are involved in the logic of relations. |
| | | | |
| − | ::
| + | =====1.4.3.1. Deductive Reasoning===== |
| | | | |
| − | R9c. {X}(u) = {Y}(u), for all u C U. :R7b
| + | In this subsection, I present a trimmed-down version of deductive reasoning in Aristotle, limiting the account to universal syllogisms, in effect, keeping to the level of propositional reasoning. Within these constraints, there are three basic "figures" of the syllogism. |
| | | | |
| − | ::
| + | In order to understand Aristotle's description of these figures, it is necessary to explain a few items of his technical terminology. In each figure of the syllogism, there are three "terms". Each term can be read as denoting either (1) a class of entities or (2) all of the members of a class of entities, depending on which interpretation the reader prefers. These terms are ranked in two ways: With respect to the "magnitudes" that they have in relation to each other, there are "major", "middle", and "minor" terms. With respect to the "positions" that they take up within the figure, there are "first", "intermediate", and "last" terms. The figures are distinguished by how the magnitudes correlate with the positions. However, the names for these rankings are not always used or translated in a rigorously systematic manner, so the reader has to be on guard to guess which type of ranking is meant. |
| | | | |
| − | R9d. ConjUu ( {X}(u) = {Y}(u) ). :R7c
| + | In addition to this terminology, it is convenient to make use of the following nomenclature: |
| | | | |
| − | ::
| + | # The ''Fact'' is the proposition that applies the term in the first position to the term in the third or last position. |
| | + | # The ''Case'' is the proposition that applies the term in the second or intermediate position to the term in the third or last position. |
| | + | # The ''Rule'' is the proposition that applies the term in the first position to the term in the second or intermediate position. |
| | | | |
| − | R9e. ConjUu ( {X}(u) <=> {Y}(u) ). :R7d
| + | Because the roles of Fact, Case, and Rule are defined with regard to positions rather than magnitudes they are insensitive to whether the proposition in question is being used as a premiss or is being drawn as a conclusion. |
| | | | |
| − | ::
| + | The ''first figure'' of the syllogism is explained as follows: |
| | | | |
| − | R9f. ConjUu (( {X}(u) , {Y}(u) )). :R7e
| + | <blockquote> |
| | + | <p>When three terms are so related to one another that the last is wholly contained in the middle and the middle is wholly contained in or excluded from the first, the extremes must admit of perfect syllogism. By "middle term" I mean that which both is contained in another and contains another in itself, and which is the middle by its position also; and by "extremes" (a) that which is contained in another, and (b) that in which another is contained. For if A is predicated of all B, and B of all C, A must necessarily be predicated of all C. ... I call this kind of figure the First.</p> |
| | | | |
| − | ::
| + | <p>(Aristotle, ''Prior Analytics'', 1.4).</p> |
| | + | </blockquote> |
| | | | |
| − | R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
| + | For example, suppose A is "animal", B is "bird", and C is "canary". Then there is a deductive conclusion to be drawn in the first figure. |
| | | | |
| | + | There is the Case: |
| | | | |
| | + | : "All canaries are birds." (C => B) |
| | | | |
| − | Rule 10 | + | There is the Rule: |
| | | | |
| − | If X, Y c U,
| + | : "All birds are animals." (B => A) |
| | | | |
| | + | One deduces the Fact: |
| | | | |
| | + | : "All canaries are animals." (C => A) |
| | | | |
| − | then the following are equivalent:
| + | The propositional content of this deduction is summarized on the right. Taken at this level of detail, deductive reasoning is nothing more than an application of the transitive rule for logical implications. |
| | | | |
| | + | The ''second figure'' of the syllogism is explained as follows: |
| | | | |
| | + | <blockquote> |
| | + | <p>When the same term applies to all of one subject and to none of the other, or to all or none of both, I call this kind of figure the Second; and in it by the middle term I mean that which is predicated of both subjects; by the extreme terms, the subjects of which the middle is predicated; by the major term, that which comes next to the middle; and by the minor that which is more distant from it. The middle is placed outside the extreme terms, and is first by position.</p> |
| | | | |
| − | R10a. X = Y. :D2a
| + | <p>(Aristotle, ''Prior Analytics'', 1.5).</p> |
| | + | </blockquote> |
| | | | |
| − | ::
| + | For example, suppose M is "mammal", N is "newt", and O is "opossum". Then there is a deductive conclusion to be drawn in the second figure. |
| | | | |
| − | R10b. u C X <=> u C Y, for all u C U. :D2b
| + | There is the Fact: |
| | | | |
| − | :R8a
| + | : "All opossums are mammals." (O => M) |
| | | | |
| − | ::
| + | There is the Rule: |
| | | | |
| − | R10c. [u C X] = [u C Y]. :R8b
| + | : "No newts are mammals." (N.M = 0) |
| | | | |
| − | ::
| + | One deduces the Case: |
| | | | |
| − | R10d. For all u C U,
| + | : "No newts are opossums." (N.O = 0) |
| | | | |
| − | [u C X](u) = [u C Y](u). :R8c
| + | The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if O c M and if N intersects M trivially, then N must also intersect O trivially. Here, I use a raised dot "." to indicate either the conjunction of two propositions or the intersection of two classes, and I use a zero "0" to indicate either the identically false proposition or the empty class, leaving the choice of interpretation to the option of the reader. |
| | | | |
| − | ::
| + | The ''third figure'' of the syllogism is explained as follows: |
| | | | |
| − | R10e. ConjUu ( [u C X](u) = [u C Y](u) ). :R8d
| + | <blockquote> |
| | + | <p>If one of the terms applies to all and the other to none of the same subject, or if both terms apply to all or none of it, I call this kind of figure the Third; and in it by the middle I mean that of which both the predications are made; by extremes the predicates; by the major term that which is [further from] the middle; and by the minor that which is nearer to it. The middle is placed outside the extremes, and is last by position.</p> |
| | | | |
| − | ::
| + | <p>(Aristotle, ''Prior Analytics'', 1.6).</p> |
| | + | </blockquote> |
| | | | |
| − | R10f. ConjUu ( [u C X](u) <=> [u C Y](u) ). :R8e
| + | It appears that this passage is only meant to mark out the limiting cases of the type. From the examples that Aristotle gives it is clear that he includes many other kinds of logical situation under this figure. Perhaps the phrase "applies to all or none" is intended to specify that a term applies "affirmatively or negatively" to another term, but is not meant to require that it applies universally so. |
| | | | |
| − | ::
| + | For example, suppose P is "poem", R is "rhapsody", and S is "sonnet". Then there is deductive conclusion to be drawn in the third figure: |
| | | | |
| − | R10g. ConjUu (( [u C X](u) , [u C Y](u) )). :R8f
| + | There is the Fact: |
| | | | |
| − | ::
| + | : "All sonnets are poems." (S => P) |
| | | | |
| − | R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
| + | There is the Case: |
| | | | |
| | + | : "Some sonnets are rhapsodies." (S.R > 0) |
| | | | |
| | + | One deduces the Rule: |
| | | | |
| − | Rule 11
| + | : "Some rhapsodies are poems." (R.P > 0) |
| | | | |
| − | If X c U
| + | The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if S c P and if R intersects S non-trivially then R must intersect P non-trivially. |
| | | | |
| | + | =====1.4.3.2. Inductive Reasoning===== |
| | | | |
| | + | (Aristotle, ''Prior Analytics'', 2.23). |
| | | | |
| − | then the following are equivalent:
| + | =====1.4.3.3. Abductive Reasoning===== |
| | | | |
| | + | A choice of method cannot be justified by deduction or by induction, at least, not wholly, but involves an element of hypothesis. In ancient times, this mode of inference to an explanatory hypothesis was described by the Greek word "apagoge", articulating an action or a process that "carries", "drives", or "leads" in a direction "away", "from", or "off". This was later translated into the Latin "abductio", and that is the source of what is today called "abduction" or "abductive reasoning". Another residue of this sense survives today in the terminology for "abductor muscles", those that "draw away (say, a limb or an eye) from a position near or parallel to the median axis of the body" (Webster's). |
| | | | |
| | + | If an image is needed, one may think of Prometheus, arrogating for the sake of an earthly purpose the divine prerogative of the gods, and then drawing the fire of their heavenly ire for the presumption of this act. This seems to sum up pretty well, not only the necessity and the utility of hypotheses, but also the risks that one incurs in making conjectures. In other guises, abductive reasoning is the mode of inference that is used to diagnose a complex situation, one that originally presents itself under a bewildering array of signs and symptoms, and fixes it subject to the terms of a succinct "nomen" or a summary predicate. Finally, by way of offering a personal speculation, I think it is likely that this entire trio of terms, "abduction", "deduction", and "induction", have reference to a style of geometric diagrams that the Ancients originally used to illustrate their reasonings. |
| | | | |
| − | R11a. X = {u C U : S}. :R5a
| + | Abductive reasoning has also been called by other names. C.S. Peirce at times called it "presumption", perhaps because it puts a plausible assumption logically prior to the observed facts, and at other times referred to it as "retroduction", because it reasons backwards from the consequent to the antecedent of a logical implication. |
| | | | |
| − | ::
| + | In its simplest form, abductive reasoning proceeds from a "fact" that A is true, using a "rule" that B => A, to presume a "case" that B is true. Thus, if A is a surprising fact that one happens to observe, and B => A is a rule to the effect that if B is true then A necessarily follows, then guessing the case that B is true is an instance of abductive reasoning. This is a backward form of reasoning, and therefore extremely fallible, but when it works it has the effect of reducing the amount of surprise in the initial observation, and thus of partially explaining the fact. |
| | | | |
| − | R11b. {X} = { {u C U : S} }. :R5e
| + | In a slightly more complicated version, abduction proceeds from a fact that C => A, using a rule that B => A, to presume a case that C => B. This is an inessential complication, since the rule of modus ponens and the rule of transitivity are essentially equivalent in their logical force, but it is often convenient to imagine that C is the "common subject" or the "current situation" that is implicit throughout the argument, namely, the existing entity that substantiates or instantiates all of the other predicates that are invoked in its course. |
| | | | |
| − | ::
| + | Suppose I have occasion to reason as follows: |
| | | | |
| − | R11c. {X} c UxB
| + | : "It looks like a duck, so I guess it is a duck." |
| | | | |
| | + | Or even more simply: |
| | | | |
| | + | : "It looks blue, therefore it is blue." |
| | | | |
| − | : {X} = {< u, v> C UxB : v = [S](u)}. :R
| + | These are instances in which I am using abductive reasoning, according to the pattern of the following schema: |
| | | | |
| − | ::
| + | I observe a Fact: |
| | | | |
| − | R11d. {X} : U -> B
| + | : "It looks like X." (X') |
| | | | |
| | + | I have in the back of my mind a general Rule: |
| | | | |
| | + | : "If it is X, then it looks like X." (X => X') |
| | | | |
| − | : {X}(u) = [S](u), for all u C U. :R | + | I reason my way back from the observed Fact and the assumed Rule to assert what I guess to be the Case: |
| | | | |
| − | ::
| + | : "It is X." (X) |
| | | | |
| − | R11e. {X} = [S]. :R
| + | The abduction is a hypothetical inference that results in a diagnostic conclusion, that is, a statement of opinion as to what is conjectured to be the case. In each case the operation of abductive reasoning starts from a complex configuration, involving a number of explicit observations in the foreground and a class of implicit assumptions in the background, and it offers a provisional statement about certain possibility, one that is typically less conspicuous, obvious, or prominent, but still potentially present in the situation, and hopefully serving to explain the surprising or the problematic aspects of the whole state of affairs. |
| | | | |
| | + | What results from the abductive inference is a concept and possibly a term, for instance, "duck" or "blue". The concept attempts to grasp a vast complex of appearances within a unitary form, and the term that connotes the concept is used to put explicit bounds on what it conveys. Working in tandem, they express an approximation or a simplification, "a reduction of the manifold of phenomena to a unified conception". Finite minds cannot operate for very long with anything more than this. |
| | | | |
| | + | The reader may have noticed some obvious distinctions between the two examples of abductive reasoning that I gave above, between the case of "looking like a duck" and the case of "looking blue". Just to mention the most glaring difference: Although a person is occasionally heard to reason out loud after the fashion of the former example, it is rare to hear anyone naturally reasoning along the lines of the latter example. Indeed, it is more likely that any appearance of doing so is always an artificial performance and a self-conscious reconstruction, if not a complete fabrication, and it is doubtful that the process of arriving at a perceptual judgment can follow this rule in just so literal a fashion. |
| | | | |
| − | An application of Rule 11 involves the recognition of an antecedent condition as a case under the Rule, that is, as a condition that matches one of the sentences in the Rule's chain of equivalents, and it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that has to be checked on input for whether it fits the antecedent condition, and there is the choice of three types of output that are generated as a consequence, only one of which is generally needed at any given time. More often than not, though, a rule is applied in only a few of its possible ways. The usual antecedent and the usual consequents for Rule 11 can be distinguished in form and specialized in practice as follows:
| + | This is true and important, but it is beside the point of the immediate discussion, which is only to identify the logical form of the inference, that is, to specify up to informational equivalence the class of conduct that is involved in each example. Thus, considering the inference as an information process, I do not care at this point whether the process is implemented by a literal-minded variety of rule-following procedure, so long as it "follows", "obeys", or "respects" these rules in the form of what it does. One can say that an information process "obeys" a set of rules in a "figurative" and a "formal" sense if the transformation that occurs in the state of information between the beginning and the end of the process has the form of a relation that can be achieved by literally following these rules with respect to the prospective class of materials. |
| | | | |
| − | a. R11a marks the usual starting place for an application of the Rule, that is, the standard form of antecedent condition that is likely to lead to an invocation of the Rule.
| + | The general drift of the strategy that is being mapped out here, the "abstract", the "formal", or the "functional" approach, is now evident. Conceptually, one partitions the space of processes into "effective", "informational", or "pragmatic" equivalence classes and then adopts the inditement of a sequence of rules as a symbolic "nomen" for the class of processes that all achieve the same class of effects. At this level of functional abstraction, the conception of a process is indifferent to the particulars of its implemenation, so long as it lives within the means of the indicated constraints. Moreover, unless there is a way to detect the nature of the "actual" process without interfering too severely with it, that is, a path-sensitive but still unobtrusive measure that can sort out a finer structure from these equivalence classes, then it is not possible to inquire any further into the supposedly "actual" details. |
| | | | |
| − | b. R11b records the trivial consequence of applying the spiny braces to both sides of the initial equation.
| + | Similar remarks apply to every case where one attributes "law-abiding" or "rule-governed" behavior to oneself, to another person, or even to a physical process. Across this diverse spectrum of cases, it ranges from likely but not certain to unlikely but still conceivable that the action in question depends on the agent "knowing" the laws that abide or the rules that are effectively being obeyed. With this in mind, I can draw this digression on appearances to a conclusion: When I say that agents are acting according to a particular pattern of rules, it only means that it "looks like" they are. In other words, they are acting "as if" they are consciously following these rules, or they are acting just like I act when I conscientiously follow such rules. A concise way to sum all of this up is to say that a pattern of rules constitutes a model of conduct, one that I can deliberately emulate, or one that I can attribute to others by way of explaining their conduct. In attributing this model to others, or even in using it to account for my own less deliberate behavior, I am making an abductive inference. |
| | | | |
| − | c. R11c gives a version of the indicator function with {X} c UxB, called its "extensional form".
| + | One way to appreciate the pertinence of this point is to notice that this entire digression, concerned with explaining the similarities between "looking like a duck" and "looking blue", is itself a form of argument, making a case of abductive inference to a case of abductive inference. In short, I am reasoning according to the following pattern: |
| | | | |
| − | d. R11d gives a version of the indicator function with {X} : U->B, called its "functional form".
| + | It appears to be the making of an abductive inference, |
| | | | |
| − | Applying Rule 9, Rule 8, and the Logical Rules to the special case where S <=> (X = Y), one obtains the following general fact.
| + | so I guess it is the making of an abductive inference. |
| | | | |
| − | Fact 1
| + | Anyone who thinks that this style of reasoning is too chancy to be tolerated ought to observe that it is only the pattern of inference that one follows in attributing minds to others, solely on the evidence that they exhibit roughly the same array of external behaviors in reaction to various external conditions as one employs to express one's experience of roughly the same conditions. |
| | | | |
| − | If X,Y c U,
| + | It goes without saying that abductive reasoning is extremely fallible. The fact that it looks like a duck does not necessarily mean that it is a duck - it might be a decoy. Moreover, in most cases of actual practice the implicit rule that serves to catalyze the abductive inference is not an absolute rule or a necessary truth in its own right but may be only a contingent rule or a probable premiss. For instance, not every case of being blue presents the fact of looking blue - the conditions of observation may be trickier than that. This brings to the fore another mark that distinguishes the two examples, highlighting a potentially important difference between "looking like a duck" and "looking blue". This is the amount of oversight, or awareness and control, that an agent has with regard to an inference, in other words, the extent to which an inference really does "go without saying". |
| | | | |
| | + | The abductive inference from "it looks blue" to "it is blue" and the abductive inference from "it looks like a duck" to "it is a duck" differ in the degrees to which they exhibit a complex of correlated properties. These variations are summed up in one sense by saying that the first, more perceptual inference is more automatic, compulsive, habitual, incorrigible, and inveterate. The correlations are summed up in the opposite sense by saying that the second, more conceptual inference is more aware, controllable, correctable, critical, deliberate, guarded, and reflective. From a fully pragmatic standpoint, these differences are naturally of critical importance. But from a purely logical standpoint, they have to be regarded as incidental aspects or secondary features of the underlying forms of inference. |
| | | | |
| | + | There is one thing yet missing from this description of abductive reasoning, and that is its creative aspect. The description so far is likely to leave the impression that the posing of a hypothesis always takes place against a narrowly circumscribed background of established terms that are available for describing cases, and thus that it amounts to nothing more original than picking out the right label for the case. Of course, the forming of a hypothesis may be bound by the generative potential of the language that is ultimately in force, but that is a far cry from a prescriptively finite list of more or less obvious choices. |
| | | | |
| − | then the following are equivalent:
| + | How does all of this bear on the choice of a method? In order to make a start toward answering that question, I need to consider the part that abductive reasoning plays in the inquiry into method, which is, after all, just another name for the inquiry into inquiry. |
| | | | |
| | + | There are times when choosing a method looks more like discovering or inventing a method, a purely spontaneous creation of a novel way to proceed, but normally the choice of a path picks its way through a landscape of familiar options and mapped out opportunities, and this presupposes a description of previously observed forms of conduct and a classification of different paths from which to choose. Hence the etymology of the word "method", indicating a review of means or a study of ways. |
| | | | |
| | + | I would now like to examine several types situations where a choice of method is involved, paying special attention to the way that abductive reasoning enters into the consideration. |
| | | | |
| − | F1a. S <=> X = Y. :R9a
| + | Example 1. |
| | | | |
| − | ::
| + | Suppose I have occasion to reason along the following lines: |
| | | | |
| − | F1b. S <=> {X} = {Y}. :R9b
| + | This situation looks like one in which this method will work, therefore I will proceed on the hypothesis that it will work. |
| | | | |
| − | ::
| + | The current situation (C) looks amenable (A') to this method, so I guess it really is amenable (A) to this method. |
| | | | |
| − | F1c. S <=> {X}(u) = {Y}(u), for all u C U. :R9c
| + | In this type of situation, my observations of the situation are reduced to a form of description that portrays it in the light of a given method, amounting to an estimate of whether the situation is a case to which the method applies. The form of the entire argument hinges on the question of whether the assurance of this application is apparent or actual. |
| | | | |
| − | ::
| + | I express my observations of the situation as a Fact: |
| | | | |
| − | F1d. S <=> ConjUu ( {X}(u) = {Y}(u) ). :R9d
| + | "The current situation looks amenable." (C => A') |
| | | | |
| − | :R8a
| + | I have in the back of my mind a general Rule: |
| | | | |
| − | ::
| + | "If it is amenable, then it looks amenable." (A => A') |
| | | | |
| − | F1e. [S] = [ ConjUu ( {X}(u) = {Y}(u) ) ]. :R8b
| + | I reason my way back from the observed Fact and the assumed Rule to assert what I guess to be the Case: |
| | | | |
| − | :???
| + | "The current situation is amenable." (C => A) |
| | | | |
| − | ::
| + | As far as it goes, this style of reasoning follows the basic pattern of abductive inference. Its obvious facticity is due to the fact that the situation is being described solely in the light of a pre-selected method. That is a relatively specious way to go about describing a situation, in spite of the fact that it may be inevitable in many of the most ultimate and limiting cases. The overall effect is noticeably strained, perhaps because it results from dictating an artificial setting, attempting to reduce a situation to the patterns that one is prepared to observe, and trying to fit what is there to see into a precut frame. A more natural way to describe a situation is in terms of the freely chosen perceptual features that inform a language of affects, impressions, and sensations. But here a situation is forced to be described in terms of the prevailing operational features that constitute a language of actions, forcing the description to be limited by the actions that are available within a prescribed framework of methods. |
| | | | |
| − | F1f. [S] = ConjUu [ {X}(u) = {Y}(u) ]. :???
| + | Instead of describing a situation solely in terms of its reactive bearing, that is, wholly in terms of how it reacts to the application of a method, one can try to describe it in terms that appear to be more its own, its independent, natural, observational, perceptual, or "proper" features. What the "proper" or "object-oriented" features are and whether they can be distinguished in the end from "reactive" or "method-oriented" features are questions that cannot be answered in the early phases of an investigation. |
| | | | |
| − | ::
| + | Example 2. |
| | | | |
| − | F1g. [S] = ConjUu (( {X}(u) , {Y}(u) )). :$1a
| + | Suppose I find myself reasoning as follows: |
| | | | |
| − | ::
| + | If the current world (C) is a blessed world (B), |
| | | | |
| − | F1h. [S] = ConjUu (( {X} , {Y} ))$(u). :$1b
| + | then it is a world in which my method works (A). |
| | | | |
| | + | Here, I call to mind an independent property of being, B, that a world or a situation can have, and I use it as a middle term to reason along the lines of the following scheme: |
| | | | |
| | + | I express my inquiry by questioning the possibility of a certain Fact, that is, by interrogating the following statement: |
| | | | |
| − | ///
| + | "The current world is amenable." (C =?> A) |
| | | | |
| − | {u C U : (f, g)$(u)}
| + | I have in the back of my mind a general Rule: |
| | | | |
| − | = {u C U : (f(u), g(u))}
| + | "What is blessed, is amenable." (B => A) |
| | | | |
| − | = {u C
| + | I reason my way back from the interrogated Fact and the assumed Rule to guess that I ought to contemplate the chances of the following Case: |
| | | | |
| − | ///
| + | "The current world is blessed." (C =?> B) |
| | | | |
| − | =====1.3.10.15 Derived Equivalence Relations=====
| + | Altogether, the argument that underlies the current question of method falls into line with the following example of abductive reasoning: |
| | | | |
| − | One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.
| + | I hope that C is A, so I guess I hope that C is B. |
| | | | |
| − | A classic way of showing that two sets are equal is to show that every element of the first belongs to the second and that every element of the second belongs to the first. The problem with this strategy is that one can exhaust a considerable amount of time trying to prove that two sets are equal before it occurs to one to look for a counterexample, that is, an element of the first that does not belong to the second or an element of the second that does not belong to the first, in cases where that is precisely what one ought to be seeking. It would be nice if there were a more balanced, impartial, neutral, or nonchalant way to go about this task, one that did not require such an undue commitment to either side, a technique that helps to pinpoint the counterexamples when they exist, and a method that keeps in mind the original relation of "proving that" and "showing that" to probing, testing, and seeing "whether".
| + | To proceed with the application of a given method on the basis of such a piece of reasoning is tantamount to the faith, the hope, or the wish that there is already the right kind of justice in the world that would make the prejudices of one's favorite method turn out to be right, that one is just lucky enough to be playing in accord with a pre-established harmony. If such a confidence is all that allows one to go on inquiring, then there is no harm in assuming it, so long as one reserves the right to question every particular of its grant, should the occasion arise. |
| | | | |
| − | A different way of seeing that two sets are equal, or of seeing whether two sets are equal, is based on the following observation:
| + | If one abstracts from the specific content of this example and examines its underlying structure, it reveals itself as the pattern of abductive reasoning that occurs in relating complex questions to simpler questions or in reducing difficult problems to easier problems. Furthermore, the iteration of this basic kind of step motivates a downward recursion from questions of fact to questions of cases, in a hopeful search for a level of cases where most of the answers are already known. |
| | | | |
| − | Two sets are equal as sets
| + | The previous examples of inquiry into method are not very satisfactory. Indeed, their schematic forms have an absurdly sketchy character about them, and they fail to convey the realistic sorts of problems that are usually involved in reasoning about the choice of a method. The first example characterizes a situation wholly in terms of a selected method. The second example characterizes a situation in terms of a property that is nominally independent of the method chosen, but the ad hoc character of this property remains obvious. In order to reason "properly" about the choice of method, it is necessary to contemplate properties of the methods themselves, and not just the situations in which they are used. |
| | | | |
| − | <=> the indicator functions of these sets are equal as functions
| + | Example 3. |
| | | | |
| − | <=> the values of these functions are equal on all domain elements.
| + | If I reason that scientific method is wise because wise people use it, then I am making the hypothesis that they use it because they are wise. Here, my reasoning can be explained according to the following pattern: |
| | | | |
| − | It is important to notice the hidden quantifier, of a universal kind, that lurks in all three equivalent statements but is only revealed in the last.
| + | I observe a fact: |
| | | | |
| − | In making the next set of definitions and in using the corresponding terminology it is taken for granted that all of the references of signs are relative to a particular sign relation R c OxSxI that either remains to be specified or is already understood. Further, I continue to assume that S = I, in which case this set is called the "syntactic domain" of R.
| + | "A certain conduct is done by wise people." (C => X) |
| | | | |
| − | In the following definitions let R c OxSxI, let S = I, and let x, y C S.
| + | I have in mind a rule: |
| | | | |
| − | Recall the definition of Con(R), the connotative component of R, in the following form:
| + | "If a wise act, then done by wise people." (A => X) |
| | | | |
| − | Con(R) = RSI = {< s, i> C SxI : <o, s, i> C R for some o C O}.
| + | I abduce the case: |
| | | | |
| − | Equivalent expressions for this concept are recorded in Definition 8.
| + | "A certain conduct is a wise act." (C => A) |
| | | | |
| − | Definition 8
| + | Example 4. |
| | | | |
| − | If R c OxSxI, | + | If I reason that scientific method is a good method on account of the fact that it works for now, then I am guessing that it works for now precisely because it is good. |
| | | | |
| | + | I observe a fact: |
| | | | |
| | + | "Scientific method works for now." (C => X) |
| | | | |
| − | then the following are identical subsets of SxI:
| + | I have in mind a rule: |
| | | | |
| | + | "What is good, works for now." (A => X) |
| | | | |
| | + | I abduce the case: |
| | | | |
| − | D8a. RSI
| + | "Scientific method is good." (C => A) |
| | | | |
| | + | As always, the abductive argument is extremely fallible. The fact that scientific method works for now can be one of its accidental features, and not due to any essential goodness that it might be thought to have. |
| | | | |
| | + | Finally, it is useful to consider an important variation on this style of argument, one that exhibits its close relation to reasoning by analogy or inference from example. Suppose that the above argument is presented in the following manner: |
| | | | |
| − | D8b. ConR
| + | Scientific method (C) has many of the features that a good method needs to have, for instance, it works for now (X), so I reason that it has all of the features of a good method, in short, that it is a good method (A). |
| | | | |
| | + | So far, the underlying argument is exactly the same. In particular, it is important to notice that the abductive argument does not depend on the prior establishment of any known cases of good methods. As of yet, the phrase "good method" is a purely hypothetical description, a term that could easily turn out to be vacuous. One has in mind a number of properties that one thinks a good method ought to have, but who knows if there is any thing that would satisfy all of these requirements? There may be some sort of subtle contradiction that is involved in the very juxtaposition of the terms "good" and "method". In sum, it can happen that scientific method is the very first method that is being considered for membership in the class of good methods, and so it is still unknown whether the class labeled "good methods" is empty or not. |
| | | | |
| | + | But what if an example of a good method is already known to exist, one that has all of the commonly accepted properties that appear to define what a good method ought to be? In this case, the abductive argument acquires the additional strength of an argument from analogy. |
| | | | |
| − | D8c. Con(R)
| + | =====1.4.3.4. Analogical Reasoning===== |
| | | | |
| | + | The classical treatment of analogical reasoning by Aristotle explains it as a combination of induction and deduction. More recently, C.S. Peirce gave two different ways of viewing the use of analogy, analyzing it into complex patterns of reasoning that involve all three types of inference. In the appropriate place, it will be useful to consider these alternative accounts of analogy in detail. At the present point, it is more useful to illustrate the different versions of analogical reasoning as they bear on the topic of choosing a method. |
| | | | |
| | + | The next example, ostensibly concerned with reasoning about a choice of method, is still too artificial to be taken seriously for this purpose, but it does serve to illustrate Aristotle's analysis of analogical reasoning as a mixed mode of inference, involving inductive and deductive phases. |
| | | | |
| − | D8d. PrSI(R)
| + | Example 5. |
| | | | |
| | + | Suppose I reason as follows. I think I can establish it as a fact that scientific method is a good method by taking it as a case of a method that always works and by using a rule that what always works is good. I think I can establish this rule, in turn, by pointing to one or more examples of methods that share the criterial property of always working and that are already acknowledged to be good. In form, this pattern of reasoning works by noticing examples of good methods, by identifying a reason why they are good, in other words, by finding a property of the examples that seems sufficient to prove them good, and by noticing that the method in question is similar to these examples precisely in the sense that it has in common this cause, criterion, property, or reason. |
| | | | |
| | + | In this situation, I am said to be reasoning by way of analogy, example, or paradigm. That is, I am drawing a conclusion about the main subject of discussion by way of its likeness to similar examples. These cases are like the main subject in the possession of a certain property, and the relation of this critical feature to the consequential feature of interest is assumed to be conclusive. The examples that exhibit the criterial property are sometimes known as "analogues" or "paradigms". For many purposes, one can imagine that the whole weight of evidence present in a body of examples is represented by a single example of the type, an exemplary or typical case, in short, an archetype or epitome. With this in mind, the overall argument can be presented as follows: |
| | | | |
| − | D8e. {< s, i> C SxI : <o, s, i> C R for some o C O}
| + | Suppose that there is an exemplary method (E) that I already know to be a good method (A). Then it pays to examine the other properties of the exemplary method, in hopes of finding a property (B) that explains why it is good. If scientific method (C) shares this property, then it can serve to establish that scientific method is good. |
| | | | |
| − | The dyadic relation RIS that constitutes the converse of the connotative relation RSI can be defined directly in the following fashion: | + | The first part of the argument is the induction of a rule: |
| | | | |
| − | Con(R)^ = RIS = {< i, s> C IxS : <o, s, i> C R for some o C O}.
| + | I notice the case: |
| | | | |
| − | A few of the many different expressions for this concept are recorded in Definition 9.
| + | "The exemplary method always works." (E => B) |
| | | | |
| − | Definition 9
| + | I observe the fact: |
| | | | |
| − | If R c OxSxI,
| + | "The exemplary method is a good method." (E => A) |
| | | | |
| | + | I induce the rule: |
| | | | |
| | + | "What always works, is good." (B => A) |
| | | | |
| − | then the following are identical subsets of IxS:
| + | The second part of the argument is the deduction of a fact: |
| | | | |
| | + | I notice the case: |
| | | | |
| | + | "Scientific method always works." (C => B) |
| | | | |
| − | D9a. RIS
| + | I recall the rule: |
| | | | |
| | + | "What always works, is good." (B => A) |
| | | | |
| | + | I deduce the fact: |
| | | | |
| − | D9b. RSI^
| + | "Scientific method is good." (C => A) |
| | | | |
| | + | Example 6. |
| | | | |
| | + | Example 7. |
| | | | |
| − | D9c. ConR^
| + | Suppose that several examples (S1, S2, S3) of a good method are already known to exist, ones that have a number of the commonly accepted properties (P1, P2, P3) that appear to define what a good method is. Then the abductive argument acquires the additional strength of an argument from analogy. |
| | | | |
| | + | The first part of the argument is the abduction of a case: |
| | | | |
| | + | I observe a set of facts: |
| | | | |
| − | D9d. Con(R)^
| + | "Scientific method is P1, P2, P3." (C => P) |
| | | | |
| | + | I recall a set of rules: |
| | | | |
| | + | "Bona fide inquiry is P1, P2, P3." (B => P) |
| | | | |
| − | D9e. PrIS(R)
| + | I abduce the case: |
| | | | |
| | + | "Scientific method is bona fide inquiry." (C => B) |
| | | | |
| | + | The second part of the argument is the induction of a rule: |
| | | | |
| − | D9f. Conv(Con(R))
| + | I notice a set of cases: |
| | | | |
| | + | "S1, S2, S3 exemplify bona fide inquiry." (S => B) |
| | | | |
| | + | I observe a set of facts: |
| | | | |
| − | D9g. {< i, s> C IxS : <o, s, i> C R for some o C O}
| + | "S1, S2, S3 exemplify good method." (S => A) |
| | | | |
| − | Recall the definition of Den(R), the denotative component of R, in the following form:
| + | I induce the rule: |
| | | | |
| − | Den(R) = ROS = {<o, s> C OxS : <o, s, i> C R for some i C I}.
| + | "Bona fide inquiry is good method." (B => A) |
| | | | |
| − | Equivalent expressions for this concept are recorded in Definition 10.
| + | The third part of the argument is the deduction of a fact: |
| | | | |
| − | Definition 10
| + | I recall the case: |
| | | | |
| − | If R c OxSxI,
| + | "Scientific method is bona fide inquiry." (C => B) |
| | | | |
| | + | I recall the rule: |
| | | | |
| | + | "Bona fide inquiry is good method." (B => A) |
| | | | |
| − | then the following are identical subsets of OxS:
| + | I deduce the fact: |
| | | | |
| | + | "Scientfic method is good method." (C => A) |
| | | | |
| | + | Now, logically and rationally in the purest sense, the argument by analogy to an example has no more force than the abductive argument, but, empirically and existentially, the example serves, not only as a model of the method to be emulated, but as an object of experimental variation and a source of further experience. |
| | | | |
| − | D10a. ROS
| + | It is time to ask the question: Why do these examples continue to maintain their unrealistic character, their comical and even ridiculous appearance, in spite of all my continuing attempts to reform them in a sensible way? It is not merely their simplicity. A simple example can be telling, if it grasps the essence of the problem, that is, so long as it captures even a single essential feature or highlights even a single critical property of the thing that one seeks to understand. It is more likely due to the circumstance that I am describing agents, methods, and situations all in one piece, that is, without any analysis, articulation, or definition of what exactly constitutes the self, the scientific method, or the world in question. It is not completely useless to consider cases of this type, since many forms of automatic, customary, and unreflective practice are underlain by arguments that are not much better that this. Of course, on reflection, their "commedius" character becomes apparent, and all deny or laugh off the suggestion that they ever think this way, but that is just the way of reflection. |
| | | | |
| | + | In order to improve the character of the discussion on this score ... |
| | | | |
| | + | ==References== |
| | | | |
| − | D10b. DenR
| + | <pre> |
| | + | Aristotle, "On The Soul", in 'Aristotle, Volume 8', |
| | + | W.S. Hett (trans.), Heinemann, London, UK, 1936, 1986. |
| | | | |
| | + | Charniak, E. & McDermott, D.V., |
| | + | 'Introduction to Artificial Intelligence', |
| | + | Addison-Wesley, Reading, MA, 1985. |
| | | | |
| | + | 2. Charniak, E., Riesbeck, C.K., & McDermott, D.V. Artificial Intelligence Programming. Lawrence Erlbaum Associates, Hillsdale, NJ, 1980. |
| | | | |
| − | D10c. Den(R)
| + | 3. Holland, J.H., Holyoak, K.J., Nisbett, R.E., & Thagard, P.R. Induction: Processes of Inference, Learning, and Discovery. MIT Press, Cambridge, MA, 1986. |
| | | | |
| | + | 4. O'Rorke, P. Review of AAAI 1990 Spring Symposium on Automated Abduction. SIGART Bulletin, Vol. 1, No. 3. ACM Press, October 1990, p. 12-17. |
| | | | |
| | + | 5. Pearl, J. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Revised 2nd printing. Morgan Kaufmann, San Mateo, CA, 1991. |
| | | | |
| − | D10d. PrOS(R)
| + | 6. Peng, Y. & Reggia, J.A. Abductive Inference Models for Diagnostic Problem-Solving. Springer-Verlag, New York, NY, 1990. |
| | | | |
| | + | 7. Sowa, J.F. Conceptual Structures: Information Processing in Mind and Machine. Addison-Wesley, Reading, MA, 1984. |
| | | | |
| | + | 8. Sowa, J.F. (ed.) Principles of Semantic Networks: Explorations in the Representation of Knowledge. Morgan Kaufmann, San Mateo, CA, 1991. |
| | | | |
| − | D10e. {<o, s> C OxS : <o, s, i> C R for some i C I}
| + | Dewey, J. (1991). How We Think. Buffalo, NY: Prometheus Books. Originally published 1910. |
| | | | |
| − | The dyadic relation RSO that constitutes the converse of the denotative relation ROS can be defined directly in the following fashion: | + | Shakespeare, Wm. (1988). William Shakespeare: The Complete Works. Compact Edition. S. Wells & G. Taylor (eds.). Oxford University Press, Oxford, UK. |
| | + | </pre> |
| | | | |
| − | Den(R)^ = RSO = {< s, o> C SxO : <o, s, i> C R for some i C I}.
| + | ==Notes== |
| | | | |
| − | A few of the many different expressions for this concept are recorded in Definition 11.
