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{{DISPLAYTITLE:Inquiry Driven Systems : Part 2}}
{{DISPLAYTITLE:Inquiry Driven Systems : Part 2}}
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<div align="center" ><big>
<div align="center">
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Contents]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Contents]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1|Part 1]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1|Part 1]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2|Part 2]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2|Part 2]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 3|Part 3]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 3|Part 3]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 4|Part 4]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5|Part 5]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6|Part 6]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 7|Part 7]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 8|Part 8]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices|Appendices]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Document History|Document History]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Document History|Document History]]
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•
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<div class="nonumtoc">__TOC__</div>
<div class="nonumtoc">__TOC__</div>
==2.==
==Part 2.==
===2.1. Reconnaissance===
===2.1. Reconnaissance===
{| align="center" cellpadding="0" cellspacing="0" width="90%"
{| align="center" cellpadding="0" cellspacing="0" width="90%"
In the process of carrying out the present reconnaissance it is useful to illustrate the pragmatic theory of signs as it bears on a series of slightly less impoverished and somewhat more interesting materials, to demonstrate a few of the ways that the theory of signs can be applied to a selection of genuinely complex and problematic texts, specifically, poetic and lyrical texts that are elicited from natural language sources through the considerable art of creative authors. In keeping with the nonchalant provenance of these texts, I let them make their appearance on the scene of the present discussion in what may seem like a purely incidental way, and only gradually to acquire an explicit recognition.
In the process of carrying out the present reconnaissance it is useful to illustrate the pragmatic theory of signs as it bears on a series of slightly less impoverished and somewhat more interesting materials, to demonstrate a few of the ways that the theory of signs can be applied to a selection of genuinely complex and problematic texts, specifically, poetic and lyrical texts that are elicited from natural language sources through the considerable art of creative authors. In keeping with the nonchalant provenance of these texts, I let them make their appearance on the scene of the present discussion in what may seem like a purely incidental way, and only gradually to acquire an explicit recognition.
====2.1.1. The Informal Context====
====2.1.1. The Informal Context====
<br>
<br>
The form of initiatory task that a certain turn of mind arrives at only toward the end of its quest is not so much to describe the tensions that exist among contexts — those between the formal arenas, bowers, courts and the informal context that surrounds them all — as it is to exhibit these forces in action and to bear up under their influences on inquiry. The task is not so much to talk about the informal context, to the point of trying to exhaust it with words, as it is to anchor one's activity in the infinitudes of its unclaimed resources, to the depth that it allows this importunity, and to buoy the significant points of one's discussion, its channels, shallows, shoals, and shores, for the time that the tide permits this opportunity.
The form of initiatory task that a certain turn of mind arrives at only toward the end of its quest is not so much to describe the tensions that exist among contexts — those between the formal arenas, bowers, courts and the informal context that surrounds them all — as it is to exhibit these forces in action and to bear up under their influences on inquiry. The task is not so much to talk about the informal context, to the point of trying to exhaust it with words, as it is to anchor one's activity in the infinitudes of its unclaimed resources, to the depth that it allows this importunity, and to buoy the significant points of one's discussion, its channels, shallows, shoals, and shores, for the time that the tide permits this opportunity.
====2.1.2. The Epitext====
====2.1.2. The Epitext====
It is time to render more explicit a feature of the text in the previous subsection, to abstract the form that it realizes from the materials that it appropriates to fill out its pattern, to extract the generic structure of its devices as a style of presentation or a standard technique, and to make this formal resource available for use as future occasions warrant. To this end, let a succession of epigraphs, incidental to a main text but having a consistent purpose all their own, and illustrating the points of the main text in an exemplary, poignant, or succinct way, be referred to as an "epitext".
It is time to render more explicit a feature of the text in the previous subsection, to abstract the form that it realizes from the materials that it appropriates to fill out its pattern, to extract the generic structure of its devices as a style of presentation or a standard technique, and to make this formal resource available for use as future occasions warrant. To this end, let a succession of epigraphs, incidental to a main text but having a consistent purpose all their own, and illustrating the points of the main text in an exemplary, poignant, or succinct way, be referred to as an "epitext".
This means that a recursive interpretation of a sign or a text can recur just so long as its interpreter has an interest in pursuing it. It can terminate, not just with the absolute extremes of an ideal object or an objective limit, that is, with states of perfect certainty or tokens of ultimate clarity, but also in the interpretive direction, that is, with forms of self-recognition and a conduct that arises from self-knowledge. In the meantime, between these points of final termination, a recursive interpretation can also pause on a temporary basis at any time that the degree of involvement of the interpreter is pushed beyond the limits of moderation, or any time that the level of interest for the interpreter drifts beyond or is driven outside the band of personal toleration.
This means that a recursive interpretation of a sign or a text can recur just so long as its interpreter has an interest in pursuing it. It can terminate, not just with the absolute extremes of an ideal object or an objective limit, that is, with states of perfect certainty or tokens of ultimate clarity, but also in the interpretive direction, that is, with forms of self-recognition and a conduct that arises from self-knowledge. In the meantime, between these points of final termination, a recursive interpretation can also pause on a temporary basis at any time that the degree of involvement of the interpreter is pushed beyond the limits of moderation, or any time that the level of interest for the interpreter drifts beyond or is driven outside the band of personal toleration.
====2.1.3. The Formative Tension====
====2.1.3. The Formative Tension====
The incidental arena or informal context is presently described in casual, derivative, or 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. 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 staging ground where everything that appears on the formal scene is first assembled for a formal presentation. In taking this view, one is stepping back a bit in one's imagination from the scene that presses on one's attention, getting a sense of its frame and its stage, and becoming accustomed to see what appears in ever dimmer lights, in short, one is learning to reflect on the more obvious actions, to read their pretexts, and to detect the motives that end in them.
The incidental arena or informal context is presently described in casual, derivative, or 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. 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 staging ground where everything that appears on the formal scene is first assembled for a formal presentation. In taking this view, one is stepping back a bit in one's imagination from the scene that presses on one's attention, getting a sense of its frame and its stage, and becoming accustomed to see what appears in ever dimmer lights, in short, 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 that is available for exercise in an informal context, that is, without being required to formalize its properties prior to their 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.
It is fair to assume that an agent of inquiry possesses a faculty of inquiry that is available for exercise in an informal context, that is, without being required to formalize its properties prior to their 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 an open forum or a formative space, it is possible to form the alignments of capacities that are indicated in Table 5.
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.
<pre>
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{Alignments of Capacities}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Formal}\!</math>
| colspan="2" | <math>\text{Formative}\!</math>
|-
| <math>\text{Objective}\!</math>
| colspan="2" | <math>\text{Instrumental}\!</math>
|-
| <math>\text{Passive}\!</math>
| colspan="2" | <math>\text{Active}\!</math>
|-
| width="33%" | <math>\text{Afforded}\!</math>
| width="33%" | <math>\text{Possessed}\!</math>
| width="33%" | <math>\text{Exercised}\!</math>
|}
<pre>
<br>
The style of this discussion, 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 fruitful to compare the current alignments with those given 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>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Alignments of Capacities in Aristotle}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Matter}\!</math>
| colspan="2" | <math>\text{Form}\!</math>
|-
|-
| valign=top | a. || The theories of the soul (psyche) handed down by our predecessors have been sufficiently discussed; now let us start afresh, as it were, and try to determine (diorisai) what the soul is, and what definition (logos) of it will be most comprehensive (koinotatos).
| <math>\text{Potentiality}\!</math>
| colspan="2" | <math>\text{Actuality}\!</math>
|-
|-
| valign=top | b. || We describe one class of existing things as substance (ousia), and this we subdivide into three: (1) matter (hyle), which in itself is not an individual thing, (2) shape (morphe) or form (eidos), in virtue of which individuality is directly attributed, and (3) the compound of the two.
| width="33%" | <math>\text{Receptivity}\!</math>
| width="33%" | <math>\text{Possession}\!</math>
| width="33%" | <math>\text{Exercise}\!</math>
|-
|-
| valign=top | c. || 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).
| <math>\text{Life}\!</math>
| <math>\text{Sleep}~\!</math>
| <math>\text{Waking}\!</math>
|-
|-
| valign=top | d. || Bodies (somata) seem to be pre-eminently substances, and most particularly those which are of natural origin (physica), for these are the sources (archai) from which the rest are derived.
| <math>\text{Wax}\!</math>
| colspan="2" | <math>\text{Impression}\!</math>
|-
|-
| valign=top | e. || But of natural bodies some have life (zoe) and some have not; by life we mean the capacity for self-sustenance, growth, and decay.
| <math>\text{Axe}\!</math>
| <math>\text{Edge}\!</math>
| <math>\text{Cutting}\!</math>
|-
|-
| valign=top | f. || Every natural body (soma physikon), then, which possesses life must be substance, and substance of the compound type (synthete).
| <math>\text{Eye}\!</math>
| <math>\text{Vision}\!</math>
| <math>\text{Seeing}\!</math>
|-
|-
| valign=top | g. || But since it is a body of a definite kind, viz., having life, the body (soma) cannot be soul (psyche), for the body is not something predicated of a subject, but rather is itself to be regarded as a subject, i.e., as matter.
| <math>\text{Body}\!</math>
| colspan="2" | <math>\text{Soul}\!</math>
|-
|-
| valign=top | h. || So the soul must be substance in the sense of being the form of a natural body, which potentially has life. And substance in this sense is actuality.
| <math>\text{Ship?}\!</math>
| colspan="2" | <math>\text{Sailor?}\!</math>
|}
<br>
| valign=top | k. || Now in one and the same person the possession of knowledge comes first.
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 alterations of case, tense, and grammatical category that occur throughout the paradigm, or perhaps it has something to do with the fact that the relationships through the middle of the Table are more analogical than categorical. For the moment I am content to leave all the paradoxes intact, taking the pattern of tensions and torsions as a puzzle for future study.
| valign=top | l. || The soul may therefore be defined as the first actuality of a natural body potentially possessing life; and such will be any body which possesses organs (organikon).
<br>
|-
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|-
|+ style="height:30px" | <math>\text{Table 7.} ~~ \text{Synthesis of Alignments}\!</math>
| valign=top | n. || If then one is to find a definition which will apply to every soul, it will be "the first actuality of a natural body possessed of organs".
|- style="height:40px; background:ghostwhite"
| <math>\text{Formal}\!</math>
| colspan="2" | <math>\text{Formative}\!</math>
|-
|-
| valign=top | p. || We have, then, given a general definition of what the soul is: it is substance in the sense of formula (logos), i.e., the essence of such-and-such a body.
| <math>\text{Objective}\!</math>
| colspan="2" | <math>\text{Instrumental}\!</math>
|-
|-
| valign=top | q. || Suppose that an implement (organon), e.g. an axe, were a natural body; the substance of the axe would be that which makes it an axe, and this would be its soul; suppose this removed, and it would no longer be an axe, except equivocally. As it is, it remains an axe, because it is not of this kind of body that the soul is the essence or formula, but only of a certain kind of natural body which has in itself a principle of movement and rest.
| <math>\text{Passive}\!</math>
| colspan="2" | <math>\text{Active}\!</math>
|-
|-
| valign=top | r. || We must, however, investigate our definition in relation to the parts of the body.
| width="33%" | <math>\text{Afforded}\!</math>
| width="33%" | <math>\text{Possessed}\!</math>
| width="33%" | <math>\text{Exercised}\!</math>
|-
|-
| valign=top | s. || If the eye were a living creature, its soul would be its vision; for this is the substance in the sense of formula of the eye. But the eye is the matter of vision, and if vision fails there is no eye, except in an equivocal sense, as for instance a stone or painted eye.
| <math>\text{To Hold}\!</math>
| <math>\text{To Have}\!</math>
| <math>\text{To Use}\!</math>
|-
|-
| valign=top | t. || Now we must apply what we have found true of the part to the whole living body. For the same relation must hold good of the whole of sensation to the whole sentient body qua sentient as obtains between their respective parts.
| <math>\text{Receptivity}\!</math>
| <math>\text{Possession}\!</math>
| <math>\text{Exercise}\!</math>
|-
|-
| valign=top | u. || That which has the capacity to live is not the body which has lost its soul, but that which possesses its soul; so seed and fruit are potentially bodies of this kind.
| <math>\text{Potentiality}\!</math>
| colspan="2" | <math>\text{Actuality}\!</math>
|-
|-
| valign=top | v. || The waking state is actuality in the same sense as the cutting of the axe or the seeing of the eye, while the soul is actuality in the same sense as the faculty of the eye for seeing, or of the implement for doing its work.
| <math>\text{Matter}\!</math>
| colspan="2" | <math>\text{Form}\!</math>
| valign=top | z. || This must suffice as an attempt to determine in rough outline the nature of the soul.
|}
|}
<br>
Due to the importance of Aristotle's account for every discussion that follows it, not to mention for the many that follow it without knowing it, and because the issues it raises arise repeatedly throughout this work, I am going to cite an extended extract from the relevant text (Aristotle, ''On the Soul'', 2.1), breaking up the argument into a number of individual premisses, stages, and examples.
<ol style="list-style-type:lower-alpha">
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.
<li>The theories of the soul (<i>psyche</i>) handed down by our predecessors have been sufficiently discussed; now let us start afresh, as it were, and try to determine (<i>diorisai</i>) what the soul is, and what definition (<i>logos</i>) of it will be most comprehensive (<i>koinotatos</i>).</li>
<li>We describe one class of existing things as substance (<i>ousia</i>), and this we subdivide into three: (1) matter (<i>hyle</i>), which in itself is not an individual thing, (2) shape (<i>morphe</i>) or form (<i>eidos</i>), in virtue of which individuality is directly attributed, and (3) the compound of the two.</li>
<li>Matter is potentiality (<i>dynamis</i>), while form is realization or actuality (<i>entelecheia</i>), and the word actuality is used in two senses, illustrated by the possession of knowledge (<i>episteme</i>) and the exercise of it (<i>theorein</i>).</li>
<li>Bodies (<i>somata</i>) seem to be pre-eminently substances, and most particularly those which are of natural origin (<i>physica</i>), for these are the sources (<i>archai</i>) from which the rest are derived.</li>
<li>But of natural bodies some have life (<i>zoe</i>) and some have not; by life we mean the capacity for self-sustenance, growth, and decay.</li>
<li>Every natural body (<i>soma physikon</i>), then, which possesses life must be substance, and substance of the compound type (<i>synthete</i>).</li>
<li>But since it is a body of a definite kind, <i>viz.</i>, having life, the body (<i>soma</i>) cannot be soul (<i>psyche</i>), for the body is not something predicated of a subject, but rather is itself to be regarded as a subject, <i>i.e.</i>, as matter.</li>
<li>So the soul must be substance in the sense of being the form of a natural body, which potentially has life. And substance in this sense is actuality.</li>
The complete sign relation involved in a situation encompasses all the things that one thinks about and all the thoughts that one thinks about them while engaged in that situation, in other words, all the signs and ideas that flit through one's mind in relation to a domain of objects. Only a rarefied sample of this complete sign relation is bound to avail itself to reflective awareness, still less of it is likely to inspire a common 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 an investigator, oneself in particular, is likely to form between two sign relations: (1) the whole sign relation that one intends to study, and (2) the selective portion of it that one is able to pin down for a formal examination.
<li>The soul, then, is the actuality of the kind of body we have described. But actuality has two senses, analogous to the possession of knowledge and the exercise of it.</li>
<li>Clearly (<i>phaneron</i>) actuality in our present sense is analogous to the possession of knowledge; for both sleep (<i>hypnos</i>) and waking (<i>egregorsis</i>) depend upon the presence of the soul, and waking is analogous to the exercise of knowledge, sleep to its possession (<i>echein</i>) but not its exercise (<i>energein</i>).</li>
<li>Now in one and the same person the possession of knowledge comes first.</li>
<li>The soul may therefore be defined as the first actuality of a natural body potentially possessing life; and such will be any body which possesses organs (<i>organikon</i>).</li>
<li>(The parts of plants are organs too, though very simple ones: <i>e.g.</i>, the leaf protects the pericarp, and the pericarp protects the seed; the roots are analogous to the mouth, for both these absorb food.)</li>
<li>If then one is to find a definition which will apply to every soul, it will be “the first actuality of a natural body possessed of organs”.</li>
<li>So one need no more ask (<i>zetein</i>) whether body and soul are one than whether the wax (<i>keros</i>) and the impression (<i>schema</i>) it receives are one, or in general whether the matter of each thing is the same as that of which it is the matter; for admitting that the terms unity and being are used in many senses, the paramount (<i>kyrios</i>) sense is that of actuality.</li>
<li>We have, then, given a general definition of what the soul is: it is substance in the sense of formula (<i>logos</i>), <i>i.e.</i>, the essence of such-and-such a body.</li>
<li>Suppose that an implement (<i>organon</i>), <i>e.g.</i> an axe, were a natural body; the substance of the axe would be that which makes it an axe, and this would be its soul; suppose this removed, and it would no longer be an axe, except equivocally. As it is, it remains an axe, because it is not of this kind of body that the soul is the essence or formula, but only of a certain kind of natural body which has in itself a principle of movement and rest.</li>
<li>We must, however, investigate our definition in relation to the parts of the body.</li>
<li>If the eye were a living creature, its soul would be its vision; for this is the substance in the sense of formula of the eye. But the eye is the matter of vision, and if vision fails there is no eye, except in an equivocal sense, as for instance a stone or painted eye.</li>
<math>\begin{array}{lll}
<li>Now we must apply what we have found true of the part to the whole living body. For the same relation must hold good of the whole of sensation to the whole sentient body <i>qua</i> sentient as obtains between their respective parts.</li>
<li>That which has the capacity to live is not the body which has lost its soul, but that which possesses its soul; so seed and fruit are potentially bodies of this kind.</li>
<li>The waking state is actuality in the same sense as the cutting of the axe or the seeing of the eye, while the soul is actuality in the same sense as the faculty of the eye for seeing, or of the implement for doing its work.</li>
<li>The body is that which exists potentially; but just as the pupil and the faculty of seeing make an eye, so in the other case the soul and body make a living creature.</li>
<li>It is quite clear, then, that neither the soul nor certain parts of it, if it has parts, can be separated from the body; for in some cases the actuality belongs to the parts themselves. Not but what there is nothing to prevent some parts being separated, because they are not actualities of any body.</li>
<li>It is also uncertain (<i>adelon</i>) whether the soul as an actuality bears the same relation to the body as the sailor (<i>ploter</i>) to the ship (<i>ploion</i>).</li>
<li>This must suffice as an attempt to determine in rough outline the nature of the soul.</li>
</ol>
===2.2. Recurring Themes===
The overall purpose of the next several Sections is threefold:
# To continue to illustrate the salient properties of sign relations in the medium of selected examples.
# To demonstrate the use of sign relations to analyze and clarify a particular order of difficult symbols and complex texts, namely, those that involve recursive, reflective, or reflexive features.
# 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 triadic 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 ''intersection'' of <math>P\!</math> and <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>^{\backprime\backprime} \, P \cap Q \, ^{\prime\prime}</math> and defined as the set of elements in <math>X\!</math> that belong to both <math>P\!</math> and <math>Q.\!</math>
The main goals of the present section are as follows:
# 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.
# To use this modicum of formalism to define a number of conceptual constructs, useful in the analysis of more general sign relations.
# To develop a proof format that is suitable for deriving facts about these constructs in a careful and potentially computational manner.
# 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.
====2.2.1. Preliminary Notions====
The present phase of discussion proceeds by recalling a series of basic definitions, refining them to deal with more specialized situations, and refitting them as necessary to cover larger families of sign relations.
The ''symmetric difference'' of <math>P\!</math> and <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>^{\backprime\backprime} \, P ~\hat{+}~ Q \, ^{\prime\prime}</math> and is defined as the set of elements in <math>X\!</math> that belong to just one of <math>P\!</math> or <math>Q.\!</math>
In this discussion the word ''semantic'' is being used as a generic adjective to describe anything concerned with or related to meaning, 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'' (SECs), ''semiotic partitions'' (SEPs), ''semiotic equations'' (SEQs), and ''semiotic equivalence relations'' (SERs), as in Segment 1.3.4.3.
The foregoing "definitions" are the bare essentials that are needed to get the rest of this discussion going, 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 ''not'', ''and'', ''or'', as these are expressed in natural language terms.
The discussion up to this point is partial to examples of sign relations that enjoy especially nice properties, in particular, whose connotative components form equivalence relations and whose denotative components conform to these equivalences, in the sense that all of the signs in a single equivalence class always denote one and the same object. By way of liberalizing this discussion to more general cases of sign relations, this subsection develops a number of additional concepts for describing the internal relations of sign relations and makes a set of definitions that do not take the aforementioned features for granted.
The complete sign relation involved in a situation encompasses all the things that one thinks about and all the thoughts that one thinks about them while engaged in that situation, in other words, all the signs and ideas that flit through one's mind in relation to a domain of objects. Only a rarefied sample of this complete sign relation is bound to avail itself to reflective awareness, still less of it is likely to inspire a common 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 an investigator, oneself in particular, is likely to form between two sign relations: (1) the whole sign relation that one intends to study, and (2) the selective portion of it that one is able to pin down for a formal examination.