| + | ===Critique Of Functional Reason : Note 78=== |
| | | | |
| − | Definition 11
| + | <pre> |
| | + | MW = Matthew West: |
| | | | |
| − | If R c OxSxI,
| + | MW: Do you have a Cactus Manual all in one place please? |
| | | | |
| | + | the documentation for my 'theme one' program |
| | + | that I wrote up for my quant psy master's |
| | + | contains the last thing like an official |
| | + | manual that I wrote, also an expository |
| | + | introduction to the cactus language and |
| | + | its application to prop calc examples. |
| | + | may still have an ancient ascii version, |
| | + | or else the medieval 'word' doc, or i can |
| | + | send the mac belle version by snail express |
| | + | if you can vouchsafe me your postal address. |
| | | | |
| | + | in the mean time, i append a few of the expositions that |
| | + | i have outlined here/elsewhere over the last year on-line. |
| | | | |
| − | then the following are identical subsets of SxO:
| + | pre-scanning this whole mess'o'messages for you, |
| | + | I find one that looks to me shortest & sweetest: |
| | | | |
| | + | http://suo.ieee.org/email/msg05694.html |
| | | | |
| | + | since this particular synopsis is mercifully short, i will copy it out here |
| | + | and use it to explain surcatenation, along with a few other thing that i am |
| | + | guessing might be puzzling at first sight about what in hey's going on here. |
| | | | |
| − | D11a. RSO
| + | o~~~~~~~~~o~~~~~~~~~o~ARCHIVE~o~~~~~~~~~o~~~~~~~~~o |
| | | | |
| | + | Reflective Extension of Logical Graphs (Ref Log) |
| | | | |
| | + | Here is a formal introduction to the RefLog Syntax. |
| | | | |
| − | D11b. ROS^
| + | Formally speaking, we have the following set-up: |
| | | | |
| | + | Set out the "alphabet of punctuation marks" $M$ = {" ", ",", "(", ")"}. |
| | + | The elements of $M$ are vocalized as "blank, "comma", "links", "right". |
| | | | |
| | + | 1. There is a parametric family of formal languages of character strings |
| | + | such that, for each set $X$ of variable names $X$ = {"x_1", ..., "x_k"}, |
| | + | there is a formal language L($X$) over the alphabet A($X$) = $M$ |_| $X$. |
| | + | The grammar can be given in gory detail, but most folks know it already. |
| | | | |
| − | D11c. DenR^
| + | | Examples. If $X$ = {"x", "y"}, then these are typical strings in L($X$): |
| | + | | |
| | + | | " ", "( )", "x", "y", "(x)", "(y)", "x y", "(x y)", "(x, y)", "((x)(y))", "((x, y))", ... |
| | | | |
| | + | 2. There is a parallel family of formal languages of graphical structures, |
| | + | generically known as "painted and rooted cacti" (PARC's), that exist in |
| | + | a one-to-one correspondence with these string expressions, being more or |
| | + | less roughly, at a suitable level of abstraction, their parse graphs as |
| | + | data structures in the computer. The PARC's for the above formulas are: |
| | | | |
| | + | | Examples. |
| | + | | x y x y |
| | + | | o o o---o |
| | + | | x y x y x y \ / \ / |
| | + | | o o o o o---o o o |
| | + | | | x y | | x y | \ / | | |
| | + | | @ @ @ @ @ @ @ @ @ @ @ ... |
| | + | | |
| | + | | " ", "( )", "x", "y", "(x)", "(y)", "x y", "(x y)", "(x, y)", "((x)(y))", "((x, y))", ... |
| | | | |
| − | D11d. Den(R)^
| + | Together, these two families of formal languages constitute a system |
| | + | that is called the "reflective extension of logical graphs" (Ref Log). |
| | | | |
| | + | Strictly speaking, Ref Log is an abstract or "uninterpreted" formal system, |
| | + | but its expressions enjoy, as a rule, two dual interpretations that assign |
| | + | them the meanings of propositions or sentences in "zeroth order logic" (ZOL), |
| | + | to wit, what Peirce called the "alpha level" of his systems of logical graphs. |
| | | | |
| | + | For example, the string expression "(x (y))" parses into the following graph: |
| | | | |
| − | D11e. PrSO(R)
| + | | x y |
| | + | | o---o |
| | + | | | |
| | + | | @ |
| | | | |
| | + | You can "deparse" the string off the graph by traversing |
| | + | it like so, reading off the marks and varnames as you go. |
| | | | |
| | + | | o---x->(--y---o |
| | + | | ^ | |
| | + | | | x ( y | |
| | + | | | o-----o v |
| | + | | | | ) ) |
| | + | | ( (|) ) |
| | + | | ^ | | |
| | + | | | @ v |
| | | | |
| − | D11f. Conv(Den(R))
| + | In the "existential" interpretation of RefLog, |
| | + | in which I do my own thinking most of the time, |
| | + | concatenation of expressions has the meaning of |
| | + | logical conjunction, while "(x)" has the meaning |
| | + | of "not x", and so the above string and graph have |
| | + | a meaning of "x => y", "x implies y", "if x then y", |
| | + | "not x without y", or anything else that's equivalent. |
| | + | The blank expression is assigned the value of "true". |
| | + | Hence, the expression "()" takes the value of "false". |
| | + | The bracket expression "(x_1, x_2, ..., x_k)" is given |
| | + | the meaning "Exactly one of the x_j is false, j=1..k". |
| | + | Therefore, "((x_1),(x_2), ...,(x_k))" partitions the |
| | + | universe of discourse, saying "Just one x_j is true". |
| | + | </pre> |
| | | | |
| | + | ===Critique Of Functional Reason : Note 83=== |
| | | | |
| | + | <pre> |
| | + | | Tantum ergo sacramentum |
| | + | | veneremur cernui, |
| | + | | et antiquum documentum |
| | + | | novo cedat ritui, |
| | + | | praestet fides supplementum |
| | + | | sensuum defectui. |
| | + | | |
| | + | | So great therefore a sacrifice |
| | + | | let us humbly adore |
| | + | | and let the old law yield |
| | + | | to the new rite; |
| | + | | let faith supplement |
| | + | | the shortcoming of the senses. |
| | + | | |
| | + | | Lyric by Thomas Aquinas, |
| | + | | Music by Amadeus Mozart, KV 142 & 197. |
| | | | |
| − | D11g. {< s, o> C SxO : <o, s, i> C R for some i C I}
| + | The increasing ossification of asciification |
| − | | + | is heaping up way too many old bones to bear. |
| − | The "denotation of x in R", written "Den(R, x)", is defined as follows:
| + | So I am going to shift my anklage a bit, and |
| − | | + | try out a new set of conventions for a while, |
| − | Den(R, x) = {o C O : <o, x> C Den(R)}.
| + | to see if I can lighten the overloading obit. |
| − | | |
| − | In other words:
| |
| | | | |
| − | Den(R, x) = {o C O : <o, x, i> C R for some i C I}.
| + | Let us try to reserve script and singly-underscored fake-fonts or formats |
| | + | for the names of sets, as in the notations !O!, !S!, !I! that I will now |
| | + | set aside and use from now on for the Object, Sign, Interpretant domains, |
| | + | respectively, of an arbitrary sign relation !L! c !O! x !S! x !I!. |
| | | | |
| − | Equivalent expressions for this concept are recorded in Definition 12.
| + | Among other benefits, this will serve to liberate the plain faced characters |
| | + | for employment as the non-terminal symbols of our formal grammars, rendering |
| | + | our formal grammatical productions far less $Capitalistic$, !Exclamatory!, |
| | + | and overbearingly prescriptive than they be otherwise hell-bent to become. |
| | | | |
| − | Definition 12
| + | So let me try out this new rite to see how it works out, |
| | + | And I will not pause to rewrite the old law in its font, |
| | + | But advise you solely of its transformed instantiations, |
| | + | And fix my faith on imagination to sense the supplement. |
| | + | </pre> |
| | | | |
| − | If R c OxSxI,
| + | ===Critique Of Functional Reason : Note 92=== |
| | | | |
| | + | <pre> |
| | + | I need to try and say some things at his point about |
| | + | why formal language theory is interesting and useful, |
| | + | but all I have at the moment are random remembrances |
| | + | and reflections that enter my mind from time to time. |
| | | | |
| | + | In many ways, the study of formal languages and grammars |
| | + | is a paradigm, more, a paragon, of the situation that we |
| | + | face whenever we inquire into a complex reality, that is, |
| | + | all of the ever-renewed sources of puzzling phenomena or |
| | + | pressing problems that we call a world. |
| | | | |
| − | and x C S, | + | The archtypical place of formal language theory is well |
| | + | understood in many quarters, and has been from the very |
| | + | outset of its constellation as an independent viewpoint. |
| | | | |
| | + | In this paradigmatic (analogical or exemplary) way of |
| | + | understanding it, a formal language is the "data" and |
| | + | a formal grammar is the "theory", and the question is, |
| | + | as always, whether a theory accounts for and explains |
| | + | the data, a "fitting" relationship that may be viewed |
| | + | in many ways, for one, the way that a theory might be |
| | + | said to "generate" the data, or perhaps better stated, |
| | + | not just to "cook" in a precociously specious fashion |
| | + | but more like to "regenerate" the form after the fact. |
| | | | |
| | + | That's all that I can manage to express at the moment, |
| | + | but maybe it will supply a grub-stake of motivational |
| | + | victuals for the grueling labors of exploration ahead. |
| | + | </pre> |
| | | | |
| − | then the following are identical subsets of O:
| + | ===IDS. Incitatory Note 1=== |
| − | | |
| − | | |
| − | | |
| − | D12a. ROS.x
| |
| − | | |
| | | | |
| | + | <pre> |
| | + | | Each ground-principle must be proved entirely |
| | + | | by that same kind of inference which it supports. |
| | + | | |
| | + | | But we cannot arrive at any conclusion |
| | + | | by mere deduction except about symbols. |
| | + | | |
| | + | | We cannot arrive at any conclusion |
| | + | | by mere induction except about things. |
| | + | | |
| | + | | And we cannot arrive at any conclusion |
| | + | | by mere hypothesis except about forms. |
| | + | | |
| | + | | C.S. Peirce, CE 1, page 290. |
| | + | | |
| | + | | Charles Sanders Peirce, "On the Logic of Science", |
| | + | | Harvard University Lectures (1865), pages 161-302 in: |
| | + | |'Writings of Charles S. Peirce: A Chronological Edition', |
| | + | |'Volume 1, 1857-1866', Peirce Edition Project, |
| | + | | Indiana University Press, Bloomington, IN, 1982. |
| | + | </pre> |
| | | | |
| − | D12b. DenR.x
| + | ===IDS. Meditative Note 1=== |
| | | | |
| | + | <pre> |
| | + | I would like to start from a "common sense practical" (CSP) point of view, |
| | + | and, indeed, never to lose sight of what appears evident from that station, |
| | + | no matter how many levels of abstract remove and abstruse mention it might |
| | + | become necessary to interpose along the way. |
| | | | |
| | + | So let's examine this initial caltrop |
| | + | "descriptive/normative/prescriptive" |
| | + | from the CSP POV, if you will. |
| | | | |
| − | D12c. DenR|x
| + | Reading "Descriptive" to mean "What it is", |
| | + | while "Normative" means "What it oughta be", |
| | + | and "Prescriptive" says "Make it so, or else", |
| | + | I will have very little to say about the last, |
| | + | and only be able to focus on the distinctions |
| | + | that may exist among the first two dimensions. |
| | | | |
| | + | From the beginning, from this point of view, difficult words, |
| | + | like "inquiry", "logic", "truth", and so on, must be taken |
| | + | as initially indexical, inchoately succeeding at little |
| | + | more than pointing to a realm of experience that may |
| | + | or may not be common to the e-mitter and re-mitter. |
| | | | |
| | + | I suspect that this stanza is likely to be controversial, |
| | + | so I'll pause at this point for the countrapunctal verse. |
| | | | |
| − | D12d. DenR(, x)
| + | Or for a rest ... |
| | + | </pre> |
| | | | |
| | + | ===IDS. Meditative Note 2=== |
| | | | |
| | + | <pre> |
| | + | So I may begin with an object and a sign in a tenuous relation, |
| | + | with the subject matter indexed under the topic name "inquiry", |
| | + | where the sign originates from a "just noticeable differential" |
| | + | of information about the object, and not a single "figit" more. |
| | + | Few would call this a foundation -- I only call it a beginning. |
| | | | |
| − | D12e. Den(R, x)
| + | Yet another of many ... |
| | | | |
| | + | But it does provide us with a clue to a signficant difference, |
| | + | however much this difference is bound by this origin to raise |
| | + | itself from egg, germ, seed, spore, or whatever it is that is |
| | + | infinitesimal in its initial condition. In this disjointness |
| | + | of an archetype where what begins, what leads, and what rules |
| | + | are not so trivially identical to one another, one encounters |
| | + | the brand of beginning that begins in the middle of the story, |
| | + | and has no need of any other foundation but the medium itself. |
| | | | |
| | + | ["sign-ficant" [stet]] |
| | + | </pre> |
| | | | |
| − | D12f. Den(R).x
| + | ===IDS. Obligatory Note 1=== |
| | | | |
| | + | While I remain compelled to remain silent on the status of the absolute fiat, the irrelative notion of the unmotivated motion and the disinterested stance, let me then turn to the other axes of description, descriptive vs. normative. Axes of description, indeed, you can almost hear one branch of the recursion already beginning to wind up its whine to the verge of a howl, but toss it a sop and try to persevere in the quest. |
| | | | |
| | + | In this view, I regard the very idea of a norm as invoking its due pragma — aim, business, concern, desire, end, function, goal, intention, interest, objective, purpose, its names are legion — and the good sense of the norm is simply to suggest what one ought to do, contingent, of course, on one's motive to achieve that pragma. |
| | | | |
| − | D12g. {o C O : <o, x> C Den(R)}
| + | If we keep in mind the kinds of ''applied research task'' (ART) that your everyday artist, designer, engineer, mathematician, scientist, or other type of technical worker has to carry out on an everyday basis, we note how these axes of description can be used to frame their activities and to depict their forms of conduct, without mistaking either the frame or the picture for the object of the picture so framed. Nor does any body imagine that the observer must flatten out into a single plane or align with a single axis, in order to make a vantage of the frame so pictured. |
| | | | |
| | + | Common sense practical wit tells us that effective action toward the achievement of a desirable result will naturally depend on acquiring good descriptions of the lay of the land in which we hope to advance. |
| | | | |
| | + | ===IDS. Projective Note 1=== |
| | | | |
| − | D12h. {o C O : <o, x, i> C R for some i C I}
| + | <pre> |
| | + | Good morning. Thanks. I had a bad night. |
| | + | I blame Bernard Morand, who wrote me this: |
| | | | |
| − | Signs are "equiferent" if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be "denotatively equivalent" or "referentially equivalent", but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumpimg to the conclusions that are implied by these latter terms.
| + | BM: But this looks as some God's view. |
| | + | What about us, finite humans, occupied |
| | + | in counting the instants of our lives? |
| | + | And thus condemned to try to improve |
| | + | the fate of our successors? |
| | | | |
| − | To define the "equiference" of signs in terms of their denotations, one says that "x is equiferent to y under R", and writes "x =R y", to mean that Den(R, x) = Den(R, y). Taken in extension, this notion of a relation between signs induces an "equiference relation" on the syntactic domain.
| + | When you think of this in the future, and of course you may never, |
| | + | you may blame him too, for in writing this he has "erged" me on |
| | + | to return to my deserted dissertation work, into which I have |
| | + | poured my life for lo! these too many years to count, truly, |
| | + | if you stop to contemplate the fact that time is relative. |
| | | | |
| − | For each sign relation R, this yields a binary relation Der(R) c SxI that is defined as follows:
| + | In that time I have come to the view that we really need |
| | + | a good "theory of inquiry" (TOI), for all sorts of very |
| | + | practical and crucial reasons, also, that we cannot get |
| | + | a good TOI without its being, at one and the same time, |
| | + | a good "theory of information" (TOI too), and also that |
| | + | an integral constituent of TOI 1 and TOI 2 would have to |
| | + | be a good "theory of representation and semiosis" (TORAS) -- |
| | + | "Bull!?", you say, well, so be it. |
| | | | |
| − | Der(R) = DerR = {<x, y> C SxI : Den(R, x) = Den(R, y)}.
| + | Further, I think that it is abundantly evident by now that |
| | + | we will get no such good theories of signs or science from |
| | + | the "establishment philosophy of science" (EPOS?) -- which |
| | + | has managed to mince and to trash the best available tries |
| | + | at such theories for over a hundred years now. But Hey! -- |
| | + | don't take my word for it -- waste a century of your own. |
| | | | |
| − | These definitions and notations are recorded in the following display.
| + | We just got our regular email back, |
| | + | so I think that I can now get going -- |
| | + | Yes, I have lost the ability to think |
| | + | if not literally writing 'to' somebody. |
| | | | |
| | + | When it begins, it begins like this: |
| | | | |
| | + | Why am I asking this question? |
| | + | </pre> |
| | | | |
| − | Definition 13
| + | ===IDS. Projective Note 2=== |
| | | | |
| − | If R c OxSxI, | + | <pre> |
| | + | So we may rest assured that we do have a "subject matter", an empirical domain, |
| | + | or a realm of experience that is indexed, however dimly, generally, or vaguely, |
| | + | by the word "inquiry", and only the question how best to describe it remains |
| | + | in doubt at this stage of the play. If we wanted to cast our net as widely |
| | + | as possible, at the risk of anticipating a bounding hypothesis, we could |
| | + | think of all the world's creatures bright and beautiful and of how they |
| | + | conduct themselves when faced with some moment of uncertainty, where |
| | + | their aim is to cope with a surprising phenomenon or to deal with |
| | + | a problematic situation that meets them in the course of their |
| | + | ever-ongoing struggles to live, to revive, and to thrive. |
| | | | |
| | + | Now, neither the fact that we begin with a descriptive task, |
| | + | nor the fact that it remains of interest for its own sake, |
| | + | necessarily means that we must end there, for it is also |
| | + | the means to a further end, of learning how to better |
| | + | our own skill at inquiry, which means in our time |
| | + | the building of tools that help with the task. |
| | | | |
| | + | I hope I have made this sound as truly and |
| | + | as trivially obvious as it ought to be. |
| | + | </pre> |
| | | | |
| − | then the following are identical subsets of SxI:
| + | ===IDS. Reflective Note 1=== |
| | | | |
| | + | <pre> |
| | + | In reflecting on what in the world a "Theory of Inquiry" (TOI) might be, |
| | + | it occurs to me that there are many different things that one might mean |
| | + | by such a theory. It could just be any number of things that one asserts |
| | + | or has a mind to assert about the ostensible subject matter. But it has |
| | + | been my experience that one can assert pretty much whatever one chooses, |
| | + | and others will choose to heed it or ignore it on many different grounds, |
| | + | the grounds themselves being a matter of choice, conditioning, or custom. |
| | | | |
| | + | But I am looking for theories that work, that is to say, theories that |
| | + | are subject to probation through proof, probability, and programming. |
| | | | |
| − | D13a. DerR
| + | Astute readers will have noticed that I've already attempted to finesse |
| | + | a very important, and most likely "infinessible" issue, to wit, that of |
| | + | the scruples dividing descriptive, normative, and prescriptive theories. |
| | | | |
| | + | I will think about that, and get back to you. |
| | + | </pre> |
| | | | |
| | + | ===IDS. Reflective Note 2=== |
| | | | |
| − | D13b. Der(R)
| + | <pre> |
| | + | | How will I approach this problem about the nature of inquiry? |
| | + | | |
| | + | | The simplest answer is this: |
| | + | | |
| | + | | I will apply the method of inquiry to the problem of inquiry's nature. |
| | + | | |
| | + | | This is the most concise and comprehensive answer that I know, but |
| | + | | it is likely to sound facetious at this point. On the other hand, |
| | + | | if I did not actually use the method of inquiry that I describe |
| | + | | as inquiry, how could the results possibly be taken seriously? |
| | + | | Accordingly, the questions of methodological self-application |
| | + | | and self-referential consistency will be found at the center |
| | + | | of this research. |
| | | | |
| | + | These lines image in compact form the crux of the problem, |
| | + | the crucible of the method, and the character that marks |
| | + | relation between the two, if indeed they really are two, |
| | + | in a form whose extended development will wind its way |
| | + | through many a later page of the present exposition. |
| | | | |
| | + | But let me just point out at this point some of |
| | + | the reasons why I have found the prerequisite |
| | + | of an inquiry into inquiry to be inescapable. |
| | | | |
| − | D13c. {<x,y> C SxI : DenR|x = DenR|y}
| + | Let us entertain the idea, for the sake of getting the inquiry started, |
| | + | if nothing else, that it is admissible to use a word like "inquiry" as |
| | + | an initially indefinite indicator of an ostensible object of inquiry. |
| | + | If we ever again find ourselves being puzzled how our reasoning can |
| | + | chastize its own entailments this way, we may remind ourselves of |
| | + | that fine old line between our "logica docens' (logic as taught) |
| | + | and our "logica utens" (logic as used). With this distinction |
| | + | in mind, we can dispell the initial puzzlement by saying that |
| | + | we are using a capacity for inquiry that we do not know how |
| | + | to formalize yet in order to examine the forms of inquiry |
| | + | that various thinkers have been able, at least partially, |
| | + | to formalize. |
| | | | |
| | + | The dilemma that we face has the following structure: |
| | | | |
| | + | If we recommend to all a method of inquiry that |
| | + | we ourselves do not use in a pinch, precisely |
| | + | in a pinch where we need to study an issue |
| | + | as important as the nature of inquiry, |
| | + | then who would take our advice? |
| | | | |
| − | D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)}
| + | So it seems that there is no choice |
| | + | but to study inquiry, the pragma, |
| | + | by way of inquiry, the praxis, |
| | + | that is to say, recursively. |
| | | | |
| − | The relation Der(R) is defined and the notation "x =R y" is meaningful in every situation where Den(-,-) makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation.
| + | Incidentally, many variations on this theme are |
| | + | thoroughly developed in Peirce's "Lectures" of |
| | + | 1865 and 1866 and recapitulated in his early |
| | + | study "On a New List of Categories" (1867). |
| | | | |
| − | 1. Reflexive property. Is it true that x =R x for every x C S = I? By definition, x =R x if and only if Den(R, x) = Den(R, x). Thus, the reflexive property holds in any setting where the denotations Den(R, x) are defined for all signs x in the syntactic domain of R.
| + | http://members.door.net/arisbe/menu/library/bycsp/newlist/nl-main.htm |
| | + | </pre> |
| | | | |
| − | 2. Symmetric property. Does x =R y => y =R x for all x, y C S? In effect, does Den(R, x) = Den(R, y) imply Den(R, y) = Den(R, x) for all signs x and y in the syntactic domain S? Yes, so long as the sets Den(R, x) and Den(R, y) are well-defined, a fact which is already being assumed.
| + | ===IDS. Work Area=== |
| | | | |
| − | 3. Transitive property. Does x =R y & y =R z => x =R z for all x, y, z C S? To belabor the point, does Den(R, x) = Den(R, y) and Den(R, y) = Den(R, z) imply Den(R, x) = Den(R, z) for all x, y, z in S? Yes, again, under the stated conditions.
| + | <pre> |
| | + | From this point of view, inquiry is form of conduct, |
| | + | an applied research task, like many others that we |
| | + | have to carry out, and that can be done either |
| | + | better or worse. |
| | + | </pre> |
| | | | |
| − | It should be clear at this point that any question about the equiference of signs reduces to a question about the equality of sets, specifically, the sets that are indexed by these signs. As a result, so long as these sets are well-defined, the issue of whether equiference relations induce equivalence relations on their syntactic domains is almost as trivial as it initially appears.
| + | ==Document History== |
| | | | |
| − | Taken in its set-theoretic extension, a relation of equiference induces a "denotative equivalence relation" (DER) on its syntactic domain S = I. This leads to the formation of "denotative equivalence classes" (DEC's), "denotative partitions" (DEP's), and "denotative equations" (DEQ's) on the syntactic domain. But what does it mean for signs to be equiferent?
| + | <pre> |
| | + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| | | | |
| − | Notice that this is not the same thing as being "semiotically equivalent", in the sense of belonging to a single "semiotic equivalence class" (SEC), falling into the same part of a "semiotic partition" (SEP), or having a "semiotic equation" (SEQ) between them. It is only when very felicitous conditions obtain, establishing a concord between the denotative and the connotative components of a sign relation, that these two ideas coalesce.
| + | | Document History |
| − | | + | | |
| − | In general, there is no necessity that the equiference of signs, that is, their denotational equivalence or their referential equivalence, induces the same equivalence relation on the syntactic domain as that defined by their semiotic equivalence, even though this state of accord seems like an especially desirable situation. This makes it necessary to find a distinctive nomenclature for these structures, for which I adopt the term "denotative equivalence relations" (DER's). In their train they bring the allied structures of "denotative equivalence classes" (DEC's) and "denotative partitions" (DEP's), while the corresponding statements of "denotative equations" (DEQ's) are expressible in the form "x =R y".
| + | | Subject: Inquiry Driven Systems: An Inquiry Into Inquiry |
| − | | + | | Contact: Jon Awbrey <jawbrey@oakland.edu> |
| − | The uses of the equal sign for denoting equations or equivalences are recalled and extended in the following ways:
| + | | Version: Draft 10.00 |
| − | | + | | Created: 23 Jun 1996 |
| − | 1. If E is an arbitrary equivalence relation,
| + | | Revised: 02 Mar 2003 |
| − | | + | | Advisor: M.A. Zohdy |
| − | then the equation "x =E y" means that <x, y> C E.
| + | | Setting: Oakland University, Rochester, Michigan, USA |
| − | | |
| − | 2. If R is a sign relation such that RSI is a SER on S = I,
| |
| − | | |
| − | then the semiotic equation "x =R y" means that <x, y> C RSI.
| |
| − | | |
| − | 3. If R is a sign relation such that F is its DER on S = I,
| |
| − | | |
| − | then the denotative equation "x =R y" means that <x, y> C F,
| |
| − | | |
| − | in other words, that Den(R, x) = Den(R, y).
| |
| − | | |
| − | The uses of square brackets for denoting equivalence classes are recalled and extended in the following ways:
| |
| − | | |
| − | 1. If E is an arbitrary equivalence relation,
| |
| − | | |
| − | then "[x]E" denotes the equivalence class of x under E.
| |
| − | | |
| − | 2. If R is a sign relation such that Con(R) is a SER on S = I,
| |
| − | | |
| − | then "[x]R" denotes the SEC of x under Con(R).
| |
| − | | |
| − | 3. If R is a sign relation such that Der(R) is a DER on S = I,
| |
| − | | |
| − | then "[x]R" denotes the DEC of x under Der(R).
| |
| − | | |
| − | By applying the form of Fact 1 to the special case where X = Den(R, x) and Y = Den(R, y), one obtains the following facts.
| |
| − | | |
| − | | |
| − | | |
| − | Fact 2.1
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | | |
| − | | |
| − | then the following are identical subsets of SxI:
| |
| − | | |
| − | | |
| − | | |
| − | F2.1a. DerR :D13a
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1b. Der(R) :D13b
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1c. {<x, y> C SxI :
| |
| − | | |
| − | Den(R, x) = Den(R, y)
| |
| − | | |
| − | } :D13c
| |
| − | | |
| − | :R9a
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1d. {<x, y> C SxI :
| |
| − | | |
| − | {Den(R, x)} = {Den(R, y)}
| |
| − | | |
| − | } :R9b
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1e. {<x, y> C SxI :
| |
| − | | |
| − | for all o C O
| |
| − | | |
| − | {Den(R, x)}(o) = {Den(R, y)}(o)
| |
| − | | |
| − | } :R9c
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1f. {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) = {Den(R, y)}(o)
| |
| − | | |
| − | } :R9d
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1g. {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | (( {Den(R, x)}(o) , {Den(R, y)}(o) ))
| |
| − | | |
| − | } :R9e
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1h. {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | (( {Den(R, x)} , {Den(R, y)} ))$(o)
| |
| − | | |
| − | } :R9f
| |
| − | | |
| − | :D12e
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1i. {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | (( {ROS.x} , {ROS.y} ))$(o)
| |
| − | | |
| − | } :D12a
| |
| − | | |
| − | | |
| − | | |
| − | | |
| − | | |
| − | Fact 2.2
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | | |
| − | | |
| − | then the following are equivalent:
| |
| − | | |
| − | | |
| − | F2.2a. DerR = {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | } :R11a
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.2b. {DerR} = { {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | }
| |
| − | | |
| − | } :R11b
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.2c. {DerR} c SxIxB
| |
| − | | |
| − | :
| |
| − | | |
| − | {DerR} = {<x, y, v> C SxIxB :
| |
| − | | |
| − | v =
| |
| − | | |
| − | [ Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | ]
| |
| − | | |
| − | } :R11c
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.2d. {DerR} = {<x, y, v> C SxIxB :
| |
| − | | |
| − | v =
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | [ {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | ]
| |
| − | | |
| − | } :Log
| |
| − | | |
| − | | |
| − | | |
| − | F2.2e. {DerR} = {<x, y, v> C SxIxB :
| |
| − | | |
| − | v =
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | (( {Den(R, x)}(o),
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | ))
| |
| − | | |
| − | } :Log
| |
| − | | |
| − | | |
| − | | |
| − | F2.2f. {DerR} = {<x, y, v> C SxIxB :
| |
| − | | |
| − | v =
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | (( {Den(R, x)},
| |
| − | | |
| − | {Den(R, y)}
| |
| − | | |
| − | ))$(o)
| |
| − | | |
| − | } :$
| |
| − | | |
| − | | |
| − | | |
| − | | |
| − | | |
| − | Fact 2.3
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | | |
| − | | |
| − | then the following are equivalent:
| |
| − | | |
| − | | |
| − | | |
| − | F2.3a. DerR = {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | } :R11a
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.3b. {DerR} : SxI -> B
| |
| − | | |
| − | :
| |
| − | | |
| − | {DerR}(x, y) = [ Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | ] :R11d
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.3c. {DerR}(x, y) = Conj(o C O)
| |
| − | | |
| − | [ {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | ] :Log
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.3d. {DerR}(x, y) = Conj(o C O)
| |
| − | | |
| − | [ {DenR}(o, x) =
| |
| − | | |
| − | {DenR}(o, y)
| |
| − | | |
| − | ] :Def
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.3e. {DerR}(x, y) = Conj(o C O)
| |
| − | | |
| − | (( {DenR}(o, x),
| |
| − | | |
| − | {DenR}(o, y)
| |
| − | | |
| − | )) :Log
| |
| − | | |
| − | :D10b
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.3f. {DerR}(x, y) = Conj(o C O)
| |
| − | | |
| − | (( {ROS}(o, x),
| |
| − | | |
| − | {ROS}(o, y)
| |
| − | | |
| − | )) :D10a
| |
| − | | |
| − | | |
| − | | |
| − | | |
| − | | |
| − | =====1.3.10.16 Digression on Derived Relations=====
| |
| − | | |
| − | A better understanding of derived equivalence relations (DER's) can be achieved by placing their constructions within a more general context, and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation R into a dyadic relation Der(R), with other types of operations on triadic relations. The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations.
| |
| − | | |
| − | To that end, let the derivation Der(R) be expressed in the following way:
| |
| − | | |
| − | {DerR}(x, y) = Conj(o C O) (( {RSO}(x, o) , {ROS}(o, y) )).
| |
| − | | |
| − | From this abstract a form of composition, temporarily notated as "P#Q", where P c XxM and Q c MxY are otherwise arbitrary dyadic relations, and where P#Q c XxY is defined as follows:
| |
| − | | |
| − | {P#Q}(x, y) = Conj(m C M) (( {P}(x, m) , {Q}(m, y) )).
| |
| − | | |
| − | Compare this with the usual form of composition, typically notated as "P.Q" and defined as follows:
| |
| − | | |
| − | {P.Q}(x, y) = Disj(m C M) ( {P}(x, m) . {Q}(m, y) ).
| |
| − | | |
| − | | |
| − | | |
| − | 1.4 Outlook of the Project: All Ways Lead to Inquiry
| |
| − | | |
| − | | |
| − | | |
| − | 1.4 Outlook of the Project: All Ways Lead to Inquiry
| |
| − | | |
| − | | |
| − | | |
| − | I am using the word "inquiry" in a way that is roughly synonymous with the
| |
| − | | |
| − | term "scientific method". Use of "inquiry" is more convenient, aside from
| |
| − | | |
| − | being the shorter term, because of the following advantages:
| |
| − | | |
| − | | |
| − | | |
| − | 1. It allows one to broaden the scope of investigation
| |
| − | | |
| − | to include any form of proceeding toward knowledge
| |
| − | | |
| − | that merely aims at such a method.
| |
| − | | |
| − | | |
| − | | |
| − | 2. It allows one to finesse the issue, for the time being,
| |
| − | | |
| − | of how much "method" there is in science.
| |
| − | | |
| − | | |
| − | | |
| − | This Subdivision and the next deal with opposite aspects of inquiry.
| |
| − | | |
| − | In many ways it might have been better to interlace the opposing points
| |
| − | | |
| − | of comparison, taking them up in a parallel fashion, but this plan was
| |
| − | | |
| − | judged to be too distracting for a first approach. In other ways, the
| |
| − | | |
| − | negative sides of each topic are prior in point of time to the positive
| |
| − | | |
| − | sides of the issue, but sensible people like to see the light at the end
| |
| − | | |
| − | of the tunnel before they trouble themselves with the obscurities of the
| |
| − | | |
| − | intervening journey. Thus, this subdivison of the text emphasizes the
| |
| − | | |
| − | positive features of inquiry and the positive qualities of its objective,
| |
| − | | |
| − | while the next Subdivision is reserved to examine the negative aspects
| |
| − | | |
| − | of each question.
| |
| − | | |
| − | | |
| − | | |
| − | In the order of nature, the absence of a feature naturally precedes the
| |
| − | | |
| − | full development of its presence. In the order of discussion, however,
| |
| − | | |
| − | positive terms must be proposed if it is desired to say anything at all.
| |
| − | | |
| − | The discussion in this Subdivision is placed to serve a primer, declaring
| |
| − | | |
| − | at least the names of enough positive concepts to propose addressing the
| |
| − | | |
| − | negative conditions of knowledge in which inquiry necessarily starts.
| |
| − | | |
| − | | |
| − | | |
| − | In this Subdivision I stand back once again from the problem of inquiry
| |
| − | | |
| − | and allow myself take a more distant view of the subject, settling into
| |
| − | | |
| − | what I think is a comfortable and a natural account of inquiry, the best
| |
| − | | |
| − | that I have at my command, and attending to the task of describing its
| |
| − | | |
| − | positive features in a positive light. I present my personal view of
| |
| − | | |
| − | inquiry as I currently understand it, without stopping to justify every
| |
| − | | |
| − | concept in detail or to examine every objection that might be made to
| |
| − | | |
| − | this view. In the next Subdivision I discuss a few of the more obvious
| |
| − | | |
| − | problems that stand in the way of this view and I try to remove a few
| |
| − | | |
| − | of the more tractable obscurities that appear ready to be cleared up.
| |
| − | | |
| − | The fact that I treat them as my "personal insights" does not mean that
| |
| − | | |
| − | all of these ideas about inquiry originate with me, but only that I have
| |
| − | | |
| − | come to adopt them for my personal use. There will be many occasions,
| |
| − | | |
| − | the next time that I go over this ground, to point out the sources of
| |
| − | | |
| − | these ideas, so far as I know them.
| |
| − | | |
| − | | |
| − | | |
| − | The reader may take my apology for this style of presentation to be
| |
| − | | |
| − | implicit in its dogmatic character. It is done this way in a first
| |
| − | | |
| − | approach for the sake of avoiding an immense number of distractions,
| |
| − | | |
| − | each of which is not being slighted but demands to be addressed in
| |
| − | | |
| − | its own good time. I want to convey the general drift of my current
| |
| − | | |
| − | model, however conjectural, naive, uncritical, and unreflective it
| |
| − | | |
| − | may seem.
| |
| − | | |
| − | ====1.4.1 The Matrix of Inquiry====
| |
| − | | |
| − | <pre>
| |
| − | | Thus when mothers have chidren suffering from sleeplessness,
| |
| − | | and want to lull them to rest, the treatment they apply is
| |
| − | | to give them, not quiet, but motion, for they rock them
| |
| − | | constantly in their arms; and instead of silence, they
| |
| − | | use a kind of crooning noise; and thus they literally
| |
| − | | cast a spell upon the children (like the victims of
| |
| − | | a Bacchic frenzy) by employing the combined movements
| |
| − | | of dance and song as a remedy.
| |
| − | |
| |
| − | | Plato, 'Laws', VII, 790D
| |
| − | </pre>
| |
| − | | |
| − | Try as I may, I've never seen a way to develop a theory of inquiry from nothing:
| |
| − | | |
| − | To take for granted nothing more than is already given, to set out from nothing
| |
| − | | |
| − | but absolutely certain beginnings, to move forward with nothing but absolutely
| |
| − | | |
| − | certain means of proceeding. In particular, the present inquiry into inquiry,
| |
| − | | |
| − | foreshadowed in the form y_0 = y·y, ought not to be misconstrued as a device
| |
| − | | |
| − | for magically generating a theory of inquiry from nothing. Like any other
| |
| − | | |
| − | inquiry, it requires an agent to invest in a conjecture, to make a guess
| |
| − | | |
| − | about the pertinent features of the subject of interest, and to choose
| |
| − | | |
| − | the actions, the aspects, and the attitudes with regard to the subject
| |
| − | | |
| − | that are critical to achieving the intended objectives of the study.
| |
| − | | |
| − | | |
| − | | |
| − | I can sum all this up by saying that an inquiry requires an inquirer to
| |
| − | | |
| − | suggest a hypothesis about the subject of interest and then to put that
| |
| − | | |
| − | particular model of the subject to the test. This in turn requires one
| |
| − | | |
| − | to devote a modicum of personal effort to the task of testing the chosen
| |
| − | | |
| − | hypothesis, to put a quantum of personal interest at stake for the sake
| |
| − | | |
| − | of finding out whether the model fits the subject, and, overall, to take
| |
| − | | |
| − | the risk of being wrong. Any model that is feasible is also defeasible,
| |
| − | | |
| − | at least, where it concerns a contingent subject of contingent inquiry.
| |
| − | | |
| − | | |
| − | | |
| − | The first step, then, of an inquiry into inquiry, is to put forth a tentative
| |
| − | | |
| − | model of inquiry, to make a hypothesis about the features of inquiry that are
| |
| − | | |
| − | essential to explaining its experienced characteristics, and thus, in a sense,
| |
| − | | |
| − | to make a guess at the very definition of inquiry. This requirement seems both
| |
| − | | |
| − | obvious and outrageous at the same time. One is perfectly justified in objecting
| |
| − | | |
| − | that there is much that precedes this so-called "first step", namely, the body of
| |
| − | | |
| − | experience that prepares one to see it and the mass of observation that prompts one
| |
| − | | |
| − | to take it. I can deal with this objection by making a distinction between mundane
| |
| − | | |
| − | experience and olympian theory, and then by saying that the making of a conjecture
| |
| − | | |
| − | is really the first "theoretical" step, but this is a hedge that covers the tracks
| |
| − | | |
| − | of theory in a very deceptive way, hiding how early in the empirical process the
| |
| − | | |
| − | "cloven hoof" of theory actually enters.