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 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.
With respect to the general variety of sampling relations there are a number of specific conceptions that are likely to be useful in this study, a few of which can now be discussed.
A ''bit'' of a sign relation is defined to be any subset of its extension, that is, an arbitrary selection from its set of ordered triples.
Described in relation to sampling relations, a bit 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 bit of a sign relation, by virtue of its appearing in evidence, can always be interpreted as a bit of evidence that some sort of sampling relation is being applied.
====2.2.2. Intermediary Notions====
A number of additional definitions are relevant to sign relations whose connotative components constitute equivalence relations, if only in part.
<br>
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. In a typical notation, the dyadic relation <math>\underline{G} = (X, G) = (G^{(1)}, G^{(2)})</math> is specified by giving the set of points <math>X = G^{(1)}\!</math> and the set of ordered pairs <math>G = G^{(2)} \subseteq X \times X</math> that go together to define the relation. In contexts where the set of points is understood, it is customary to call the whole relation <math>\underline{G}</math> by the name of the set <math>G.\!</math>
A ''subrelation'' of a dyadic relation <math>\underline{G} = (X, G) = (G^{(1)}, G^{(2)})</math> is a dyadic relation <math>\underline{H} = (Y, H) = (H^{(1)}, H^{(2)})</math> that has all of its points and pairs in <math>\underline{G}</math> more precisely, that has all of its points <math>Y \subseteq X</math> and all of its pairs <math>H \subseteq G.</math>
<br>
The ''induced subrelation on a subset'' (ISOS), taken with respect to the dyadic relation <math>G \subseteq X \times X</math> and the subset <math>Y \subseteq X,</math> is the maximal subrelation of <math>G\!</math> whose points belong to <math>Y.\!</math> In other words, it is the dyadic relation on <math>Y\!</math> whose extension contains all of the pairs of <math>Y \times Y</math> that appear in <math>G.\!</math> Since the construction of an ISOS is uniquely determined by the data of <math>G\!</math> and <math>Y,\!</math> it can be represented as a function of these arguments, as in the notation <math>\operatorname{ISOS} (G, Y),</math> which can be denoted more briefly as <math>\underline{G}_Y.\!</math> Using the symbol <math>\bigcap</math> to indicate the intersection of a pair of sets, the construction of <math>\underline{G}_Y = \operatorname{ISOS} (G, Y)</math> can be defined as follows:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
\underline{G}_Y & = & (Y, \ G_Y)
\\ \\
& = & (G_Y^{(1)}, \ G_Y^{(2)})
\\ \\
& = & (Y, \ \{ (x, y) \in Y\!\times\!Y : (x, y) \in G^{(2)} \})
\\ \\
& = & (Y, \ Y\!\times\!Y \, \bigcap \, G^{(2)}).
\\
\end{array}</math>
|}
These definitions for dyadic relations can now be applied in a context where each bit of a sign relation that is being considered satisfies a special set of conditions, namely, if <math>R\!</math> is the relational bit under consideration:
<br>
# Syntactic domain <math>{X}\!</math> = Sign domain <math>{S}\!</math> = Interpretant domain <math>{I}.\!</math>
# Connotative component = <math>{R_{XX}}\!</math> = <math>{R_{SI}}\!</math> = Equivalence relation <math>{E}.\!</math>
Under these assumptions, and with regard to bits of sign relations that satisfy these conditions, it is useful to consider further selections of a specialized sort, namely, those that keep equivalent signs synonymous.
An ''arbit'' of a sign relation is a slightly more judicious bit of it, preserving a semblance of whatever SEP happens to rule over its signs, and respecting the semiotic parts of the sampled sign relation, when it has such parts. In other words, an arbit suggests an act of selection that represents the parts of the original SEP by means of the parts of the resulting SEP, that extracts an ISOS of each clique in the SER that it bothers to select any points at all from, and that manages to portray in at least this partial fashion all or none of every SEC that appears in the original sign relation.
====2.2.3. Propositions and Sentences====
<br>
The concept of a sign relation is typically extended as a set <math>L \subseteq O \times S \times I.\!</math> 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 in which it is likely to be encountered, a good amount of set-theoretic machinery is necessary 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 <math>Q,\!</math> that comprises a subject of interest in a particular discussion or that constitutes a topic of interest in a particular moment of discussion, is a subset of a set <math>X,\!</math> one that is sufficiently universal relative to that discussion or big enough to cover everything that is being talked about in that moment. In a setting like this it is possible to make a number of useful definitions, to which we now turn.
<br>
The ''negation'' of a sentence <math>s\!</math>, written as <math>{}^{\backprime\backprime} \texttt{(} s \texttt{)} \, {}^{\prime\prime}</math> and read as <math>{}^{\backprime\backprime} \, \operatorname{not}\ s \, {}^{\prime\prime},\!</math> is a sentence that is true when <math>s\!</math> is false and false when <math>s\!</math> is true.
The ''complement'' of a set <math>Q\!</math> with respect to the universe <math>X\!</math> is denoted by <math>{}^{\backprime\backprime} \, X\!-\!Q \, {}^{\prime\prime}\!</math> and is defined as the set of elements in <math>X\!</math> that do not belong to <math>Q.\!</math> When the universe <math>X\!</math> is fixed throughout a given discussion, the complement <math>X\!-\!Q</math> may be denoted either by <math>{}^{\backprime\backprime} \thicksim \! Q \, {}^{\prime\prime}\!</math> or by <math>{}^{\backprime\backprime} \, \tilde{Q} \, {}^{\prime\prime}.\!</math> Thus we have the following series of equivalences:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lllllll}
\tilde{Q}
& = & \thicksim\!Q
& = & X\!-\!Q
& = & \{ x \in X : \texttt{(} x \in Q \texttt{)} \}.
\end{array}</math>
|}
The ''relative complement'' of <math>P\!</math> in <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>{}^{\backprime\backprime} \, Q\!-\!P \, {}^{\prime\prime}</math> and defined as the set of elements in <math>Q\!</math> that do not belong to <math>P,\!</math> that is:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
Q\!-\!P
& = & \{ x \in X : x \in Q ~\operatorname{and}~ \texttt{(} x \in P \texttt{)} \}.
\end{array}</math>
|}
The ''indicator function'' or the ''characteristic function'' of the set <math>Q \subseteq X,</math> written <math>f_Q,\!</math> is the map from the universe <math>X\!</math> to the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> that is defined in the following ways:
The ''intersection'' of <math>P\!</math> and <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>{}^{\backprime\backprime} \, P \cap Q \, {}^{\prime\prime}</math> and defined as the set of elements in <math>X\!</math> that belong to both <math>P\!</math> and <math>Q.\!</math>
<ol style="list-style-type:decimal">
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
P \cap Q
& = & \{ x \in X : x \in P ~\operatorname{and}~ x \in Q \}.
\end{array}</math>
|}
<li>
The ''union'' of <math>P\!</math> and <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>{}^{\backprime\backprime} \, P \cup Q \, {}^{\prime\prime}</math> and defined as the set of elements in <math>X\!</math> that belong to at least one of <math>P\!</math> or <math>Q.\!</math>
<p>Considered in extensional form, <math>f_Q\!</math> is the subset of <math>X \times \underline\mathbb{B}</math> that is given by the following formula:</p>
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
P \cup Q
& = & \{ x \in X : x \in P ~\operatorname{or}~ x \in Q \}.
\end{array}</math>
|}
<li>
The ''symmetric difference'' of <math>P\!</math> and <math>Q,\!</math> for two sets <math>P, Q \subseteq X,</math> is denoted by <math>{}^{\backprime\backprime} \, P ~\hat{+}~ Q \, {}^{\prime\prime}\!</math> and is defined as the set of elements in <math>X\!</math> that belong to just one of <math>P\!</math> or <math>Q.\!</math>
<p>Considered in functional form, <math>f_Q\!</math> is the map from <math>X\!</math> to <math>\underline\mathbb{B}</math> that is given by the following condition:</p>
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
P ~\hat{+}~ Q
& = & \{ x \in X : x \in P\!-\!Q ~\operatorname{or}~ x \in Q\!-\!P \}.
\end{array}</math>
|}
The foregoing “definitions” are the bare essentials that are needed to get the rest of this discussion going, 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 ''not'', ''and'', ''or'', as these are expressed in natural language terms.
As formative definitions, these initial postulations neither acquire the privileged status of untouchable axioms and infallible intuitions nor do they deserve any special suspicion, at least, nothing over and above the reflective critique that one ought to apply to all important definitions. These dim beginnings of anything approaching genuine definitions also serve to accustom the mind's eye to a particular style of observation, 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 <math>\{ x \in X : \underline{~~~} \}\!</math> functions like the ''eye of the needle'' through which one is trying to transport a suitably rich import of mathematics.
Part the task of the remaining discussion is gradually to formalize the promissory notes that are represented by these terms and stipulations and to see whether their casual comprehension can be converted into an explicit subject matter, one that depends on grasping the corresponding collection of almost wholly, if still partially formalized conceptions. To this we now turn.
The ''fiber'' of a codomain element <math>y \in Y\!</math> under a function <math>f : X \to Y</math> is the subset of the domain <math>X\!</math> that is mapped onto <math>y,\!</math> in short, it is <math>f^{-1} (y) \subseteq X.</math> In other language that is often used, the fiber of <math>y\!</math> under <math>f\!</math> is called the ''antecedent set'', the ''inverse image'', the ''level set'', or the ''pre-image'' of <math>y\!</math> under <math>f.\!</math> All of these equivalent concepts are defined as follows:
The ''binary domain'' is the set <math>{\mathbb{B} = \{ 0, 1 \}}\!</math> of two algebraic values, whose arithmetic operations obey the rules of <math>\operatorname{GF}(2),\!</math> the ''galois field'' of exactly two elements, whose addition and multiplication tables are tantamount to addition and multiplication of integers modulo 2.
The ''boolean domain'' is the set <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}\!</math> of two logical values, 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 domains is slight or significant, and so this question must evolve its 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 <math>\underline\mathbb{B}</math> 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 <math>\mathbb{B},</math> and operations that are isomorphic to the rest of the boolean operations in <math>\underline\mathbb{B}</math> can always be built on the binary basis of <math>\mathbb{B}.</math>
Of course, as sets of the same cardinality, the domains <math>\mathbb{B}\!</math> and <math>\underline\mathbb{B}\!</math> and all of the structures that can be built on them become isomorphic at a high enough level of abstraction. Consequently, the main reason for making this distinction in the present context appears to be a matter more of grammar than an issue of logical and mathematical substance, namely, so that the signs <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime}\!</math> can appear with some semblance of syntactic legitimacy in linguistic contexts that call for a grammatical sentence or a sentence surrogate to represent the classes of sentences that are ''always false'' and ''always true'', respectively. The signs <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} 1 {}^{\prime\prime},\!</math> 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 particular choice of natural language context, are ultimately important, is another thing that remains to be determined.
{| align="center" cellpadding="8" width="90%"
The ''negation'' of a value <math>x\!</math> in <math>\underline\mathbb{B},</math> written <math>{}^{\backprime\backprime} \texttt{(} x \texttt{)} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \lnot x {}^{\prime\prime}</math> and read as <math>{}^{\backprime\backprime} \operatorname{not}\ x {}^{\prime\prime},</math> is the boolean value <math>\texttt{(} x \texttt{)} \in \underline\mathbb{B}</math> that is <math>\underline{1}</math> when <math>x\!</math> is <math>\underline{0}</math> and <math>\underline{0}</math> when <math>x\!</math> is <math>\underline{1}.</math> Negation is a monadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \to \underline\mathbb{B},</math> as shown in Table 8.
<br>
{| align="center" cellpadding="8" width="90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
| <math>\operatorname{If}~ f : X \to \underline\mathbb{B},</math>
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Negation Operation for the Boolean Domain}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>x\!</math>
| <math>\texttt{(} x \texttt{)}</math>
|-
|-
| <math>\operatorname{then}~ [| f |] ~=~ f^{-1} (\underline{1}) ~=~ \{ x \in X : f(x) = \underline{1} \}.</math>
| <math>\underline{0}</math>
| <math>\underline{1}</math>
| <math>\operatorname{If}~ f : X \to \underline\mathbb{B},</math>
|-
|-
| <math>\operatorname{then}~ [| f |] ~=~ f^{-1} (\underline{1}) ~=~ \{ x \in X : f(x) \}.</math>
| <math>\underline{1}</math>
| <math>\underline{0}</math>
|}
|}
<br>
It is convenient to transport the product and the sum operations of <math>\mathbb{B}</math> into the logical setting of <math>\underline\mathbb{B},</math> where they can be symbolized by signs of the same character. This yields the following definitions of a ''product'' and a ''sum'' in <math>\underline\mathbb{B}</math> and leads to the following forms of multiplication and addition tables.
The ''product'' <math>x \cdot y</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given by Table 9. As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the product corresponds to the logical operation of ''conjunction'', written <math>{}^{\backprime\backprime} x \land y {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} x\!\And\!y {}^{\prime\prime}</math> and read as <math>{}^{\backprime\backprime} x ~\operatorname{and}~ y {}^{\prime\prime}.</math> In accord with common practice, the multiplication sign is frequently omitted from written expressions of the product.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Product Operation for the Boolean Domain}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>\cdot\!</math>
| <math>\underline{0}</math>
| <math>\underline{1}</math>
|-
| style="background:ghostwhite" | <math>\underline{0}</math>
| <math>\underline{0}</math>
| <math>\underline{0}</math>
|-
| style="background:ghostwhite" | <math>\underline{1}</math>
| <math>\underline{0}</math>
| <math>\underline{1}</math>
|}
<br>
The ''sum'' <math>x + y\!</math> of two values <math>x\!</math> and <math>y\!</math> in <math>\underline\mathbb{B}</math> is given in Table 10. As a dyadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> the sum corresponds to the logical operation of ''exclusive disjunction'', usually read as <math>{}^{\backprime\backprime} x ~\text{or}~ y ~\text{but not both} {}^{\prime\prime}.\!</math> Depending on the context, other signs and readings that invoke this operation are: <math>{}^{\backprime\backprime} x \ne y {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} x \not\Leftrightarrow y {}^{\prime\prime},</math> read as <math>{}^{\backprime\backprime} x ~\text{is not equal to}~ y {}^{\prime\prime},</math> <math>{}^{\backprime\backprime} x ~\text{is not equivalent to}~ y {}^{\prime\prime},</math> or <math>{}^{\backprime\backprime} \text{exactly one of}~ x, y ~\text{is true} {}^{\prime\prime}.\!</math>
<br>
The upshot of this whole discussion of evaluation is that it allows us to rewrite the definitions of indicator functions and their fibers as follows:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{Sum Operation for the Boolean Domain}\!</math>
The ''indicator function'' or the ''characteristic function'' of a set <math>Q \subseteq X,</math> written <math>f_Q,\!</math> is the map from <math>X\!</math> to the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> that is defined in the following ways:
|- style="height:40px; background:ghostwhite"
| <math>+\!</math>
| <math>\underline{0}</math>
| <math>\underline{1}</math>
|-
| style="background:ghostwhite" | <math>\underline{0}</math>
| <math>\underline{0}</math>
| <math>\underline{1}</math>
|-
| style="background:ghostwhite" | <math>\underline{1}</math>
| <math>\underline{1}</math>
| <math>\underline{0}</math>
|}
<br>
For sentences, the signs of equality <math>(=)\!</math> and inequality <math>(\ne)\!</math> are reserved to signify the syntactic identity and non-identity, respectively, of the literal strings of characters that make up the sentences in question, while the signs of equivalence <math>(\Leftrightarrow)</math> and inequivalence <math>(\not\Leftrightarrow)</math> refer to the logical values, if any, of these strings, and thus they signify the equality and inequality, respectively, of their conceivable boolean values. For the logical values themselves, the two pairs of symbols collapse in their senses to a single pair, signifying a single form of coincidence or a single form of distinction, respectively, between the boolean values of the entities involved.
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. Although the remainder of the dyadic operations on boolean values, in other words, the rest of the sixteen functions of the form <math>f : \underline\mathbb{B} \times \underline\mathbb{B} \to \underline\mathbb{B},</math> 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.
The utility of a suitable calculus would involve, among other things:
# Finding the values of given functions for given arguments.
# Inverting boolean functions, that is, ''finding the fibers'' of boolean functions, or solving logical equations that are expressed in terms of boolean functions.
# Facilitating the recognition of invariant forms that take boolean functions as their functional components.
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, 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 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.
The ''indicator function'' or the ''characteristic function'' of the set <math>Q \subseteq X,</math> written <math>f_Q,\!</math> is the map from the universe <math>X\!</math> to the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> that is defined in the following ways:
<ol style="list-style-type:decimal">
<ol style="list-style-type:decimal">
<li>
<li>
<p>Considered in extensional form, <math>f_Q\!</math> is the subset of <math>X \times \underline\mathbb{B}</math> that is given by the following formula:</p>
<p>Considered in extensional form, <math>f_Q\!</math> is the subset of <math>X \times \underline\mathbb{B}\!</math> that is given by the following formula:</p>
<p><math>f_Q ~=~ \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~\Leftrightarrow~ x \in Q \}.</math></p></li>
<p><math>f_Q ~=~ \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y = \underline{1} ~\Leftrightarrow~ x \in Q \}.</math></p></li>
<li>
<li>
<p>Considered in functional form, <math>f_Q\!</math> is the map from <math>X\!</math> to <math>\underline\mathbb{B}</math> that is given by the following condition:</p>
<p>Considered in functional form, <math>f_Q\!</math> is the map from <math>X\!</math> to <math>\underline\mathbb{B}</math> that is given by the following condition:</p>
<p><math>f_Q ~\Leftrightarrow~ x \in Q.</math></p></li>
<p><math>f_Q (x) ~=~ \underline{1} ~\Leftrightarrow~ x \in Q.</math></p></li>
</ol>
</ol>
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 <math>f : X \to \underline\mathbb{B}.</math> 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 probably or eventually want to know exactly what subset that is.
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 imagined to "indicate" a set can be expressed in terms of the following concepts.
The ''fiber'' of a codomain element <math>y \in Y\!</math> under a function <math>f : X \to Y</math> is the subset of the domain <math>X\!</math> that is mapped onto <math>y,\!</math> in short, it is <math>f^{-1} (y) \subseteq X.</math> In other language that is often used, the fiber of <math>y\!</math> under <math>f\!</math> is called the ''antecedent set'', the ''inverse image'', the ''level set'', or the ''pre-image'' of <math>y\!</math> under <math>f.\!</math> All of these equivalent concepts are defined as follows:
{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{Fiber~of}~ y ~\operatorname{under}~ f ~=~ f^{-1} (y) ~=~ \{ x \in X : f(x) = y \}.</math>
|}
In the special case where <math>f\!</math> is the indicator function <math>f_Q\!</math> of a set <math>Q \subseteq X,</math> the fiber of <math>\underline{1}</math> under <math>f_Q\!</math> is just the set <math>Q\!</math> back again:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|
| <math>\operatorname{Fiber~of}~ \underline{1} ~\operatorname{under}~ f_Q ~=~ f_Q ^{-1} (\underline{1}) ~=~ \{ x \in X : f_Q (x) = \underline{1} \} ~=~ Q.</math>
<math>\begin{array}{lll}
|}
|}
In this specifically boolean setting, as in the more generally logical context, where ''truth'' under any name is especially valued, it is worth devoting a specialized notation to the ''fiber of truth'' in a proposition, to mark with particular ease and explicitness the set that it indicates. For this purpose, I introduce the use of ''fiber bars'' or ''ground signs'', written as a frame of the form <math>[| \, \ldots \, |]</math> around a sentence or the sign of a proposition, and whose application is defined as follows:
{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{If}~ f : X \to \underline\mathbb{B},</math>
|-
| <math>\operatorname{then}~ [| f |] ~=~ f^{-1} (\underline{1}) ~=~ \{ x \in X : f(x) = \underline{1} \}.</math>
|}
Some may recognize here fledgling efforts to reinforce flights of Fregean semantics with impish pitches of Peircean semiotics. Some may deem it Icarean, all too Icarean.
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 an operator on propositions, signified by framing the signs of propositions with fiber bars or ground signs, remains a purely notational device, and yet the notion of a fiber in a logical context serves to raise an interesting point. By way of illustration, it is legitimate to rewrite the above definition in the following form:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|
| <math>\operatorname{If}~ f : X \to \underline\mathbb{B},</math>
<math>\begin{array}{lllclllcl}
|-
| <math>\operatorname{then}~ [| f |] ~=~ f^{-1} (\underline{1}) ~=~ \{ x \in X : f(x) \}.</math>
|}
|}
The set-builder frame <math>\{ x \in X : \underline{~~~} \}\!</math> requires a grammatical sentence or a sentential clause to fill in the blank, as with the sentence <math>{}^{\backprime\backprime} f(x) = \underline{1} {}^{\prime\prime}</math> 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 <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> and the sentence <math>{}^{\backprime\backprime} f(x) = \underline{1} {}^{\prime\prime}</math> represent the very same value to this context, for all <math>x\!</math> in <math>X,\!</math> that is, they will appear equal in their truth or falsity to any reasonable interpreter of signs or 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, frame, and reception.
The sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> manifestly names the value <math>f(x).\!</math> This is a value that can be seen in many lights. It is, at turns:
# The value that the proposition <math>f\!</math> has at the point <math>x,\!</math> in other words, the value that <math>f\!</math> bears at the point <math>x\!</math> where <math>f\!</math> is being evaluated, the value that <math>f\!</math> takes on with respect to the argument or the object <math>x\!</math> that the whole proposition is taken to be about.
# The value that the proposition <math>f\!</math> not only takes up at the point <math>x,\!</math> but that it carries, conveys, transfers, or transports into the setting <math>{}^{\backprime\backprime} \{ x \in X : \underline{~~~} \} {}^{\prime\prime}</math> or into any other context of discourse where <math>f\!</math> is meant to be evaluated.
# The value that the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> 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 <math>f\!</math> and the same object <math>x\!</math> are borne in mind.
# The value that the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> represents to its full interpretive context as being its own logical interpretant, namely, the value that it signifies as its canonical connotation to any interpreter of the sign that is cognizant of the context in which it appears.
The sentence <math>{}^{\backprime\backprime} f(x) = \underline{1} {}^{\prime\prime}</math> indirectly names what the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> more directly names, that is, the value <math>f(x).\!</math> In other words, the sentence <math>{}^{\backprime\backprime} f(x) = \underline{1} {}^{\prime\prime}</math> has the same value to its interpretive context that the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> imparts to any comparable context, each by way of its respective evaluation for the same <math>x \in X.</math>
What is the relation among connoting, denoting, and ''evaluing'', where the last term is coined to describe all the ways of bearing, conveying, 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 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, the question 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.
Introducing the realm of ''values'' is a stopgap measure that temporarily permits the discussion to avoid certain singularities in the embedding sign relation, and allowing the process of ''evaluation'' as a compromise mode of signification between connotation and denotation only manages to 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.
With this degree of flexibility in mind, one can say that the sentence <math>{}^{\backprime\backprime} f(x) = \underline{1} {}^{\prime\prime}</math> latently connotes what the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime}</math> patently connotes. Taken in abstraction, both syntactic entities fall into an equivalence class of signs that constitutes an abstract object, a thing of value that is ''identified by'' the sign <math>{}^{\backprime\backprime} f(x) {}^{\prime\prime},</math> and thus an object that might as well be ''identified with'' the value <math>f(x).\!</math>
The upshot of this whole discussion of evaluation is that it allows us to rewrite the definitions of indicator functions and their fibers as follows:
The ''indicator function'' or the ''characteristic function'' of a set <math>Q \subseteq X,</math> written <math>f_Q,\!</math> is the map from <math>X\!</math> to the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> that is defined in the following ways:
<ol style="list-style-type:decimal">
<li>
<p>Considered in extensional form, <math>f_Q\!</math> is the subset of <math>X \times \underline\mathbb{B}\!</math> that is given by the following formula:</p>
<p><math>f_Q ~=~ \{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~\Leftrightarrow~ x \in Q \}.</math></p></li>
<li>
<p>Considered in functional form, <math>f_Q\!</math> is the map from <math>X\!</math> to <math>\underline\mathbb{B}</math> that is given by the following condition:</p>
<p><math>f_Q ~\Leftrightarrow~ x \in Q.</math></p></li>
</ol>
The ''fibers'' of truth and falsity under a proposition <math>f : X \to \underline\mathbb{B}</math> are subsets of <math>X\!</math> that are variously described as follows:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
\text{The fiber of}~ \underline{1} ~\text{under}~ f
& = & [| f |]
\\
& = & f^{-1} (\underline{1})
\\
& = & \{ x \in X ~:~ f(x) = \underline{1} \}
\\
& = & \{ x \in X ~:~ f(x) \}.
\\
\\
\text{The fiber of}~ \underline{0} ~\text{under}~ f
& = & {}^{_\sim} [| f |]
\\
& = & f^{-1} (\underline{0})
\\
& = & \{ x \in X ~:~ f(x) = \underline{0} \}
\\
& = & \{ x \in X ~:~ \texttt{(} f(x) \texttt{)} \, \}.
\end{array}</math>
|}
====2.2.4. Empirical Types and Rational Types====
Perhaps this looks like a lot of work for the sake of what seems to be 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 <math>{}^{\backprime\backprime} \Leftrightarrow {}^{\prime\prime},</math> as written between logical sentences, and the sign of equality <math>{}^{\backprime\backprime} = {}^{\prime\prime},</math> as written between their logical values, or else between propositions and their boolean values, respectively. Doing this removes a longstanding but wholly unnecessary conceptual confound between the idea of an ''assertion'' and the notion of an ''equation'', and it allows one to treat logical equality on a par with the other logical operations.
As a purely informal aid to interpretation, I frequently use the letters <math>{}^{\backprime\backprime} p {}^{\prime\prime}, {}^{\backprime\backprime} q {}^{\prime\prime}</math> to denote propositions. This can serve to tip off the reader that a function is intended as the indicator function of a set, and thus it saves us the trouble of declaring the type <math>f : X \to \underline\mathbb{B}</math> each time that a function is introduced as a proposition.
Another convention of use in this context is to let underscored letters stand for <math>k\!</math>-tuples, lists, or sequences of objects. Typically, the elements of the <math>k\!</math>-tuple, list, or sequence are all of one type, and the underscored letter is typically the same basic character as the letters that are indexed or subscripted to denote the individual components of the <math>k\!</math>-tuple, list, or sequence. When the dimension of the <math>k\!</math>-tuple, list, or sequence is clear from context, the underscoring may be omitted. For example, the following patterns of construction are very often seen:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lllclllcl}
1.
& \text{If} & x_1, \dots, x_k & \in & X
& \text{then} & \underline{x} = (x_1, \ldots, x_k) & \in & X^k.
\\
2.
& \text{If} & x_1, \dots, x_k & : & X
& \text{then} & \underline{x} = (x_1, \ldots, x_k) & : & X^k.
\\
3.
& \text{If} & f_1, \dots, f_k & : & X \to Y
& \text{then} & \underline{f} = (f_1, \ldots, f_k) & : & (X \to Y)^k.
\\
\end{array}\!</math>
|}
There is usually felt to be a slight but significant distinction between a ''membership statement'' of the form <math>{}^{\backprime\backprime} x \in X \, {}^{\prime\prime}</math> and a ''type indication'' of the form <math>{}^{\backprime\backprime} x : X \, {}^{\prime\prime},</math> for instance, as they are used in the examples above. The difference that appears to be perceived in categorical statements, when those of the form <math>{}^{\backprime\backprime} x \in X \, {}^{\prime\prime}</math> and those of the form <math>{}^{\backprime\backprime} x : X \, {}^{\prime\prime}</math> 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 regarded as the more serious matter, one that concerns the reality of an object and the substance of a predicate, than the question of falling under a type, that may depend only on 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 the present instance, those of the forms <math>{}^{\backprime\backprime} x \in X ~\Leftrightarrow~ \underline{x} \in \underline{X} \, {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} x : X ~\Leftrightarrow~ \underline{x} : \underline{X} \, {}^{\prime\prime},</math> 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 abbreviate and summarize what is already stated in the protasis.
A sentence is ''articulate'' (1) if it has a significant form, a compound constitution, or a non-trivial structure as a sign, and (2) if there is an informative relationship that exists between its structure as a sign and the proposition that it happens to denote. A sentence of this kind is typically given in the form of a ''description'', an ''expression'', or a ''formula'', in other words, as an articulated sign or a well-structured element of a formal language. As a general rule, the class of sentences that one is willing to contemplate is compiled from a particular brand of complex signs and syntactic strings, those that are put together from the basic building blocks of a formal language and held in a special esteem for the roles that they play within its grammar. However, even if a typical sentence is a sign 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.
A ''boolean connection'' of degree <math>k,\!</math> also known as a ''boolean function'' on <math>k\!</math> variables, is a map of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.</math> In other words, a boolean connection of degree <math>k\!</math> is a proposition about things in the universe <math>X = \underline\mathbb{B}^k.</math>
An ''imagination'' of degree <math>k\!</math> on <math>X\!</math> is a <math>k\!</math>-tuple of propositions about things in the universe <math>X.\!</math> By way of displaying the kinds of notation that are used to express this idea, the imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> is given as a sequence of indicator functions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = {}_1^k.</math> All of these features of the typical imagination <math>\underline{f}\!</math> can be summed up in either one of two ways: either in the form of a membership statement, to the effect that <math>\underline{f} \in (X \to \underline\mathbb{B})^k,</math> or in the form of a type statement, to the effect that <math>\underline{f} : (X \to \underline\mathbb{B})^k,</math> though perhaps the latter form is slightly more precise than the former.
The ''play of images'' determined by <math>\underline{f}\!</math> and <math>x,\!</math> more specifically, the play of the imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> that has to do with the element <math>x \in X,</math> is the <math>k\!</math>-tuple <math>\underline{y} = (y_1, \ldots, y_k)</math> of values in <math>\underline\mathbb{B}</math> that satisfies the equations <math>y_j = f_j (x),\!</math> for <math>j = 1 ~\text{to}~ k.</math>
A ''projection'' of <math>\underline\mathbb{B}^k,\!</math> written <math>\pi_j\!</math> or <math>\operatorname{pr}_j,\!</math> is one of the maps <math>\pi_j : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> that is defined as follows:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{cccccc}
\text{If} & \underline{y} & = & (y_1, \ldots, y_k) & \in & \underline\mathbb{B}^k, \\
\\
\text{then} & \pi_j (\underline{y}) & = & \pi_j (y_1, \ldots, y_k) & = & y_j. \\
\end{array}</math>
|}
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 <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> or <math>^{\backprime\backprime} \underline{1} ^{\prime\prime},</math> can be taken to denote a constant proposition of the form <math>c : X \to \underline\mathbb{B}.</math> 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 <math>X.\!</math>
The ''projective imagination'' of <math>\underline\mathbb{B}^k</math> is the imagination <math>(\pi_1, \ldots, \pi_k).\!</math>
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 <math>f : X \to \underline\mathbb{B}.</math>
To assist the reading of informal examples, I frequently use the letters <math>^{\backprime\backprime} s ^{\prime\prime}</math> and <math>^{\backprime\backprime} t ^{\prime\prime},</math> to denote sentences. Thus, it is conceivable to have a situation where <math>s ~=~ ^{\backprime\backprime} p ^{\prime\prime}</math> and where <math>p : X \to \underline\mathbb{B}.</math> Altogether, this means that the sign <math>^{\backprime\backprime} s ^{\prime\prime}</math> denotes the sentence <math>s,\!</math> that the sentence <math>s\!</math> is the sentence <math>^{\backprime\backprime} p ^{\prime\prime},</math> and that the sentence <math>^{\backprime\backprime} p ^{\prime\prime}</math> denotes the proposition or the indicator function <math>p : X \to \underline\mathbb{B}.</math> In settings where it is necessary to keep track of a large number of sentences, I use subscripted letters like <math>{}^{\backprime\backprime} e_1 {}^{\prime\prime}, \, \ldots, \, {}^{\backprime\backprime} e_n {}^{\prime\prime}</math> to refer to the various expressions.