| |
| − | | |
| − | | |
| − | | |
| − | Leaving behind the mythical states of "pure" experience and "naive" observation,
| |
| − | | |
| − | and at least by the time that one has come to give a name to the subject of the
| |
| − | | |
| − | investigation, one's trek through the data is already half-shod, half-fettered
| |
| − | | |
| − | by the connotations of the name, and in their turn by all of the concepts that
| |
| − | | |
| − | it invokes in its train. That name, the concepts that it suggests, and the
| |
| − | | |
| − | tacit but vague definition of the subject that this complex of associations
| |
| − | | |
| − | is already beginning to constellate, to attract certain experiences to the
| |
| − | | |
| − | complex, and to filter out other observations from having any bearing on
| |
| − | | |
| − | the subject matter. By this time, one is already busy translating one's
| |
| − | | |
| − | empirical acquaintance with the subject into an arrangement of concepts
| |
| − | | |
| − | that is intended to define its essential nature.
| |
| − | | |
| − | | |
| − | | |
| − | An array of concepts that is set up in order to capture the essence
| |
| − | | |
| − | of a subject is a provisional definition of it, an implicit model
| |
| − | | |
| − | of the subject that contains the makings of an explicit theory.
| |
| − | | |
| − | It amounts to a selection from the phenomenal aspects of the
| |
| − | | |
| − | subject, expresses a guess about its relevant features, and
| |
| − | | |
| − | constitutes a hypothesis in explanation of its experienced
| |
| − | | |
| − | characteristics. This incipient order of model or theory
| |
| − | | |
| − | is tantamount to a definition because it sets bounds on
| |
| − | | |
| − | the "stretches" and the "holds" of a term -- that is,
| |
| − | | |
| − | the extension, intension, and intention of the term --
| |
| − | | |
| − | but this is not the kind of definition that has to
| |
| − | | |
| − | be taken on faith, that constitutes the first and
| |
| − | | |
| − | the last word on the subject. In other words,
| |
| − | | |
| − | it is an empirical definition, one that is
| |
| − | | |
| − | subject to being falsified in reference
| |
| − | | |
| − | to its intended subject, by failing to
| |
| − | | |
| − | indicate the necessary, the pertinent,
| |
| − | | |
| − | or the relevant features that account
| |
| − | | |
| − | for the presence of its phenomena or
| |
| − | | |
| − | the persistence of its process.
| |
| − | | |
| − | | |
| − | | |
| − | If I reflect on the conduct of inquiry,
| |
| − | | |
| − | seeking to fix it in a fitting image
| |
| − | | |
| − | and trying to cast it in a positive
| |
| − | | |
| − | light, the best I can do is this:
| |
| − | | |
| − | | |
| − | | |
| − | Inquiry is a process that aims at achieving belief or knowledge.
| |
| − | | |
| − | | |
| − | | |
| − | But even this simple a description already plunges the discussion deep into
| |
| − | | |
| − | a number of obscurities. Most prominently, there is the disjunction between
| |
| − | | |
| − | belief and knowledge that cries out to be explained or resolved. Stirring a
| |
| − | | |
| − | little beneath the surface, and not quite fading into the background, many of
| |
| − | | |
| − | the other terms that are invoked in the description are capable of hiding the
| |
| − | | |
| − | entire contents of the original ignorance that the image as a whole is aimed
| |
| − | | |
| − | to dispell. And yet there is nothing that I can do in this avowedly positive
| |
| − | | |
| − | context but to mark these points down as topics for future discussion.
| |
| − | | |
| − | | |
| − | | |
| − | There is already a model of inquiry that is implicit,
| |
| − | | |
| − | at least partially, in the text of the above description.
| |
| − | | |
| − | Let me see if I can tease out a few of its tacit assumptions.
| |
| − | | |
| − | | |
| − | | |
| − | o20:56, 26 May 2007 (PDT)[[User:Jon Awbrey|Jon Awbrey]] 20:56, 26 May 2007 (PDT)o20:56, 26 May 2007 (PDT)[[User:Jon Awbrey|Jon Awbrey]] 20:56, 26 May 2007 (PDT)o20:56, 26 May 2007 (PDT)[[User:Jon Awbrey|Jon Awbrey]] 20:56, 26 May 2007 (PDT)o20:56, 26 May 2007 (PDT)[[User:Jon Awbrey|Jon Awbrey]] 20:56, 26 May 2007 (PDT)o20:56, 26 May 2007 (PDT)[[User:Jon Awbrey|Jon Awbrey]] 20:56, 26 May 2007 (PDT)o
| |
| − | | |
| − | | |
| − | | |
| − | I am using the word "inquiry" in a way that is roughly synonymous with the term "scientific method". Use of "inquiry" is more convenient, aside from being the shorter term, because of the following advantages: (1) It allows one to broaden the scope of investigation to include any form of proceeding toward knowledge that merely aims at such a method. (2) It allows one to finesse the issue, for the time being, of how much "method" there is in science.
| |
| − | | |
| − | This subdivision and the next deal with opposite aspects of inquiry. In many ways it might have been better to interlace the opposing points of comparison, taking them up in a parallel fashion, but this plan was judged to be too distracting for a first approach. In other ways, the negative sides of each topic are prior in point of time to the positive sides of the issue, but sensible people like to see the light at the end of the tunnel before they trouble themselves with the obscurities of the intervening journey. Thus, this subdivison of the text emphasizes the positive features of inquiry and the positive qualities of its objective, while the next subdivision is reserved to examine the negative aspects of each question.
| |
| − | | |
| − | In the order of nature, the absence of a feature naturally precedes the full development of its presence. In the order of discussion, however, positive terms must be proposed if it is desired to say anything at all. The discussion in this subdivision is placed to serve a primer, declaring at least the names of enough positive concepts to propose addressing the negative conditions of knowledge in which inquiry necessarily starts.
| |
| − | | |
| − | In this subdivision I stand back once again from the problem of inquiry and allow myself take a more distant view of the subject, settling into what I think is a comfortable and a natural account of inquiry, the best that I have at my command, and attending to the task of describing its positive features in a positive light. I present my personal view of inquiry as I currently understand it, without stopping to justify every concept in detail or to examine every objection that might be made to this view. In the next subdivision I discuss a few of the more obvious problems that stand in the way of this view and I try to remove a few of the more tractable obscurities that appear ready to be cleared up. The fact that I treat them as my "personal insights" does not mean that all of these ideas about inquiry originate with me, but only that I have come to adopt them for my personal use. There will be many occasions, the next time that I go over this ground, to point out the sources of these ideas, so far as I know them.
| |
| − | | |
| − | The reader may take my apology for this style of presentation to be implicit in its dogmatic character. It is done this way in a first approach for the sake of avoiding an immense number of distractions, each of which is not being slighted but demands to be addressed in its own good time. I want to convey the general drift of my current model, however conjectural, naive, uncritical, and unreflective it may seem.
| |
| − | | |
| − | ====1.4.1 The Matrix of Inquiry (2)====
| |
| − | | |
| − | <blockquote>
| |
| − | <p>Thus when mothers have chidren suffering from sleeplessness, and want to lull them to rest, the treatment they apply is to give them, not quiet, but motion, for they rock them constantly in their arms; and instead of silence, they use a kind of crooning noise; and thus they literally cast a spell upon the children (like the victims of a Bacchic frenzy) by employing the combined movements of dance and song as a remedy.</p>
| |
| − | | |
| − | <p>(Plato, Laws, VII, 790D).</p>
| |
| − | </blockquote>
| |
| − | | |
| − | Try as I might, I do not see a way to develop a theory of inquiry from nothing: To take for granted nothing more than is already given, to set out from nothing but absolutely certain beginnings, or to move forward with nothing but absolutely certain means of proceeding. In particular, the present inquiry into inquiry, y0 = y.y, ought not to be misconstrued as a device for magically generating a theory of inquiry from nothing. Like any other inquiry, it requires an agent to invest in a conjecture, to make a guess about the relevant features of the subject of interest, and to choose the actions, the aspects, and the attitudes with regard to the subject that are critical to achieving the objectives of the study.
| |
| − | | |
| − | I can sum all this up by saying that an inquiry requires an inquirer to suggest a hypothesis about the subject of interest and then to put that particular model of the subject to the test. This in turn requires one to devote a modicum of personal effort to the task of testing the chosen hypothesis, to put a quantum of personal interest at stake for the sake of finding out whether the model fits the subject, and, overall, to take the risk of being wrong. Any model that is feasible is also defeasible, at least, where it concerns a contingent subject of inquiry.
| |
| − | | |
| − | The first step, then, of an inquiry into inquiry, is to put forth a tentative model of inquiry, to make a hypothesis about the features of inquiry that are essential to explaining its experienced characteristics, and thus, in a sense, to make a guess at the very definition of inquiry. This requirement seems both obvious and outrageous at the same time. One is perfectly justified in objecting that there is much that precedes this so-called "first step", namely, the body of experience that prepares one to see it and the mass of observation that prompts one to take it. I can deal with this objection by making a distinction between mundane experience and olympian theory, and then by saying that the making of a conjecture is really the first "theoretical" step, but this is a hedge that covers the tracks of theory in a deceptive way, hiding how early in the empirical process the "cloven hoof" of theory actually enters.
| |
| − | | |
| − | Leaving behind the mythical conditions of pure experience and naive observation, and at least by the time that one comes to give a name to the subject of investigation, one's trek through the data is already half-shod, half-fettered by the connotations of the name, and in turn by all of the concepts that it invokes in its train. The name, the concepts that it suggests, and the tacit but vague definition of the subject that this complex of associations is already beginning to constellate, attract certain experiences to the complex and filter out other observations from having any bearing on the subject matter. By this point, one is already busy translating one's empirical acquaintance with the subject into an arrangement of concepts that is intended to define its essential nature.
| |
| − | | |
| − | An array of concepts that is set up to capture the essence of a subject is a provisional definition of it, an implicit model of the subject that contains the makings of an explicit theory. It amounts to a selection from the phenomenal aspects of the subject, expresses a guess about its relevant features, and constitutes a hypothesis in explanation of its experienced characteristics. This incipient order of model or theory is tantamount to a definition because it sets bounds on the "stretches" and the "holds" of a term - its extension, intension, and intention - but this is not the kind of definition that has to be taken on faith, or that constitutes the first and the last word on the subject. In other words, it is an empirical definition, one that is subject to being falsified in reference to its intended subject, by failing to indicate the necessary, the pertinent, or the relevant features that account for the presence of its phenomena or the persistence of its process.
| |
| − | | |
| − | If I reflect on the conduct of inquiry, seeking to fix it in a fitting image and trying to cast it in a positive light, the best I can do is this:
| |
| − | | |
| − | Inquiry is a process that aims at achieving belief or knowledge.
| |
| − | | |
| − | But even this simple a description already plunges the discussion deep into a number of obscurities. Most prominently, there is the disjunction between belief and knowledge that cries out to be explained or resolved. Stirring beneath the surface, and not quite fading into the background, many of the other terms that are invoked in the description are capable of hiding the entire contents of the original ignorance that the image as a whole is aimed to dispell. And yet, there is nothing that I can do in this avowedly positive context but to mark these points down as topics for future discussion.
| |
| − | | |
| − | There is already a model of inquiry that is implicit, at least partially, in the text of the above description. Let me see if I can tease out a few of its tacit assumptions.
| |
| − | | |
| − | =====1.4.1.1 Inquiry as Conduct=====
| |
| − | | |
| − | First of all, inquiry is conceived to be a form of conduct.
| |
| − | This invokes the technical term "conduct", referring to the
| |
| − | species of prototypically human action that is both dynamic
| |
| − | and deliberate, or conceived to fall under a form of purposeful
| |
| − | control, usually conscious but possibly not. For the sake of
| |
| − | clarity, it helps to seek a more formal definition of conduct,
| |
| − | one that expresses the concept in terms of abstract features
| |
| − | rather than trying to suggest it by means of typical examples.
| |
| − | | |
| − | Conduct is action with respect to an object. The distinction between action and conduct, reduced to the level of the most abstract formal relations that are involved, can be described in the following manner.
| |
| − | | |
| − | Action is a matter of going from A to B, whereas conduct is matter of going from A to B in relation to C. In describing particular cases and types of conduct, the phrase "in relation to" can be filled out in more detail as "on account of", "in the cause of", "in order to bring about", "for the sake of", "in the interests of", or in many other ways. Thus, action by itself has a dyadic character, involving transitions through pairs of states, while conduct has a triadic character, involving the kinds of transactions between states that relate throughout to an object.
| |
| − | | |
| − | With regard to this distinction, notice that "action" is used inclusively, to name the genus of which "conduct" names a species, and thus depicts whatever has the aspect of action, even if it is actually more complex.
| |
| − | | |
| − | This creates the difficulty that the reputed "genus" is less than fully "generative", "generic", "genetic", or even "genuine" -- and so it is necessary to remain on guard against this source of misunderstanding.
| |
| − | | |
| − | What does this definition of conduct say about the temporal ordering of the object with respect to the states? The states are conceived to be ordered in time, but so far nothing has been said to pin down where in relation to these states the object must be conceived to fall in time. Nor does the definition make any particular specification necessary. This makes the question of relative time a secular parameter of the definition, allowing the consideration of the following options:
| |
| − | | |
| − | 1. If the object is thought to precede the action of the conduct, then it tends to be regarded as a creative act, an initial intention, an original stimulus, a principal cause, or a prime mover.
| |
| − | | |
| − | 2. If the object is thought to succeed the action of the conduct, then it tends to be regarded as an end, a goal, or a purpose, in other words, a state envisioned to be fulfilled.
| |
| − | | |
| − | 3. If the object is thought to be concurrent, immanent, or transcendent throughout the action of the conduct, then it tends to be regarded as falling under one of the following possibilities: a prevailing value, a controlling parameter, a universal system of effective forces, a pervasive field of potentials, a ruling law, or a governing principle.
| |
| − | | |
| − | A prevailing value or a controlling parameter, which guides the temporal development of a system, is a term that fits into a law or a principle, which governs the system at a higher level. The existence of a value or a law that rules a system, and the information that an agent of the system has about its parameters and its principles, are two different matters. Indeed, a major task of development for an inquiring agent is to inform itself about the values and the laws that form its own system. Thus, one of the objects of the conduct of inquiry is a description in terms of laws and values of the rules that govern and guide inquiry.
| |
| − | | |
| − | The elaboration of an object in terms of this rich vocabulary -- as a cause, end, field, force, goal, intention, law, parameter, principle, purpose, system, or value -- adds colorful detail and concrete sensation to the account, and it helps to establish connections with the arrays of terminology that are widely used to discuss these issues. From a formal and relational point of view, however, all of these concepts are simply different ways of describing, at possibly different levels of generality, the object of a form of conduct. With that in mind, I find it useful to return to the simpler form of description as often as possible.
| |
| − | | |
| − | This account of conduct brings to the fore a number of issues, some of them new and some of them familiar, but each of them allowing itself to be approached from a fresh direction by treating it as an implication of a critical thesis just laid down. I next examine these issues in accord with the tenets from which they stem.
| |
| − | | |
| − | 1. Inquiry is a form of conduct.
| |
| − | | |
| − | This makes inquiry into inquiry a special case of inquiry into conduct.
| |
| − | | |
| − | Certainly, it must be possible to reason about conduct in general, especially if forms of conduct need to be learned, examined, modified, and improved.
| |
| − | | |
| − | Placing the subject of inquiry within the subject of conduct and making the inquiry into inquiry a subordinate part of the inquiry into conduct does not automatically further the investigation, especially if it turns out that the general subject of conduct is more difficult to understand than the specialized subject of inquiry. But in those realms of inquiry where it is feasible to proceed hypothetically and recursively, stretching the appropriate sort of hypothesis over a wider subject area can act to prime the pump of mathematical induction all the more generously, and actually increase the power of the recursion. Of course, the use of a recursive strategy comes at the expense of having to establish a more extended result at the base.
| |
| − | | |
| − | 2. The existence of an object that rules a form of conduct and the information that an agent of the conduct has about the object are two different matters.
| |
| − | | |
| − | This means that the exact specification of the object can demand an order of information that the agent does not have available, at least, not for use in reflective action, or even require an amount of information that the agent lacks the capacity to store. No matter how true it is that the actual course of the agent's conduct exactly reflects the influence of the object, and thus, in a sense, represents the object exactly, the question is whether the agent possesses the equivalent of this information in any kind of accessible, exploitable, reflective, surveyable, or usable form of representation, in effect, in any mode of information that the agent can use to forsee, to modify, or to temper its own temporal course.
| |
| − | | |
| − | This issue may seem familiar as a repetition of the "meta" question.
| |
| − | | |
| − | Once again, there is a distinction between (a) the properties of an action, agent, conduct, or system, as expressible by the agent that is engaged in the conduct, or as representable within the system that is undergoing the action, and (b) the properties of the same entities, as evident from an "external viewpoint", or as statable by the equivalent of an "outside observer".
| |
| − | | |
| − | 3. Reflection is a part of inquiry.
| |
| − | | |
| − | Reflection is a form of conduct.
| |
| − | | |
| − | The task of reflection on conduct is to pass from a purely interior view of one's own conduct to an outlook that is, effectively, an exterior view.
| |
| − | | |
| − | What is sought is a wider perspective, one that is able to incorporate the sort of information that might be available to an outside observer, that ought to be evident from an external vantage point, or that one reasonably imagines might be obvious from an independent viewpoint. I am tempted to refer to such a view as a "quasi-objective perspective", but only so long as it possible to keep in mind that there is no such thing as a "completely outside perspective", at least, not one that a finite and mortal agent can hope to achieve, nor one that a reasonably socialized member of a community can wish to take up as a permanent station in life.
| |
| − | | |
| − | With these qualifications, reflection is a form of conduct that can serve inquiry into conduct. Inquiry and its component reflection, applied to a form of conduct, are intended to provide information that can be used to develop the conduct in question. The "reflective development" that occurs depends on the nature of the case. It can be the continuation, the correction, or the complete cessation of the conduct in question.
| |
| − | | |
| − | If it is to have the properties that it is commonly thought to have, then reflection must be capable of running in parallel, and not interfering too severely, with the conduct on which it reflects. If this turns out to be an illusion of reflection that is not really possible in actuality, then reflection must be capable, at the very least, of reviewing the memory record of the conduct in question, in ways that appear concurrent with a replay of its action. But these are the abilities that reflection is "pre-reflectively" thought to have, that is, before the reflection on reflection can get under way. If reflection is truly a form of conduct, then it becomes conceivable as a project to reflect on reflection itself, and this reflection can even lead to the conclusion that reflection does not have all of the powers that it is commonly portrayed to have.
| |
| − | | |
| − | First of all, inquiry is conceived to be a form of conduct. This invokes the technical term "conduct", referring to the species of prototypically human action that is both dynamic and deliberate, or conceived to fall under a form of purposeful control, usually conscious but possibly not. For the sake of clarity, it helps to seek a more formal definition of conduct, one that expresses the concept in terms of abstract features rather than trying to suggest it by means of typical examples.
| |
| − | | |
| − | Conduct is action with respect to an object. The distinction between action and conduct, reduced to the level of the most abstract formal relations that are involved, can be described in the following manner. Action is a matter of going from A to B, whereas conduct is matter of going from A to B in relation to C. In describing particular cases and types of conduct, the phrase "in relation to" can be filled out in more detail as "on account of", "in the cause of", "in order to bring about", "for the sake of", "in the interests of", or in many other ways. Thus, action by itself has a dyadic character, involving transitions through pairs of states, while conduct has a triadic character, involving the kinds of transactions between states that relate throughout to an object.
| |
| − | | |
| − | With regard to this distinction, notice that "action" is used inclusively, to name the genus of which "conduct" names a species, and thus depicts whatever has the aspect of action, even if it is actually more complex. This creates the difficulty that the reputed "genus" is less than fully "generative", "generic", "genetic", or even "genuine" - and so it is necessary to remain on guard against this source of misunderstanding.
| |
| − | | |
| − | What does this definition of conduct say about the temporal ordering of the object with respect to the states? The states are conceived to be ordered in time, but so far nothing has been said to pin down where in relation to these states the object must be conceived to fall in time. Nor does the definition make any particular specification necessary. This makes the question of relative time a secular parameter of the definition, allowing the consideration of the following options:
| |
| − | | |
| − | 1. If the object is thought to precede the action of the conduct, then it tends to be regarded as a creative act, an initial intention, an original stimulus, a principal cause, or a prime mover.
| |
| − | | |
| − | 2. If the object is thought to succeed the action of the conduct, then it tends to be regarded as an end, a goal, or a purpose, in other words, a state envisioned to be fulfilled.
| |
| − | | |
| − | 3. If the object is thought to be concurrent, immanent, or transcendent throughout the action of the conduct, then it tends to be regarded as falling under one of the following possibilities: a prevailing value, a controlling parameter, a universal system of effective forces, a pervasive field of potentials, a ruling law, or a governing principle.
| |
| − | | |
| − | | |
| − | A prevailing value or a controlling parameter, which guides the temporal development of a system, is a term that fits into a law or a principle, which governs the system at a higher level. The existence of a value or a law that rules a system, and the information that an agent of the system has about its parameters and its principles, are two different matters. Indeed, a major task of development for an inquiring agent is to inform itself about the values and the laws that form its own system. Thus, one of the objects of the conduct of inquiry is a description in terms of laws and values of the rules that govern and guide inquiry.
| |
| − | | |
| − | The elaboration of an object in terms of this rich vocabulary - as a cause, end, field, force, goal, intention, law, parameter, principle, purpose, system, or value - adds colorful detail and concrete sensation to the account, and it helps to establish connections with the arrays of terminology that are widely used to discuss these issues. From a formal and relational point of view, however, all of these concepts are simply different ways of describing, at possibly different levels of generality, the object of a form of conduct. With that in mind, I find it useful to return to the simpler form of description as often as possible.
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| − | | |
| − | This account of conduct brings to the fore a number of issues, some of them new and some of them familiar, but each of them allowing itself to be approached from a fresh direction by treating it as an implication of a critical thesis just laid down. I next examine these issues in accord with the tenets from which they stem.
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| − | | |
| − | 1. Inquiry is a form of conduct.
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| − | | |
| − | This makes inquiry into inquiry a special case of inquiry into conduct. Certainly, it must be possible to reason about conduct in general, especially if forms of conduct need to be learned, examined, modified, and improved.
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| − | | |
| − | Placing the subject of inquiry within the subject of conduct and making the inquiry into inquiry a subordinate part of the inquiry into conduct does not automatically further the investigation, especially if it turns out that the general subject of conduct is more difficult to understand than the specialized subject of inquiry. But in those realms of inquiry where it is feasible to proceed hypothetically and recursively, stretching the appropriate sort of hypothesis over a wider subject area can act to prime the pump of mathematical induction all the more generously, and actually increase the power of the recursion. Of course, the use of a recursive strategy comes at the expense of having to establish a more extended result at the base.
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| − | | |
| − | 2. The existence of an object that rules a form of conduct and the information that an agent of the conduct has about the object are two different matters.
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| − | | |
| − | This means that the exact specification of the object can require an order of information that the agent does not have available, at least, not for use in reflective action, or even an amount of information that the agent lacks the capacity to store. No matter how true it is that the actual course of the agent's conduct exactly reflects the influence of the object, and thus, in a sense, represents the object exactly, the question is whether the agent possesses the equivalent of this information in any kind of accessible, exploitable, reflective, surveyable, or usable form of representation, in effect, any mode of information that the agent can use to forsee, to modify, or to temper its own temporal course.
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| − | | |
| − | This issue may seem familiar as a repetition of the "meta" question. Once again, there is a distinction between (a) the properties of an action, agent, conduct, or system, as expressible by the agent that is engaged in the conduct, or as representable within the system that is undergoing the action, and (b) the properties of the same entities, as evident from an "external viewpoint", or as statable by the equivalent of an "outside observer".
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| − | | |
| − | 3. Reflection is a part of inquiry. Reflection is a form of conduct.
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| − | | |
| − | The task of reflection on conduct is to pass from a purely interior view of one's own conduct to an outlook that is, effectively, an exterior view. What is sought is a wider perspective, one that is able to incorporate the sort of information that might be available to an outside observer, that ought to be evident from an external vantage point, or that one reasonably imagines might be obvious from an independent viewpoint. I am tempted to refer to such a view as a "quasi-objective perspective", but only so long as it possible to keep in mind that there is no such thing as a "completely outside perspective", at least, not one that a finite and mortal agent can hope to achieve, nor one that a reasonably socialized member of a community can wish to take up as a permanent station in life.
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| − | | |
| − | With these qualifications, reflection is a form of conduct that can serve inquiry into conduct. Inquiry and its component reflection, applied to a form of conduct, are intended to provide information that can be used to develop the conduct in question. The "reflective development" that occurs depends on the nature of the case. It can be the continuation, the correction, or the complete cessation of the conduct in question.
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| − | | |
| − | If it is to have the properties that it is commonly thought to have, then reflection must be capable of running in parallel, and not interfering too severely, with the conduct on which it reflects. If this turns out to be an illusion of reflection that is not really possible in actuality, then reflection must be capable, at the very least, of reviewing the memory record of the conduct in question, in ways that appear concurrent with a replay of its action. But these are the abilities that reflection is "pre-reflectively" thought to have, that is, before the reflection on reflection can get under way. If reflection is truly a form of conduct, then it becomes conceivable as a project to reflect on reflection itself, and this reflection can even lead to the conclusion that reflection does not have all of the powers that it is commonly portrayed to have.
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| − | | |
| − | =====1.4.1.2 Types of Conduct=====
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| − | | |
| − | The chief distinction that applies to different forms of conduct is whether the object is the same sort of thing as the states or whether it is something entirely different, a thing apart, of a wholly other order. Although I am using different words for objects and states, it is always possible that these words are indicative of different roles in a formal relation and not indicative of substantially different types of things. If objects and states are but formal points and naturally belong to the same domain, then it is conceivable that a temporal sequence of states can include the object in its succession, in other words, that a path through a state space can reach or pass through an object of conduct. But if a form of conduct has an object that is completely different from any one of its temporal states, then the role of the object in regard to the action cannot be like the end or goal of a temporal development.
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| − | | |
| − | What names can be given to these two orders of conduct?
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| − | | |
| − | =====1.4.1.3 Perils of Inquiry=====
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| − | | |
| − | Now suppose that making a hypothesis is a kind of action, no matter how covert, or that testing a hypothesis takes an action that is more overt. If entertaining a hypothesis in any serious way requires action, and if action is capable of altering the situation in which it acts, then what prevents this action from interfering with the subject of inquiry in a way that undermines, with positive or negative intentions, the very aim of inquiry, namely, to understand the situation as it is in itself?
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| − | | |
| − | That making a hypothesis is a type of action may seem like a hypothesis that is too far-fetched, but it appears to follow without exception from thinking that thinking is a form of conduct, in other words, an activity with a purpose or an action that wants an end. The justification of a hypothesis is not to be found in a rational pedigree, by searching back through a deductive genealogy, or determined by that which precedes it in the logical order, since a perfectly trivial tautology caps them all. Since a logical tautology, that conveys no empirical information, finds every proposition appearing to implicate it, in other words, since it is an ultimate implication of every proposition and a conceivable conclusion that is implicit in every piece of reasoning, it is obvious that seeking logical precedents is the wrong way to go for empirical content.
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| − | | |
| − | In making a hypothesis or choosing a model, one appears to select from a vaster number of conceivable possibilities than a finite agent could ever enumerate in complete detail or consider as an articulate totality. As the very nature of a contingent description and the very character of a discriminate action is to apply in some cases but not in others, there is no escaping the making of a risky hypothesis or a speculative interpretation, even in the realm of a purely mental action. Thus, all significant thought, even thinking to any purpose about thought itself, demands a guess at the subject or a grasp of the situation that is contingent, dubious, fallible, and uncertain.
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| − | | |
| − | If all this is true - if inquiry begins with doubt, if every significant hypothesis is itself a dubious proposition, if the making and the testing of a hypothesis are instances of equally doubtful actions, and if every action has the potential to alter the very situation and the very subject matter that are being addressed - then it leads to the critical question: How is the conduct of inquiry, that begins by making a hypothesis and that continues by testing this description in action, supposed to help with the situation of uncertainty that incites it in the first place and that is supposed to maintain its motivation until the end is reached? The danger is that the posing of a hypothesis may literally introduce an irreversible change in the situation or the subject matter in question. The fear is that this change might be one that too conveniently fulfills or too perversely subverts the very hypothesis that engenders it, that it may obstruct the hypothesis from ever being viewed with equanimity again, and thus prevent the order of reflection that is needed to amend or discard the hypothesis when the occasion to do so arises.
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| − | | |
| − | If one fears that merely contemplating a special hypothesis is enough to admit a spurious demonstration into the foundations of one's reasoning, even to allow a specious demon to subvert all one's hopes of a future rationality and to destroy all one's chances of a reasonable share of knowledge, then one is hardly in a state of mind that can tolerate the tensions of a full-fledged, genuine inquiry. If one is beset with such radical doubts, then all inquiry is no more comfort than pure enchoiry. Sometimes it seems like the best you can do is sing yourself a song that soothes your doubts. Perhaps it is even quite literally true that all inquiry comes back at last to a form of "enchoiry", the invocation of a nomos, a way of life, or a song and a dance. But even if this is the ultimate case, it does no harm and it does not seem like a bad idea to store up in this song one or two bits of useful lore, and to weave into its lyric a few suggestions of a practical character.
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| − | | |
| − | Let us now put aside these more radical doubts. This putting aside of doubts is itself a form of inquiry, that is, a way of allaying doubts. The fact that I appear to do this by fiat, and to beg for tacit assent, tends to make me suspect the validity of this particular tactic. Still, it is not too inanely dismissive, as its appeal is based on an argument, the argument that continuing to entertain this type of doubt leads to a paralysis of the reason, and that paralyzing the ability to think is not in the interests of the agent concerned. Thus, I adopt the hypothesis that the relationship between the world and the mind is not so perverse that merely making a hypothesis is enough to alter the nature of either. If, in future, I or anyone sees the need to reconsider this hypothesis, then I see nothing about making it that prevents anyone from doing so. Indeed, making it explicit only renders it more subject to reflection.
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| − | | |
| − | Of course, a finite person can only take up so many causes in a single lifetime, and so there is always the excuse of time for not chasing down every conceivable hypothesis that comes to mind.
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| − | | |
| − | =====1.4.1.4 Forms of Relations=====
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| − | | |
| − | The next distingishing trait that I can draw out of this incipient treatise is its emphasis on the forms of relations. From a sufficiently "formal and relational" (FAR) point of view, many of the complexities that arise from throwing intentions, objectives, and purposes into the mix of discussion are conceivably due to the greater arity of triadic relations over dyadic relations, and do not necessarily implicate any differences of essence inhering in the entities and the states invoked. As far as this question goes, whether a dynamic object is essentially different from a deliberate object, I intend to remain as neutral as possible, at least, until forced by some good reason to do otherwise. In the meantime, the factors that are traceable to formal differences among relations are ready to be investigated and useful to examine. With this in mind, it it useful to make the following definition:
| |
| − | | |
| − | A "conduct relation" is a triadic relation involving a domain of objects and two domains of states. When a shorter term is desired, I refer to a conduct relation as a "conduit". A conduit is given in terms of its extension as a subset C c XxYxZ, where X is the "object domain" and where Y and Z are the "state domains". Typically, Y = Z.
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| − | | |
| − | In general, a conduct relation serves as a "model of conduct" (MOC), not always the kind of model that is meant to be emulated, but the type of model that captures an aspect of structure in a form of conduct.
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| − | | |
| − | The question arises: What is the relationship between signs and states? On the assumption that signs and states are comparable in their levels of generality, consider the following possibilities:
| |
| − | | |
| − | 1. Signs are special cases of states.
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| − | | |
| − | 2. Signs and states are the same sorts of things.
| |
| − | | |
| − | 3. States are special cases of signs.
| |
| − | | |
| − | Depending on how one answers this question, one is also choosing among the following options:
| |
| − | | |
| − | 1. Sign relations are special cases of conduct relations.
| |
| − | | |
| − | 2. Sign relations and conduct relations are the same sorts of things.
| |
| − | | |
| − | 3. Conduct relations are special cases of sign relations.
| |
| − | | |
| − | I doubt if there is any hard and fast answer to this question, but think that it depends on particular interpreters and particular observers, to what extent each one interprets a state as a sign, and to what degree each one recognizes a sign as a component of a state.
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| − | | |
| − | =====1.4.1.5 Models of Inquiry=====
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| − | | |
| − | The value of a hypothesis, or the worth of a model, is not to be given a prior justification, as by a deductive proof, but has to be examined in practice, as by an empirical probation. It is not intended to be taken for granted or to go untested, but its meaning in practice has to be articulated before its usefulness can be judged. This means that the conceivable practical import of the hypothesis or the model has to be developed in terms of its predicted and its promised consequences, after which it is judged by the comparison of these speculative consequences with the actual results. But this is not the end of the matter, for it can be a useful piece of information to discover that a particular kind of conception fails a particular kind of comparison. Thus, the final justification for a hypothesis or a model is contained in the order of work that it leads one to do, and the value of this work is often the same whether or not its premiss is true. Indeed, the fruitfulness of a suggestion can lie in the work that proves it untrue.
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| − | | |
| − | My plan then has to be, rather than trying to derive a model of inquiry in a deductive fashion from a number of conditions like y0 = y.y, only to propose a plausible model, and then to test it under such conditions. Each of these tests is a "two-edged sword", and the result of applying a particular test to a proposed model can have either one of two effects. If one believes that a particular test is a hard and fast rule of inquiry, or a condition that any inquiry is required to satisfy, then the failure of a model to live up to its standard tends only to rule out that model. If one has reason to believe that a particular model of inquiry covers a significant number of genuine examples, then the failure of these models to follow the prescribed rule can reflect badly on the test itself.
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| − | | |
| − | In order to prime the pump, therefore, let me offer the following account of inquiry in general, the whole of which can be taken as a plausible hypothesis about the nature of inquiry in general.
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| − | | |
| − | My observations of inquiry in general, together with a few suggestions that seem apt to me, have led me to believe that inquiry begins with a "surprise" or a "problem". The way I understand these words, they refer to departures, differences, or discrepancies among various modalities of experience, in particular, among "observations", "expectations", and "intentions".
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| − | | |
| − | 1. A "surprise" is a departure of an observation from an expectation, and thus it invokes a comparison between present experience and past experience, since expectations are based on the remembered disposition of past experience.
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| − | | |
| − | 2. A "problem" is a departure of an observation from an intention, and thus it invokes a comparison between present experience and future experience, since intentions choose from the envisioned disposition of future experience.
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| − | | |
| − | With respect to these
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| − | | |
| − | With respect to this hypothetical
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| − | | |
| − | I now test this model of inquiry under the conditions of an inquiry into inquiry, asking whether it is consistent in its application to itself. This leaves others to test the models they like best under the same conditions, should they ever see the need to do so.
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| − | | |
| − | Does the inquiry into inquiry begin with a surprise or a problem concerning the process or the conduct of inquiry? In other words, does the inquiry into inquiry start with one of the following forms of departure: (1) a surprising difference between what is expected of inquiry and what is observed about it, or (2) a problematic difference between what is observed about inquiry and what is intended for it?
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| − | | |
| − | ====1.4.2 The Moment of Inquiry====
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| − | | |
| − | <blockquote>
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| − | <p>Every young man - not to speak of old men - on hearing or seeing anything unusual and strange, is likely to avoid jumping to a hasty and impulsive solution of his doubts about it, and to stand still; just as a man who has come to a crossroads and is not quite sure of his way, if he be travelling alone, will question himself, or if travelling with others, will question them too about the matter in doubt, and refuse to proceed until he has made sure by investigation of the direction of his path.</p>
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| − | | |
| − | <p>(Plato, Laws, VII, 799C).</p>
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| − | </blockquote>
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| − | | |
| − | Observe the paradox of this precise ambiguity: That both the occasion and the impulse of inquiry are instances of a negative moment. But the immediate discussion is aimed at the positive aspects of inquiry, and so I convert this issue into its corresponding positive form.
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| − | | |
| − | The positive aim of inquiry is a state of belief, certainty, or knowledge. There are distinctions that can be made in the use of these words, but the question remains as to what kind of distinctions these are. In my opinion, the differences that arise in practice have more to do with the purely grammatical distinctions of "case", "mood", "number", "person", and "voice", and thus raise the issues of plurality and point of view, as opposed to indicating substantial differences in the relevant features of state, as actually experienced by the agent concerned.
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| − | It is often claimed that there are signficant differences between the conditions of belief and knowledge, but the way that I understand the distinction is as follows. One says that a person "knows" something when that person believes exactly the same thing that one believes. When one is none other than the person in question, then one says that one "knows" exactly what one believes. Differences arise between the invocations of "belief" and "knowledge" only when more than one person is involved in the issue. Thus, there is no occasion for a difference between belief and knowledge unless there is more than one person that is being consulted about the matter in question, or else a single person in a divided state of opinion, in any case, when there is more than one impulse, moment, or occasion that currently falls under consideration.
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| − | | |
| − | In any case, belief or knowledge is the feature of state that an agent of inquiry lacks at the moment of setting out. Inquiry begins in a state of impoverishment, need, or privation, a state that is absent the quality of certainty. It is due to this feature that the agent is motivated, and it is on account of its continuing absence that the agent keeps on striving to achieve it, at least, with respect to the subject in question, and, at any rate, in sufficient measure to make action possible.
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| − | | |
| − | ====1.4.3 The Modes of Inquiry====
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| − | | |
| − | <blockquote>
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| − | <p>Let the strange fact be granted, we say, that our hymns are now made into "nomes" (laws), just as the men of old, it would seem, gave this name to harp-tunes, - so that they, too, perhaps, would not wholly disagree with our present suggestion, but one of them may have divined it vaguely, as in a dream by night or a waking vision: anyhow, let this be the decree on the matter:- In violation of public tunes and sacred songs and the whole choristry of the young, just as in violation of any other "nome" (law), no person shall utter a note or move a limb in the dance.</p>
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| − | | |
| − | <p>(Plato, Laws, VII, 799E-800A).</p>
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| − | </blockquote>
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| − | | |
| − | In the present section, I am concerned with the kinds of reasoning that might be involved in the choice of a method, that is, in discovering a way to go about inquiry, in constructing a way to carry it through, and in justifying the way that one chooses. If the choice of a method can be established on the basis of reasoning, if it can be rationalized or reconstructed on grounds that are commonly thought to be sensible, or if it is likely to be affected or influenced in any way by a rational argument, then there is reason to examine the kinds of reasoning that go into this choice. All of this requires a minimal discussion of different modes of reasoning.