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 <math>f : X \to \underline\mathbb{B}.</math>
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 <math>\underline\mathbb{B},</math> 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 the consideration of what things in what universe the corresponding proposition happens to indicate.
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 are they can be rendered in artificially contrived arenas.
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.
An ''assertion'' is just a sentence that is being used in a certain way, namely, to indicate the indication of the indicator function that the sentence is usually used to denote. In other words, an assertion is a sentence that is being converted to a certain use or 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 of discourse for which the sentence is held to be true.
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.
Taken in a context of communication, an assertion invites the interpreter to consider the things for which the sentence is true, in other words, to find the fiber of truth in the associated proposition, or yet again, to invert the indicator function denoted by the sentence with respect to its possible value of truth.
A ''denial'' of a sentence <math>s\!</math> is an assertion of its negation <math>{}^{\backprime\backprime} \, \texttt{(} s \texttt{)} \, {}^{\prime\prime}.</math> 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 denoted by the sentence with respect to its possible value of falsity.
According to this manner of definition, any sign that happens to denote a proposition, any sign that is taken as denoting an indicator function, by that very fact alone successfully qualifies as a sentence. That is, a sentence is any sign that actually succeeds in denoting a proposition, any sign that one way or another brings to mind, as its actual object, a function of the form <math>f : X \to \underline\mathbb{B}.</math>
There are many features of this definition that need to be understood. Indeed, there are problems involved in this whole style of definition that need to be discussed, and doing this requires a slight excursion.
====2.2.4. Empirical Types and Rational Types====
In this Segment, I want to examine the style of definition that I used 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.
I defined a sentence in terms of what it denotes, and not 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 happens 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 import 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 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 them.
Notice the general character of this development. I start by defining a type of sign according to the type of object that it happens to denote, ignoring at first the structural potential that the sign itself 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.
====2.2.5. Articulate Sentences====
A sentence is ''articulate'' (1) if it has a significant form, a compound constitution, or a non-trivial structure as a sign, and (2) if there is an informative relationship that exists between its structure as a sign and the proposition that it happens to denote. A sentence of this kind is typically given in the form of a ''description'', an ''expression'', or a ''formula'', in other words, as an articulated sign or a well-structured element of a formal language. As a general rule, the class of sentences that one is willing to contemplate is compiled from a particular brand of complex signs and syntactic strings, those that are put together from the basic building blocks of a formal language and held in a special esteem for the roles that they play within its grammar. However, even if a typical sentence is a sign 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 universe <math>X.\!</math> A variable is an ambiguous or equivocal sign that can be interpreted as denoting any element of the set that it ranges over.
If a sentence denotes a proposition <math>f : X \to \underline\mathbb{B},</math> then the ''value'' of the sentence with regard to <math>x \in X</math> is the value <math>f(x)\!</math> of the proposition at <math>x,\!</math> where <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}</math> is interpreted as ''false'' and <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime}</math> is interpreted as ''true''.
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 usual to say that a sentence or a proposition ''refers'' to a universe and to its elements, though perhaps in a variety of different senses. Furthermore, a proposition, acting in the role of as 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 possible 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 senses of ''reference'' that arise in this setting is to make the following sorts of distinctions among them. 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. Let the references that an indicator function <math>f\!</math> has to the elements on which it evaluates to <math>\underline{0}</math> be called its ''negative references''. Let the references that an indicator function <math>f\!</math> has to the elements on which it evaluates to <math>\underline{1}</math> be called its ''positive references'' or its ''indications''. Finally, unspecified references to the "references" of a sentence, a proposition, or an indicator function can be taken by default as references to their specific, positive references.
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 <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime},</math> can be taken to denote a constant proposition of the form <math>c : X \to \underline\mathbb{B}.</math> 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 <math>X.\!</math>
Notice that the letters <math>{}^{\backprime\backprime} p {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} q {}^{\prime\prime},</math> interpreted as signs that denote indicator functions <math>p, q : X \to \underline\mathbb{B},</math> 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 discussions. 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 <math>{}^{\backprime\backprime} s {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} t {}^{\prime\prime},</math> to denote sentences. Thus, it is conceivable to have a situation where <math>s ~=~ {}^{\backprime\backprime} p {}^{\prime\prime}</math> and where <math>p : X \to \underline\mathbb{B}.</math> Altogether, this means that the sign <math>{}^{\backprime\backprime} s {}^{\prime\prime}</math> denotes the sentence <math>s,\!</math> that the sentence <math>s\!</math> is the sentence <math>{}^{\backprime\backprime} p {}^{\prime\prime},</math> and that the sentence <math>{}^{\backprime\backprime} p {}^{\prime\prime}</math> denotes the proposition or the indicator function <math>p : X \to \underline\mathbb{B}.</math> In settings where it is necessary to keep track of a large number of sentences, I use subscripted letters like <math>{}^{\backprime\backprime} e_1 {}^{\prime\prime}, \, \ldots, \, {}^{\backprime\backprime} e_n {}^{\prime\prime}</math> to refer to the various expressions.
A ''sentential connective'' is a sign, a coordinated sequence of signs, a significant pattern of 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 <math>k\!</math> is the number of sentences that are connected, then the connective is said to be of order <math>k.\!</math> If the sentences acquire a logical relationship by this means, and are not just strung together by this mechanism, 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 is called a ''propositional connective''.
====2.2.6. Stretching Principles====
There is a principle, of constant use in this work, that needs to be made explicit. In order to give it a name, I refer to this idea as the ''stretching principle''. Expressed in different ways, it says that:
# Any relation of values extends to a relation of what is valued.
# Any statement about values says something about the things that are given these values.
# Any association among a range of values establishes an association among the domains of things that these values are the values of.
# Any connection between two values can be stretched to create a connection, of analogous form, between the objects, persons, qualities, or relationships that are valued in these connections.
# 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.
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.
In particular, a connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> can be understood to indicate a relation among boolean values, namely, the <math>k\!</math>-ary relation <math>F^{-1} (\underline{1}) \subseteq \underline\mathbb{B}^k.</math> If these <math>k\!</math> values are values of things in a universe <math>X,\!</math> that is, if one imagines each value in a <math>k\!</math>-tuple of values to be the functional image that results from evaluating an element of <math>X\!</math> under one of its possible aspects of value, then one has in mind the <math>k\!</math> propositions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> in sum, one embodies the imagination <math>\underline{f} = (f_1, \ldots, f_k).</math> Together, the imagination <math>\underline{f} \in (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> stretch each other to cover the universe <math>X,\!</math> yielding a new proposition <math>p : X \to \underline\mathbb{B}.</math>
In particular, a connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> can be understood to indicate a relation among boolean values, namely, the <math>k\!</math>-ary relation <math>F^{-1} (\underline{1}) \subseteq \underline\mathbb{B}^k.</math> If these <math>k\!</math> values are values of things in a universe <math>X,\!</math> that is, if one imagines each value in a <math>k\!</math>-tuple of values to be the functional image that results from evaluating an element of <math>X\!</math> under one of its possible aspects of value, then one has in mind the <math>k\!</math> propositions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> in sum, one embodies the imagination <math>\underline{f} = (f_1, \ldots, f_k).</math> Together, the imagination <math>\underline{f} \in (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> stretch each other to cover the universe <math>X,\!</math> yielding a new proposition <math>p : X \to \underline\mathbb{B}.</math>
To encapsulate the form of this general result, I define a composition that takes an imagination <math>\underline{f} = (f_1, \ldots, f_k) \in (X \to \underline\mathbb{B})^k</math> and a boolean connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> and gives a proposition <math>p : X \to \underline\mathbb{B}.</math> Depending on the situation, specifically, according to whether many <math>F\!</math> and many <math>\underline{f},</math> a single <math>F\!</math> and many <math>\underline{f},</math> or many <math>F\!</math> and a single <math>\underline{f}</math> are being considered, respectively, the proposition <math>p\!</math> thus constructed may be referred to under one of three descriptions:
To encapsulate the form of this general result, I define a composition that takes an imagination <math>\underline{f} = (f_1, \ldots, f_k) \in (X \to \underline\mathbb{B})^k\!</math> and a boolean connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}\!</math> and gives a proposition <math>p : X \to \underline\mathbb{B}.\!</math> Depending on the situation, specifically, according to whether many <math>F\!</math> and many <math>\underline{f},\!</math> a single <math>F\!</math> and many <math>\underline{f},\!</math> or many <math>F\!</math> and a single <math>\underline{f}\!</math> are being considered, respectively, the proposition <math>p\!</math> thus constructed may be referred to under one of three descriptions:
<ol style="list-style-type:decimal">
<ol style="list-style-type:decimal">
<li>
<li>
<p>In a general setting, where the connection <math>F\!</math> and the imagination <math>\underline{f}</math> are both permitted to take up a variety of concrete possibilities, call <math>p\!</math> the ''stretch of <math>F\!</math> and <math>\underline{f}</math> from <math>X\!</math> to <math>\underline\mathbb{B},</math>'', and write it in the style of a composition as <math>F ~\$~ \underline{f}.</math> This is meant to suggest that the symbol <math>^{\backprime\backprime} $ ^{\prime\prime},</math> here read as ''stretch'', denotes an operator of the form:</p>
<p>In a general setting, where the connection <math>F\!</math> and the imagination <math>\underline{f}\!</math> are both permitted to take up a variety of concrete possibilities, call <math>p\!</math> the ''stretch of <math>F\!</math> and <math>\underline{f}\!</math> from <math>X\!</math> to <math>\underline\mathbb{B},\!</math>'' and write it in the style of a composition as <math>F ~\$~ \underline{f}.\!</math> This is meant to suggest that the symbol <math>{}^{\backprime\backprime} $ {}^{\prime\prime},\!</math> here read as ''stretch'', denotes an operator of the form:</p>
<p><math>\$ : (\underline\mathbb{B}^k \to \underline\mathbb{B}) \times (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B}).</math></p></li>
<p><math>\$ : (\underline\mathbb{B}^k \to \underline\mathbb{B}) \times (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B}).</math></p></li>
<li>
<li>
<p>In a setting where the connection <math>F\!</math> is fixed but the imagination <math>\underline{f}</math> is allowed to vary over a wide range of possibilities, call <math>p\!</math> the ''stretch of <math>F\!</math> to <math>\underline{f}</math> on <math>X,\!</math>'' and write it in the style <math>F^\$ \underline{f},</math> exactly as if <math>^{\backprime\backprime} F^\$ \, ^{\prime\prime}</math> denotes an operator <math>F^\$ : (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B})</math> that is derived from <math>F\!</math> and applied to <math>\underline{f},</math> ultimately yielding a proposition <math>F^\$ \underline{f} : X \to \underline\mathbb{B}.</math></p></li>
<p>In a setting where the connection <math>F\!</math> is fixed but the imagination <math>\underline{f}\!</math> is allowed to vary over a wide range of possibilities, call <math>p\!</math> the ''stretch of <math>F\!</math> to <math>\underline{f}\!</math> on <math>X,\!</math>'' and write it in the style <math>F^\$ \underline{f},\!</math> exactly as if <math>{}^{\backprime\backprime} F^\$ \, {}^{\prime\prime}\!</math> denotes an operator <math>F^\$ : (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B})\!</math> that is derived from <math>F\!</math> and applied to <math>\underline{f},\!</math> ultimately yielding a proposition <math>F^\$ \underline{f} : X \to \underline\mathbb{B}.\!</math></p></li>
<li>
<li>
<p>In a setting where the imagination<math>\underline{f}</math> is fixed but the connection <math>F\!</math> is allowed to range over wide variety of possibilities, call <math>p\!</math> the ''stretch of <math>\underline{f}</math> by <math>F\!</math> to <math>\underline\mathbb{B},</math>'' and write it in the style <math>\underline{f}^\$ F,</math> exactly as if <math>^{\backprime\backprime} \underline{f}^\$ \, ^{\prime\prime}</math> denotes an operator <math>\underline{f}^\$ : (\underline\mathbb{B}^k \to \underline\mathbb{B}) \to (X \to \underline\mathbb{B}</math> that is derived from <math>\underline{f}</math> and applied to <math>F,\!</math> ultimately yielding a proposition <math>\underline{f}^\$ F : X \to \underline\mathbb{B}.</math></p></li>
<p>In a setting where the imagination<math>\underline{f}\!</math> is fixed but the connection <math>F\!</math> is allowed to range over wide variety of possibilities, call <math>p\!</math> the ''stretch of <math>\underline{f}\!</math> by <math>F\!</math> to <math>\underline\mathbb{B},\!</math>'' and write it in the style <math>\underline{f}^\$ F,\!</math> exactly as if <math>{}^{\backprime\backprime} \underline{f}^\$ \, {}^{\prime\prime}\!</math> denotes an operator <math>\underline{f}^\$ : (\underline\mathbb{B}^k \to \underline\mathbb{B}) \to (X \to \underline\mathbb{B}\!</math> that is derived from <math>\underline{f}\!</math> and applied to <math>F,\!</math> ultimately yielding a proposition <math>\underline{f}^\$ F : X \to \underline\mathbb{B}.\!</math></p></li>
</ol>
</ol>
Because this notation is only used in settings where the imagination <math>\underline{f} : (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> are distinguished by their types, it does not really matter whether one writes <math>^{\backprime\backprime} F ~\$~ \underline{f} \, ^{\prime\prime}</math> or <math>^{\backprime\backprime} \underline{f} ~\$~ F \, ^{\prime\prime}</math> for the initial composition.
Because this notation is only used in settings where the imagination <math>\underline{f} : (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> are distinguished by their types, it does not really matter whether one writes <math>{}^{\backprime\backprime} F ~\$~ \underline{f} \, {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \underline{f} ~\$~ F \, {}^{\prime\prime}</math> for the initial composition.
Just as a sentence is a sign that denotes a proposition, 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:
Just as a sentence is a sign that denotes a proposition, 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:
# An imagination of degree <math>k\!</math> on <math>X,\!</math> in other words, a <math>k\!</math>-tuple of propositions <math>f_j : X \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> or an object of the form <math>\underline{f} = (f_1, \ldots, f_k) : (X \to \underline\mathbb{B})^k.</math>
# An imagination of degree <math>k\!</math> on <math>X,\!</math> in other words, a <math>k\!</math>-tuple of propositions <math>f_j : X \to \underline\mathbb{B},\!</math> for <math>j = 1 ~\text{to}~ k,\!</math> or an object of the form <math>\underline{f} = (f_1, \ldots, f_k) : (X \to \underline\mathbb{B})^k.\!</math>
# A connection of degree <math>k,\!</math> in other words, a proposition about things in <math>\underline\mathbb{B}^k,</math> or a boolean function of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.</math>
# A connection of degree <math>k,\!</math> in other words, a proposition about things in <math>\underline\mathbb{B}^k,\!</math> or a boolean function of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.\!</math>
From these materials, it is required to construct a proposition <math>p : X \to \underline\mathbb{B}</math> such that <math>p(x) = F(f_1 (x), \ldots, f_k (x)),</math> for all <math>x \in X.</math> The desired construction is determined as follows:
From these materials, it is required to construct a proposition <math>p : X \to \underline\mathbb{B}\!</math> such that <math>p(x) = F(f_1 (x), \ldots, f_k (x)),\!</math> for all <math>x \in X.\!</math> The desired construction is determined as follows:
The cartesian power <math>\underline\mathbb{B}^k,</math> as a cartesian product, is characterized by the possession of a projective imagination <math>\pi = (\pi_1, \ldots, \pi_k)</math> of degree <math>k\!</math> on <math>\underline\mathbb{B}^k,</math> along with the property that any imagination <math>\underline{f} = (f_1, \ldots, f_k)</math> of degree <math>k\!</math> on an arbitrary set <math>W\!</math> determines a unique map <math>f! : W \to \underline\mathbb{B}^k,</math> the play of whose projective images <math>(\pi_1 (f!(w), \ldots, \pi_k (f!(w))</math> on the functional image <math>f!(w)\!</math> matches the play of images <math>(f_1 (w), \ldots, f_k (w))</math> under <math>\underline{f},</math> term for term and at every element <math>w\!</math> in <math>W.\!</math>
The cartesian power <math>\underline\mathbb{B}^k,\!</math> as a cartesian product, is characterized by the possession of a projective imagination <math>\pi = (\pi_1, \ldots, \pi_k)\!</math> of degree <math>k\!</math> on <math>\underline\mathbb{B}^k,\!</math> along with the property that any imagination <math>\underline{f} = (f_1, \ldots, f_k)\!</math> of degree <math>k\!</math> on an arbitrary set <math>W\!</math> determines a unique map <math>f! : W \to \underline\mathbb{B}^k,\!</math> the play of whose projective images <math>(\pi_1 (f!(w), \ldots, \pi_k (f!(w))\!</math> on the functional image <math>{f!(w)}\!</math> matches the play of images <math>(f_1 (w), \ldots, f_k (w))\!</math> under <math>\underline{f},\!</math> term for term and at every element <math>w\!</math> in <math>W.\!</math>
Just to be on the safe side, I state this again in more standard terms. The cartesian power <math>\underline\mathbb{B}^k,</math> as a cartesian product, is characterized by the possession of <math>k\!</math> projection maps <math>\pi_j : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>j = 1 ~\text{to}~ k,</math> along with the property that any <math>k\!</math> maps <math>f_j : W \to \underline\mathbb{B},</math> from an arbitrary set <math>W\!</math> to <math>\underline\mathbb{B},</math> determine a unique map <math>f! : W \to \underline\mathbb{B}^k</math> such that <math>\pi_j (f!(w)) = f_j (w),\!</math> for all <math>j = 1 ~\text{to}~ k,</math> and for all <math>w \in W.</math>
Just to be on the safe side, I state this again in more standard terms. The cartesian power <math>\underline\mathbb{B}^k,\!</math> as a cartesian product, is characterized by the possession of <math>k\!</math> projection maps <math>\pi_j : \underline\mathbb{B}^k \to \underline\mathbb{B},\!</math> for <math>j = 1 ~\text{to}~ k,\!</math> along with the property that any <math>k\!</math> maps <math>f_j : W \to \underline\mathbb{B},\!</math> from an arbitrary set <math>W\!</math> to <math>\underline\mathbb{B},\!</math> determine a unique map <math>f! : W \to \underline\mathbb{B}^k\!</math> such that <math>\pi_j (f!(w)) = f_j (w),\!</math> for all <math>j = 1 ~\text{to}~ k,\!</math> and for all <math>w \in W.\!</math>
Now suppose that the arbitrary set <math>W\!</math> in this construction is just the relevant universe <math>X.\!</math> Given that the function <math>f! : X \to \underline\mathbb{B}^k</math> is uniquely determined by the imagination <math>\underline{f} : (X \to \underline\mathbb{B})^k,</math> that is, by the <math>k\!</math>-tuple of propositions <math>\underline{f} = (f_1, \ldots, f_k),</math> it is safe to identify <math>f!\!</math> and <math>\underline{f}</math> 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 <math>^{\backprime\backprime} (f_1, \ldots, f_k) \, ^{\prime\prime}.</math> This facility of address is especially appropriate whenever a concrete term or a constructive precision is demanded by the context of discussion.
Now suppose that the arbitrary set <math>W\!</math> in this construction is just the relevant universe <math>X.\!</math> Given that the function <math>f! : X \to \underline\mathbb{B}^k\!</math> is uniquely determined by the imagination <math>\underline{f} : (X \to \underline\mathbb{B})^k,\!</math> that is, by the <math>k\!</math>-tuple of propositions <math>\underline{f} = (f_1, \ldots, f_k),\!</math> it is safe to identify <math>f!\!</math> and <math>\underline{f}\!</math> 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 <math>{}^{\backprime\backprime} (f_1, \ldots, f_k) \, {}^{\prime\prime}.\!</math> This facility of address is especially appropriate whenever a concrete term or a constructive precision is demanded by the context of discussion.
====2.2.7. Stretching Operations====
====2.2.7. Stretching Operations====
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.
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.
Thus, <math>F^\$</math> is what 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.