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| − | | |
| − | In this work as a whole, each instance of inquiry is analyzed in accord with various modes of reasoning, the prospective "elements of inquiry", and its structure as an object of inquiry is articulated, rationalized, and reconstructed with respect to the corresponding "form of analysis", "form of synthesis", or "objective genre" (OG).
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| − | | |
| − | According to my current understanding, the elements of inquiry can be found to rest on three types of steps, called "abductive", "deductive", and "inductive" modes of inference. As a result of this opinion, I do not believe that I can do any better at present than to articulate the structure of each instance of learning or reasoning according to these three types of motions of the mind. But since this work as a whole is nowhere near complete, I cannot dictate these steps in a dogmatic style, nor will it do for me to to call the tune of this form of analysis in a purely ritual or a wholly routine fashion.
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| − | | |
| − | Since the complexity of reasoning about different modes of reasoning is enough of a complication to occupy my attention at the present stage of development in this work, it is proably best to restrain this discussion along the majority of its other dimensions. A convenient way to do this is to limit its scope to simple examples and concrete situations, just enough to illustrate the selected modes of reasoning.
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| − | | |
| − | With all of these considerations in mind, the best plan that I can find for addressing the tasks of the present section is to proceed as follows: I make it my primary aim to examine only a few of the simplest settings in which these different modes of reasoning are able to appear, and I try to plot my path through this domain by way of concrete examples. Along the way, I discuss a few of the problems that are associated with reasoning about different modes of reasoning. Given the present stage of development, the majority of these issues have to be put aside almost as quickly as they are taken up. If they are ever going to be subject to resolution, it is not within reach of the present moment of discussion. In the body of this section, I therefore return to the initial strategy: to examine a few of the simplest cases and situations that can serve to illustrate the distinctions among the chosen modes of reasoning.
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| − | | |
| − | In trying to initiate a general discussion of the different modes of reasoning that might be available, and thus to motivate a model of this subject matter that makes an initial kind of sense to me, I meet once again with all the old "difficulties at the beginning", the kinds of obstructions that always seem to arise on trying to open up any new subject for discussion or in trying to introduce any new model of an old subject area. Much of this gratuitous bedevilment is probably due to the inherent conservatism of the human mind. Everything familiar is taken for granted, but each new picture of the situation is immediately subjected to the severest suspicions.
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| − | Now, I cannot reason with necessary force that the mind must use these particular modes of reasoning, any more than I can say that it must use a given language in order to express itself. But I can argue, relative to a particular model of thinking that must be proposed hypothetically, that certain modes of reasoning are available to the mind and are likely to be evident in its operation, if one only takes the trouble to look.
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| − | Ultimately, the model of thinking that I plan to propose makes use of the proposition that all thinking takes place in signs, and thus that inquiry is the transformation of a sign relation. Relative to this hypothesis, it would be possible to discharge the current assumptions about the basic modes of reasoning, that is, to derive the elementary modes of inquiry from a sign relational model of inquiry, and then to compare them with the current suggestions. Until this work is done, however, the assumption that these really are the most basic modes of reasoning has to be treated as a still more tentative hypothesis.
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| − | | |
| − | When a subject matter is so familiar that the logical connections between its parts are known both forwards and backwards, then it is reasonable and convenient to organize its presentation in an axiomatic fashion. This would not be such a bad idea, if it did not make it so easy to forget the nature of the reorganization that goes into a representation, and it would not constitute such a deceptive conception of the subject, if it did not mean that the exposition of the subject matter is just as often the falsification of its actual development and the covering up of its real excavation. Indeed, the logical order of axioms and theorems may have little to do with the original order of discovery and invention. In practice, the deepest axioms are often the last to come to light.
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| − | | |
| − | Once again, the structure of a reflective context means that each mode of reasoning is able to appear in a double role, once as an object and once as an instrument of the same extended discussion. And once again, the discussion runs into an array of obstructions, whose structures are becoming, if not more clear, at least, more familiar with each encounter. In particular, a description of different modes of reasoning involves a classification, and a classification presupposes a basis of distinctive features that cannot be treated as categorical, or objectively neutral, but has to be regarded as hypothetical, or potentially biased. In other words, the language that I use to describe different modes of reasoning may already have a particular model of reasoning built into it, and this disposition to a particular conception of logic may be lodged in such a way that it makes it nearly impossible to reflect on the operations and the limitations of this model.
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| − | | |
| − | Inquiry begins when a law is violated. It marks a time when a certain peace of mind is breached, it reigns all the while that a common accord is broken, disturbed, forgotten, or lost, and it rules right up until the time when a former condition of harmony is restored or until the moment when a new state of accord is established. Of course, the word "law" is a highly equivocal choice, especially to convey the sense of a founding principle. It renders not just its own meaning irrevocably subject to interpretation, but delivers into a similar subjection all the forms of understanding that depend on it. But the letter must release its hold on the spirit, if the word "law" is meant to evoke the requisite variety of connotations, and yet to maintain a sensible degree of order among their concrete meanings. Only in this way can it rise above the many different kinds of law that come into play.
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| − | | |
| − | There are descriptive laws, that organize experiences into expectations. There are prescriptive laws, that organize performances into intentions.
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| − | | |
| − | Other names for descriptive laws are "declarative" or "empirical" laws. Other names for prescriptive laws are "procedural" or "normative" laws.
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| − | | |
| − | Implicit in a descriptive law is the connection to be found or made, discovered or created, between past experience and present expectation. What one knows about these connections is kept in a descrptive model.
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| − | | |
| − | Implicit in a prescriptive law is the connection to be found or made, discovered or created, between current conduct and future experience. What one knows about these connections is kept in a prescriptive model.
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| − | | |
| − | A violation of an expectation, the contravention of a descriptive law, occurs when a present experience departs from a predicted experience, which is what a past expectation or description projected to be present. This is a "surprise", a state of affairs that calls for an explanation. An explanation points to other descriptions that better predict the actual experience, and suggests an alteration to the descriptive model that generated the expectation from a past experience.
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| − | | |
| − | A violation of an intention, the contravention of a prescriptive law, occurs when a present experience departs from a desired experience, which is what a past intention or prescription projected to be present. This is a "problem", a state of affairs that calls for a plan of action. , A plan of action points to other actions that better achieve the desired experience, and suggests an alteration to the prescriptive model that generated the conduct toward a prospective experience.
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| − | | |
| − | In the rest of this section, I treat the different modes of reasoning according to the forms that Aristotle gave them, collectively referred to as the "syllogistic" model. The discussion is kept within the bounds of propositional reasoning by considering only those "figures of syllogism" that are "purely universal", that is, the forms of argument all of whose premisses, and therefore all of whose conclusions, involve nothing but universal quantifications.
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| − | | |
| − | If it were only a matter of doing propositional reasoning as efficiently as possible, I would simply use the cactus language and be done with it, but there are several other reasons for revisiting the syllogistic model. Treating the discipline that is commonly called "logic" as a cultural subject with a rich and varied history of development, and attending to the thread of tradition in which I currently find myself, I observe what looks like a critical transition that occurs between the classical and the modern ages. Aside from supplying the barest essentials of a historical approach to the subject, a consideration of this elder standard makes it easier to appreciate the nature and the character of this transformation. In addition, and surprisingly enough to warrant further attention, there appear to be a number of cryptic relationships that exist between the syllogistic patterns of reasoning and the ostensibly more advanced forms of analysis and synthesis that are involved in the logic of relations.
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| − | | |
| − | =====1.4.3.1 Deductive Reasoning=====
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| − | | |
| − | In this subsection, I present a trimmed-down version of deductive reasoning in Aristotle, limiting the account to universal syllogisms, in effect, keeping to the level of propositional reasoning. Within these constraints, there are three basic "figures" of the syllogism.
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| − | | |
| − | In order to understand Aristotle's description of these figures, it is necessary to explain a few items of his technical terminology. In each figure of the syllogism, there are three "terms". Each term can be read as denoting either (1) a class of entities or (2) all of the members of a class of entities, depending on which interpretation the reader prefers. These terms are ranked in two ways: With respect to the "magnitudes" that they have in relation to each other, there are "major", "middle", and "minor" terms. With respect to the "positions" that they take up within the figure, there are "first", "intermediate", and "last" terms. The figures are distinguished by how the magnitudes correlate with the positions. However, the names for these rankings are not always used or translated in a rigorously systematic manner, so the reader has to be on guard to guess which type of ranking is meant.
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| − | | |
| − | In addition to this terminology, it is convenient to make use of the following nomenclature:
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| − | | |
| − | 1. The "Fact" is the proposition that applies the term in the first position to the term in the third or last position.
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| − | | |
| − | 2. The "Case" is the proposition that applies the term in the second or intermediate position to the term in the third or last position.
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| − | | |
| − | 3. The "Rule" is the proposition that applies the term in the first position to the term in the second or intermediate position.
| |
| − | | |
| − | Because the roles of Fact, Case, and Rule are defined with regard to positions rather than magnitudes they are insensitive to whether the proposition in question is being used as a premiss or is being drawn as a conclusion.
| |
| − | | |
| − | The "first figure" of the syllogism is explained as follows:
| |
| − | | |
| − | <blockquote>
| |
| − | <p>When three terms are so related to one another that the last is wholly contained in the middle and the middle is wholly contained in or excluded from the first, the extremes must admit of perfect syllogism. By "middle term" I mean that which both is contained in another and contains another in itself, and which is the middle by its position also; and by "extremes" (a) that which is contained in another, and (b) that in which another is contained. For if A is predicated of all B, and B of all C, A must necessarily be predicated of all C. ... I call this kind of figure the First.</p>
| |
| − | | |
| − | <p>(Aristotle, Prior Analytics, 1.4).</p>
| |
| − | </blockquote>
| |
| − | | |
| − | For example, suppose A is "animal", B is "bird", and C is "canary". Then there is a deductive conclusion to be drawn in the first figure.
| |
| − | | |
| − | There is the Case:
| |
| − | | |
| − | "All canaries are birds." (C => B)
| |
| − | | |
| − | There is the Rule:
| |
| − | | |
| − | "All birds are animals." (B => A)
| |
| − | | |
| − | One deduces the Fact:
| |
| − | | |
| − | "All canaries are animals." (C => A)
| |
| − | | |
| − | The propositional content of this deduction is summarized on the right. Taken at this level of detail, deductive reasoning is nothing more than an application of the transitive rule for logical implications.
| |
| − | | |
| − | The "second figure" of the syllogism is explained as follows:
| |
| − | | |
| − | <blockquote>
| |
| − | When the same term applies to all of one subject and to none of the other, or to all or none of both, I call this kind of figure the Second; and in it by the middle term I mean that which is predicated of both subjects; by the extreme terms, the subjects of which the middle is predicated; by the major term, that which comes next to the middle; and by the minor that which is more distant from it. The middle is placed outside the extreme terms, and is first by position. (Aristotle, Prior Analytics, 1.5).
| |
| − | </blockquote>
| |
| − | | |
| − | For example, suppose M is "mammal", N is "newt", and O is "opossum". Then there is a deductive conclusion to be drawn in the second figure.
| |
| − | | |
| − | There is the Fact:
| |
| − | | |
| − | "All opossums are mammals." (O => M)
| |
| − | | |
| − | There is the Rule:
| |
| − | | |
| − | "No newts are mammals." (N.M = 0)
| |
| − | | |
| − | One deduces the Case:
| |
| − | | |
| − | "No newts are opossums." (N.O = 0)
| |
| − | | |
| − | The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if O c M and if N intersects M trivially, then N must also intersect O trivially. Here, I use a raised dot "." to indicate either the conjunction of two propositions or the intersection of two classes, and I use a zero "0" to indicate either the identically false proposition or the empty class, leaving the choice of interpretation to the option of the reader.
| |
| − | | |
| − | The "third figure" of the syllogism is explained as follows:
| |
| − | | |
| − | <blockquote>
| |
| − | If one of the terms applies to all and the other to none of the same subject, or if both terms apply to all or none of it, I call this kind of figure the Third; and in it by the middle I mean that of which both the predications are made; by extremes the predicates; by the major term that which is [further from] the middle; and by the minor that which is nearer to it. The middle is placed outside the extremes, and is last by position. Aristotle, Prior Analytics, 1.6).
| |
| − | </blockquote>
| |
| − | | |
| − | It appears that this passage is only meant to mark out the limiting cases of the type. From the examples that Aristotle gives it is clear that he includes many other kinds of logical situation under this figure. Perhaps the phrase "applies to all or none" is intended to specify that a term applies "affirmatively or negatively" to another term, but is not meant to require that it applies universally so.
| |
| − | | |
| − | For example, suppose P is "poem", R is "rhapsody", and S is "sonnet". Then there is deductive conclusion to be drawn in the third figure:
| |
| − | | |
| − | There is the Fact:
| |
| − | | |
| − | "All sonnets are poems." (S => P)
| |
| − | | |
| − | There is the Case:
| |
| − | | |
| − | "Some sonnets are rhapsodies." (S.R > 0)
| |
| − | | |
| − | One deduces the Rule:
| |
| − | | |
| − | "Some rhapsodies are poems." (R.P > 0)
| |
| − | | |
| − | The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if S c P and if R intersects S non-trivially then R must intersect P non-trivially.
| |
| − | | |
| − | =====1.4.3.2 Inductive Reasoning=====
| |
| − | | |
| − | (Aristotle, Prior Analytics, 2.23).
| |
| − | | |
| − | =====1.4.3.3 Abductive Reasoning=====
| |
| − | | |
| − | A choice of method cannot be justified by deduction or by induction, at least, not wholly, but involves an element of hypothesis. In ancient times, this mode of inference to an explanatory hypothesis was described by the Greek word "apagoge", articulating an action or a process that "carries", "drives", or "leads" in a direction "away", "from", or "off". This was later translated into the Latin "abductio", and that is the source of what is today called "abduction" or "abductive reasoning". Another residue of this sense survives today in the terminology for "abductor muscles", those that "draw away (say, a limb or an eye) from a position near or parallel to the median axis of the body" (Webster's).
| |
| − | | |
| − | If an image is needed, one may think of Prometheus, arrogating for the sake of an earthly purpose the divine prerogative of the gods, and then drawing the fire of their heavenly ire for the presumption of this act. This seems to sum up pretty well, not only the necessity and the utility of hypotheses, but also the risks that one incurs in making conjectures. In other guises, abductive reasoning is the mode of inference that is used to diagnose a complex situation, one that originally presents itself under a bewildering array of signs and symptoms, and fixes it subject to the terms of a succinct "nomen" or a summary predicate. Finally, by way of offering a personal speculation, I think it is likely that this entire trio of terms, "abduction", "deduction", and "induction", have reference to a style of geometric diagrams that the Ancients originally used to illustrate their reasonings.
| |
| − | | |
| − | Abductive reasoning has also been called by other names. C.S. Peirce at times called it "presumption", perhaps because it puts a plausible assumption logically prior to the observed facts, and at other times referred to it as "retroduction", because it reasons backwards from the consequent to the antecedent of a logical implication.
| |
| − | | |
| − | In its simplest form, abductive reasoning proceeds from a "fact" that A is true, using a "rule" that B => A, to presume a "case" that B is true. Thus, if A is a surprising fact that one happens to observe, and B => A is a rule to the effect that if B is true then A necessarily follows, then guessing the case that B is true is an instance of abductive reasoning. This is a backward form of reasoning, and therefore extremely fallible, but when it works it has the effect of reducing the amount of surprise in the initial observation, and thus of partially explaining the fact.
| |
| − | | |
| − | In a slightly more complicated version, abduction proceeds from a fact that C => A, using a rule that B => A, to presume a case that C => B. This is an inessential complication, since the rule of modus ponens and the rule of transitivity are essentially equivalent in their logical force, but it is often convenient to imagine that C is the "common subject" or the "current situation" that is implicit throughout the argument, namely, the existing entity that substantiates or instantiates all of the other predicates that are invoked in its course.
| |
| − | | |
| − | Suppose I have occasion to reason as follows:
| |
| − | | |
| − | "It looks like a duck, so I guess it is a duck."
| |
| − | | |
| − | Or even more simply:
| |
| − | | |
| − | "It looks blue, therefore it is blue."
| |
| − | | |
| − | These are instances in which I am using abductive reasoning, according to the pattern of the following schema:
| |
| − | | |
| − | I observe a Fact:
| |
| − | | |
| − | "It looks like X." (X')
| |
| − | | |
| − | I have in the back of my mind a general Rule:
| |
| − | | |
| − | "If it is X, then it looks like X." (X => X')
| |
| − | | |
| − | I reason my way back from the observed Fact and the assumed Rule to assert what I guess to be the Case:
| |
| − | | |
| − | "It is X." (X)
| |
| − | | |
| − | The abduction is a hypothetical inference that results in a diagnostic conclusion, that is, a statement of opinion as to what is conjectured to be the case. In each case the operation of abductive reasoning starts from a complex configuration, involving a number of explicit observations in the foreground and a class of implicit assumptions in the background, and it offers a provisional statement about certain possibility, one that is typically less conspicuous, obvious, or prominent, but still potentially present in the situation, and hopefully serving to explain the surprising or the problematic aspects of the whole state of affairs.
| |
| − | | |
| − | What results from the abductive inference is a concept and possibly a term, for instance, "duck" or "blue". The concept attempts to grasp a vast complex of appearances within a unitary form, and the term that connotes the concept is used to put explicit bounds on what it conveys. Working in tandem, they express an approximation or a simplification, "a reduction of the manifold of phenomena to a unified conception". Finite minds cannot operate for very long with anything more than this.
| |
| − | | |
| − | The reader may have noticed some obvious distinctions between the two examples of abductive reasoning that I gave above, between the case of "looking like a duck" and the case of "looking blue". Just to mention the most glaring difference: Although a person is occasionally heard to reason out loud after the fashion of the former example, it is rare to hear anyone naturally reasoning along the lines of the latter example. Indeed, it is more likely that any appearance of doing so is always an artificial performance and a self-conscious reconstruction, if not a complete fabrication, and it is doubtful that the process of arriving at a perceptual judgment can follow this rule in just so literal a fashion.
| |
| − | | |
| − | This is true and important, but it is beside the point of the immediate discussion, which is only to identify the logical form of the inference, that is, to specify up to informational equivalence the class of conduct that is involved in each example. Thus, considering the inference as an information process, I do not care at this point whether the process is implemented by a literal-minded variety of rule-following procedure, so long as it "follows", "obeys", or "respects" these rules in the form of what it does. One can say that an information process "obeys" a set of rules in a "figurative" and a "formal" sense if the transformation that occurs in the state of information between the beginning and the end of the process has the form of a relation that can be achieved by literally following these rules with respect to the prospective class of materials.
| |
| − | | |
| − | The general drift of the strategy that is being mapped out here, the "abstract", the "formal", or the "functional" approach, is now evident. Conceptually, one partitions the space of processes into "effective", "informational", or "pragmatic" equivalence classes and then adopts the inditement of a sequence of rules as a symbolic "nomen" for the class of processes that all achieve the same class of effects. At this level of functional abstraction, the conception of a process is indifferent to the particulars of its implemenation, so long as it lives within the means of the indicated constraints. Moreover, unless there is a way to detect the nature of the "actual" process without interfering too severely with it, that is, a path-sensitive but still unobtrusive measure that can sort out a finer structure from these equivalence classes, then it is not possible to inquire any further into the supposedly "actual" details.
| |
| − | | |
| − | Similar remarks apply to every case where one attributes "law-abiding" or "rule-governed" behavior to oneself, to another person, or even to a physical process. Across this diverse spectrum of cases, it ranges from likely but not certain to unlikely but still conceivable that the action in question depends on the agent "knowing" the laws that abide or the rules that are effectively being obeyed. With this in mind, I can draw this digression on appearances to a conclusion: When I say that agents are acting according to a particular pattern of rules, it only means that it "looks like" they are. In other words, they are acting "as if" they are consciously following these rules, or they are acting just like I act when I conscientiously follow such rules. A concise way to sum all of this up is to say that a pattern of rules constitutes a model of conduct, one that I can deliberately emulate, or one that I can attribute to others by way of explaining their conduct. In attributing this model to others, or even in using it to account for my own less deliberate behavior, I am making an abductive inference.
| |
| − | | |
| − | One way to appreciate the pertinence of this point is to notice that this entire digression, concerned with explaining the similarities between "looking like a duck" and "looking blue", is itself a form of argument, making a case of abductive inference to a case of abductive inference. In short, I am reasoning according to the following pattern:
| |
| − | | |
| − | It appears to be the making of an abductive inference,
| |
| − | | |
| − | so I guess it is the making of an abductive inference.
| |
| − | | |
| − | Anyone who thinks that this style of reasoning is too chancy to be tolerated ought to observe that it is only the pattern of inference that one follows in attributing minds to others, solely on the evidence that they exhibit roughly the same array of external behaviors in reaction to various external conditions as one employs to express one's experience of roughly the same conditions.
| |
| − | | |
| − | It goes without saying that abductive reasoning is extremely fallible. The fact that it looks like a duck does not necessarily mean that it is a duck - it might be a decoy. Moreover, in most cases of actual practice the implicit rule that serves to catalyze the abductive inference is not an absolute rule or a necessary truth in its own right but may be only a contingent rule or a probable premiss. For instance, not every case of being blue presents the fact of looking blue - the conditions of observation may be trickier than that. This brings to the fore another mark that distinguishes the two examples, highlighting a potentially important difference between "looking like a duck" and "looking blue". This is the amount of oversight, or awareness and control, that an agent has with regard to an inference, in other words, the extent to which an inference really does "go without saying".
| |
| − | | |
| − | The abductive inference from "it looks blue" to "it is blue" and the abductive inference from "it looks like a duck" to "it is a duck" differ in the degrees to which they exhibit a complex of correlated properties. These variations are summed up in one sense by saying that the first, more perceptual inference is more automatic, compulsive, habitual, incorrigible, and inveterate. The correlations are summed up in the opposite sense by saying that the second, more conceptual inference is more aware, controllable, correctable, critical, deliberate, guarded, and reflective. From a fully pragmatic standpoint, these differences are naturally of critical importance. But from a purely logical standpoint, they have to be regarded as incidental aspects or secondary features of the underlying forms of inference.
| |
| − | | |
| − | There is one thing yet missing from this description of abductive reasoning, and that is its creative aspect. The description so far is likely to leave the impression that the posing of a hypothesis always takes place against a narrowly circumscribed background of established terms that are available for describing cases, and thus that it amounts to nothing more original than picking out the right label for the case. Of course, the forming of a hypothesis may be bound by the generative potential of the language that is ultimately in force, but that is a far cry from a prescriptively finite list of more or less obvious choices.
| |
| − | | |
| − | How does all of this bear on the choice of a method? In order to make a start toward answering that question, I need to consider the part that abductive reasoning plays in the inquiry into method, which is, after all, just another name for the inquiry into inquiry.
| |
| − | | |
| − | There are times when choosing a method looks more like discovering or inventing a method, a purely spontaneous creation of a novel way to proceed, but normally the choice of a path picks its way through a landscape of familiar options and mapped out opportunities, and this presupposes a description of previously observed forms of conduct and a classification of different paths from which to choose. Hence the etymology of the word "method", indicating a review of means or a study of ways.
| |
| − | | |
| − | I would now like to examine several types situations where a choice of method is involved, paying special attention to the way that abductive reasoning enters into the consideration.
| |
| − | | |
| − | Example 1.
| |
| − | | |
| − | Suppose I have occasion to reason along the following lines:
| |
| − | | |
| − | This situation looks like one in which this method will work, therefore I will proceed on the hypothesis that it will work.
| |
| − | | |
| − | The current situation (C) looks amenable (A') to this method, so I guess it really is amenable (A) to this method.
| |
| − | | |
| − | In this type of situation, my observations of the situation are reduced to a form of description that portrays it in the light of a given method, amounting to an estimate of whether the situation is a case to which the method applies. The form of the entire argument hinges on the question of whether the assurance of this application is apparent or actual.
| |
| − | | |
| − | I express my observations of the situation as a Fact:
| |
| − | | |
| − | "The current situation looks amenable." (C => A')
| |
| − | | |
| − | I have in the back of my mind a general Rule:
| |
| − | | |
| − | "If it is amenable, then it looks amenable." (A => A')
| |
| − | | |
| − | I reason my way back from the observed Fact and the assumed Rule to assert what I guess to be the Case:
| |
| − | | |
| − | "The current situation is amenable." (C => A)
| |
| − | | |
| − | As far as it goes, this style of reasoning follows the basic pattern of abductive inference. Its obvious facticity is due to the fact that the situation is being described solely in the light of a pre-selected method. That is a relatively specious way to go about describing a situation, in spite of the fact that it may be inevitable in many of the most ultimate and limiting cases. The overall effect is noticeably strained, perhaps because it results from dictating an artificial setting, attempting to reduce a situation to the patterns that one is prepared to observe, and trying to fit what is there to see into a precut frame. A more natural way to describe a situation is in terms of the freely chosen perceptual features that inform a language of affects, impressions, and sensations. But here a situation is forced to be described in terms of the prevailing operational features that constitute a language of actions, forcing the description to be limited by the actions that are available within a prescribed framework of methods.
| |
| − | | |
| − | Instead of describing a situation solely in terms of its reactive bearing, that is, wholly in terms of how it reacts to the application of a method, one can try to describe it in terms that appear to be more its own, its independent, natural, observational, perceptual, or "proper" features. What the "proper" or "object-oriented" features are and whether they can be distinguished in the end from "reactive" or "method-oriented" features are questions that cannot be answered in the early phases of an investigation.
| |
| − | | |
| − | Example 2.
| |
| − | | |
| − | Suppose I find myself reasoning as follows:
| |
| − | | |
| − | If the current world (C) is a blessed world (B),
| |
| − | | |
| − | then it is a world in which my method works (A).
| |
| − | | |
| − | Here, I call to mind an independent property of being, B, that a world or a situation can have, and I use it as a middle term to reason along the lines of the following scheme:
| |
| − | | |
| − | I express my inquiry by questioning the possibility of a certain Fact, that is, by interrogating the following statement:
| |
| − | | |
| − | "The current world is amenable." (C =?> A)
| |
| − | | |
| − | I have in the back of my mind a general Rule:
| |
| − | | |
| − | "What is blessed, is amenable." (B => A)
| |
| − | | |
| − | I reason my way back from the interrogated Fact and the assumed Rule to guess that I ought to contemplate the chances of the following Case:
| |
| − | | |
| − | "The current world is blessed." (C =?> B)
| |
| − | | |
| − | Altogether, the argument that underlies the current question of method falls into line with the following example of abductive reasoning:
| |
| − | | |
| − | I hope that C is A, so I guess I hope that C is B.
| |
| − | | |
| − | To proceed with the application of a given method on the basis of such a piece of reasoning is tantamount to the faith, the hope, or the wish that there is already the right kind of justice in the world that would make the prejudices of one's favorite method turn out to be right, that one is just lucky enough to be playing in accord with a pre-established harmony. If such a confidence is all that allows one to go on inquiring, then there is no harm in assuming it, so long as one reserves the right to question every particular of its grant, should the occasion arise.
| |
| − | | |
| − | If one abstracts from the specific content of this example and examines its underlying structure, it reveals itself as the pattern of abductive reasoning that occurs in relating complex questions to simpler questions or in reducing difficult problems to easier problems. Furthermore, the iteration of this basic kind of step motivates a downward recursion from questions of fact to questions of cases, in a hopeful search for a level of cases where most of the answers are already known.
| |
| − | | |
| − | The previous examples of inquiry into method are not very satisfactory. Indeed, their schematic forms have an absurdly sketchy character about them, and they fail to convey the realistic sorts of problems that are usually involved in reasoning about the choice of a method. The first example characterizes a situation wholly in terms of a selected method. The second example characterizes a situation in terms of a property that is nominally independent of the method chosen, but the ad hoc character of this property remains obvious. In order to reason "properly" about the choice of method, it is necessary to contemplate properties of the methods themselves, and not just the situations in which they are used.
| |
| − | | |
| − | Example 3.
| |
| − | | |
| − | If I reason that scientific method is wise because wise people use it, then I am making the hypothesis that they use it because they are wise. Here, my reasoning can be explained according to the following pattern:
| |
| − | | |
| − | I observe a fact:
| |
| − | | |
| − | "A certain conduct is done by wise people." (C => X)
| |
| − | | |
| − | I have in mind a rule:
| |
| − | | |
| − | "If a wise act, then done by wise people." (A => X)
| |
| − | | |
| − | I abduce the case:
| |
| − | | |
| − | "A certain conduct is a wise act." (C => A)
| |
| − | | |
| − | Example 4.
| |
| − | | |
| − | If I reason that scientific method is a good method on account of the fact that it works for now, then I am guessing that it works for now precisely because it is good.
| |
| − | | |
| − | I observe a fact:
| |
| − | | |
| − | "Scientific method works for now." (C => X)
| |
| − | | |
| − | I have in mind a rule:
| |
| − | | |
| − | "What is good, works for now." (A => X)
| |
| − | | |
| − | I abduce the case:
| |
| − | | |
| − | "Scientific method is good." (C => A)
| |
| − | | |
| − | As always, the abductive argument is extremely fallible. The fact that scientific method works for now can be one of its accidental features, and not due to any essential goodness that it might be thought to have.
| |
| − | | |
| − | Finally, it is useful to consider an important variation on this style of argument, one that exhibits its close relation to reasoning by analogy or inference from example. Suppose that the above argument is presented in the following manner:
| |
| − | | |
| − | Scientific method (C) has many of the features that a good method needs to have, for instance, it works for now (X), so I reason that it has all of the features of a good method, in short, that it is a good method (A).
| |
| − | | |
| − | So far, the underlying argument is exactly the same. In particular, it is important to notice that the abductive argument does not depend on the prior establishment of any known cases of good methods. As of yet, the phrase "good method" is a purely hypothetical description, a term that could easily turn out to be vacuous. One has in mind a number of properties that one thinks a good method ought to have, but who knows if there is any thing that would satisfy all of these requirements? There may be some sort of subtle contradiction that is involved in the very juxtaposition of the terms "good" and "method". In sum, it can happen that scientific method is the very first method that is being considered for membership in the class of good methods, and so it is still unknown whether the class labeled "good methods" is empty or not.
| |
| − | | |
| − | But what if an example of a good method is already known to exist, one that has all of the commonly accepted properties that appear to define what a good method ought to be? In this case, the abductive argument acquires the additional strength of an argument from analogy.
| |
| − | | |
| − | =====1.4.3.4 Analogical Reasoning=====
| |
| − | | |
| − | The classical treatment of analogical reasoning by Aristotle explains it as a combination of induction and deduction. More recently, C.S. Peirce gave two different ways of viewing the use of analogy, analyzing it into complex patterns of reasoning that involve all three types of inference. In the appropriate place, it will be useful to consider these alternative accounts of analogy in detail. At the present point, it is more useful to illustrate the different versions of analogical reasoning as they bear on the topic of choosing a method.
| |
| − | | |
| − | The next example, ostensibly concerned with reasoning about a choice of method, is still too artificial to be taken seriously for this purpose, but it does serve to illustrate Aristotle's analysis of analogical reasoning as a mixed mode of inference, involving inductive and deductive phases.
| |
| − | | |
| − | Example 5.
| |
| − | | |
| − | Suppose I reason as follows. I think I can establish it as a fact that scientific method is a good method by taking it as a case of a method that always works and by using a rule that what always works is good. I think I can establish this rule, in turn, by pointing to one or more examples of methods that share the criterial property of always working and that are already acknowledged to be good. In form, this pattern of reasoning works by noticing examples of good methods, by identifying a reason why they are good, in other words, by finding a property of the examples that seems sufficient to prove them good, and by noticing that the method in question is similar to these examples precisely in the sense that it has in common this cause, criterion, property, or reason.
| |
| − | | |
| − | In this situation, I am said to be reasoning by way of analogy, example, or paradigm. That is, I am drawing a conclusion about the main subject of discussion by way of its likeness to similar examples. These cases are like the main subject in the possession of a certain property, and the relation of this critical feature to the consequential feature of interest is assumed to be conclusive. The examples that exhibit the criterial property are sometimes known as "analogues" or "paradigms". For many purposes, one can imagine that the whole weight of evidence present in a body of examples is represented by a single example of the type, an exemplary or typical case, in short, an archetype or epitome. With this in mind, the overall argument can be presented as follows:
| |
| − | | |
| − | Suppose that there is an exemplary method (E) that I already know to be a good method (A). Then it pays to examine the other properties of the exemplary method, in hopes of finding a property (B) that explains why it is good. If scientific method (C) shares this property, then it can serve to establish that scientific method is good.
| |
| − | | |
| − | The first part of the argument is the induction of a rule:
| |
| − | | |
| − | I notice the case:
| |
| − | | |
| − | "The exemplary method always works." (E => B)
| |
| − | | |
| − | I observe the fact:
| |
| − | | |
| − | "The exemplary method is a good method." (E => A)
| |
| − | | |
| − | I induce the rule:
| |
| − | | |
| − | "What always works, is good." (B => A)
| |
| − | | |
| − | The second part of the argument is the deduction of a fact:
| |
| − | | |
| − | I notice the case:
| |
| − | | |
| − | "Scientific method always works." (C => B)
| |
| − | | |
| − | I recall the rule:
| |
| − | | |
| − | "What always works, is good." (B => A)
| |
| − | | |
| − | I deduce the fact:
| |
| − | | |
| − | "Scientific method is good." (C => A)
| |
| − | | |
| − | Example 6.
| |
| − | | |
| − | | |
| − | | |
| − | Example 7.
| |
| − | | |
| − | Suppose that several examples (S1, S2, S3) of a good method are already known to exist, ones that have a number of the commonly accepted properties (P1, P2, P3) that appear to define what a good method is. Then the abductive argument acquires the additional strength of an argument from analogy.
| |
| − | | |
| − | The first part of the argument is the abduction of a case:
| |
| − | | |
| − | I observe a set of facts:
| |
| − | | |
| − | "Scientific method is P1, P2, P3." (C => P)
| |
| − | | |
| − | I recall a set of rules:
| |
| − | | |
| − | "Bona fide inquiry is P1, P2, P3." (B => P)
| |
| − | | |
| − | I abduce the case:
| |
| − | | |
| − | "Scientific method is bona fide inquiry." (C => B)
| |
| − | | |
| − | The second part of the argument is the induction of a rule:
| |
| − | | |
| − | I notice a set of cases:
| |
| − | | |
| − | "S1, S2, S3 exemplify bona fide inquiry." (S => B)
| |
| − | | |
| − | I observe a set of facts:
| |
| − | | |
| − | "S1, S2, S3 exemplify good method." (S => A)
| |
| − | | |
| − | I induce the rule:
| |
| − | | |
| − | "Bona fide inquiry is good method." (B => A)
| |
| − | | |
| − | The third part of the argument is the deduction of a fact:
| |
| − | | |
| − | I recall the case:
| |
| − | | |
| − | "Scientific method is bona fide inquiry." (C => B)
| |
| − | | |
| − | I recall the rule:
| |
| − | | |
| − | "Bona fide inquiry is good method." (B => A)
| |
| − | | |
| − | I deduce the fact:
| |
| − | | |
| − | "Scientfic method is good method." (C => A)
| |
| − | | |
| − | Now, logically and rationally in the purest sense, the argument by analogy to an example has no more force than the abductive argument, but, empirically and existentially, the example serves, not only as a model of the method to be emulated, but as an object of experimental variation and a source of further experience.
| |
| − | | |
| − | It is time to ask the question: Why do these examples continue to maintain their unrealistic character, their comical and even ridiculous appearance, in spite of all my continuing attempts to reform them in a sensible way? It is not merely their simplicity. A simple example can be telling, if it grasps the essence of the problem, that is, so long as it captures even a single essential feature or highlights even a single critical property of the thing that one seeks to understand. It is more likely due to the circumstance that I am describing agents, methods, and situations all in one piece, that is, without any analysis, articulation, or definition of what exactly constitutes the self, the scientific method, or the world in question. It is not completely useless to consider cases of this type, since many forms of automatic, customary, and unreflective practice are underlain by arguments that are not much better that this. Of course, on reflection, their "commedius" character becomes apparent, and all deny or laugh off the suggestion that they ever think this way, but that is just the way of reflection.
| |
| − | | |
| − | In order to improve the character of the discussion on this score ...
| |
| − | | |
| − | <pre>
| |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Additional Notes
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | CFR. Note 78
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | MW = Matthew West:
| |
| − | | |
| − | MW: Do you have a Cactus Manual all in one place please?
| |
| − | | |
| − | the documentation for my 'theme one' program
| |
| − | that I wrote up for my quant psy master's
| |
| − | contains the last thing like an official
| |
| − | manual that I wrote, also an expository
| |
| − | introduction to the cactus language and
| |
| − | its application to prop calc examples.
| |
| − | may still have an ancient ascii version,
| |
| − | or else the medieval 'word' doc, or i can
| |
| − | send the mac belle version by snail express
| |
| − | if you can vouchsafe me your postal address.
| |
| − | | |
| − | in the mean time, i append a few of the expositions that
| |
| − | i have outlined here/elsewhere over the last year on-line.
| |
| − | | |
| − | pre-scanning this whole mess'o'messages for you,
| |
| − | I find one that looks to me shortest & sweetest:
| |
| − | | |
| − | http://suo.ieee.org/email/msg05694.html
| |
| − | | |
| − | since this particular synopsis is mercifully short, i will copy it out here
| |
| − | and use it to explain surcatenation, along with a few other thing that i am
| |
| − | guessing might be puzzling at first sight about what in hey's going on here.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~ARCHIVE~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | Reflective Extension of Logical Graphs (Ref Log)
| |
| − | | |
| − | Here is a formal introduction to the RefLog Syntax.
| |
| − | | |
| − | Formally speaking, we have the following set-up:
| |
| − | | |
| − | Set out the "alphabet of punctuation marks" $M$ = {" ", ",", "(", ")"}.
| |
| − | The elements of $M$ are vocalized as "blank, "comma", "links", "right".
| |
| − | | |
| − | 1. There is a parametric family of formal languages of character strings
| |
| − | such that, for each set $X$ of variable names $X$ = {"x_1", ..., "x_k"},
| |
| − | there is a formal language L($X$) over the alphabet A($X$) = $M$ |_| $X$.
| |
| − | The grammar can be given in gory detail, but most folks know it already.
| |
| − | | |
| − | | Examples. If $X$ = {"x", "y"}, then these are typical strings in L($X$):
| |
| − | |
| |
| − | | " ", "( )", "x", "y", "(x)", "(y)", "x y", "(x y)", "(x, y)", "((x)(y))", "((x, y))", ...
| |
| − | | |
| − | 2. There is a parallel family of formal languages of graphical structures,
| |
| − | generically known as "painted and rooted cacti" (PARC's), that exist in
| |
| − | a one-to-one correspondence with these string expressions, being more or
| |
| − | less roughly, at a suitable level of abstraction, their parse graphs as
| |
| − | data structures in the computer. The PARC's for the above formulas are:
| |
| − | | |
| − | | Examples.
| |
| − | | x y x y
| |
| − | | o o o---o
| |
| − | | x y x y x y \ / \ /
| |
| − | | o o o o o---o o o
| |
| − | | | x y | | x y | \ / | |
| |
| − | | @ @ @ @ @ @ @ @ @ @ @ ...
| |
| − | |
| |
| − | | " ", "( )", "x", "y", "(x)", "(y)", "x y", "(x y)", "(x, y)", "((x)(y))", "((x, y))", ...
| |
| − | | |
| − | Together, these two families of formal languages constitute a system
| |
| − | that is called the "reflective extension of logical graphs" (Ref Log).
| |
| − | | |
| − | Strictly speaking, Ref Log is an abstract or "uninterpreted" formal system,
| |
| − | but its expressions enjoy, as a rule, two dual interpretations that assign
| |
| − | them the meanings of propositions or sentences in "zeroth order logic" (ZOL),
| |
| − | to wit, what Peirce called the "alpha level" of his systems of logical graphs.
| |
| − | | |
| − | For example, the string expression "(x (y))" parses into the following graph:
| |
| − | | |
| − | | x y
| |
| − | | o---o
| |
| − | | |
| |
| − | | @
| |
| − | | |
| − | You can "deparse" the string off the graph by traversing
| |
| − | it like so, reading off the marks and varnames as you go.
| |
| − | | |
| − | | o---x->(--y---o
| |
| − | | ^ |
| |
| − | | | x ( y |
| |
| − | | | o-----o v
| |
| − | | | | ) )
| |
| − | | ( (|) )
| |
| − | | ^ | |
| |
| − | | | @ v
| |
| − | | |
| − | In the "existential" interpretation of RefLog,
| |
| − | in which I do my own thinking most of the time,
| |
| − | concatenation of expressions has the meaning of
| |
| − | logical conjunction, while "(x)" has the meaning
| |
| − | of "not x", and so the above string and graph have
| |
| − | a meaning of "x => y", "x implies y", "if x then y",
| |
| − | "not x without y", or anything else that's equivalent.
| |
| − | The blank expression is assigned the value of "true".
| |
| − | Hence, the expression "()" takes the value of "false".
| |
| − | The bracket expression "(x_1, x_2, ..., x_k)" is given
| |
| − | the meaning "Exactly one of the x_j is false, j=1..k".
| |
| − | Therefore, "((x_1),(x_2), ...,(x_k))" partitions the
| |
| − | universe of discourse, saying "Just one x_j is true".
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | CFR. Note 83
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | | Tantum ergo sacramentum
| |
| − | | veneremur cernui,
| |
| − | | et antiquum documentum
| |
| − | | novo cedat ritui,
| |
| − | | praestet fides supplementum
| |
| − | | sensuum defectui.
| |
| − | |
| |
| − | | So great therefore a sacrifice
| |
| − | | let us humbly adore
| |
| − | | and let the old law yield
| |
| − | | to the new rite;
| |
| − | | let faith supplement
| |
| − | | the shortcoming of the senses.
| |
| − | |
| |
| − | | Lyric by Thomas Aquinas,
| |
| − | | Music by Amadeus Mozart, KV 142 & 197.
| |
| − | | |
| − | The increasing ossification of asciification
| |
| − | is heaping up way too many old bones to bear.
| |
| − | So I am going to shift my anklage a bit, and
| |
| − | try out a new set of conventions for a while,
| |
| − | to see if I can lighten the overloading obit.
| |
| − | | |
| − | Let us try to reserve script and singly-underscored fake-fonts or formats
| |
| − | for the names of sets, as in the notations !O!, !S!, !I! that I will now
| |
| − | set aside and use from now on for the Object, Sign, Interpretant domains,
| |
| − | respectively, of an arbitrary sign relation !L! c !O! x !S! x !I!.
| |
| − | | |
| − | Among other benefits, this will serve to liberate the plain faced characters
| |
| − | for employment as the non-terminal symbols of our formal grammars, rendering
| |
| − | our formal grammatical productions far less $Capitalistic$, !Exclamatory!,
| |
| − | and overbearingly prescriptive than they be otherwise hell-bent to become.
| |
| − | | |
| − | So let me try out this new rite to see how it works out,
| |
| − | And I will not pause to rewrite the old law in its font,
| |
| − | But advise you solely of its transformed instantiations,
| |
| − | And fix my faith on imagination to sense the supplement.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | CFR. Note 92
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | I need to try and say some things at his point about
| |
| − | why formal language theory is interesting and useful,
| |
| − | but all I have at the moment are random remembrances
| |
| − | and reflections that enter my mind from time to time.
| |
| − | | |
| − | In many ways, the study of formal languages and grammars
| |
| − | is a paradigm, more, a paragon, of the situation that we
| |
| − | face whenever we inquire into a complex reality, that is,
| |
| − | all of the ever-renewed sources of puzzling phenomena or
| |
| − | pressing problems that we call a world.
| |
| − | | |
| − | The archtypical place of formal language theory is well
| |
| − | understood in many quarters, and has been from the very
| |
| − | outset of its constellation as an independent viewpoint.
| |
| − | | |
| − | In this paradigmatic (analogical or exemplary) way of
| |
| − | understanding it, a formal language is the "data" and
| |
| − | a formal grammar is the "theory", and the question is,
| |
| − | as always, whether a theory accounts for and explains
| |
| − | the data, a "fitting" relationship that may be viewed
| |
| − | in many ways, for one, the way that a theory might be
| |
| − | said to "generate" the data, or perhaps better stated,
| |
| − | not just to "cook" in a precociously specious fashion
| |
| − | but more like to "regenerate" the form after the fact.
| |
| − | | |
| − | That's all that I can manage to express at the moment,
| |
| − | but maybe it will supply a grub-stake of motivational
| |
| − | victuals for the grueling labors of exploration ahead.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Outline
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | Inquiry Driven Systems
| |
| − | | |
| − | 1. Research Proposal
| |
| − | | |
| − | 1.1. Outline of the Project: Inquiry Into Inquiry
| |
| − | | |
| − | 1.1.1. Problem
| |
| − | | |
| − | 1.1.2. Method
| |
| − | | |
| − | 1.1.2.1. The Paradigmatic & Process-Analytic Phase
| |
| − | | |
| − | 1.1.2.2. The Paraphrastic & Faculty-Synthetic Phase
| |
| − | | |
| − | 1.1.2.3. Reprise of Methods
| |
| − | | |
| − | 1.1.3. Criterion
| |
| − | | |
| − | 1.1.4. Application
| |
| − | | |
| − | 1.2. Onus of the Project: No Way But Inquiry
| |
| − | | |
| − | 1.2.1. A Modulating Prelude
| |
| − | | |
| − | 1.2.2. A Fugitive Canon
| |
| − | | |
| − | | |
| − | | |
| − | 1.3. Option of the Project: A Way Up To Inquiry
| |
| − | | |
| − | 1.3.1. Initial Analysis of Inquiry Allegro Aperto
| |
| − | | |
| − | 1.3.2. Discussion of Discussion
| |
| − | | |
| − | 1.3.3. Discussion of Formalization: General Topics
| |
| − | | |
| − | 1.3.3.1. A Formal Charge
| |
| − | | |
| − | 1.3.3.2. A Formalization of Formalization?
| |
| − | | |
| − | 1.3.3.3. A Formalization of Discussion?
| |
| − | | |
| − | 1.3.3.4. A Concept of Formalization
| |
| − | | |
| − | 1.3.3.5. A Formal Approach
| |
| − | | |
| − | 1.3.3.6. A Formal Development
| |
| − | | |
| − | 1.3.3.7. A Formal Perasion
| |
| − | | |
| − | 1.3.4. Discussion of Formalization: Concrete Examples
| |
| − | | |
| − | 1.3.4.1. Formal Models: A Sketch
| |
| − | | |
| − | 1.3.4.2. Sign Relations: A Primer
| |
| − | | |
| − | 1.3.4.3. Semiotic Equivalence Relations
| |
| − | | |
| − | 1.3.4.4. Graphical Representations
| |
| − | | |
| − | 1.3.4.5. Taking Stock
| |
| − | | |
| − | 1.3.4.6. The "Meta" Question
| |
| − | | |
| − | 1.3.4.7. Iconic Signs
| |
| − | | |
| − | 1.3.4.8. The Conflict of Interpretations
| |
| − | | |
| − | 1.3.4.9. Indexical Signs
| |
| − | | |
| − | 1.3.4.10. Sundry Problems
| |
| − | | |
| − | 1.3.4.11. Review & Prospect
| |
| − | | |
| − | 1.3.4.12. Objective Plans & Levels
| |
| − | | |
| − | 1.3.4.13. Formalization of OF: Objective Levels
| |
| − | | |
| − | 1.3.4.14. Application of OF: Generic Level
| |
| − | | |
| − | 1.3.4.15. Application of OF: Motive Level
| |
| − | | |
| − | 1.3.4.16. The Integration of Frameworks
| |
| − | | |
| − | 1.3.4.17. Recapitulation: A Brush with Symbols
| |
| − | | |
| − | 1.3.4.18. C'est Moi
| |
| − | | |
| − | 1.3.4.19. Entr'acte
| |
| − | | |
| − | 1.3.5. Discussion of Formalization: Specific Objects
| |
| − | | |
| − | 1.3.5.1. The Will to Form
| |
| − | | |
| − | 1.3.5.2. The Forms of Reasoning
| |
| − | | |
| − | 1.3.5.3. A Fork in the Road
| |
| − | | |
| − | 1.3.5.4. A Forged Bond
| |
| − | | |
| − | 1.3.5.5. A Formal Account
| |
| − | | |
| − | 1.3.5.6. Analogs, Icons, Models, Surrogates
| |
| − | | |
| − | 1.3.5.7. Steps & Tests of Formalization
| |
| − | | |
| − | 1.3.5.8. Puck, the Ref
| |
| − | | |
| − | 1.3.5.9. Partial Formalizations
| |
| − | | |
| − | 1.3.5.10. A Formal Utility
| |
| − | | |
| − | 1.3.5.11. A Formal Aesthetic
| |
| − | | |
| − | 1.3.5.12. A Formal Apology
| |
| − | | |
| − | 1.3.5.13. A Formal Suspicion
| |
| − | | |
| − | 1.3.5.14. The Double Aspect of Concepts
| |
| − | | |
| − | 1.3.5.15. A Formal Permission
| |
| − | | |
| − | 1.3.5.16. A Formal Invention
| |
| − | | |
| − | 1.3.6. Recursion in Perpetuity
| |
| − | | |
| − | 1.3.7. Processus, Regressus, Progressus
| |
| − | | |
| − | 1.3.8. Rondeau Tempo di Menuetto
| |
| − | | |
| − | 1.3.9. Reconnaissance
| |
| − | | |
| − | 1.3.9.1. The Informal Context
| |
| − | | |
| − | 1.3.9.2. The Epitext
| |
| − | | |
| − | 1.3.9.3. The Formative Tension
| |
| − | | |
| − | 1.3.10. Recurring Themes
| |
| − | | |
| − | 1.3.10.1. Preliminary Notions
| |
| − | | |
| − | 1.3.10.2. Intermediary Notions
| |
| − | | |
| − | 1.3.10.3. Propositions & Sentences
| |
| − | | |
| − | 1.3.10.4. Empirical Types & Rational Types
| |
| − | | |
| − | 1.3.10.5. Articulate Sentences
| |
| − | | |
| − | 1.3.10.6. Stretching Principles
| |
| − | | |
| − | 1.3.10.7. Stretching Operations
| |
| − | | |
| − | 1.3.10.8. The Cactus Patch
| |
| − | | |
| − | 1.3.10.9. The Cactus Language: Syntax
| |
| − | | |
| − | 1.3.10.10. The Cactus Language: Stylistics
| |
| − | | |
| − | 1.3.10.11. The Cactus Language: Mechanics
| |
| − | | |
| − | 1.3.10.12. The Cactus Language: Semantics
| |
| − | | |
| − | 1.3.10.13. Stretching Exercises
| |
| − | | |
| − | 1.3.10.14. Syntactic Transformations
| |
| − | | |
| − | 1.3.10.15. Derived Equivalence Relations
| |
| − | | |
| − | 1.3.10.16. Digression on Derived Relations
| |
| − | | |
| − | | |
| − | | |
| − | 1.4. Outlook of the Project: All Ways Lead to Inquiry
| |
| − | | |
| − | 1.4.1. The Matrix of Inquiry
| |
| − | | |
| − | 1.4.1.1. Inquiry as Conduct
| |
| − | | |
| − | 1.4.1.2. Types of Conduct
| |
| − | | |
| − | 1.4.1.3. Perils of Inquiry
| |
| − | | |
| − | 1.4.1.4. Forms of Relations
| |
| − | | |
| − | 1.4.1.5. Models of Inquiry
| |
| − | | |
| − | 1.4.2. The Moment of Inquiry
| |
| − | | |
| − | 1.4.3. The Modes of Inquiry
| |
| − | | |
| − | 1.4.3.1. Deductive Reasoning
| |
| − | | |
| − | 1.4.3.2. Inductive Reasoning
| |
| − | | |
| − | 1.4.3.3. Abductive Reasoning
| |
| − | | |
| − | 1.4.3.4. Analogical Reasoning
| |
| − | | |
| − | | |
| − | | |
| − | 1.5.Obstacles to the Project: In the Way of Inquiry
| |
| − | | |
| − | 1.5.1. The Initial Unpleasantness
| |
| − | | |
| − | 1.5.2. The Justification Trap
| |
| − | | |
| − | 1.5.3. A Formal Apology
| |
| − | | |
| − | 1.5.3.1. Category Double-Takes
| |
| − | | |
| − | 1.5.3.2. Conceptual Extensions
| |
| − | | |
| − | 1.5.3.3. Explosional Recombinations
| |
| − | | |
| − | 1.5.3.4. Interpretive Frameworks
| |
| − | | |
| − | 1.5.4. A Material Exigency
| |
| − | | |
| − | 1.5.5. A Reconciliation of Accounts
| |
| − | | |
| − | 1.5.6. Objections to Reflexive Inquiry
| |
| − | | |
| − | 1.5.7. Empirical Considerations
| |
| − | | |
| − | 1.5.8. Computational Considerations
| |
| − | | |
| − | 1.5.8.1. A Form of Recursion
| |
| − | | |
| − | 1.5.8.2. A Power of Abstraction
| |
| − | | |
| − | | |
| − | | |
| − | 1.6. Orientation of the Project: A Way Into Inquiry
| |
| − | | |
| − | 1.6.1. Initial Description of Inquiry
| |
| − | | |
| − | 1.6.2. Terms of Analysis
| |
| − | | |
| − | 1.6.2.1. Digression on Signs
| |
| − | | |
| − | 1.6.2.2. Empirical Status of ID
| |
| − | | |
| − | 1.6.3. Expansion of Terms
| |
| − | | |
| − | 1.6.3.1. Agency
| |
| − | | |
| − | 1.6.3.2. Abstraction
| |
| − | | |
| − | 1.6.3.3. Analogy
| |
| − | | |
| − | 1.6.3.4. Accuracy
| |
| − | | |
| − | 1.6.3.5. Authenticity
| |
| − | | |
| − | 1.6.4. Anchoring Terms in Phenomena
| |
| − | | |
| − | 1.6.4.1. A Mistaken ID
| |
| − | | |
| − | 1.6.4.2. Phenomenology of Doubt
| |
| − | | |
| − | 1.6.4.3. Modalities of Knowledge
| |
| − | | |
| − | 1.6.5. Sets, Systems, & Substantive Agents
| |
| − | | |
| − | 1.6.6. Interpretive Systems
| |
| − | | |
| − | 1.6.6.1. Syntactic Systems
| |
| − | | |
| − | 1.6.6.2. Semantic Systems
| |
| − | | |
| − | 1.6.6.3. Pragmatic Systems
| |
| − | | |
| − | 1.6.7. Inquiry Driven Systems
| |
| − | | |
| − | 1.6.7.1. A Definition of Inquiry
| |
| − | | |
| − | 1.6.7.2. The Faculty of Inquiry
| |
| − | | |
| − | 1.6.7.3. A Definition of Determination
| |
| − | | |
| − | 1.6.7.4. A Definition of Definition
| |
| − | | |
| − | | |
| − | | |
| − | 1.7. Organization of the Project: A Way Through Inquiry
| |
| − | | |
| − | 1.7.1. The Problem: Inquiry Found as an Object of Study
| |
| − | | |
| − | 1.7.2. The Method: Inquiry Found as a Means of Study
| |
| − | | |
| − | 1.7.2.1. Conditions for the Possibility of Inquiry into Inquiry
| |
| − | | |
| − | 1.7.2.2. Conditions for the Success of Inquiry into Inquiry
| |
| − | | |
| − | 1.7.3. The Criterion: Inquiry in Search of a Sensible End
| |
| − | | |
| − | 1.7.3.1. The Irritation of Doubt, and The Scratch Test.
| |
| − | | |
| − | 1.7.3.2. Enabling Provision 1: The Scenes & Context of Inquiry.
| |
| − | | |
| − | 1.7.3.3. Enabling Provision 2: The Stages & Content of Inquiry.
| |
| − | | |
| − | 1.8. Objectives of the Project: Inquiry All the Way
| |
| − | | |
| − | 1.8.1. Substantial Objective
| |
| − | | |
| − | 1.8.1.1. Objective 1a: The Propositions as Types Analogy.
| |
| − | | |
| − | 1.8.1.2. Objective 1b: The Styles of Proof Development.
| |
| − | | |
| − | 1.8.1.3. Objective 1c: The Analysis of Interpreters, or A Problem with Authority.
| |
| − | | |
| − | 1.8.2. Instrumental Objective
| |
| − | | |
| − | 1.8.3. Coordination of Objectives
| |
| − | | |
| − | 1.8.4. Recapitulation: Da Capo, Al Segno
| |
| − | | |
| − | | |
| − | | |
| − | 2. Discussion of Inquiry
| |
| − | | |
| − | 2.1. Approaches to Inquiry
| |
| − | | |
| − | 2.1.1. The Classical Framework: Syllogistic Approaches
| |
| − | | |
| − | 2.1.2. The Pragmatic Framework: Sign-Theoretic Approaches
| |
| − | | |
| − | 2.1.3. The Dynamical Framework: System-Theoretic Approaches
| |
| − | | |
| − | 2.1.3.1. Inquiry & Computation
| |
| − | | |
| − | 2.1.3.2. Inquiry Driven Systems
| |
| − | | |
| − | 2.2. The Context of Inquiry
| |
| − | | |
| − | 2.2.1. The Field of Observation
| |
| − | | |
| − | 2.2.2. The Problem of Reflection
| |
| − | | |
| − | 2.2.3. The Problem of Reconstruction
| |
| − | | |
| − | 2.2.4. The Trivializing of Integration
| |
| − | | |
| − | 2.2.5. Tensions in the Field of Observation
| |
| − | | |
| − | 2.2.6. Problems of Representation & Communication
| |
| − | | |
| − | | |
| − | | |
| − | 2.3. The Conduct of Inquiry
| |
| − | | |
| − | 2.3.1. Introduction
| |
| − | | |
| − | 2.3.2. The Types of Reasoning
| |
| − | | |
| − | 2.3.2.1. Deduction
| |
| − | | |
| − | 2.3.2.2. Induction
| |
| − | | |
| − | 2.3.2.3. Abduction
| |
| − | | |
| − | 2.3.3. Hybrid Types of Inference
| |
| − | | |
| − | 2.3.3.1. Analogy
| |
| − | | |
| − | 2.3.3.2. Inquiry
| |
| − | | |
| − | 2.3.4. Details of Induction
| |
| − | | |
| − | 2.3.4.1. Learning
| |
| − | | |
| − | 2.3.4.2. Transfer
| |
| − | | |
| − | 2.3.4.3. Testing
| |
| − | | |
| − | 2.3.5. The Stages of Inquiry
| |
| − | | |
| − | | |
| − | | |
| − | 3. The Medium & Its Message
| |
| − | | |
| − | 3.1. Reflective Expression
| |
| − | | |
| − | 3.1.1. Casual Reflection
| |
| − | | |
| − | 3.1.1.1. Ostensibly Recursive Texts
| |
| − | | |
| − | 3.1.1.2. Analogical Recursion
| |
| − | | |
| − | 3.1.2. Conscious Reflection
| |
| − | | |
| − | 3.1.2.1. The Signal Moment
| |
| − | | |
| − | 3.1.2.2. The Symbolic Object
| |
| − | | |
| − | 3.1.2.3. The Endeavor to Communicate
| |
| − | | |
| − | 3.1.2.4. The Medium of Communication
| |
| − | | |
| − | 3.1.2.5. The Ark of Types: The Order of Things to Come.
| |
| − | | |
| − | 3.1.2.6. The Epitext
| |
| − | | |
| − | 3.1.2.7. The Context of Interpretation
| |
| − | | |
| − | 3.1.2.8. The Formative Tension
| |
| − | | |
| − | 3.1.2.9. The Vehicle of Communication: Reflection on the Scene, Reflection on the Self.
| |
| − | | |
| − | | |
| − | 3.1.2.10. (7)
| |
| − | | |
| − | 3.1.2.11. (6)
| |
| − | | |
| − | 3.1.2.12. Recursions: Possible, Actual, Necessary
| |
| − | | |
| − | 3.1.2.13. Ostensibly Recursive Texts
| |
| − | | |
| − | 3.1.2.14. (3)
| |
| − | | |
| − | 3.1.2.15. The Freedom of Interpretation
| |
| − | | |
| − | 3.1.2.16. The Eternal Return
| |
| − | | |
| − | 3.1.2.17. (1)
| |
| − | | |
| − | 3.1.2.18. Information in Formation
| |
| − | | |
| − | 3.1.2.19. Reflectively Indexical Texts
| |
| − | | |
| − | 3.1.2.20. (4)
| |
| − | | |
| − | 3.1.2.21. (5)
| |
| − | | |
| − | 3.1.2.22. (6)
| |
| − | | |
| − | 3.1.2.23. (7)
| |
| − | | |
| − | 3.1.2.24. (8)
| |
| − | | |
| − | 3.1.2.25. The Discursive Universe
| |
| − | | |
| − | 3.1.2.26. (7)
| |
| − | | |
| − | 3.1.2.27. (6)
| |
| − | | |
| − | 3.1.2.28. (5)
| |
| − | | |
| − | 3.1.2.29. (4)
| |
| − | | |
| − | 3.1.2.30. (3)
| |
| − | | |
| − | 3.1.2.31. (2)
| |
| − | | |
| − | 3.1.2.32. (1)
| |
| − | | |
| − | | |
| − | | |
| − | 3.2. Reflective Inquiry
| |
| − | | |
| − | 3.2.1. Integrity & Unity of Inquiry
| |
| − | | |
| − | 3.2.2. Apparitions & Allegations
| |
| − | | |
| − | 3.2.3. A Reflective Heuristic
| |
| − | | |
| − | 3.2.4. Either/Or: A Sense of Absence
| |
| − | | |
| − | 3.2.5. Apparent, Occasional, & Practical Necessity
| |
| − | | |
| − | 3.2.6. Approaches, Aspects, Exposures, Fronts
| |
| − | | |
| − | 3.2.7. Synthetic A Priori Truths
| |
| − | | |
| − | 3.2.8. Priorisms of Normative Sciences
| |
| − | | |
| − | 3.2.9. Principle of Rational Action
| |
| − | | |
| − | 3.2.10. The Pragmatic Cosmos
| |
| − | | |
| − | 3.2.11. Reflective Interpretive Frameworks
| |
| − | | |
| − | 3.2.11.1. Principals Versus Principles
| |
| − | | |
| − | 3.2.11.2. The Initial Description of Inquiry
| |
| − | | |
| − | 3.2.11.3. An Early Description of Interpretation
| |
| − | | |
| − | 3.2.11.4. Descriptions of the Mind
| |
| − | | |
| − | 3.2.11.5. Of Signs & the Mind
| |
| − | | |
| − | 3.2.11.6. Questions of Justification
| |
| − | | |
| − | 3.2.11.7. The Experience of Satisfaction
| |
| − | | |
| − | 3.2.11.8. An Organizational Difficulty
| |
| − | | |
| − | 3.2.11.9. Pragmatic Certainties
| |
| − | | |
| − | 3.2.11.10. Problems & Methods
| |
| − | | |
| − | | |
| − | | |
| − | 3.3. Reflection on Reflection
| |
| − | | |
| − | 3.4. Reflective Interpretive Frameworks
| |
| − | | |
| − | 3.4.1. The Phenomenology of Reflection
| |
| − | | |
| − | 3.4.2. A Candid Point of View
| |
| − | | |
| − | 3.4.3. A Projective Point of View
| |
| − | | |
| − | 3.4.4. A Formal Point of View
| |
| − | | |
| − | 3.4.5. Three Styles of Linguistic Usage
| |
| − | | |
| − | 3.4.6. Basic Notions of Group Theory
| |
| − | | |
| − | 3.4.7. Basic Notions of Formal Language Theory
| |
| − | | |
| − | 3.4.8. A Perspective on Computation
| |
| − | | |
| − | 3.4.9. Higher Order Sign Relations: Introduction
| |
| − | | |
| − | 3.4.10. Higher Order Sign Relations: Examples
| |
| − | | |
| − | 3.4.11. Higher Order Sign Relations: Application
| |
| − | | |
| − | 3.4.12. Issue 1: The Status of Signs
| |
| − | | |
| − | 3.4.13. Issue 2: The Status of Sets
| |
| − | | |
| − | 3.4.14. Issue 3: The Status of Variables
| |
| − | | |
| − | 3.4.15. Propositional Calculus
| |
| − | | |
| − | 3.4.16. Recursive Aspects
| |
| − | | |
| − | 3.4.17. Patterns of Self-Reference
| |
| − | | |
| − | 3.4.18. Practical Intuitions
| |
| − | | |
| − | 3.4.19. Examples of Self-Reference
| |
| − | | |
| − | 3.4.20. Three Views of Systems
| |
| − | | |
| − | 3.4.21. Building Bridges Between Representations
| |
| − | | |
| − | 3.4.22. Extensional Representations of Sign Relations
| |
| − | | |
| − | 3.4.23. Intensional Representations of Sign Relations
| |
| − | | |
| − | 3.4.24. Literal Intensional Representations
| |
| − | | |
| − | | |
| − | 3.4.25. Analytic Intensional Representations
| |
| − | | |
| − | 3.4.26. Differential Logic & Directed Graphs
| |
| − | | |
| − | 3.4.27. Differential Logic & Group Operations
| |
| − | | |
| − | 3.4.28. The Bridge: From Obstruction to Opportunity
| |
| − | | |
| − | 3.4.29. Projects of Representation
| |
| − | | |
| − | 3.4.30. Connected, Integrated, Reflective Symbols
| |
| − | | |
| − | 3.4.31. Generic Orders of Relations
| |
| − | | |
| − | 3.4.32. Partiality: Selective Operations
| |
| − | | |
| − | 3.4.33. Sign Relational Complexes
| |
| − | | |
| − | 3.4.34. Set-Theoretic Constructions
| |
| − | | |
| − | 3.4.35. Reducibility of Sign Relations
| |
| − | | |
| − | 3.4.36. Irreducibly Triadic Relations
| |
| − | | |
| − | 3.4.37. Propositional Types
| |
| − | | |
| − | 3.4.38. Considering the Source
| |
| − | | |
| − | 3.4.39. Prospective Indices: Pointers to Future Work
| |
| − | | |
| − | 3.4.40. Dynamic & Evaluative Frameworks
| |
| − | | |
| − | 3.4.41. Elective & Motive Forces
| |
| − | | |
| − | 3.4.42. Sign Processes: A Start
| |
| − | | |
| − | 3.4.43. Reflective Extensions
| |
| − | | |
| − | 3.4.44. Reflections on Closure
| |
| − | | |
| − | 3.4.45. Intelligence => Critical Reflection
| |
| − | | |
| − | 3.4.46. Looking Ahead
| |
| − | | |
| − | 3.4.47. Mutually Intelligible Codes
| |
| − | | |
| − | 3.4.48. Discourse Analysis: Ways & Means
| |
| − | | |
| − | 3.4.49. Combinations of Sign Relations
| |
| − | | |
| − | 3.4.50. Revisiting the Source
| |
| − | | |
| − | | |
| − | | |
| − | 3.5. Divertimento: Eternity in Love with the Creatures of Time
| |
| − | | |
| − | 3.5.1. Reflections on the Presentation of Examples
| |
| − | | |
| − | 3.5.2. Searching for Parameters
| |
| − | | |
| − | 3.5.3. Defect Analysis
| |
| − | | |
| − | 3.5.4. The Pragmatic Critique
| |
| − | | |
| − | 3.5.5. Pragmatic Operating Notions
| |
| − | | |
| − | 3.5.6. Defects of Presentation
| |
| − | | |
| − | 3.5.7. Dues to Process
| |
| − | | |
| − | 3.5.8. Duties to Purpose
| |
| − | | |
| − | | |
| − | | |
| − | 3.6. Computational Design Philosophy
| |
| − | | |
| − | 3.6.1. Intentional Objects & Attitudes
| |
| − | | |
| − | 3.6.2. Imperfect Design & Persistent Error
| |
| − | | |
| − | 3.6.3. Propositional Reasoning About Relations
| |
| − | | |
| − | 3.6.4. Dynamic & Evaluative Frameworks
| |
| − | | |
| − | 3.6.5. Discussion of Examples
| |
| − | | |
| − | 3.6.6. Information & Inquiry
| |
| − | | |
| − | | |
| − | | |
| − | 4. Overview of the Domain: Interpretive Inquiry
| |
| − | | |
| − | 4.1. Interpretive Bearings: Conceptual & Descriptive Frameworks
| |
| − | | |
| − | 4.1.1. Catwalks: Flexible Frameworks & Peripatetic Categories
| |
| − | | |
| − | 4.1.1.1. Eponymous Ancestors: The Precursors of Abstraction?
| |
| − | | |
| − | 4.1.1.2 Reticles: Interpretive Flexibility as a Design Issue
| |
| − | | |
| − | 4.1.2. Heuristic Inclinations & Regulative Principles
| |
| − | | |
| − | 4.2. Features of Inquiry Driven Systems
| |
| − | | |
| − | 4.2.1. The Pragmatic Theory of Signs
| |
| − | | |
| − | 4.2.1.1. Sign Relations
| |
| − | | |
| − | 4.2.1.2. Types of Signs
| |
| − | | |
| − | 4.2.2. The Pragmatic Theory of Inquiry
| |
| − | | |
| − | 4.2.2.1. Abduction
| |
| − | | |
| − | 4.2.2.2. Deduction
| |
| − | | |
| − | 4.2.2.3. Induction
| |
| − | | |
| − | 4.3. Examples of Inquiry Driven Systems
| |
| − | | |
| − | 4.3.1. "Index": A Program for Learning Formal Languages
| |
| − | | |
| − | 4.3.2. "Study": A Program for Reasoning with Propositions
| |
| − | | |
| − | 5. Discussion & Development of Objectives
| |
| − | | |
| − | 5.1. Objective 1a: Propositions as Types
| |
| − | | |
| − | 5.2. Objective 1b: Proof Styles & Developments
| |
| − | | |
| − | 5.3. Objective 1c: Interpretation & Authority
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. References
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | Aristotle, "On The Soul", in 'Aristotle, Volume 8',
| |
| − | W.S. Hett (trans.), Heinemann, London, UK, 1936, 1986.
| |
| − | | |
| − | Charniak, E. & McDermott, D.V.,
| |
| − | 'Introduction to Artificial Intelligence',
| |
| − | Addison-Wesley, Reading, MA, 1985.
| |
| − | | |
| − | 2. Charniak, E., Riesbeck, C.K., & McDermott, D.V. Artificial Intelligence Programming. Lawrence Erlbaum Associates, Hillsdale, NJ, 1980.
| |
| − | | |
| − | 3. Holland, J.H., Holyoak, K.J., Nisbett, R.E., & Thagard, P.R. Induction: Processes of Inference, Learning, and Discovery. MIT Press, Cambridge, MA, 1986.
| |
| − | | |
| − | 4. O'Rorke, P. Review of AAAI 1990 Spring Symposium on Automated Abduction. SIGART Bulletin, Vol. 1, No. 3. ACM Press, October 1990, p. 12-17.
| |
| − | | |
| − | 5. Pearl, J. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Revised 2nd printing. Morgan Kaufmann, San Mateo, CA, 1991.
| |
| − | | |
| − | 6. Peng, Y. & Reggia, J.A. Abductive Inference Models for Diagnostic Problem-Solving. Springer-Verlag, New York, NY, 1990.
| |
| − | | |
| − | 7. Sowa, J.F. Conceptual Structures: Information Processing in Mind and Machine. Addison-Wesley, Reading, MA, 1984.
| |
| − | | |
| − | 8. Sowa, J.F. (ed.) Principles of Semantic Networks: Explorations in the Representation of Knowledge. Morgan Kaufmann, San Mateo, CA, 1991.
| |
| − | | |
| − | Dewey, J. (1991). How We Think. Buffalo, NY: Prometheus Books. Originally published 1910.
| |
| − | | |
| − | Shakespeare, Wm. (1988). William Shakespeare: The Complete Works. Compact Edition. S. Wells & G. Taylor (eds.). Oxford University Press, Oxford, UK.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Email Format
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | | Document History | |
| − | | | |
| − | | Subject: Inquiry Driven Systems: An Inquiry Into Inquiry | |
| − | | Contact: Jon Awbrey <jawbrey@oakland.edu> | |
| − | | Version: Draft 10.00 | |
| − | | Created: 23 Jun 1996 | |
| − | | Revised: 02 Mar 2003 | |
| − | | Advisor: M.A. Zohdy | |
| − | | Setting: Oakland University, Rochester, Michigan, USA | |
| − | | |
| − | http://members.door.net/arisbe/menu/library/aboutcsp/awbrey/inquiry.htm
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Incitatory Note 1
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | | Each ground-principle must be proved entirely
| |
| − | | by that same kind of inference which it supports.
| |
| − | |
| |
| − | | But we cannot arrive at any conclusion
| |
| − | | by mere deduction except about symbols.
| |
| − | |
| |
| − | | We cannot arrive at any conclusion
| |
| − | | by mere induction except about things.
| |
| − | |
| |
| − | | And we cannot arrive at any conclusion
| |
| − | | by mere hypothesis except about forms.
| |
| − | |
| |
| − | | C.S. Peirce, CE 1, page 290.
| |
| − | |
| |
| − | | Charles Sanders Peirce, "On the Logic of Science",
| |
| − | | Harvard University Lectures (1865), pages 161-302 in:
| |
| − | |'Writings of Charles S. Peirce: A Chronological Edition',
| |
| − | |'Volume 1, 1857-1866', Peirce Edition Project,
| |
| − | | Indiana University Press, Bloomington, IN, 1982.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Incitatory Note 2
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | | |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Meditative Note 1
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | I would like to start from a "common sense practical" (CSP) point of view,
| |
| − | and, indeed, never to lose sight of what appears evident from that station,
| |
| − | no matter how many levels of abstract remove and abstruse mention it might
| |
| − | become necessary to interpose along the way.
| |
| − | | |
| − | So let's examine this initial caltrop
| |
| − | "descriptive/normative/prescriptive"
| |
| − | from the CSP POV, if you will.
| |
| − | | |
| − | Reading "Descriptive" to mean "What it is",
| |
| − | while "Normative" means "What it oughta be",
| |
| − | and "Prescriptive" says "Make it so, or else",
| |
| − | I will have very little to say about the last,
| |
| − | and only be able to focus on the distinctions
| |
| − | that may exist among the first two dimensions.
| |
| − | | |
| − | From the beginning, from this point of view, difficult words,
| |
| − | like "inquiry", "logic", "truth", and so on, must be taken
| |
| − | as initially indexical, inchoately succeeding at little
| |
| − | more than pointing to a realm of experience that may
| |
| − | or may not be common to the e-mitter and re-mitter.
| |
| − | | |
| − | I suspect that this stanza is likely to be controversial,
| |
| − | so I'll pause at this point for the countrapunctal verse.
| |
| − | | |
| − | Or for a rest ...
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Meditative Note 2
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | So I may begin with an object and a sign in a tenuous relation,
| |
| − | with the subject matter indexed under the topic name "inquiry",
| |
| − | where the sign originates from a "just noticeable differential"
| |
| − | of information about the object, and not a single "figit" more.
| |
| − | Few would call this a foundation -- I only call it a beginning.
| |
| − | | |
| − | Yet another of many ...
| |
| − | | |
| − | But it does provide us with a clue to a signficant difference,
| |
| − | however much this difference is bound by this origin to raise
| |
| − | itself from egg, germ, seed, spore, or whatever it is that is
| |
| − | infinitesimal in its initial condition. In this disjointness
| |
| − | of an archetype where what begins, what leads, and what rules
| |
| − | are not so trivially identical to one another, one encounters
| |
| − | the brand of beginning that begins in the middle of the story,
| |
| − | and has no need of any other foundation but the medium itself.
| |
| − | | |
| − | ["sign-ficant" [stet]]
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Meditative Note 3
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | | |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Obligatory Note 1
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | While I remain compelled to remain silent on the status of the absolute fiat,
| |
| − | the irrelative notion of the unmotivated motion and the disinterested stance,
| |
| − | let me then turn to the other axes of description, descriptive vs. normative.
| |
| − | Axes of description, indeed, you can almost hear one branch of the recursion
| |
| − | already beginning to wind up its whine to the verge of a howl, but toss it a
| |
| − | sop and try to persevere in the quest.
| |
| − | | |
| − | In this view, I regard the very idea of a norm as invoking its due pragma --
| |
| − | aim, business, concern, desire, end, function, goal, intention, interest,
| |
| − | objective, purpose, its names are legion -- and the good sense of the
| |
| − | norm is simply to suggest what one ought to do, contingent, of course,
| |
| − | on one's motive to achieve that pragma.
| |
| − | | |
| − | If we keep in mind the kinds of "applied research task" (ART) that your
| |
| − | everyday artist, designer, engineer, mathematician, scientist, or other
| |
| − | type of technical worker has to carry out on an everyday basis, we note
| |
| − | how these axes of description can be used to frame their activities and
| |
| − | to depict their forms of conduct, without mistaking either the frame or
| |
| − | the picture for the object of the picture so framed. Nor does any body
| |
| − | imagine that the observer must flatten out into a single plane or align
| |
| − | with a single axis, in order to make a vantage of the frame so pictured.
| |
| − | | |
| − | Common sense practical wit tells us that effective action toward the
| |
| − | achievement of a desirable result will naturally depend on acquiring
| |
| − | good descriptions of the lay of the land in which we hope to advance.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Obligatory Note 2
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | | |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Projective Note 1
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | Good morning. Thanks. I had a bad night.
| |
| − | I blame Bernard Morand, who wrote me this:
| |
| − | | |
| − | BM: But this looks as some God's view.
| |
| − | What about us, finite humans, occupied
| |
| − | in counting the instants of our lives?
| |
| − | And thus condemned to try to improve
| |
| − | the fate of our successors?
| |
| − | | |
| − | When you think of this in the future, and of course you may never,
| |
| − | you may blame him too, for in writing this he has "erged" me on
| |
| − | to return to my deserted dissertation work, into which I have
| |
| − | poured my life for lo! these too many years to count, truly,
| |
| − | if you stop to contemplate the fact that time is relative.
| |
| − | | |
| − | In that time I have come to the view that we really need
| |
| − | a good "theory of inquiry" (TOI), for all sorts of very
| |
| − | practical and crucial reasons, also, that we cannot get
| |
| − | a good TOI without its being, at one and the same time,
| |
| − | a good "theory of information" (TOI too), and also that
| |
| − | an integral constituent of TOI 1 and TOI 2 would have to
| |
| − | be a good "theory of representation and semiosis" (TORAS) --
| |
| − | "Bull!?", you say, well, so be it.
| |
| − | | |
| − | Further, I think that it is abundantly evident by now that
| |
| − | we will get no such good theories of signs or science from
| |
| − | the "establishment philosophy of science" (EPOS?) -- which
| |
| − | has managed to mince and to trash the best available tries
| |
| − | at such theories for over a hundred years now. But Hey! --
| |
| − | don't take my word for it -- waste a century of your own.
| |
| − | | |
| − | We just got our regular email back,
| |
| − | so I think that I can now get going --
| |
| − | Yes, I have lost the ability to think
| |
| − | if not literally writing 'to' somebody.
| |
| − | | |
| − | When it begins, it begins like this:
| |
| − | | |
| − | Why am I asking this question?
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Projective Note 2
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | So we may rest assured that we do have a "subject matter", an empirical domain,
| |
| − | or a realm of experience that is indexed, however dimly, generally, or vaguely,
| |
| − | by the word "inquiry", and only the question how best to describe it remains
| |
| − | in doubt at this stage of the play. If we wanted to cast our net as widely
| |
| − | as possible, at the risk of anticipating a bounding hypothesis, we could
| |
| − | think of all the world's creatures bright and beautiful and of how they
| |
| − | conduct themselves when faced with some moment of uncertainty, where
| |
| − | their aim is to cope with a surprising phenomenon or to deal with
| |
| − | a problematic situation that meets them in the course of their
| |
| − | ever-ongoing struggles to live, to revive, and to thrive.
| |
| − | | |
| − | Now, neither the fact that we begin with a descriptive task,
| |
| − | nor the fact that it remains of interest for its own sake,
| |
| − | necessarily means that we must end there, for it is also
| |
| − | the means to a further end, of learning how to better
| |
| − | our own skill at inquiry, which means in our time
| |
| − | the building of tools that help with the task.
| |
| − | | |
| − | I hope I have made this sound as truly and
| |
| − | as trivially obvious as it ought to be.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Reflective Note 1
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | In reflecting on what in the world a "Theory of Inquiry" (TOI) might be,
| |
| − | it occurs to me that there are many different things that one might mean
| |
| − | by such a theory. It could just be any number of things that one asserts
| |
| − | or has a mind to assert about the ostensible subject matter. But it has
| |
| − | been my experience that one can assert pretty much whatever one chooses,
| |
| − | and others will choose to heed it or ignore it on many different grounds,
| |
| − | the grounds themselves being a matter of choice, conditioning, or custom.
| |
| − | | |
| − | But I am looking for theories that work, that is to say, theories that
| |
| − | are subject to probation through proof, probability, and programming.
| |
| − | | |
| − | Astute readers will have noticed that I've already attempted to finesse
| |
| − | a very important, and most likely "infinessible" issue, to wit, that of
| |
| − | the scruples dividing descriptive, normative, and prescriptive theories.
| |
| − | | |
| − | I will think about that, and get back to you.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Reflective Note 2
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | | How will I approach this problem about the nature of inquiry?
| |
| − | |
| |
| − | | The simplest answer is this:
| |
| − | |
| |
| − | | I will apply the method of inquiry to the problem of inquiry's nature.
| |
| − | |
| |
| − | | This is the most concise and comprehensive answer that I know, but
| |
| − | | it is likely to sound facetious at this point. On the other hand,
| |
| − | | if I did not actually use the method of inquiry that I describe
| |
| − | | as inquiry, how could the results possibly be taken seriously?
| |
| − | | Accordingly, the questions of methodological self-application
| |
| − | | and self-referential consistency will be found at the center
| |
| − | | of this research.
| |
| − | | |
| − | These lines image in compact form the crux of the problem,
| |
| − | the crucible of the method, and the character that marks
| |
| − | relation between the two, if indeed they really are two,
| |
| − | in a form whose extended development will wind its way
| |
| − | through many a later page of the present exposition.
| |
| − | | |
| − | But let me just point out at this point some of
| |
| − | the reasons why I have found the prerequisite
| |
| − | of an inquiry into inquiry to be inescapable.
| |
| − | | |
| − | Let us entertain the idea, for the sake of getting the inquiry started,
| |
| − | if nothing else, that it is admissible to use a word like "inquiry" as
| |
| − | an initially indefinite indicator of an ostensible object of inquiry.
| |
| − | If we ever again find ourselves being puzzled how our reasoning can
| |
| − | chastize its own entailments this way, we may remind ourselves of
| |
| − | that fine old line between our "logica docens' (logic as taught)
| |
| − | and our "logica utens" (logic as used). With this distinction
| |
| − | in mind, we can dispell the initial puzzlement by saying that
| |
| − | we are using a capacity for inquiry that we do not know how
| |
| − | to formalize yet in order to examine the forms of inquiry
| |
| − | that various thinkers have been able, at least partially,
| |
| − | to formalize.
| |
| − | | |
| − | The dilemma that we face has the following structure:
| |
| − | | |
| − | If we recommend to all a method of inquiry that
| |
| − | we ourselves do not use in a pinch, precisely
| |
| − | in a pinch where we need to study an issue
| |
| − | as important as the nature of inquiry,
| |
| − | then who would take our advice?
| |
| − | | |
| − | So it seems that there is no choice
| |
| − | but to study inquiry, the pragma,
| |
| − | by way of inquiry, the praxis,
| |
| − | that is to say, recursively.
| |
| − | | |
| − | Incidentally, many variations on this theme are
| |
| − | thoroughly developed in Peirce's "Lectures" of
| |
| − | 1865 and 1866 and recapitulated in his early
| |
| − | study "On a New List of Categories" (1867).
| |
| − | | |
| − | http://members.door.net/arisbe/menu/library/bycsp/newlist/nl-main.htm
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Reflective Note 3
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | | |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Work Area
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | From this point of view, inquiry is form of conduct,
| |
| − | an applied research task, like may others that we
| |
| − | have to carry out, and that can be done either
| |
| − | better or worse.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Outline
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | Inquiry Driven Systems
| |
| − | | |
| − | 1. Introduction
| |
| − | 1.1. Outline of the Project: Inquiry Into Inquiry
| |
| − | 1.1.1. Problem
| |
| − | 1.1.2. Method
| |
| − | 1.1.2.1. The Paradigmatic and Process-Analytic Phase
| |
| − | 1.1.2.2. The Paraphrastic and Faculty-Synthetic Phase
| |
| − | 1.1.2.3. Reprise of Methods
| |
| − | 1.1.3. Criterion
| |
| − | 1.1.4. Application
| |
| − | | |
| − | 1.2. Onus of the Project: No Way But Inquiry
| |
| − | 1.2.1. A Modulating Prelude
| |
| − | 1.2.2. A Fugitive Canon
| |
| − | | |
| − | 1.3. Opening of the Project: A Way Up To Inquiry
| |
| − | 1.3.1. Initial Analysis of Inquiry Allegro Aperto
| |
| − | 1.3.2. Discussion of Discussion
| |
| − | 1.3.3. Discussion of Formalization: General Topics
| |
| − | 1.3.3.1. A Formal Charge
| |
| − | 1.3.3.2. A Formalization of Formalization?
| |
| − | 1.3.3.3. A Formalization of Discussion?
| |
| − | 1.3.3.4. A Concept of Formalization
| |
| − | 1.3.3.5. A Formal Approach
| |
| − | 1.3.3.6. A Formal Development
| |
| − | 1.3.3.7 A Formal Persuasion
| |
| − | 1.3.4. Discussion of Formalization: Concrete Examples
| |
| − | 1.3.4.1. Formal Models: A Sketch
| |
| − | 1.3.4.2. Sign Relations: A Primer
| |
| − | 1.3.4.3. Semiotic Equivalence Relations
| |
| − | 1.3.4.4. Graphical Representations
| |
| − | 1.3.4.5. Taking Stock
| |
| − | 1.3.4.6. The "Meta" Question
| |
| − | 1.3.4.7. Iconic Signs
| |
| − | 1.3.4.8. The Conflict of Interpretations
| |
| − | 1.3.4.9. Indexical Signs
| |
| − | 1.3.4.10. Sundry Problems
| |
| − | 1.3.4.11. Review and Prospect
| |
| − | 1.3.4.12. Objective Plans & Levels
| |
| − | 1.3.4.13. Formalization of OF: Objective Levels
| |
| − | 1.3.4.14. Application of OF: Generic Level
| |
| − | 1.3.4.15. Application of OF: Motive Level
| |
| − | 1.3.4.16. The Integration of Frameworks
| |
| − | 1.3.4.17. Recapitulation: A Brush with Symbols
| |
| − | 1.3.4.18. C'est Moi
| |
| − | 1.3.4.19. Entr'acte
| |
| − | | |
| − | 1.3.5 Discussion of Formalization: Specific Objects
| |
| − | 1.3.5.1 The Will to Form
| |
| − | 1.3.5.2 The Forms of Reasoning
| |
| − | 1.3.5.3 A Fork in the Road
| |
| − | 1.3.5.4 A Forged Bond
| |
| − | 1.3.5.5 A Formal Account
| |
| − | 1.3.5.6 Analogs, Icons, Models, Surrogates
| |
| − | 1.3.5.7 Steps and Tests of Formalization
| |
| − | 1.3.5.8 Puck, the Ref
| |
| − | 1.3.5.9 Partial Formalizations
| |
| − | 1.3.5.10 A Formal Utility
| |
| − | 1.3.5.11 A Formal Aesthetic
| |
| − | 1.3.5.12 A Formal Apology
| |
| − | 1.3.5.13 A Formal Suspicion
| |
| − | 1.3.5.14 The Double Aspect of Concepts
| |
| − | 1.3.5.15 A Formal Permission
| |
| − | 1.3.5.16 A Formal Invention
| |
| − | 1.3.6 Recursion in Perpetuity
| |
| − | 1.3.7 Processus, Regressus, Progressus
| |
| − | 1.3.8 Rondeau Tempo di Menuetto
| |
| − | 1.3.9 Reconnaissance
| |
| − | 1.3.9.1 The Informal Context
| |
| − | 1.3.9.2 The Epitext
| |
| − | 1.3.9.3 The Formative Tension
| |
| − | 1.3.10 Recurring Themes
| |
| − | 1.3.10.1 Preliminary Notions
| |
| − | 1.3.10.2 Intermediary Notions
| |
| − | 1.3.10.3 Propositions and Sentences
| |
| − | 1.3.10.4 Empirical Types and Rational Types
| |
| − | 1.3.10.5 Articulate Sentences
| |
| − | 1.3.10.6 Stretching Principles
| |
| − | 1.3.10.7 Stretching Operations
| |
| − | 1.3.10.8 The Cactus Patch
| |
| − | 1.3.10.9 The Cactus Language: Syntax
| |
| − | 1.3.10.10 The Cactus Language: Stylistics
| |
| − | 1.3.10.11 The Cactus Language: Mechanics
| |
| − | 1.3.10.12 The Cactus Language: Semantics
| |
| − | 1.3.10.13 Stretching Exercises
| |
| − | 1.3.10.14 Syntactic Transformations
| |
| − | 1.3.10.15 Derived Equivalence Relations
| |
| − | 1.3.10.16 Digression on Derived Relations
| |
| − | | |
| − | 1.4 Outlook of the Project: All Ways Lead to Inquiry
| |
| − | 1.4.1 The Matrix of Inquiry
| |
| − | 1.4.1.1 Inquiry as Conduct
| |
| − | 1.4.1.2 Types of Conduct
| |
| − | 1.4.1.3 Perils of Inquiry
| |
| − | 1.4.1.4 Forms of Relations
| |
| − | 1.4.1.5 Models of Inquiry
| |
| − | 1.4.2 The Moment of Inquiry
| |
| − | 1.4.3 The Modes of Inquiry
| |
| − | 1.4.3.1 Deductive Reasoning
| |
| − | 1.4.3.2 Inductive Reasoning
| |
| − | 1.4.3.3 Abductive Reasoning
| |
| − | 1.4.3.4 Analogical Reasoning
| |
| − | | |
| − | 1.5 Obstacles to the Project: In the Way of Inquiry
| |
| − | 1.5.1 The Initial Unpleasantness
| |
| − | 1.5.2 The Justification Trap
| |
| − | 1.5.3 A Formal Apology
| |
| − | 1.5.3.1 Category Double-Takes
| |
| − | 1.5.3.2 Conceptual Extensions
| |
| − | 1.5.3.3 Explosional Recombinations
| |
| − | 1.5.3.4 Interpretive Frameworks
| |
| − | 1.5.4 A Material Exigency
| |
| − | 1.5.5 A Reconciliation of Accounts
| |
| − | 1.5.6 Objections to Reflexive Inquiry
| |
| − | 1.5.7 Empirical Considerations
| |
| − | 1.5.8 Computational Considerations
| |
| − | 1.5.8.1 A Form of Recursion
| |
| − | 1.5.8.2 A Power of Abstraction
| |
| − | | |
| − | 1.6 Orientation of the Project: A Way Into Inquiry
| |
| − | 1.6.1 Initial Description of Inquiry
| |
| − | 1.6.2 Terms of Analysis
| |
| − | 1.6.2.1 Digression on Signs
| |
| − | 1.6.2.2 Empirical Status of ID
| |
| − | 1.6.3 Expansion of Terms
| |
| − | 1.6.3.1 Agency
| |
| − | 1.6.3.2 Abstraction
| |
| − | 1.6.3.3 Analogy
| |
| − | 1.6.3.4 Accuracy
| |
| − | 1.6.3.5 Authenticity
| |
| − | 1.6.4 Anchoring Terms in Phenomena
| |
| − | 1.6.4.1 A Mistaken ID
| |
| − | 1.6.4.2 Phenomenology of Doubt
| |
| − | 1.6.4.3 Modalities of Knowledge
| |
| − | 1.6.5 Sets, Systems, & Substantive Agents
| |
| − | 1.6.6 Interpretive Systems
| |
| − | 1.6.6.1 Syntactic Systems
| |
| − | 1.6.6.2 Semantic Systems
| |
| − | 1.6.6.3 Pragmatic Systems
| |
| − | 1.6.7 Inquiry Driven Systems
| |
| − | 1.6.7.1 A Definition of Inquiry
| |
| − | 1.6.7.2 The Faculty of Inquiry
| |
| − | 1.6.7.3 A Definition of Determination
| |
| − | 1.6.7.4 A Definition of Definition
| |
| − | | |
| − | 1.7 Organization of the Project: A Way Through Inquiry
| |
| − | 1.7.1 The Problem: Inquiry Found as an Object of Study
| |
| − | 1.7.2 The Method: Inquiry Found as a Means of Study
| |
| − | 1.7.2.1 Conditions for the Possibility
| |
| − | of Inquiry into Inquiry
| |
| − | 1.7.2.2 Conditions for the Success of Inquiry into Inquiry
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| − | 1.7.3 The Criterion: Inquiry in Search of a Sensible End
| |
| − | 1.7.3.1 The Irritation of Doubt, and The Scratch Test
| |
| − | 1.7.3.2 Enabling Provision 1: The Scenes & Context of Inquiry
| |
| − | 1.7.3.3 Enabling Provision 2: The Stages & Content of Inquiry
| |
| − | 1.8 Objectives of the Project: Inquiry All the Way
| |
| − | 1.8.1 Substantial Objective
| |
| − | 1.8.1.1 Objective 1a: The Propositions as Types Analogy
| |
| − | 1.8.1.2 Objective 1b: The Styles of Proof Development
| |
| − | 1.8.1.3 Objective 1c: The Analysis of Interpreters, or A Problem with Authority
| |
| − | 1.8.2 Instrumental Objective
| |
| − | 1.8.3 Coordination of Objectives
| |
| − | 1.8.4 Recapitulation -- Da Capo, Al Segno
| |
| − | | |
| − | 2. Discussion of Inquiry
| |
| − | 2.1 Approaches to Inquiry
| |
| − | 2.1.1 The Classical Framework: Syllogistic Approaches
| |
| − | 2.1.2 The Pragmatic Framework: Sign-Theoretic Approaches
| |
| − | 2.1.3 The Dynamical Framework: System-Theoretic Approaches
| |
| − | 2.1.3.1 Inquiry & Computation
| |
| − | 2.1.3.2 Inquiry Driven Systems
| |
| − | 2.2 The Context of Inquiry
| |
| − | 2.2.1 The Field of Observation
| |
| − | 2.2.2 The Problem of Reflection
| |
| − | 2.2.3 The Problem of Reconstruction
| |
| − | 2.2.4 The Trivializing of Integration
| |
| − | 2.2.5 Tensions in the Field of Observation
| |
| − | 2.2.6 Problems of Representation & Communication
| |
| − | | |
| − | 2.3 The Conduct of Inquiry
| |
| − | 2.3.1 Introduction
| |
| − | 2.3.2 The Types of Reasoning
| |
| − | 2.3.2.1 Deduction
| |
| − | 2.3.2.2 Induction
| |
| − | 2.3.2.3 Abduction
| |
| − | 2.3.3 Hybrid Types of Inference
| |
| − | 2.3.3.1 Analogy
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| − | 2.3.3.2 Inquiry
| |
| − | 2.3.4 Details of Induction
| |
| − | 2.3.4.1 Learning
| |
| − | 2.3.4.2 Transfer
| |
| − | 2.3.4.3 Testing
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| − | 2.3.5 The Stages of Inquiry
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| − | | |
| − | 3. The Medium & Its Message
| |
| − | 3.1 Reflective Expression
| |
| − | 3.1.1 Casual Reflection
| |
| − | 3.1.1.1 Ostensibly Recursive Texts
| |
| − | 3.1.1.2 Analogical Recursion
| |
| − | 3.1.2 Conscious Reflection
| |
| − | 3.1.2.1 The Signal Moment
| |
| − | 3.1.2.2 The Symbolic Object
| |
| − | 3.1.2.3 The Endeavor to Communicate
| |
| − | 3.1.2.4 The Medium of Communication
| |
| − | 3.1.2.5 The Ark of Types:
| |
| − | The Order of Things to Come.
| |
| − | 3.1.2.6 The Epitext
| |
| − | 3.1.2.7 The Context of Interpretation
| |
| − | 3.1.2.8 The Formative Tension
| |
| − | 3.1.2.9 The Vehicle of Communication:
| |
| − | Reflection on the Scene,
| |
| − | Reflection on the Self.
| |
| − | 3.1.2.10 (7)
| |
| − | 3.1.2.11 (6)
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| − | 3.1.2.12 Recursions: Possible, Actual, Necessary
| |
| − | 3.1.2.13 Ostensibly Recursive Texts
| |
| − | 3.1.2.14 (3)
| |
| − | 3.1.2.15 The Freedom of Interpretation
| |
| − | 3.1.2.16 The Eternal Return
| |
| − | 3.1.2.17 (1)
| |
| − | 3.1.2.18 Information in Formation
| |
| − | 3.1.2.19 Reflectively Indexical Texts
| |
| − | 3.1.2.20 (4)
| |
| − | 3.1.2.21 (5)
| |
| − | 3.1.2.22 (6)
| |
| − | 3.1.2.23 (7)
| |
| − | 3.1.2.24 (8)
| |
| − | 3.1.2.25 The Discursive Universe
| |
| − | 3.1.2.26 (7)
| |
| − | 3.1.2.27 (6)
| |
| − | 3.1.2.28 (5)
| |
| − | 3.1.2.29 (4)
| |
| − | 3.1.2.30 (3)
| |
| − | 3.1.2.31 (2)
| |
| − | 3.1.2.32 (1)
| |
| − | | |
| − | 3.2 Reflective Inquiry
| |
| − | 3.2.1 Integrity and Unity of Inquiry
| |
| − | 3.2.2 Apparitions & Allegations
| |
| − | 3.2.3 A Reflective Heuristic
| |
| − | 3.2.4 Either/Or: A Sense of Absence
| |
| − | 3.2.5 Apparent, Occasional, & Practical Necessity
| |
| − | 3.2.6 Approaches, Aspects, Exposures, Fronts
| |
| − | 3.2.7 Synthetic A Priori Truths
| |
| − | 3.2.8 Priorisms of Normative Sciences
| |
| − | 3.2.9 Principle of Rational Action
| |
| − | 3.2.10 The Pragmatic Cosmos
| |
| − | 3.2.11 Reflective Interpretive Frameworks
| |
| − | 3.2.11.1 Principals Versus Principles
| |
| − | 3.2.11.2 The Initial Description of Inquiry
| |
| − | 3.2.11.3 An Early Description of Interpretation
| |
| − | 3.2.11.4 Descriptions of the Mind
| |
| − | 3.2.11.5 Of Signs & the Mind
| |
| − | 3.2.11.6 Questions of Justification
| |
| − | 3.2.11.7 The Experience of Satisfaction
| |
| − | 3.2.11.8 An Organizational Difficulty
| |
| − | 3.2.11.9 Pragmatic Certainties
| |
| − | 3.2.11.10 Problems & Methods
| |
| − | | |
| − | 3.3 Reflection on Reflection
| |
| − | 3.4 Reflective Interpretive Frameworks
| |
| − | 3.4.1 The Phenomenology of Reflection
| |
| − | 3.4.2 A Candid Point of View
| |
| − | 3.4.3 A Projective Point of View
| |
| − | 3.4.4 A Formal Point of View
| |
| − | 3.4.5 Three Styles of Linguistic Usage
| |
| − | 3.4.6 Basic Notions of Group Theory
| |
| − | 3.4.7 Basic Notions of Formal Language Theory
| |
| − | 3.4.8 A Perspective on Computation
| |
| − | 3.4.9 Higher Order Sign Relations: Introduction
| |
| − | 3.4.10 Higher Order Sign Relations: Examples
| |
| − | 3.4.11 Higher Order Sign Relations: Application
| |
| − | 3.4.12 Issue 1: The Status of Signs
| |
| − | 3.4.13 Issue 2: The Status of Sets
| |
| − | 3.4.14 Issue 3: The Status of Variables
| |
| − | 3.4.15 Propositional Calculus
| |
| − | 3.4.16 Recursive Aspects
| |
| − | 3.4.17 Patterns of Self-Reference
| |
| − | 3.4.18 Practical Intuitions
| |
| − | 3.4.19 Examples of Self-Reference
| |
| − | 3.4.20 Three Views of Systems
| |
| − | 3.4.21 Building Bridges Between Representations
| |
| − | 3.4.22 Extensional Representations of Sign Relations
| |
| − | 3.4.23 Intensional Representations of Sign Relations
| |
| − | 3.4.24 Literal Intensional Representations
| |
| − | 3.4.25 Analytic Intensional Representations
| |
| − | 3.4.26 Differential Logic & Directed Graphs
| |
| − | 3.4.27 Differential Logic & Group Operations
| |
| − | 3.4.28 The Bridge: From Obstruction to Opportunity
| |
| − | 3.4.29 Projects of Representation
| |
| − | 3.4.30 Connected, Integrated, Reflective Symbols
| |
| − | 3.4.31 Generic Orders of Relations
| |
| − | 3.4.32 Partiality: Selective Operations
| |
| − | 3.4.33 Sign Relational Complexes
| |
| − | 3.4.34 Set-Theoretic Constructions
| |
| − | 3.4.35 Reducibility of Sign Relations
| |
| − | 3.4.36 Irreducibly Triadic Relations
| |
| − | 3.4.37 Propositional Types
| |
| − | 3.4.38 Considering the Source
| |
| − | 3.4.39 Prospective Indices: Pointers to Future Work
| |
| − | 3.4.40 Dynamic & Evaluative Frameworks
| |
| − | 3.4.41 Elective & Motive Forces
| |
| − | 3.4.42 Sign Processes: A Start
| |
| − | 3.4.43 Reflective Extensions
| |
| − | 3.4.44 Reflections on Closure
| |
| − | 3.4.45 Intelligence => Critical Reflection
| |
| − | 3.4.46 Looking Ahead
| |
| − | 3.4.47 Mutually Intelligible Codes
| |
| − | 3.4.48 Discourse Analysis: Ways & Means
| |
| − | 3.4.49 Combinations of Sign Relations
| |
| − | 3.4.50 Revisiting the Source
| |
| − | 3.5 Divertimento:
| |
| − | Eternity in Love with the Creatures of Time
| |
| − | 3.5.1 Reflections on the Presentation of Examples
| |
| − | 3.5.2 Searching for Parameters
| |
| − | 3.5.3 Defect Analysis
| |
| − | 3.5.4 The Pragmatic Critique
| |
| − | 3.5.5 Pragmatic Operating Notions
| |
| − | 3.5.6 Defects of Presentation
| |
| − | 3.5.7 Dues to Process
| |
| − | 3.5.8 Duties to Purpose
| |
| − | 3.6 Computational Design Philosophy
| |
| − | 3.6.1 Intentional Objects & Attitudes
| |
| − | 3.6.2 Imperfect Design & Persistent Error
| |
| − | 3.6.3 Propositional Reasoning About Relations
| |
| − | 3.6.4 Dynamic & Evaluative Frameworks
| |
| − | 3.6.5 Discussion of Examples
| |
| − | 3.6.6 Information & Inquiry
| |
| − | | |
| − | 4. Overview of the Domain: Interpretive Inquiry
| |
| − | 4.1 Interpretive Bearings: Conceptual & Descriptive Frameworks
| |
| − | 4.1.1 Catwalks: Flexible Frameworks & Peripatetic Categories
| |
| − | 4.1.1.1 Eponymous Ancestors:
| |
| − | The Precursors of Abstraction?
| |
| − | 4.1.1.2 Reticles:
| |
| − | Interpretive Flexibility as a Design Issue.
| |
| − | 4.1.2 Heuristic Inclinations & Regulative Principles
| |
| − | 4.2 Features of Inquiry Driven Systems
| |
| − | 4.2.1 The Pragmatic Theory of Signs
| |
| − | 4.2.1.1 Sign Relations
| |
| − | 4.2.1.2 Types of Signs
| |
| − | 4.2.2 The Pragmatic Theory of Inquiry
| |
| − | 4.2.2.1 Abduction
| |
| − | 4.2.2.2 Deduction
| |
| − | 4.2.2.3 Induction
| |
| − | 4.3 Examples of Inquiry Driven Systems
| |
| − | 4.3.1 "Index": A Program for Learning Formal Languages
| |
| − | 4.3.2 "Study": A Program for Reasoning with Propositions
| |
| − | 5. Discussion & Development of Objectives
| |
| − | 5.1 Objective 1a: Propositions as Types
| |
| − | 5.2 Objective 1b: Proof Styles & Developments
| |
| − | 5.3 Objective 1c: Interpretation & Authority
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| − | | |
| − | IDS. Inquiry Driven Systems -- Ontology List
| |
| − | | |
| − | 01. http://suo.ieee.org/ontology/msg04618.html
| |
| − | 02. http://suo.ieee.org/ontology/msg04621.html
| |
| − | 03. http://suo.ieee.org/ontology/msg04626.html
| |
| − | 04.
| |
| − | | |
| − | IDS. Inquiry Driven Systems -- Incitatory Notes
| |
| − | | |
| − | 01. http://suo.ieee.org/ontology/msg04637.html
| |
| − | 02.
| |
| − | | |
| − | IDS. Inquiry Driven Systems -- Meditative Notes
| |
| − | | |
| − | 01. http://suo.ieee.org/ontology/msg04622.html
| |
| − | 02. http://suo.ieee.org/ontology/msg04636.html
| |
| − | 03.
| |
| − | | |
| − | IDS. Inquiry Driven Systems -- Obligatory Notes
| |
| − | | |
| − | 01. http://suo.ieee.org/ontology/msg04623.html
| |
| − | 02.
| |
| − | | |
| − | IDS. Inquiry Driven Systems -- Projective Notes
| |
| − | | |
| − | 01. http://suo.ieee.org/ontology/msg04619.html
| |
| − | 02. http://suo.ieee.org/ontology/msg04625.html
| |
| − | 03.
| |
| − | | |
| − | IDS. Inquiry Driven Systems -- Reflective Notes
| |
| − | | |
| − | 01. http://suo.ieee.org/ontology/msg04620.html
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| − | 02. http://suo.ieee.org/ontology/msg04631.html
| |
| − | 03.
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| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| − | | |
| − | IDS. Inquiry Driven Systems -- Inquiry List
| |
| − | | |
| − | 01.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| − | | |
| − | IDS. Email Label
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| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| − | | |
| − | | Title: Inquiry Driven Systems
| |
| − | | Author: Jon Awbrey <jawbrey@oakland.edu>
| |
| − | | Version: Draft 10.01
| |
| − | | Created: 23 Jun 1996
| |
| − | | Revised: 07 Apr 2003
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| − | | |
| − | IDS. Inquiry Driven Systems
| |
| − | | |
| − | SUO List -- 04 Jan 2001
| |
| − | | |
| − | 01. http://suo.ieee.org/email/msg02678.html
| |
| − | 02. http://suo.ieee.org/email/msg02679.html
| |
| − | 03. http://suo.ieee.org/email/msg02682.html
| |
| − | 04. http://suo.ieee.org/email/msg02685.html
| |
| − | 05. http://suo.ieee.org/email/msg02695.html
| |
| − | 06. http://suo.ieee.org/email/msg02697.html
| |
| − | 07. http://suo.ieee.org/email/msg02720.html
| |
| − | 08. http://suo.ieee.org/email/msg03943.html
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| − | | |
| − | IDS. Inquiry Driven Systems
| |
| − | | |
| − | Ontology List -- Jan-Aug 2001
| |
| − | | |
| − | Systems Engineering Interest Statement
| |
| − | | |
| − | 00. http://suo.ieee.org/ontology/thrd103.html#00272
| |
| − | 01. http://suo.ieee.org/ontology/msg00272.html
| |
| − | 02. http://suo.ieee.org/ontology/msg00273.html
| |
| − | 03. http://suo.ieee.org/ontology/msg00276.html
| |
| − | 04. http://suo.ieee.org/ontology/msg00279.html
| |
| − | 05. http://suo.ieee.org/ontology/msg00289.html
| |
| − | 06. http://suo.ieee.org/ontology/msg00291.html
| |
| − | 07. http://suo.ieee.org/ontology/msg00314.html
| |
| − | | |
| − | Inquiry Driven Systems Essay 1
| |
| − | | |
| − | 08. http://suo.ieee.org/ontology/msg01535.html
| |
| − | | |
| − | Systems Engineering Dissertation
| |
| − | | |
| − | 00. http://suo.ieee.org/ontology/thrd103.html#03071
| |
| − | 09. http://suo.ieee.org/ontology/msg03071.html
| |
| − | 10. http://suo.ieee.org/ontology/msg03136.html
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| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| − | | |
| − | IDS. Inquiry Driven Systems
| |
| − | | |
| − | Arisbe List -- Jan 2002
| |
| − | | |
| − | http://stderr.org/pipermail/arisbe/2002-January/thread.html#1247
| |
| − | | |
| − | Ontology List -- Jan 2002
| |
| − | | |
| − | http://suo.ieee.org/ontology/thrd36.html#03604
| |
| − | http://suo.ieee.org/ontology/msg03604.html
| |
| − | | |
| − | 1.3.4.5. Taking Stock
| |
| − | http://suo.ieee.org/ontology/msg03605.html
| |
| − | | |
| − | 1.3.4.6. The "Meta" Question
| |
| − | http://suo.ieee.org/ontology/msg03607.html
| |
| − | | |
| − | 1.3.4.7. Iconic Signs
| |
| − | http://suo.ieee.org/ontology/msg03608.html
| |
| − | | |
| − | 1.3.4.8. The Conflict of Interpretations
| |
| − | http://suo.ieee.org/ontology/msg03609.html
| |
| − | | |
| − | Comment
| |
| − | http://suo.ieee.org/ontology/msg03613.html
| |
| − | | |
| − | 1.3.4.9. Indexical Signs
| |
| − | http://suo.ieee.org/ontology/msg03610.html
| |
| − | | |
| − | 1.3.4.10. Sundry Problems
| |
| − | http://suo.ieee.org/ontology/msg03611.html
| |
| − | | |
| − | 1.3.4.11. Review and Prospect
| |
| − | http://suo.ieee.org/ontology/msg03614.html
| |
| − | | |
| − | 1.3.4.12. Objective Plans and Levels
| |
| − | http://suo.ieee.org/ontology/msg03615.html
| |
| − | http://suo.ieee.org/ontology/msg03616.html
| |
| − | | |
| − | 1.3.4.13. Formalization of OF: Objective Levels
| |
| − | http://suo.ieee.org/ontology/msg03617.html
| |
| − | http://suo.ieee.org/ontology/msg03618.html
| |
| − | http://suo.ieee.org/ontology/msg03619.html
| |
| − | | |
| − | 1.3.4.14. Application of OF: Generic Level
| |
| − | http://suo.ieee.org/ontology/msg03620.html
| |
| − | http://suo.ieee.org/ontology/msg03621.html
| |
| − | http://suo.ieee.org/ontology/msg03622.html
| |
| − | http://suo.ieee.org/ontology/msg03623.html
| |
| − | | |
| − | 1.3.4.15. Application of OF: Motive Level
| |
| − | http://suo.ieee.org/ontology/msg03624.html
| |
| − | | |
| − | Comment
| |
| − | http://suo.ieee.org/ontology/msg03625.html
| |
| − | http://suo.ieee.org/ontology/msg03626.html
| |
| − | | |
| − | 1.3.4.16. Integration of Frameworks
| |
| − | http://suo.ieee.org/ontology/msg03627.html
| |
| − | | |
| − | Comment
| |
| − | http://suo.ieee.org/ontology/msg03629.html
| |
| − | | |
| − | 1.3.4.17 Recapitulation: A Brush with Symbols
| |
| − | http://suo.ieee.org/ontology/msg03630.html
| |
| − | | |
| − | Comment
| |
| − | http://suo.ieee.org/ontology/msg03631.html
| |
| − | http://suo.ieee.org/ontology/msg03634.html
| |
| − | http://suo.ieee.org/ontology/msg03636.html
| |
| − | http://suo.ieee.org/ontology/msg03638.html
| |
| − | http://suo.ieee.org/ontology/msg03639.html
| |
| − | | |
| − | 1.3.4.18. C'est Moi
| |
| − | http://suo.ieee.org/ontology/msg03640.html
| |
| − | | |
| − | 1.3.4.19 Entr'acte
| |
| − | http://suo.ieee.org/ontology/msg03642.html
| |
| − | | |
| − | Comment
| |
| − | http://suo.ieee.org/ontology/msg03645.html
| |
| − | http://suo.ieee.org/ontology/msg03647.html
| |
| − | http://suo.ieee.org/ontology/msg03648.html
| |
| − | http://suo.ieee.org/ontology/msg03649.html
| |
| − | http://suo.ieee.org/ontology/msg03650.html
| |
| − | http://suo.ieee.org/ontology/msg03652.html
| |
| − | http://suo.ieee.org/ontology/msg03657.html
| |
| − | http://suo.ieee.org/ontology/msg03659.html
| |
| − | http://suo.ieee.org/ontology/msg03660.html
| |
| − | http://suo.ieee.org/ontology/msg03661.html
| |
| − | http://suo.ieee.org/ontology/msg03662.html
| |
| − | http://suo.ieee.org/ontology/msg03663.html
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| − | http://suo.ieee.org/ontology/msg03664.html
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| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| − | | |
| − | Inquiry Into Inquiry (I^3)
| |
| − | | |
| − | 01. http://suo.ieee.org/ontology/msg02959.html
| |
| − | 02. http://suo.ieee.org/ontology/msg02961.html
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| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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| − | | |
| − | JA: 1.3.10.3 Propositions & Sentences
| |
| − | 01: http://suo.ieee.org/email/msg07444.html
| |
| − | 02: http://suo.ieee.org/email/msg07409.html
| |
| − | 03: http://suo.ieee.org/email/msg07416.html
| |
| − | 04: http://suo.ieee.org/email/msg07435.html
| |
| − | 05: http://suo.ieee.org/email/msg07443.html
| |
| − | 06: http://suo.ieee.org/email/msg07449.html
| |
| − | | |
| − | JA: 1.3.10.4 Empirical Types & Rational Types
| |
| − | 07: http://suo.ieee.org/email/msg07455.html
| |
| − | | |
| − | JA: 1.3.10.5 Articulate Sentences
| |
| − | 08: http://suo.ieee.org/email/msg07459.html
| |
| − | 09: http://suo.ieee.org/email/msg07461.html
| |
| − | | |
| − | JA: 1.3.10.6 Stretching Principles
| |
| − | 10: http://suo.ieee.org/email/msg07466.html
| |
| − | 11: http://suo.ieee.org/email/msg07469.html
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Discussion Notes
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Discussion Note 0
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | JA = Jon Awbrey
| |
| − | SZ = Steven Ericsson-Zenith
| |
| − | | |
| − | SZ: You generate a seemingly endless stream of "inquiry" -- some
| |
| − | of which seems to ramble and some of which is quite facinating.
| |
| − | | |
| − | SZ: Now you have started to torture me with Nietzsche :)
| |
| − | | |
| − | SZ: I catch just enough of the stream to want to keep watching but
| |
| − | I find I need a statement of systematic intent. I know it is
| |
| − | inquiry into inquiry but can you summarise for me in brief
| |
| − | where you want to go and how you intend to get there.
| |
| − | | |
| − | SZ: Are these the endless streets of Eurpoean cities in which we can
| |
| − | occassionally find ourself lost, or do we wander a US city that
| |
| − | has had the luxury of laying down a grid first?
| |
| − | | |
| − | This is the document formerly known as my dissertation proposal --
| |
| − | in a system engineering program that I returned to school to do
| |
| − | as a kind of capstone / 2nd childhood / unfinished symphony,
| |
| − | mostly from '91 to '99. The formal beginning of it can be
| |
| − | found starting here:
| |
| − | | |
| − | http://stderr.org/pipermail/inquiry/2004-May/thread.html#1434
| |
| − | | |
| − | But I think most folks on the SemioCom List had seen the earlier parts
| |
| − | a couple of years ago, so I started at a point where I was starting
| |
| − | to re-write some things slghtly clearer than the last time, I hope.
| |
| − | | |
| − | The immediate excuse/occasion of my thinking on this stuff again was
| |
| − | the intermittent/interminable discussion that Bernard and I have been
| |
| − | having on the nature of the "formalization arrow", plus many questions
| |
| − | about what would constitute non-trivial examples of sign relations or
| |
| − | truly significant applications of the pragmatic theory of signs, and
| |
| − | what kind of conceptual/software architecture it would take to support
| |
| − | thinking about this level of complexity. So I was trying to bring folks
| |
| − | up to date with the "state of my art" (SOMA) circa 1996 before I ventured
| |
| − | to return to those issues.
| |
| − | | |
| − | Don't worry overmuch about the Nietzsche -- the stuff that I put in epigraphs
| |
| − | is called the "epitext", and it is often intended to serve more as an exercise
| |
| − | in counterpoint, if not fugue, than as a statement of the main theme. Still,
| |
| − | Freddy Nightmare was being remarkably Apollonian in these passages, I think.
| |
| − | | |
| − | Back to N'Orleans ...
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Discussion Note 1
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | JA = Jon Awbrey
| |
| − | BM = Bernard Morand
| |
| − | | |
| − | Re: IDS 118. http://stderr.org/pipermail/inquiry/2004-May/001557.html
| |
| − | In: IDS. http://stderr.org/pipermail/inquiry/2004-May/thread.html#1434
| |
| − | | |
| − | I will go ahead and start a reply but I have to be on the road
| |
| − | to memorial day visits with out of town family in a little while,
| |
| − | so I will continue later tonight.
| |
| − | | |
| − | I should explain that this document arose out of the communication
| |
| − | situation with my advisor, committee, and other professors over the
| |
| − | better part of a decade. These people had very good backgrounds in
| |
| − | computer science, (control and optimal) systems engineering, and also
| |
| − | mathematics. So they already had a sense of how scientific method and
| |
| − | the formal sciences work, a sense of how they are applied in practical
| |
| − | settings, and a sense of how one uses empirical and statistical methods
| |
| − | to test the fitness of these applications on a recurring, incrementally
| |
| − | self-correcting basis. So the train of inquiry is already in motion,
| |
| − | and does not wait at the station for a good theory of how it works.
| |
| − | No one is going to stop the train and fire up the boilers again
| |
| − | from scratch. If I think that C.S. Peirce would make a better
| |
| − | conductor or engineer for the locomotion of inquiry, and not
| |
| − | just another "featherbedder" philosophy of science, I have
| |
| − | to show what he contributes to what is already under way.
| |
| − | That is to be contrasted with the epi-cartesian method
| |
| − | of flagging down the train, tearing up the rails, and
| |
| − | trying to justify its existence and motivation from
| |
| − | a standstill.
| |
| − | | |
| − | At any rate, this is the problem that I continually faced
| |
| − | in trying to write this erstwhile dissertation proposal,
| |
| − | and it forced me to work in a very different way from
| |
| − | anything that I had ever tried before, for instance,
| |
| − | where I could pretend to begin by just writing down
| |
| − | a bunch of axiomatic definitions as if it were the
| |
| − | first day of creation, and then following up their
| |
| − | consequences as best I could. Instead of doing
| |
| − | that, I had to write my opera 'in medias res'.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Discussion Note 2
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | JA = Jon Awbrey
| |
| − | BM = Bernard Morand
| |
| − | | |
| − | Re: IDS 118. http://stderr.org/pipermail/inquiry/2004-May/001557.html
| |
| − | In: IDS. http://stderr.org/pipermail/inquiry/2004-May/thread.html#1434
| |
| − | | |
| − | JA: It is important to realize that a "sampling relation", to express it
| |
| − | roughly, is a special case of a sign relation. Aside from acting on
| |
| − | sign relations and creating an association between sign relations, a
| |
| − | sampling relation is also involved in a larger sign relation, at least,
| |
| − | it can be subsumed within a general order of sign relations that allows
| |
| − | sign relations themselves to be taken as the objects, the signs, and the
| |
| − | interpretants of what can be called a "higher order" (HO) sign relation.
| |
| − | Considered with respect to its full potential, its use, and its purpose,
| |
| − | a sampling relation does not fall outside the closure of sign relations.
| |
| − | To be precise, a sampling relation falls within the denotative component
| |
| − | of a higher order sign relation, since the sign relation sampled is the
| |
| − | object of study and the sample is taken as a sign of it.
| |
| − | | |
| − | BM: I was away for the last whole week and I could not read your previous notes.
| |
| − | This paragraph of what seems to be some prolegomena for further explanations
| |
| − | caught my attention. A "sampling relation" can be subsumed within a general
| |
| − | order of sign relations: well, you seem to define the sampling case as some
| |
| − | kind of reverted hypostatic abstraction.
| |
| − | | |
| − | I may have to wait for you to explain what you mean by
| |
| − | this "reverted hypostatic abstraction". In the meantime,
| |
| − | what I am trying to say is this: If we approach "inquiry"
| |
| − | as an empirical domain or a quasi-natural phenomenon, taking
| |
| − | the word "inquiry" as a pointer to a certain field of activity
| |
| − | going on in the world, then whatever theory of inquiry we may
| |
| − | form will be based on our local sample of experience with this
| |
| − | domain of practice. At least, this would be the starting gate
| |
| − | in any other empirical domain. So the object is "all inquiry"
| |
| − | and the sign is "our sample of experience with all inquiry".
| |
| − | Indeed, we will ask whether the sample is "representative"
| |
| − | of the object domain, and a sensible method will try to
| |
| − | take steps to ensure that it is. Recall that the
| |
| − | root "sem-" in Hippocrates, from whom Aristotle
| |
| − | learned to appreciate abductive or diagnostic
| |
| − | reasoning, connotes "sample" or "specimen".
| |
| − | | |
| − | BM: Or to refer to the replica device between a legisign and its
| |
| − | sinsigns. If this is really your intend, you are missing a
| |
| − | third, I think. Namely the fact that sampling involves to
| |
| − | my sense particularizing much more than singularizing a
| |
| − | general type.
| |
| − | | |
| − | Yes, we know that the sample is more particular than the object domain
| |
| − | of interest, and thus gives us partial information. Indeed, since the
| |
| − | notion of "inquiry" is a rational concept, the domain "inquiry" is not
| |
| − | bounded by any finite experience or by all human experience together.
| |
| − | Thus we have to take measures that give us confidence of collecting a
| |
| − | "fair", "representative", or "typical" sample. This is only possible
| |
| − | in the long run, of course. Our initial sample is likely to be wholly
| |
| − | opportunistic and thus full of biases and "partialities".
| |
| − | | |
| − | This was partly the point of reverting to Aristotle's 'Peri Psyche' --
| |
| − | we possess and exercise an aptitude for inquiry long before we have
| |
| − | reflected on it sufficiently to formalize or objectify the smallest
| |
| − | sample of it.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Discussion Note 3
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | JA = Jon Awbrey
| |
| − | BM = Bernard Morand
| |
| − | | |
| − | Re: IDS 118. http://stderr.org/pipermail/inquiry/2004-May/001557.html
| |
| − | In: IDS. http://stderr.org/pipermail/inquiry/2004-May/thread.html#1434
| |
| − | | |
| − | BM: Doing so, you surely get the benefit of applying the signhood properties
| |
| − | to sample relations because you assume from the beginning the idea that
| |
| − | they don't fall outside the closure of sign relations. This is the great
| |
| − | deductive machinery used in inquiry. But aren't you dismissing from the
| |
| − | start his other friends, induction and abduction? As your text appears
| |
| − | to be a basic framework in order to inquire into inquiry, this would be
| |
| − | a too severe restriction.
| |
| − | | |
| − | I am merely saying that we can learn about X in general
| |
| − | by looking at examples of X, whether X is "inquiries" or
| |
| − | "sign relations" or anything else. Reasoning by way of
| |
| − | examples, analogies, or "paradigms" was classified by
| |
| − | Aristotle as a mixed form of reasoning that combined
| |
| − | induction of a rule and deduction of a similar fact,
| |
| − | while Peirce gave a couple of different analyses of
| |
| − | analogy that involved all three types of reasoning.
| |
| − | So I do not know why you say it is all deductive.
| |
| − | If one took the definition of a sign relation on
| |
| − | the basis of an a priori dictate, or authority,
| |
| − | then it might be so, but all sorts of abstract
| |
| − | definitions turn out to be useless for a given
| |
| − | purpose, and so Peirce's definition of a sign
| |
| − | has to prove its usefulness in the effort to
| |
| − | understand the object phenomena in question.
| |
| − | | |
| − | BM: From another side, it would throw tychism out of the picture:
| |
| − | you know, these samples which have absolutely not any subsumers.
| |
| − | | |
| − | I did not understand this comment fully. But choosing
| |
| − | random samples is a favorite way of getting fair ones.
| |
| − | | |
| − | BM: Your precision in the last sentence of the paragraph doesn't make
| |
| − | it more convenient in restricting sample relation to fall into the
| |
| − | denotative component of HO. The term "component" would deserve to
| |
| − | be itself defined: a restriction onto the !O! x !S! columns in L?
| |
| − | | |
| − | I made what seems like a simple observation, and hardly a novel one
| |
| − | if one considers the etymology and a host of classical discussions.
| |
| − | What use we make of the observation is another thing. I agree that
| |
| − | the word "component" is very multi-purpose -- here I conformed to
| |
| − | the usage that refers to factors of a product as "components",
| |
| − | as distinct from the sense used in relational "composition".
| |
| − | I plead the poverty of language.
| |
| − | | |
| − | BM: While I think that the component idea is at work in
| |
| − | sign relations, splitting them into a denotative part
| |
| − | and into another (?) connotative part would amount to
| |
| − | presume the problem at hand already solved. Reference to
| |
| − | components opens the difficult question (at least for me)
| |
| − | of the effective properties of composition relationship.
| |
| − | | |
| − | I don't understand this. We are operating in a situation of
| |
| − | partial information. We have focal sign relations that we can
| |
| − | objectify enough to study in detail, learning at least something
| |
| − | about the properties and variety of sign relations, at least some
| |
| − | of which learning will apply to classes of sign relations beyond
| |
| − | our immediate focus, perhaps even a little to the sign relation
| |
| − | in which we are embedded when we consider the relation of these
| |
| − | focal sign relations to the general class. Some people would
| |
| − | call this a "hermeneutic circle", I think.
| |
| − | | |
| − | BM: In short, the sign relation sampled is not the object of study
| |
| − | if it is not at the same time its interpretant, I think (and
| |
| − | then the concept of model is just newly born!)
| |
| − | | |
| − | The objective class of interest is "all" sign relations.
| |
| − | The sample that we have under the microscope is taken to
| |
| − | provide us with information about the object domain of
| |
| − | all sign relations, which it can do by virtue of the
| |
| − | fact that it "represents" the object domain more or
| |
| − | less well. If we transform the sample in some way,
| |
| − | or act on the information that it provides, then
| |
| − | we generate an interpretant sign of the sample.
| |
| − | Yes, I agree with that. I will have to ask
| |
| − | what sense of the word "model" you mean in
| |
| − | this context, though.
| |
| − | | |
| − | BM: Finally, what does it mean for a sample to be TAKEN AS a sign of
| |
| − | some study? The difficulty seems to me that the answer presupposes
| |
| − | the whole semiosis theory itself. I am not arguing here against the
| |
| − | possibility of this method, I am just trying to say that it would be
| |
| − | inaccurate to pretend escaping its complexity as a starting point.
| |
| − | On the contrary, I think that to begin with the assumption of
| |
| − | complexity will end (perhaps) into simplicity.
| |
| − | | |
| − | Yes, there is a shade of difference between passive experience,
| |
| − | where we take the samples that come our way, willy nilly, and
| |
| − | active experimentation, where we contrive to gather samples
| |
| − | under more contrived or controlled conditions, but none of
| |
| − | these variations are unique to the theory of signs or the
| |
| − | theory of inquiry. I do not know how I can presuppose
| |
| − | something that I am still in the middle of supposing.
| |
| − | That is, I do not view the theory of sign relations
| |
| − | or the theory of inquiry as finished products that
| |
| − | I might presuppose, or what would be the point of
| |
| − | an inquiry into their nature? I do have my pet
| |
| − | hypotheses, of course, but they are uncertain.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Discussion Note 4
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | JA = Jon Awbrey
| |
| − | BM = Bernard Morand
| |
| − | | |
| − | Re: IDS Discuss 1. http://stderr.org/pipermail/inquiry/2004-May/001560.html
| |
| − | In: IDS Discuss. http://stderr.org/pipermail/inquiry/2004-May/thread.html#1560
| |
| − | | |
| − | JA: I should explain that this document arose out of the communication
| |
| − | situation with my advisor, committee, and other professors over the
| |
| − | better part of a decade. These people had very good backgrounds in
| |
| − | computer science, (control and optimal) systems engineering, and also
| |
| − | mathematics. So they already had a sense of how scientific method and
| |
| − | the formal sciences work, a sense of how they are applied in practical
| |
| − | settings, and a sense of how one uses empirical and statistical methods
| |
| − | to test the fitness of these applications on a recurring, incrementally
| |
| − | self-correcting basis. So the train of inquiry is already in motion,
| |
| − | and does not wait at the station for a good theory of how it works.
| |
| − | No one is going to stop the train and fire up the boilers again
| |
| − | from scratch. If I think that C.S. Peirce would make a better
| |
| − | conductor or engineer for the locomotion of inquiry, and not
| |
| − | just another "featherbedder" philosophy of science, I have
| |
| − | to show what he contributes to what is already under way.
| |
| − | That is to be contrasted with the epi-cartesian method
| |
| − | of flagging down the train, tearing up the rails, and
| |
| − | trying to justify its existence and motivation from
| |
| − | a standstill.
| |
| − | | |
| − | BM: Agreed on the whole and the details, Jon. Except my suspicion
| |
| − | for "empirical method" for which I prefer "experimental" but
| |
| − | we already discussed that.
| |
| − | | |
| − | Okay. Those are basically synonyms to me. But I make no
| |
| − | inference from "empirical" to "radically naive empiricism",
| |
| − | or anything like that. Indeed, one of the principal jobs of
| |
| − | this whole project, that began long before I started trying to
| |
| − | document what I had been doing all along, was to integrate the
| |
| − | empirical data-driven and rational concept-driven modes of work.
| |
| − | Perhaps we could agree just between us -- I have already given up
| |
| − | trying to convert the masses (= effete minds) -- that "empiricism"
| |
| − | and "rationalism" are the names of heuristic attitudes, angles of
| |
| − | approach to be adopted on alternate weekdays, not the brands of
| |
| − | jealous religions that demand a fear and trembling either-or.
| |
| − | | |
| − | But I admit that I still see a residue of difference
| |
| − | between passive and active experience that comes up
| |
| − | all the time in the actual practice of research.
| |
| − | It is why we have consent forms, for example.
| |
| − | I had been meaning for a while now to take
| |
| − | it up under a separate thread, entitled
| |
| − | the "Lessons Of Play" (LOP), but I am
| |
| − | not ready to say what I think yet.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Discussion Note 5
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | JA = Jon Awbrey
| |
| − | BM = Bernard Morand
| |
| − | | |
| − | Re: IDS Discuss 3. http://stderr.org/pipermail/inquiry/2004-May/001562.html
| |
| − | In: IDS Discuss. http://stderr.org/pipermail/inquiry/2004-May/thread.html#1560
| |
| − | | |
| − | BM: Doing so, you surely get the benefit of applying the signhood properties
| |
| − | to sample relations because you assume from the beginning the idea that
| |
| − | they don't fall outside the closure of sign relations. This is the great
| |
| − | deductive machinery used in inquiry. But aren't you dismissing from the
| |
| − | start his other friends, induction and abduction? As your text appears
| |
| − | to be a basic framework in order to inquire into inquiry, this would be
| |
| − | a too severe restriction.
| |
| − | | |
| − | JA: I am merely saying that we can learn about X in general
| |
| − | by looking at examples of X, whether X is "inquiries" or
| |
| − | "sign relations" or anything else.
| |
| − | | |
| − | BM: All the difficulty resides in the meaning of "example" I think.
| |
| − | From which place can we judge that x is an example of some
| |
| − | (partially determined) X? This will become even more
| |
| − | difficult when we will have to make a selection
| |
| − | between several candidates x_i for learning X.
| |
| − | | |
| − | That is why we begin with easy examples.
| |
| − | | |
| − | It looks like some kind of Zenoesque "impossibility
| |
| − | of getting out of the starting blocks" difficulty here --
| |
| − | before Achilles can take a step he has to take a semi-step,
| |
| − | before he can take a semi-step he has to take a demi-semi-step,
| |
| − | before he can take a demi-semi-step he has to take a hemi-demi-semi-step, ...
| |
| − | | |
| − | This is the influence of epi-cartesion thinking again,
| |
| − | and I used to be sorely afflicted with it, so I know,
| |
| − | but Peirce, and already Aristotle before him, gave us
| |
| − | the way out with the abductive step of making a guess.
| |
| − | We want to minimize our risk, of course, but there is
| |
| − | an irreducible minimum of uncertainty that has to be
| |
| − | tolerated if thought and action are not to remain in
| |
| − | a state of utter paralysis.
| |
| − | | |
| − | So, can we read Peirce's definition of a sign relation and
| |
| − | use it to pick out some concrete and simple examples of
| |
| − | sign relations, or not? It's not much use if we can't.
| |
| − | Can I point to some examples of "inquiry" that are so
| |
| − | clearly examples of what we want to talk about that
| |
| − | both I and my committee will agree that they fit
| |
| − | the general description? Yes, though I might
| |
| − | have to defer to their language to do so,
| |
| − | calling it "research" or "applications
| |
| − | of scientific method" just by way of
| |
| − | getting out of the starting blocks.
| |
| − | | |
| − | If we get good at thinking about the simple examples,
| |
| − | then it may be worth the trouble to try and tackle the
| |
| − | harder cases. From my experience, 3-adic relations are
| |
| − | so difficult to think about that it will take some help
| |
| − | from software e-virons before we get much better at it.
| |
| − | | |
| − | JA: Reasoning by way of examples, analogies, or "paradigms" was classified by
| |
| − | Aristotle as a mixed form of reasoning that combined induction of a rule
| |
| − | and deduction of a similar fact, while Peirce gave a couple of different
| |
| − | analyses of analogy that involved all three types of reasoning. So I do
| |
| − | not know why you say it is all deductive. If one took the definition of
| |
| − | a sign relation on the basis of an a priori dictate, or authority, then it
| |
| − | might be so, but all sorts of abstract definitions turn out to be useless
| |
| − | for a given purpose, and so Peirce's definition of a sign has to prove its
| |
| − | usefulness in the effort to understand the object phenomena in question.
| |
| − | | |
| − | BM: Agreed
| |
| − | | |
| − | BM: From another side, it would throw tychism out of the picture:
| |
| − | you know, these samples which have absolutely not any subsumers.
| |
| − | | |
| − | JA: I did not understand this comment fully. But choosing
| |
| − | random samples is a favorite way of getting fair ones.
| |
| − | | |
| − | BM: This is a very complex problem but it is at the heart of the question of
| |
| − | induction. You certainly know that we can't elaborate true (absolute)
| |
| − | random samples. It seems that we have to admit in consequence that
| |
| − | probabilities fall into the domain of mathematics. The sampling
| |
| − | procedures of statisticians fall into the domain of experimental
| |
| − | sciences and both have to be not confused.
| |
| − | | |
| − | Yes, there is no reason to expect that inquiry into inquiry
| |
| − | will be any less complex than inquiry into anything else,
| |
| − | but I sense that I may have misunderstood your comment.
| |
| − | | |
| − | There are, of course, complications arising here over the difference
| |
| − | between descriptive sciences and normative sciences and what mix of
| |
| − | the two a particular person wants to focus on. But later, maybe.
| |
| − | | |
| − | BM: Your precision in the last sentence of the paragraph doesn't make
| |
| − | it more convenient in restricting sample relation to fall into the
| |
| − | denotative component of HO. The term "component" would deserve to
| |
| − | be itself defined: a restriction onto the !O! x !S! columns in L?
| |
| − | | |
| − | JA: I made what seems like a simple observation, and hardly a novel one
| |
| − | if one considers the etymology and a host of classical discussions.
| |
| − | What use we make of the observation is another thing. I agree that
| |
| − | the word "component" is very multi-purpose -- here I conformed to
| |
| − | the usage that refers to factors of a product as "components",
| |
| − | as distinct from the sense used in relational "composition".
| |
| − | I plead the poverty of language.
| |
| − | | |
| − | BM: Hum. Could you expand a little bit? This is not very familiar to me.
| |
| − | In relational composition, why does the relations couldn't be seen as
| |
| − | the factors of a product?
| |
| − | | |
| − | All I can do here is note the variety of usage. People will often
| |
| − | call the domains in a cartesian product or a direct product by the
| |
| − | name of "components" and they will speak of the "decomposition" of
| |
| − | a space X into the form of a product X = X_1 x ... x X_k, but not
| |
| − | be thinking of functional composition or relational composition
| |
| − | when they say this. I don't know any way around this, except
| |
| − | to use adjectives in front of the ambiguous words whenever
| |
| − | there's a chance of confusion.
| |
| − | | |
| − | If I have a 2-adic relation L that happens to be a composite of
| |
| − | two other 2-adic relations, L = M o N, then I'd tend to say that
| |
| − | L factors into M and N, or that M o N is "a" decomposition of L,
| |
| − | but M and N are not "the" factors of L or "the" components of L,
| |
| − | because we have no "unique factorization" theorem for relations
| |
| − | in general. So maybe that explains the nuance of usage. Maybe.
| |
| − | | |
| − | BM: While I think that the component idea is at work in
| |
| − | sign relations, splitting them into a denotative part
| |
| − | and into another (?) connotative part would amount to
| |
| − | presume the problem at hand already solved. Reference to
| |
| − | components opens the difficult question (at least for me)
| |
| − | of the effective properties of composition relationship.
| |
| − | | |
| − | JA: I don't understand this. We are operating in a situation of
| |
| − | partial information. We have focal sign relations that we can
| |
| − | objectify enough to study in detail, learning at least something
| |
| − | about the properties and variety of sign relations, at least some
| |
| − | of which learning will apply to classes of sign relations beyond
| |
| − | our immediate focus, perhaps even a little to the sign relation
| |
| − | in which we are embedded when we consider the relation of these
| |
| − | focal sign relations to the general class. Some people would
| |
| − | call this a "hermeneutic circle", I think.
| |
| − | | |
| − | BM: Yes, this is the strategy of learning which amounts for me to what
| |
| − | I poorly call synthesis. But there is its opposite too, analysis
| |
| − | that goes backward and allows to explain facts. The whole secret
| |
| − | of the method is articulating both of them together. Proceeding
| |
| − | this way, there is no more circle but something like a spiral.
| |
| − | | |
| − | Yes, not all circles are vicious. I understand all these things
| |
| − | on the model of recursive descent down to some basis that is so
| |
| − | simple as to be immediate -- what we do in top-down programming
| |
| − | or stepwise refinement -- and a spiral is a good image of that.
| |
| − | | |
| − | BM: On this point I am actually reading a book from K-O Apel
| |
| − | "Expliquer-Comprendre: La controverse centrale des sciences
| |
| − | humaines". It is a very fine book the first chapters of which
| |
| − | are difficult to read but it's a very great illumination when
| |
| − | arriving at the middle of the book. It is a French translation
| |
| − | from German. I don't know if there is an English one.
| |
| − | | |
| − | I will see if I can find it.
| |
| − | | |
| − | BM: In short, the sign relation sampled is not the object of study
| |
| − | if it is not at the same time its interpretant, I think (and then
| |
| − | the concept of model is just newly born!)
| |
| − | | |
| − | JA: The objective class of interest is "all" sign relations.
| |
| − | The sample that we have under the microscope is taken to
| |
| − | provide us with information about the object domain of
| |
| − | all sign relations, which it can do by virtue of the
| |
| − | fact that it "represents" the object domain more or
| |
| − | less well. If we transform the sample in some way,
| |
| − | or act on the information that it provides, then
| |
| − | we generate an interpretant sign of the sample.
| |
| − | Yes, I agree with that. I will have to ask
| |
| − | what sense of the word "model" you mean in
| |
| − | this context, though.
| |
| − | | |
| − | BM: I think of it as a pure synonym for sign, in all contexts.
| |
| − | And as for the case of signs there are 10 or 66 cases of
| |
| − | models. This is just an intuition of mine, not a theorem :-)
| |
| − | | |
| − | BM: Finally, what does it mean for a sample to be TAKEN AS a sign of
| |
| − | some study? The difficulty seems to me that the answer presupposes
| |
| − | the whole semiosis theory itself. I am not arguing here against the
| |
| − | possibility of this method, I am just trying to say that it would be
| |
| − | inaccurate to pretend escaping its complexity as a starting point.
| |
| − | On the contrary, I think that to begin with the assumption of
| |
| − | complexity will end (perhaps) into simplicity.
| |
| − | | |
| − | JA: Yes, there is a shade of difference between passive experience,
| |
| − | where we take the samples that come our way, willy nilly, and
| |
| − | active experimentation, where we contrive to gather samples
| |
| − | under more contrived or controlled conditions, but none of
| |
| − | these variations are unique to the theory of signs or the
| |
| − | theory of inquiry. I do not know how I can presuppose
| |
| − | something that I am still in the middle of supposing.
| |
| − | That is, I do not view the theory of sign relations
| |
| − | or the theory of inquiry as finished products that
| |
| − | I might presuppose, or what would be the point of
| |
| − | an inquiry into their nature? I do have my pet
| |
| − | hypotheses, of course, but they are uncertain.
| |
| − | | |
| − | BM: Yes. However I would add the following amendment. What any individual
| |
| − | inquirer (you, me or him) necessarily presupposes is the totality of
| |
| − | the previous inquiries. As such they aren't personal hypotheses,
| |
| − | and they have to be rendered explicit. If it was not the case
| |
| − | inquiry couldn't grow. We need them in order to be able to
| |
| − | experiment with samples. Perhaps it is there that we are
| |
| − | quite departing the one from the other.
| |
| − | | |
| − | That's kind of what I mean by 'in medias res'. But now the distinction
| |
| − | between "consciously presuppose" and "unconsciously presuppose" raises
| |
| − | its head. I recently had to invoke the term "quasi-belief" to discuss
| |
| − | this issue. It can take a considerable effort of critical reflection
| |
| − | to recognize that we are acting just as if certain propositions hold.
| |
| − | Again, consider Aristotle's 3-fold:
| |
| − | | |
| − | | Matter is potentiality (dynamis), while form is
| |
| − | | realization or actuality (entelecheia), and the
| |
| − | | word actuality is used in two senses, illustrated
| |
| − | | by the possession of knowledge (episteme) and the
| |
| − | | exercise of it (theorein).
| |
| − | | |
| − | I think that his reputation as dichotomous thinker is greatly exaggerated.
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Discussion Note 6
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | | |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | Inquiry Driven Systems
| |
| − | | |
| − | 1. Introduction
| |
| − | 1.1. Outline of the Project: Inquiry Into Inquiry
| |
| − | 1.1.1. Problem
| |
| − | 1.1.2. Method
| |
| − | 1.1.2.1. The Paradigmatic and Process-Analytic Phase
| |
| − | 1.1.2.2. The Paraphrastic and Faculty-Synthetic Phase
| |
| − | 1.1.2.3. Reprise of Methods
| |
| − | 1.1.3. Criterion
| |
| − | 1.1.4. Application
| |
| − | | |
| − | 1.2. Onus of the Project: No Way But Inquiry
| |
| − | 1.2.1. A Modulating Prelude
| |
| − | 1.2.2. A Fugitive Canon
| |
| − | | |
| − | 1.3. Opening of the Project: A Way Up To Inquiry
| |
| − | 1.3.1. Initial Analysis of Inquiry Allegro Aperto
| |
| − | 1.3.2. Discussion of Discussion
| |
| − | 1.3.3. Discussion of Formalization: General Topics
| |
| − | 1.3.3.1. A Formal Charge
| |
| − | 1.3.3.2. A Formalization of Formalization?
| |
| − | 1.3.3.3. A Formalization of Discussion?
| |
| − | 1.3.3.4. A Concept of Formalization
| |
| − | 1.3.3.5. A Formal Approach
| |
| − | 1.3.3.6. A Formal Development
| |
| − | 1.3.3.7 A Formal Persuasion
| |
| − | 1.3.4. Discussion of Formalization: Concrete Examples
| |
| − | 1.3.4.1. Formal Models: A Sketch
| |
| − | 1.3.4.2. Sign Relations: A Primer
| |
| − | 1.3.4.3. Semiotic Equivalence Relations
| |
| − | 1.3.4.4. Graphical Representations
| |
| − | 1.3.4.5. Taking Stock
| |
| − | 1.3.4.6. The "Meta" Question
| |
| − | 1.3.4.7. Iconic Signs
| |
| − | 1.3.4.8. The Conflict of Interpretations
| |
| − | 1.3.4.9. Indexical Signs
| |
| − | 1.3.4.10. Sundry Problems
| |
| − | 1.3.4.11. Review and Prospect
| |
| − | 1.3.4.12. Objective Plans & Levels
| |
| − | 1.3.4.13. Formalization of OF: Objective Levels
| |
| − | 1.3.4.14. Application of OF: Generic Level
| |
| − | 1.3.4.15. Application of OF: Motive Level
| |
| − | 1.3.4.16. The Integration of Frameworks
| |
| − | 1.3.4.17. Recapitulation: A Brush with Symbols
| |
| − | 1.3.4.18. C'est Moi
| |
| − | 1.3.4.19. Entr'acte
| |
| − | | |
| − | 1.3.5 Discussion of Formalization: Specific Objects
| |
| − | 1.3.5.1 The Will to Form
| |
| − | 1.3.5.2 The Forms of Reasoning
| |
| − | 1.3.5.3 A Fork in the Road
| |
| − | 1.3.5.4 A Forged Bond
| |
| − | 1.3.5.5 A Formal Account
| |
| − | 1.3.5.6 Analogs, Icons, Models, Surrogates
| |
| − | 1.3.5.7 Steps and Tests of Formalization
| |
| − | 1.3.5.8 Puck, the Ref
| |
| − | 1.3.5.9 Partial Formalizations
| |
| − | 1.3.5.10 A Formal Utility
| |
| − | 1.3.5.11 A Formal Aesthetic
| |
| − | 1.3.5.12 A Formal Apology
| |
| − | 1.3.5.13 A Formal Suspicion
| |
| − | 1.3.5.14 The Double Aspect of Concepts
| |
| − | 1.3.5.15 A Formal Permission
| |
| − | 1.3.5.16 A Formal Invention
| |
| − | 1.3.6 Recursion in Perpetuity
| |
| − | 1.3.7 Processus, Regressus, Progressus
| |
| − | 1.3.8 Rondeau Tempo di Menuetto
| |
| − | 1.3.9 Reconnaissance
| |
| − | 1.3.9.1 The Informal Context
| |
| − | 1.3.9.2 The Epitext
| |
| − | 1.3.9.3 The Formative Tension
| |
| − | 1.3.10 Recurring Themes
| |
| − | 1.3.10.1 Preliminary Notions
| |
| − | 1.3.10.2 Intermediary Notions
| |
| − | 1.3.10.3 Propositions and Sentences
| |
| − | 1.3.10.4 Empirical Types and Rational Types
| |
| − | 1.3.10.5 Articulate Sentences
| |
| − | 1.3.10.6 Stretching Principles
| |
| − | 1.3.10.7 Stretching Operations
| |
| − | 1.3.10.8 The Cactus Patch
| |
| − | 1.3.10.9 The Cactus Language: Syntax
| |
| − | 1.3.10.10 The Cactus Language: Stylistics
| |
| − | 1.3.10.11 The Cactus Language: Mechanics
| |
| − | 1.3.10.12 The Cactus Language: Semantics
| |
| − | 1.3.10.13 Stretching Exercises
| |
| − | 1.3.10.14 Syntactic Transformations
| |
| − | 1.3.10.15 Derived Equivalence Relations
| |
| − | 1.3.10.16 Digression on Derived Relations
| |
| − | | |
| − | 1.4 Outlook of the Project: All Ways Lead to Inquiry
| |
| − | 1.4.1 The Matrix of Inquiry
| |
| − | 1.4.1.1 Inquiry as Conduct
| |
| − | 1.4.1.2 Types of Conduct
| |
| − | 1.4.1.3 Perils of Inquiry
| |
| − | 1.4.1.4 Forms of Relations
| |
| − | 1.4.1.5 Models of Inquiry
| |
| − | 1.4.2 The Moment of Inquiry
| |
| − | 1.4.3 The Modes of Inquiry
| |
| − | 1.4.3.1 Deductive Reasoning
| |
| − | 1.4.3.2 Inductive Reasoning
| |
| − | 1.4.3.3 Abductive Reasoning
| |
| − | 1.4.3.4 Analogical Reasoning
| |
| − | | |
| − | 1.5 Obstacles to the Project: In the Way of Inquiry
| |
| − | 1.5.1 The Initial Unpleasantness
| |
| − | 1.5.2 The Justification Trap
| |
| − | 1.5.3 A Formal Apology
| |
| − | 1.5.3.1 Category Double-Takes
| |
| − | 1.5.3.2 Conceptual Extensions
| |
| − | 1.5.3.3 Explosional Recombinations
| |
| − | 1.5.3.4 Interpretive Frameworks
| |
| − | 1.5.4 A Material Exigency
| |
| − | 1.5.5 A Reconciliation of Accounts
| |
| − | 1.5.6 Objections to Reflexive Inquiry
| |
| − | 1.5.7 Empirical Considerations
| |
| − | 1.5.8 Computational Considerations
| |
| − | 1.5.8.1 A Form of Recursion
| |
| − | 1.5.8.2 A Power of Abstraction
| |
| − | | |
| − | 1.6 Orientation of the Project: A Way Into Inquiry
| |
| − | 1.6.1 Initial Description of Inquiry
| |
| − | 1.6.2 Terms of Analysis
| |
| − | 1.6.2.1 Digression on Signs
| |
| − | 1.6.2.2 Empirical Status of ID
| |
| − | 1.6.3 Expansion of Terms
| |
| − | 1.6.3.1 Agency
| |
| − | 1.6.3.2 Abstraction
| |
| − | 1.6.3.3 Analogy
| |
| − | 1.6.3.4 Accuracy
| |
| − | 1.6.3.5 Authenticity
| |
| − | 1.6.4 Anchoring Terms in Phenomena
| |
| − | 1.6.4.1 A Mistaken ID
| |
| − | 1.6.4.2 Phenomenology of Doubt
| |
| − | 1.6.4.3 Modalities of Knowledge
| |
| − | 1.6.5 Sets, Systems, & Substantive Agents
| |
| − | 1.6.6 Interpretive Systems
| |
| − | 1.6.6.1 Syntactic Systems
| |
| − | 1.6.6.2 Semantic Systems
| |
| − | 1.6.6.3 Pragmatic Systems
| |
| − | 1.6.7 Inquiry Driven Systems
| |
| − | 1.6.7.1 A Definition of Inquiry
| |
| − | 1.6.7.2 The Faculty of Inquiry
| |
| − | 1.6.7.3 A Definition of Determination
| |
| − | 1.6.7.4 A Definition of Definition
| |
| − | | |
| − | 1.7 Organization of the Project: A Way Through Inquiry
| |
| − | 1.7.1 The Problem: Inquiry Found as an Object of Study
| |
| − | 1.7.2 The Method: Inquiry Found as a Means of Study
| |
| − | 1.7.2.1 Conditions for the Possibility
| |
| − | of Inquiry into Inquiry
| |
| − | 1.7.2.2 Conditions for the Success of Inquiry into Inquiry
| |
| − | 1.7.3 The Criterion: Inquiry in Search of a Sensible End
| |
| − | 1.7.3.1 The Irritation of Doubt, and The Scratch Test
| |
| − | 1.7.3.2 Enabling Provision 1: The Scenes & Context of Inquiry
| |
| − | 1.7.3.3 Enabling Provision 2: The Stages & Content of Inquiry
| |
| − | 1.8 Objectives of the Project: Inquiry All the Way
| |
| − | 1.8.1 Substantial Objective
| |
| − | 1.8.1.1 Objective 1a: The Propositions as Types Analogy
| |
| − | 1.8.1.2 Objective 1b: The Styles of Proof Development
| |
| − | 1.8.1.3 Objective 1c: The Analysis of Interpreters, or A Problem with Authority
| |
| − | 1.8.2 Instrumental Objective
| |
| − | 1.8.3 Coordination of Objectives
| |
| − | 1.8.4 Recapitulation -- Da Capo, Al Segno
| |
| − | | |
| − | 2. Discussion of Inquiry
| |
| − | 2.1 Approaches to Inquiry
| |
| − | 2.1.1 The Classical Framework: Syllogistic Approaches
| |
| − | 2.1.2 The Pragmatic Framework: Sign-Theoretic Approaches
| |
| − | 2.1.3 The Dynamical Framework: System-Theoretic Approaches
| |
| − | 2.1.3.1 Inquiry & Computation
| |
| − | 2.1.3.2 Inquiry Driven Systems
| |
| − | 2.2 The Context of Inquiry
| |
| − | 2.2.1 The Field of Observation
| |
| − | 2.2.2 The Problem of Reflection
| |
| − | 2.2.3 The Problem of Reconstruction
| |
| − | 2.2.4 The Trivializing of Integration
| |
| − | 2.2.5 Tensions in the Field of Observation
| |
| − | 2.2.6 Problems of Representation & Communication
| |
| − | | |
| − | 2.3 The Conduct of Inquiry
| |
| − | 2.3.1 Introduction
| |
| − | 2.3.2 The Types of Reasoning
| |
| − | 2.3.2.1 Deduction
| |
| − | 2.3.2.2 Induction
| |
| − | 2.3.2.3 Abduction
| |
| − | 2.3.3 Hybrid Types of Inference
| |
| − | 2.3.3.1 Analogy
| |
| − | 2.3.3.2 Inquiry
| |
| − | 2.3.4 Details of Induction
| |
| − | 2.3.4.1 Learning
| |
| − | 2.3.4.2 Transfer
| |
| − | 2.3.4.3 Testing
| |
| − | 2.3.5 The Stages of Inquiry
| |
| − | | |
| − | 3. The Medium & Its Message
| |
| − | 3.1 Reflective Expression
| |
| − | 3.1.1 Casual Reflection
| |
| − | 3.1.1.1 Ostensibly Recursive Texts
| |
| − | 3.1.1.2 Analogical Recursion
| |
| − | 3.1.2 Conscious Reflection
| |
| − | 3.1.2.1 The Signal Moment
| |
| − | 3.1.2.2 The Symbolic Object
| |
| − | 3.1.2.3 The Endeavor to Communicate
| |
| − | 3.1.2.4 The Medium of Communication
| |
| − | 3.1.2.5 The Ark of Types:
| |
| − | The Order of Things to Come.
| |
| − | 3.1.2.6 The Epitext
| |
| − | 3.1.2.7 The Context of Interpretation
| |
| − | 3.1.2.8 The Formative Tension
| |
| − | 3.1.2.9 The Vehicle of Communication:
| |
| − | Reflection on the Scene,
| |
| − | Reflection on the Self.
| |
| − | 3.1.2.10 (7)
| |
| − | 3.1.2.11 (6)
| |
| − | 3.1.2.12 Recursions: Possible, Actual, Necessary
| |
| − | 3.1.2.13 Ostensibly Recursive Texts
| |
| − | 3.1.2.14 (3)
| |
| − | 3.1.2.15 The Freedom of Interpretation
| |
| − | 3.1.2.16 The Eternal Return
| |
| − | 3.1.2.17 (1)
| |
| − | 3.1.2.18 Information in Formation
| |
| − | 3.1.2.19 Reflectively Indexical Texts
| |
| − | 3.1.2.20 (4)
| |
| − | 3.1.2.21 (5)
| |
| − | 3.1.2.22 (6)
| |
| − | 3.1.2.23 (7)
| |
| − | 3.1.2.24 (8)
| |
| − | 3.1.2.25 The Discursive Universe
| |
| − | 3.1.2.26 (7)
| |
| − | 3.1.2.27 (6)
| |
| − | 3.1.2.28 (5)
| |
| − | 3.1.2.29 (4)
| |
| − | 3.1.2.30 (3)
| |
| − | 3.1.2.31 (2)
| |
| − | 3.1.2.32 (1)
| |
| − | | |
| − | 3.2 Reflective Inquiry
| |
| − | 3.2.1 Integrity and Unity of Inquiry
| |
| − | 3.2.2 Apparitions & Allegations
| |
| − | 3.2.3 A Reflective Heuristic
| |
| − | 3.2.4 Either/Or: A Sense of Absence
| |
| − | 3.2.5 Apparent, Occasional, & Practical Necessity
| |
| − | 3.2.6 Approaches, Aspects, Exposures, Fronts
| |
| − | 3.2.7 Synthetic A Priori Truths
| |
| − | 3.2.8 Priorisms of Normative Sciences
| |
| − | 3.2.9 Principle of Rational Action
| |
| − | 3.2.10 The Pragmatic Cosmos
| |
| − | 3.2.11 Reflective Interpretive Frameworks
| |
| − | 3.2.11.1 Principals Versus Principles
| |
| − | 3.2.11.2 The Initial Description of Inquiry
| |
| − | 3.2.11.3 An Early Description of Interpretation
| |
| − | 3.2.11.4 Descriptions of the Mind
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| − | 3.2.11.5 Of Signs & the Mind
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| − | 3.2.11.6 Questions of Justification
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| − | 3.2.11.7 The Experience of Satisfaction
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| − | 3.2.11.8 An Organizational Difficulty
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| − | 3.2.11.9 Pragmatic Certainties
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| − | 3.2.11.10 Problems & Methods
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| − | | |
| − | 3.3 Reflection on Reflection
| |
| − | 3.4 Reflective Interpretive Frameworks
| |
| − | 3.4.1 The Phenomenology of Reflection
| |
| − | 3.4.2 A Candid Point of View
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| − | 3.4.3 A Projective Point of View
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| − | 3.4.4 A Formal Point of View
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| − | 3.4.5 Three Styles of Linguistic Usage
| |
| − | 3.4.6 Basic Notions of Group Theory
| |
| − | 3.4.7 Basic Notions of Formal Language Theory
| |
| − | 3.4.8 A Perspective on Computation
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| − | 3.4.9 Higher Order Sign Relations: Introduction
| |
| − | 3.4.10 Higher Order Sign Relations: Examples
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| − | 3.4.11 Higher Order Sign Relations: Application
| |
| − | 3.4.12 Issue 1: The Status of Signs
| |
| − | 3.4.13 Issue 2: The Status of Sets
| |
| − | 3.4.14 Issue 3: The Status of Variables
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| − | 3.4.15 Propositional Calculus
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| − | 3.4.16 Recursive Aspects
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| − | 3.4.17 Patterns of Self-Reference
| |
| − | 3.4.18 Practical Intuitions
| |
| − | 3.4.19 Examples of Self-Reference
| |
| − | 3.4.20 Three Views of Systems
| |
| − | 3.4.21 Building Bridges Between Representations
| |
| − | 3.4.22 Extensional Representations of Sign Relations
| |
| − | 3.4.23 Intensional Representations of Sign Relations
| |
| − | 3.4.24 Literal Intensional Representations
| |
| − | 3.4.25 Analytic Intensional Representations
| |
| − | 3.4.26 Differential Logic & Directed Graphs
| |
| − | 3.4.27 Differential Logic & Group Operations
| |
| − | 3.4.28 The Bridge: From Obstruction to Opportunity
| |
| − | 3.4.29 Projects of Representation
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| − | 3.4.30 Connected, Integrated, Reflective Symbols
| |
| − | 3.4.31 Generic Orders of Relations
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| − | 3.4.32 Partiality: Selective Operations
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| − | 3.4.33 Sign Relational Complexes
| |
| − | 3.4.34 Set-Theoretic Constructions
| |
| − | 3.4.35 Reducibility of Sign Relations
| |
| − | 3.4.36 Irreducibly Triadic Relations
| |
| − | 3.4.37 Propositional Types
| |
| − | 3.4.38 Considering the Source
| |
| − | 3.4.39 Prospective Indices: Pointers to Future Work
| |
| − | 3.4.40 Dynamic & Evaluative Frameworks
| |
| − | 3.4.41 Elective & Motive Forces
| |
| − | 3.4.42 Sign Processes: A Start
| |
| − | 3.4.43 Reflective Extensions
| |
| − | 3.4.44 Reflections on Closure
| |
| − | 3.4.45 Intelligence => Critical Reflection
| |
| − | 3.4.46 Looking Ahead
| |
| − | 3.4.47 Mutually Intelligible Codes
| |
| − | 3.4.48 Discourse Analysis: Ways & Means
| |
| − | 3.4.49 Combinations of Sign Relations
| |
| − | 3.4.50 Revisiting the Source
| |
| − | 3.5 Divertimento:
| |
| − | Eternity in Love with the Creatures of Time
| |
| − | 3.5.1 Reflections on the Presentation of Examples
| |
| − | 3.5.2 Searching for Parameters
| |
| − | 3.5.3 Defect Analysis
| |
| − | 3.5.4 The Pragmatic Critique
| |
| − | 3.5.5 Pragmatic Operating Notions
| |
| − | 3.5.6 Defects of Presentation
| |
| − | 3.5.7 Dues to Process
| |
| − | 3.5.8 Duties to Purpose
| |
| − | 3.6 Computational Design Philosophy
| |
| − | 3.6.1 Intentional Objects & Attitudes
| |
| − | 3.6.2 Imperfect Design & Persistent Error
| |
| − | 3.6.3 Propositional Reasoning About Relations
| |
| − | 3.6.4 Dynamic & Evaluative Frameworks
| |
| − | 3.6.5 Discussion of Examples
| |
| − | 3.6.6 Information & Inquiry
| |
| − | | |
| − | 4. Overview of the Domain: Interpretive Inquiry
| |
| − | 4.1 Interpretive Bearings: Conceptual & Descriptive Frameworks
| |
| − | 4.1.1 Catwalks: Flexible Frameworks & Peripatetic Categories
| |
| − | 4.1.1.1 Eponymous Ancestors:
| |
| − | The Precursors of Abstraction?
| |
| − | 4.1.1.2 Reticles:
| |
| − | Interpretive Flexibility as a Design Issue.
| |
| − | 4.1.2 Heuristic Inclinations & Regulative Principles
| |
| − | 4.2 Features of Inquiry Driven Systems
| |
| − | 4.2.1 The Pragmatic Theory of Signs
| |
| − | 4.2.1.1 Sign Relations
| |
| − | 4.2.1.2 Types of Signs
| |
| − | 4.2.2 The Pragmatic Theory of Inquiry
| |
| − | 4.2.2.1 Abduction
| |
| − | 4.2.2.2 Deduction
| |
| − | 4.2.2.3 Induction
| |
| − | 4.3 Examples of Inquiry Driven Systems
| |
| − | 4.3.1 "Index": A Program for Learning Formal Languages
| |
| − | 4.3.2 "Study": A Program for Reasoning with Propositions
| |
| − | 5. Discussion & Development of Objectives
| |
| − | 5.1 Objective 1a: Propositions as Types
| |
| − | 5.2 Objective 1b: Proof Styles & Developments
| |
| − | 5.3 Objective 1c: Interpretation & Authority
| |
| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | Jon Awbrey, "Inquiry Driven Systems: Inquiry Into Inquiry"
| |
| − | IDS. http://stderr.org/pipermail/inquiry/2004-May/thread.html#1434
| |
| − | IDS. http://stderr.org/pipermail/inquiry/2004-May/thread.html#1564
| |
| − | IDS. http://stderr.org/pipermail/inquiry/2004-June/thread.html#1574
| |
| − | IDS. http://members.door.net/arisbe/menu/library/aboutcsp/awbrey/inquiry.htm
| |
| − | | |
| − | 1.3.4.12. Objective Plans and Levels
| |
| − | IDS 46. http://stderr.org/pipermail/inquiry/2004-May/001485.html
| |
| − | IDS 47. http://stderr.org/pipermail/inquiry/2004-May/001486.html
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| − | IDS 48. http://stderr.org/pipermail/inquiry/2004-May/001487.html
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| − | IDS 49. http://stderr.org/pipermail/inquiry/2004-May/001488.html
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| − | | |
| − | 1.3.4.13. Formalization of OF: Objective Levels
| |
| − | IDS 50. http://stderr.org/pipermail/inquiry/2004-May/001489.html
| |
| − | IDS 51. http://stderr.org/pipermail/inquiry/2004-May/001490.html
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| − | IDS 52. http://stderr.org/pipermail/inquiry/2004-May/001491.html
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| − | IDS 53. http://stderr.org/pipermail/inquiry/2004-May/001492.html
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| − | IDS 54. http://stderr.org/pipermail/inquiry/2004-May/001493.html
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| − | IDS 55. http://stderr.org/pipermail/inquiry/2004-May/001494.html
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| − | | |
| − | 1.3.4.14. Application of OF: Generic Level
| |
| − | IDS 56. http://stderr.org/pipermail/inquiry/2004-May/001495.html
| |
| − | IDS 57. http://stderr.org/pipermail/inquiry/2004-May/001496.html
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| − | IDS 58. http://stderr.org/pipermail/inquiry/2004-May/001497.html
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| − | IDS 59. http://stderr.org/pipermail/inquiry/2004-May/001498.html
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| − | IDS 60. http://stderr.org/pipermail/inquiry/2004-May/001499.html
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| − | IDS 61. http://stderr.org/pipermail/inquiry/2004-May/001500.html
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| − | IDS 62. http://stderr.org/pipermail/inquiry/2004-May/001501.html
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| − | IDS 63. http://stderr.org/pipermail/inquiry/2004-May/001502.html
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| − | | |
| − | 1.3.4.15. Application of OF: Motive Level
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| − | IDS 64. http://stderr.org/pipermail/inquiry/2004-May/001503.html
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| − | IDS 65. http://stderr.org/pipermail/inquiry/2004-May/001504.html
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| − | | |
| − | 1.3.4.16. The Integration of Frameworks
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| − | IDS 66. http://stderr.org/pipermail/inquiry/2004-May/001505.html
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| − | IDS 67. http://stderr.org/pipermail/inquiry/2004-May/001506.html
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| − | | |
| − | 1.3.4.17. Recapitulation: A Brush with Symbols
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| − | IDS 68. http://stderr.org/pipermail/inquiry/2004-May/001507.html
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| − | IDS 69. http://stderr.org/pipermail/inquiry/2004-May/001508.html
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| − | | |
| − | 1.3.4.18. C'est Moi
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| − | IDS 70. http://stderr.org/pipermail/inquiry/2004-May/001509.html
| |
| − | IDS 71. http://stderr.org/pipermail/inquiry/2004-May/001510.html
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| − | | |
| − | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| |
| − | | |
| − | IDS. Inquiry Driven Systems -- 2004
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| − | | |
| − | 000. http://stderr.org/pipermail/inquiry/2004-May/thread.html#1434
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| − | 000. http://stderr.org/pipermail/inquiry/2004-May/thread.html#1564
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| − | 000. http://stderr.org/pipermail/inquiry/2004-June/thread.html#1574
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| − | | |
| − | 001. http://stderr.org/pipermail/inquiry/2004-May/001434.html
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| − | 002. http://stderr.org/pipermail/inquiry/2004-May/001435.html
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| − | 003. http://stderr.org/pipermail/inquiry/2004-May/001436.html
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| − | 004. http://stderr.org/pipermail/inquiry/2004-May/001437.html
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| − | 005. http://stderr.org/pipermail/inquiry/2004-May/001438.html
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| − | 006. http://stderr.org/pipermail/inquiry/2004-May/001439.html
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| − | 007. http://stderr.org/pipermail/inquiry/2004-May/001440.html
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| − | 148. http://stderr.org/pipermail/inquiry/2004-June/001593.html
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| − | 149. http://stderr.org/pipermail/inquiry/2004-June/001594.html
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| − | 150. http://stderr.org/pipermail/inquiry/2004-June/001595.html
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| − | 151. http://stderr.org/pipermail/inquiry/2004-June/001596.html
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| − | 152. http://stderr.org/pipermail/inquiry/2004-June/001597.html
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| − | 153. http://stderr.org/pipermail/inquiry/2004-June/001598.html
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| − | 154. http://stderr.org/pipermail/inquiry/2004-June/001599.html
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| − | 155. http://stderr.org/pipermail/inquiry/2004-June/001600.html
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| − | 156. http://stderr.org/pipermail/inquiry/2004-June/001601.html
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| − | 157. http://stderr.org/pipermail/inquiry/2004-June/001602.html
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| − | 158. http://stderr.org/pipermail/inquiry/2004-June/001603.html
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| − | 159. http://stderr.org/pipermail/inquiry/2004-June/001604.html
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| − | 160. http://stderr.org/pipermail/inquiry/2004-June/001605.html
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| − | 161. http://stderr.org/pipermail/inquiry/2004-June/001606.html
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| − | 162. http://stderr.org/pipermail/inquiry/2004-June/001607.html
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| − | 163. http://stderr.org/pipermail/inquiry/2004-June/001608.html
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| − | 164. http://stderr.org/pipermail/inquiry/2004-June/001609.html
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| − | 165. http://stderr.org/pipermail/inquiry/2004-June/001610.html
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| − | 166. http://stderr.org/pipermail/inquiry/2004-June/001611.html
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| − | 167. http://stderr.org/pipermail/inquiry/2004-June/001612.html
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| − | 168. http://stderr.org/pipermail/inquiry/2004-June/001613.html
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| − | 169. http://stderr.org/pipermail/inquiry/2004-June/001614.html
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| − | 170. http://stderr.org/pipermail/inquiry/2004-June/001615.html
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| − | 171. http://stderr.org/pipermail/inquiry/2004-June/001616.html
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| − | 172. http://stderr.org/pipermail/inquiry/2004-June/001617.html
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| − | 173. http://stderr.org/pipermail/inquiry/2004-June/001618.html
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| − | 174. http://stderr.org/pipermail/inquiry/2004-June/001623.html
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| − | 175. http://stderr.org/pipermail/inquiry/2004-June/001629.html
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| − | 176.
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| − | | |
| − | IDS. Inquiry Driven Systems -- Discussion
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| − | | |
| − | 00. http://stderr.org/pipermail/inquiry/2004-May/thread.html#1560
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| − | 00. http://stderr.org/pipermail/inquiry/2004-June/thread.html#1576
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| − | | |
| − | 01. http://stderr.org/pipermail/inquiry/2004-May/001560.html
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| − | 02. http://stderr.org/pipermail/inquiry/2004-May/001561.html
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| − | 03. http://stderr.org/pipermail/inquiry/2004-May/001562.html
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| − | 04. http://stderr.org/pipermail/inquiry/2004-May/001563.html
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| − | 05. http://stderr.org/pipermail/inquiry/2004-June/001576.html
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| − | 06.
| |
| | | | |
| | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
| | </pre> | | </pre> |
| | | | |
| − | ==Outline== | + | ===Inquiry List=== |
| | | | |
| − | <pre>
| + | * http://stderr.org/pipermail/inquiry/2004-May/thread.html#1434 |
| − | Inquiry Driven Systems (07 Apr 2003)
| + | * http://stderr.org/pipermail/inquiry/2004-May/thread.html#1564 |
| − | 1. Research Proposal
| + | * http://stderr.org/pipermail/inquiry/2004-June/thread.html#1574 |
| − | 1.1 Outline of the Project : Inquiry Driven Systems
| |
| − | 1.1.1 Problem
| |
| − | 1.1.2 Method
| |
| − | 1.1.2.1 The Paradigmatic & Process-Analytic Phase.
| |
| − | 1.1.2.2 The Paraphrastic & Faculty-Synthetic Phase.
| |
| − | 1.1.2.3 Reprise of Methods
| |
| − | 1.1.3 Criterion
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| − | 1.1.4 Application
| |
| − | 1.2 Onus of the Project : No Way But Inquiry
| |
| − | 1.2.1 A Modulating Prelude
| |
| − | 1.2.2 A Fugitive Canon
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| − | | |
| − | 1.3 Option of the Project : A Way Up To Inquiry
| |
| − | 1.3.1 Initial Analysis of Inquiry : Allegro Aperto
| |
| − | 1.3.2 Discussion of Discussion
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| − | 1.3.3 Discussion of Formalization : General Topics
| |
| − | 1.3.3.1 A Formal Charge
| |
| − | 1.3.3.2 A Formalization of Formalization?
| |
| − | 1.3.3.3 A Formalization of Discussion?
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| − | 1.3.3.4 A Concept of Formalization
| |
| − | 1.3.3.5 A Formal Approach
| |
| − | 1.3.3.6 A Formal Development
| |
| − | 1.3.3.7 A Formal Persuasion
| |
| − | 1.3.4 Discussion of Formalization : Concrete Examples
| |
| − | 1.3.4.1 Formal Models : A Sketch
| |
| − | 1.3.4.2 Sign Relations : A Primer
| |
| − | 1.3.4.3 Semiotic Equivalence Relations
| |
| − | 1.3.4.4 Graphical Representations
| |
| − | 1.3.4.5 Taking Stock
| |
| − | 1.3.4.6 The "Meta" Question
| |
| − | 1.3.4.7 Iconic Signs
| |
| − | 1.3.4.8 The Conflict of Interpretations
| |
| − | 1.3.4.9 Indexical Signs
| |
| − | 1.3.4.10 Sundry Problems
| |
| − | 1.3.4.11 Review & Prospect
| |
| − | 1.3.4.12 Objective Plans & Levels
| |
| − | 1.3.4.13 Formalization of OF : Objective Levels
| |
| − | 1.3.4.14 Application of OF : Generic Level
| |
| − | 1.3.4.15 Application of OF : Motive Level
| |
| − | 1.3.4.16 The Integration of Frameworks
| |
| − | 1.3.4.17 Recapitulation : A Brush with Symbols
| |
| − | 1.3.4.18 C'est Moi
| |
| − | 1.3.4.19 Entr'acte
| |
| − | 1.3.5 Discussion of Formalization : Specific Objects
| |
| − | 1.3.5.1 The Will to Form
| |
| − | 1.3.5.2 The Forms of Reasoning
| |
| − | 1.3.5.3 A Fork in the Road
| |
| − | 1.3.5.4 A Forged Bond
| |
| − | 1.3.5.5 A Formal Account
| |
| − | 1.3.5.6 Analogs, Icons, Models, Surrogates
| |
| − | 1.3.5.7 Steps & Tests of Formalization
| |
| − | 1.3.5.8 Puck, the Ref
| |
| − | 1.3.5.9 Partial Formalizations
| |
| − | 1.3.5.10 A Formal Utility
| |
| − | 1.3.5.11 A Formal Aesthetic
| |
| − | 1.3.5.12 A Formal Apology
| |
| − | 1.3.5.13 A Formal Suspicion
| |
| − | 1.3.5.14 The Double Aspect of Concepts
| |
| − | 1.3.5.15 A Formal Permission
| |
| − | 1.3.5.16 A Formal Invention
| |
| − | 1.3.6 Recursion in Perpetuity
| |
| − | 1.3.7 Processus, Regressus, Progressus
| |
| − | 1.3.8 Rondeau : Tempo di Menuetto
| |
| − | 1.3.9 Reconnaissance
| |
| − | 1.3.9.1 The Informal Context
| |
| − | 1.3.9.2 The Epitext
| |
| − | 1.3.9.3 The Formative Tension
| |
| − | 1.3.10 Recurring Themes
| |
| − | 1.3.10.1 Preliminary Notions
| |
| − | 1.3.10.2 Intermediary Notions
| |
| − | 1.3.10.3 Propositions & Sentences
| |
| − | 1.3.10.4 Empirical Types & Rational Types
| |
| − | 1.3.10.5 Articulate Sentences
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| − | 1.3.10.6 Stretching Principles
| |
| − | 1.3.10.7 Stretching Operations
| |
| − | 1.3.10.8 The Cactus Patch
| |
| − | 1.3.10.9 The Cactus Language : Syntax
| |
| − | 1.3.10.10 The Cactus Language : Stylistics
| |
| − | 1.3.10.11 The Cactus Language : Mechanics
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| − | 1.3.10.12 The Cactus Language : Semantics
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| − | 1.3.10.13 Stretching Exercises
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| − | 1.3.10.14 Syntactic Transformations
| |
| − | 1.3.10.15 Derived Equivalence Relations
| |
| − | 1.3.10.16 Digression on Derived Relations
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| − | | |
| − | 1.4 Outlook of the Project : All Ways Lead to Inquiry
| |
| − | 1.4.1 The Matrix of Inquiry
| |
| − | 1.4.1.1 Inquiry as Conduct
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| − | 1.4.1.2 Types of Conduct
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| − | 1.4.1.3 Perils of Inquiry
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| − | 1.4.1.4 Forms of Relations
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| − | 1.4.1.5 Models of Inquiry
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| − | 1.4.2 The Moment of Inquiry
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| − | 1.4.3 The Modes of Inquiry
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| − | 1.4.3.1 Deductive Reasoning
| |
| − | 1.4.3.2 Inductive Reasoning
| |
| − | 1.4.3.3 Abductive Reasoning
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| − | 1.4.3.4 Analogical Reasoning
| |
| − | ...
| |
| − | | |
| − | 1.5 Obstacles to the Project : In the Way of Inquiry
| |
| − | 1.5.1 The Initial Unpleasantness
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| − | 1.5.2 The Justification Trap
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| − | 1.5.3 A Formal Apology
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| − | 1.5.3.1 Category Double-Takes
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| − | 1.5.3.2 Conceptual Extensions
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| − | 1.5.3.3 Explosional Recombinations
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| − | 1.5.3.4 Interpretive Frameworks
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| − | 1.5.4 A Material Exigency
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| − | 1.5.5 A Reconciliation of Accounts
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| − | 1.5.6 Objections to Reflexive Inquiry
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| − | 1.5.7 Empirical Considerations
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| − | 1.5.8 Computational Considerations
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| − | 1.5.8.1 A Form of Recursion
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| − | 1.5.8.2 A Power of Abstraction
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| − | | |
| − | 1.6 Orientation of the Project : A Way Into Inquiry
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| − | 1.6.1 Initial Description of Inquiry
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| − | 1.6.2 Terms of Analysis
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| − | 1.6.2.1 Digression on Signs
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| − | 1.6.2.2 Empirical Status of ID
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| − | 1.6.3 Expansion of Terms
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| − | 1.6.3.1 Agency
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| − | 1.6.3.2 Abstraction
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| − | 1.6.3.3 Analogy
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| − | 1.6.3.4 Accuracy
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| − | 1.6.3.5 Authenticity
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| − | 1.6.4 Anchoring Terms in Phenomena
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| − | 1.6.4.1 A Mistaken ID
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| − | 1.6.4.2 Phenomenology of Doubt
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| − | 1.6.4.3 Modalities of Knowledge
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| − | 1.6.5 Sets, Systems, & Substantive Agents
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| − | 1.6.6 Interpretive Systems
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| − | 1.6.6.1 Syntactic Systems
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| − | 1.6.6.2 Semantic Systems
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| − | 1.6.6.3 Pragmatic Systems
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| − | 1.6.7 Inquiry Driven Systems
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| − | 1.6.7.1 A Definition of Inquiry
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| − | 1.6.7.2 The Faculty of Inquiry
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| − | 1.6.7.3 A Definition of Determination
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| − | 1.6.7.4 A Definition of Definition
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| − | | |
| − | 1.7 Organization of the Project : A Way Through Inquiry
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| − | 1.7.1 The Problem : Inquiry Found as an Object of Study
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| − | 1.7.2 The Method : Inquiry Found as a Means of Study
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| − | 1.7.2.1 Conditions for the Possibility
| |
| − | of Inquiry into Inquiry
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| − | 1.7.2.2 Conditions for the Success
| |
| − | of Inquiry into Inquiry
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| − | 1.7.3 The Criterion : Inquiry in Search of a Sensible End
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| − | 1.7.3.1 The Irritation of Doubt, and The Scratch Test.
| |
| − | 1.7.3.2 Enabling Provision 1 : The Scenes & Context of Inquiry.
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| − | 1.7.3.3 Enabling Provision 2 : The Stages & Content of Inquiry.
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| − | 1.8 Objectives of the Project : Inquiry All the Way
| |
| − | 1.8.1 Substantial Objective
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| − | 1.8.1.1 Objective 1a : The Propositions as Types Analogy.
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| − | 1.8.1.2 Objective 1b : The Styles of Proof Development.
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| − | 1.8.1.3 Objective 1c : The Analysis of Interpreters, or A Problem with Authority.
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| − | 1.8.2 Instrumental Objective
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| − | 1.8.3 Coordination of Objectives
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| − | 1.8.4 Recapitulation : Da Capo, Al Segno
| |
| − | | |
| − | 2. Discussion of Inquiry
| |
| − | 2.1 Approaches to Inquiry
| |
| − | 2.1.1 The Classical Framework : Syllogistic Approaches
| |
| − | 2.1.2 The Pragmatic Framework : Sign-Theoretic Approaches
| |
| − | 2.1.3 The Dynamical Framework : System-Theoretic Approaches
| |
| − | 2.1.3.1 Inquiry & Computation
| |
| − | 2.1.3.2 Inquiry Driven Systems
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| − | 2.2 The Context of Inquiry
| |
| − | 2.2.1 The Field of Observation
| |
| − | 2.2.2 The Problem of Reflection
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| − | 2.2.3 The Problem of Reconstruction
| |
| − | 2.2.4 The Trivializing of Integration
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| − | 2.2.5 Tensions in the Field of Observation
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| − | 2.2.6 Problems of Representation & Communication
| |
| − | | |
| − | 2.3 The Conduct of Inquiry
| |
| − | 2.3.1 Introduction
| |
| − | 2.3.2 The Types of Reasoning
| |
| − | 2.3.2.1 Deduction
| |
| − | 2.3.2.2 Induction
| |
| − | 2.3.2.3 Abduction
| |
| − | 2.3.3 Hybrid Types of Inference
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| − | 2.3.3.1 Analogy
| |
| − | 2.3.3.2 Inquiry
| |
| − | 2.3.4 Details of Induction
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| − | 2.3.4.1 Learning
| |
| − | 2.3.4.2 Transfer
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| − | 2.3.4.3 Testing
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| − | 2.3.5 The Stages of Inquiry
| |
| − | | |
| − | 3. The Medium & Its Message
| |
| − | 3.1 Reflective Expression
| |
| − | 3.1.1 Casual Reflection
| |
| − | 3.1.1.1 Ostensibly Recursive Texts
| |
| − | 3.1.1.2 Analogical Recursion
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| − | 3.1.2 Conscious Reflection
| |
| − | 3.1.2.1 The Signal Moment
| |
| − | 3.1.2.2 The Symbolic Object
| |
| − | 3.1.2.3 The Endeavor to Communicate
| |
| − | 3.1.2.4 The Medium of Communication
| |
| − | 3.1.2.5 The Ark of Types : The Order of Things to Come.
| |
| − | 3.1.2.6 The Epitext
| |
| − | 3.1.2.7 The Context of Interpretation
| |
| − | 3.1.2.8 The Formative Tension
| |
| − | 3.1.2.9 The Vehicle of Communication :
| |
| − | Reflection on the Scene,
| |
| − | Reflection on the Self.
| |
| − | 3.1.2.10 (7)
| |
| − | 3.1.2.11 (6)
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| − | 3.1.2.12 Recursions : Possible, Actual, Necessary
| |
| − | 3.1.2.13 Ostensibly Recursive Texts
| |
| − | 3.1.2.14 (3)
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| − | 3.1.2.15 The Freedom of Interpretation
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| − | 3.1.2.16 The Eternal Return
| |
| − | 3.1.2.17 (1)
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| − | 3.1.2.18 Information in Formation
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| − | 3.1.2.19 Reflectively Indexical Texts
| |
| − | 3.1.2.20 (4)
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| − | 3.1.2.21 (5)
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| − | 3.1.2.22 (6)
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| − | 3.1.2.23 (7)
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| − | 3.1.2.24 (8)
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| − | 3.1.2.25 The Discursive Universe
| |
| − | 3.1.2.26 (7)
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| − | 3.1.2.27 (6)
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| − | 3.1.2.28 (5)
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| − | 3.1.2.29 (4)
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| − | 3.1.2.30 (3)
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| − | 3.1.2.31 (2)
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| − | 3.1.2.32 (1)
| |
| − | | |
| − | 3.2 Reflective Inquiry
| |
| − | 3.2.1 Integrity & Unity of Inquiry
| |
| − | 3.2.2 Apparitions & Allegations
| |
| − | 3.2.3 A Reflective Heuristic
| |
| − | 3.2.4 Either/Or : A Sense of Absence
| |
| − | 3.2.5 Apparent, Occasional, & Practical Necessity
| |
| − | 3.2.6 Approaches, Aspects, Exposures, Fronts
| |
| − | 3.2.7 Synthetic A Priori Truths
| |
| − | 3.2.8 Priorisms of Normative Sciences
| |
| − | 3.2.9 Principle of Rational Action
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| − | 3.2.10 The Pragmatic Cosmos
| |
| − | 3.2.11 Reflective Interpretive Frameworks
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| − | 3.2.11.1 Principals Versus Principles
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| − | 3.2.11.2 The Initial Description of Inquiry
| |
| − | 3.2.11.3 An Early Description of Interpretation
| |
| − | 3.2.11.4 Descriptions of the Mind
| |
| − | 3.2.11.5 Of Signs & the Mind
| |
| − | 3.2.11.6 Questions of Justification
| |
| − | 3.2.11.7 The Experience of Satisfaction
| |
| − | 3.2.11.8 An Organizational Difficulty
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| − | 3.2.11.9 Pragmatic Certainties
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| − | 3.2.11.10 Problems & Methods
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| − | | |
| − | 3.3 Reflection on Reflection
| |
| − | 3.4 Reflective Interpretive Frameworks
| |
| − | 3.4.1 The Phenomenology of Reflection
| |
| − | 3.4.2 A Candid Point of View
| |
| − | 3.4.3 A Projective Point of View
| |
| − | 3.4.4 A Formal Point of View
| |
| − | 3.4.5 Three Styles of Linguistic Usage
| |
| − | 3.4.6 Basic Notions of Group Theory
| |
| − | 3.4.7 Basic Notions of Formal Language Theory
| |
| − | 3.4.8 A Perspective on Computation
| |
| − | 3.4.9 Higher Order Sign Relations : Introduction
| |
| − | 3.4.10 Higher Order Sign Relations : Examples
| |
| − | 3.4.11 Higher Order Sign Relations : Application
| |
| − | 3.4.12 Issue 1 : The Status of Signs
| |
| − | 3.4.13 Issue 2 : The Status of Sets
| |
| − | 3.4.14 Issue 3 : The Status of Variables
| |
| − | 3.4.15 Propositional Calculus
| |
| − | 3.4.16 Recursive Aspects
| |
| − | 3.4.17 Patterns of Self-Reference
| |
| − | 3.4.18 Practical Intuitions
| |
| − | 3.4.19 Examples of Self-Reference
| |
| − | 3.4.20 Three Views of Systems
| |
| − | 3.4.21 Building Bridges Between Representations
| |
| − | 3.4.22 Extensional Representations of Sign Relations
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| − | 3.4.23 Intensional Representations of Sign Relations
| |
| − | 3.4.24 Literal Intensional Representations
| |
| − | 3.4.25 Analytic Intensional Representations
| |
| − | 3.4.26 Differential Logic & Directed Graphs
| |
| − | 3.4.27 Differential Logic & Group Operations
| |
| − | 3.4.28 The Bridge : From Obstruction to Opportunity
| |
| − | 3.4.29 Projects of Representation
| |
| − | 3.4.30 Connected, Integrated, Reflective Symbols
| |
| − | 3.4.31 Generic Orders of Relations
| |
| − | 3.4.32 Partiality : Selective Operations
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| − | 3.4.33 Sign Relational Complexes
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| − | 3.4.34 Set-Theoretic Constructions
| |
| − | 3.4.35 Reducibility of Sign Relations
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| − | 3.4.36 Irreducibly Triadic Relations
| |
| − | 3.4.37 Propositional Types
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| − | 3.4.38 Considering the Source
| |
| − | 3.4.39 Prospective Indices : Pointers to Future Work
| |
| − | 3.4.40 Dynamic & Evaluative Frameworks
| |
| − | 3.4.41 Elective & Motive Forces
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| − | 3.4.42 Sign Processes : A Start
| |
| − | 3.4.43 Reflective Extensions
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| − | 3.4.44 Reflections on Closure
| |
| − | 3.4.45 Intelligence => Critical Reflection
| |
| − | 3.4.46 Looking Ahead
| |
| − | 3.4.47 Mutually Intelligible Codes
| |
| − | 3.4.48 Discourse Analysis : Ways & Means
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| − | 3.4.49 Combinations of Sign Relations
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| − | 3.4.50 Revisiting the Source
| |
| − | 3.5 Divertimento : Eternity in Love with the Creatures of Time
| |
| − | 3.5.1 Reflections on the Presentation of Examples
| |
| − | 3.5.2 Searching for Parameters
| |
| − | 3.5.3 Defect Analysis
| |
| − | 3.5.4 The Pragmatic Critique
| |
| − | 3.5.5 Pragmatic Operating Notions
| |
| − | 3.5.6 Defects of Presentation
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| − | 3.5.7 Dues to Process
| |
| − | 3.5.8 Duties to Purpose
| |
| − | 3.6 Computational Design Philosophy
| |
| − | 3.6.1 Intentional Objects & Attitudes
| |
| − | 3.6.2 Imperfect Design & Persistent Error
| |
| − | 3.6.3 Propositional Reasoning About Relations
| |
| − | 3.6.4 Dynamic & Evaluative Frameworks
| |
| − | 3.6.5 Discussion of Examples
| |
| − | 3.6.6 Information & Inquiry
| |
| − | | |
| − | 4. Overview of the Domain : Interpretive Inquiry
| |
| − | 4.1 Interpretive Bearings : Conceptual & Descriptive Frameworks
| |
| − | 4.1.1 Catwalks : Flexible Frameworks & Peripatetic Categories
| |
| − | 4.1.1.1 Eponymous Ancestors : The Precursors of Abstraction?
| |
| − | 4.1.1.2 Reticles : Interpretive Flexibility as a Design Issue.
| |
| − | 4.1.2 Heuristic Inclinations & Regulative Principles
| |
| − | 4.2 Features of Inquiry Driven Systems
| |
| − | 4.2.1 The Pragmatic Theory of Signs
| |
| − | 4.2.1.1 Sign Relations
| |
| − | 4.2.1.2 Types of Signs
| |
| − | 4.2.2 The Pragmatic Theory of Inquiry
| |
| − | 4.2.2.1 Abduction
| |
| − | 4.2.2.2 Deduction
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| − | 4.2.2.3 Induction
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| − | 4.3 Examples of Inquiry Driven Systems
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| − | 4.3.1 "Index" : A Program for Learning Formal Languages
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| − | 4.3.2 "Study" : A Program for Reasoning with Propositions
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| − | 5. Discussion & Development of Objectives
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| − | 5.1 Objective 1a : Propositions as Types
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| − | 5.2 Objective 1b : Proof Styles & Developments
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| − | 5.3 Objective 1c : Interpretation & Authority
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| − | </pre>
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