Thus, <math>F^\$</math> is what 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.
Now <math>^{\backprime\backprime} f_Q \, ^{\prime\prime}</math> is sign that denotes the proposition <math>f_Q,\!</math> and it certainly seems like a sufficient sign for it. Why is there is a need to recognize any other signs of it?
Now <math>{}^{\backprime\backprime} f_Q \, {}^{\prime\prime}</math> is sign that denotes the proposition <math>f_Q,\!</math> and it certainly seems like a sufficient sign for it. Why is there is a need to recognize any other signs of it?
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.
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.
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ '''Table 11. Levels of Indication'''
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{Levels of Indication}\!</math>
|- style="background:whitesmoke"
|- style="height:40px; background:ghostwhite"
| width="33%" | <math>\operatorname{Object}</math>
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\operatorname{Sign}</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="34%" | <math>\operatorname{Higher~Order~Sign}</math>
| width="34%" | <math>\text{Higher Order Sign}\!</math>
|- style="background:whitesmoke"
|- style="background:ghostwhite"
| <math>\text{Set}\!</math>
| <math>\text{Proposition}\!</math>
| <math>\text{Sentence}\!</math>
|-
|-
| <math>f^{-1} (y)\!</math>
| <math>f^{-1} (y)\!</math>
| <math>f\!</math>
| <math>f\!</math>
| <math>^{\backprime\backprime} f \, ^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} f \, {}^{\prime\prime}\!</math>
|-
|-
| <math>Q\!</math>
| <math>Q\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \underline{1} {}^{\prime\prime}\!</math>
|-
|-
| <math>{}^{_\sim} Q</math>
| <math>{}^{_\sim} Q\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \underline{0} {}^{\prime\prime}\!</math>
|}
|}
In order to make these notations useful in practice, it is necessary to note of a couple of their finer points, points that might otherwise seem too fine to take much trouble over. For this reason, I express their usage a bit more carefully as follows:
In order to make these notations useful in practice, it is necessary to note of a couple of their finer points, points that might otherwise seem too fine to take much trouble over. For this reason, I express their usage a bit more carefully as follows:
# Let the ''down hooks'' <math>\downharpoonleft \cdots \downharpoonright</math> be placed around the name of a sentence <math>s,\!</math> as in the expression <math>^{\backprime\backprime} \downharpoonleft s \downharpoonright \, ^{\prime\prime},</math> or else around a token appearance of the sentence itself, to serve as a name for the proposition that <math>s\!</math> denotes.
# Let the ''down hooks'' <math>\downharpoonleft \cdots \downharpoonright</math> be placed around the name of a sentence <math>s,\!</math> as in the expression <math>{}^{\backprime\backprime} \downharpoonleft s \downharpoonright \, {}^{\prime\prime},</math> or else around a token appearance of the sentence itself, to serve as a name for the proposition that <math>s\!</math> denotes.
# Let the ''up hooks'' <math>\upharpoonleft \cdots \upharpoonright</math> be placed around a name of a set <math>Q,\!</math> as in the expression <math>^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, ^{\prime\prime},</math> to serve as a name for the indicator function <math>f_Q.\!</math>
# Let the ''up hooks'' <math>\upharpoonleft \cdots \upharpoonright</math> be placed around a name of a set <math>Q,\!</math> as in the expression <math>{}^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, {}^{\prime\prime},</math> to serve as a name for the indicator function <math>f_Q.\!</math>
Table 12 illustrates the use of this notation, listing in each column several different but equivalent ways of referring to the same entity.
Table 12 illustrates the use of this notation, listing in each column several different but equivalent ways of referring to the same entity.
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
|+ '''Table 12. Ilustrations of Notation'''
|+ style="height:30px" | <math>\text{Table 12.} ~~ \text{Illustrations of Notation}\!</math>
|- style="background:whitesmoke"
|- style="height:40px; background:ghostwhite"
| width="33%" | <math>\operatorname{Object}</math>
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\operatorname{Sign}</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="34%" | <math>\operatorname{Higher~Order~Sign}</math>
| width="34%" | <math>\text{Higher Order Sign}\!</math>
|- style="background:whitesmoke"
|- style="background:ghostwhite"
| <math>\text{Set}\!</math>
| <math>\text{Proposition}\!</math>
| <math>\text{Sentence}\!</math>
|-
|-
| <math>Q\!</math>
| <math>Q\!</math>
| <math>s\!</math>
| <math>s\!</math>
|-
|-
| <math>[| \downharpoonleft s \downharpoonright |]</math>
| <math>[| \downharpoonleft s \downharpoonright |]\!</math>
| <math>\downharpoonleft s \downharpoonright</math>
| <math>\downharpoonleft s \downharpoonright\!</math>
| <math>s\!</math>
| <math>s\!</math>
|-
|-
| <math>[| q |]\!</math>
| <math>[| q |]\!</math>
| <math>q\!</math>
| <math>q\!</math>
| <math>^{\backprime\backprime} q \, ^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} q \, {}^{\prime\prime}~\!</math>
|-
|-
| <math>[| f_Q |]\!</math>
| <math>[| f_Q |]\!</math>
| <math>f_Q\!</math>
| <math>f_Q\!</math>
| <math>^{\backprime\backprime} f_Q \, ^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} f_Q \, {}^{\prime\prime}\!</math>
|-
|-
| <math>Q\!</math>
| <math>Q\!</math>
| <math>\upharpoonleft Q \upharpoonright</math>
| <math>\upharpoonleft Q \upharpoonright\!</math>
| <math>^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, ^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \upharpoonleft Q \upharpoonright \, {}^{\prime\prime}\!</math>
|}
|}
|-
|-
| || || where
| || || where
| <math>q : X \to \underline\mathbb{B}.</math>
| <math>q : X \to \underline\mathbb{B}.\!</math>
|-
|-
|
|
|-
|-
| ||
| ||
| colspan="2" | <math>\downharpoonleft s \downharpoonright ~=~ q.</math>
| colspan="2" | <math>\downharpoonleft s \downharpoonright ~=~ q.\!</math>
|-
|-
| valign="top" | 2.
| valign="top" | 2.
|-
|-
| || || where
| || || where
| <math>q : X \to \underline\mathbb{B}</math>
| <math>q : X \to \underline\mathbb{B}\!</math>
|-
|-
| || || and
| || || and
| <math>[| q |] ~=~ q^{-1} (\underline{1}) ~=~ Q \subseteq X.</math>
| <math>[| q |] ~=~ q^{-1} (\underline{1}) ~=~ Q \subseteq X.\!</math>
|-
|-
|
|
|-
|-
| ||
| ||
| colspan="2" | <math>\downharpoonleft s \downharpoonright ~=~ q ~=~ f_Q ~=~ \upharpoonleft Q \upharpoonright.</math>
| colspan="2" | <math>\downharpoonleft s \downharpoonright ~=~ q ~=~ f_Q ~=~ \upharpoonleft Q \upharpoonright.\!</math>
|-
|-
| valign="top" | 3.
| valign="top" | 3.
|
|
| align="center" | <math>=\!</math>
| align="center" | <math>=\!</math>
| <math>[| \upharpoonleft X \upharpoonright |] ~=~ \upharpoonleft X \upharpoonright^{-1} (\underline{1})</math>
| <math>[| \upharpoonleft X \upharpoonright |] ~=~ \upharpoonleft X \upharpoonright^{-1} (\underline{1})\!</math>
|-
|-
|
|
|
|
| align="center" | <math>=\!</math>
| align="center" | <math>=\!</math>
| <math>[| f_Q |] ~=~ f_Q^{-1} (\underline{1}).</math>
| <math>[| f_Q |] ~=~ f_Q^{-1} (\underline{1}).\!</math>
|-
|-
| valign="top" | 4.
| valign="top" | 4.
| align="center" | <math>\upharpoonleft Q \upharpoonright</math>
| align="center" | <math>\upharpoonleft Q \upharpoonright\!</math>
| align="center" | <math>=\!</math>
| align="center" | <math>=\!</math>
| <math>\upharpoonleft \{ x \in X : x \in Q \} \upharpoonright</math>
| <math>\upharpoonleft \{ x \in X : x \in Q \} \upharpoonright\!</math>
|-
|-
|
|
|
|
| align="center" | <math>=\!</math>
| align="center" | <math>=\!</math>
| <math>\downharpoonleft x \in Q \downharpoonright</math>
| <math>\downharpoonleft x \in Q \downharpoonright\!</math>
|-
|-
|
|
|}
|}
Now if a sentence <math>s\!</math> really denotes a proposition <math>q,\!</math> and if the notation <math>^{\backprime\backprime} \downharpoonleft s \downharpoonright \, ^{\prime\prime}</math> is merely meant to supply another name for the proposition that <math>s\!</math> already denotes, then why is there any need for the additional notation? It is because the interpretive mind habitually races from the sentence <math>s,\!</math> through the proposition <math>q\!</math> that it denotes, and on to the set <math>Q = q^{-1} (\underline{1})</math> that the proposition <math>q\!</math> indicates, often jumping to the conclusion that the set <math>Q\!</math> is the only thing that the sentence <math>s\!</math> is intended to denote. This higher order sign situation and the mind's inclination when placed in 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 typically takes place from <math>s\!</math> to <math>q\!</math> to <math>Q.\!</math>
Now if a sentence <math>s\!</math> really denotes a proposition <math>q,\!</math> and if the notation <math>{}^{\backprime\backprime} \downharpoonleft s \downharpoonright \, {}^{\prime\prime}\!</math> is merely meant to supply another name for the proposition that <math>s\!</math> already denotes, then why is there any need for the additional notation? It is because the interpretive mind habitually races from the sentence <math>s,\!</math> through the proposition <math>q\!</math> that it denotes, and on to the set <math>Q = q^{-1} (\underline{1})\!</math> that the proposition <math>q\!</math> indicates, often jumping to the conclusion that the set <math>Q\!</math> is the only thing that the sentence <math>s\!</math> is intended to denote. This higher order sign situation and the mind's inclination when placed in 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 typically takes place from <math>s\!</math> to <math>q\!</math> to <math>Q.\!</math>
===2.3. The Cactus Patch===
===2.3. The Cactus Patch===
{| align="center" cellpadding="0" cellspacing="0" width="90%"
{| align="center" cellpadding="0" cellspacing="0" width="90%"
In the usual way of proceeding on formal grounds, meaning is added by giving 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 is carried out in Subsection 1.3.10.12.
In the usual way of proceeding on formal grounds, meaning is added by giving 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 is carried out in Subsection 1.3.10.12.
====2.3.1. The Cactus Language : Syntax====
====2.3.1. The Cactus Language : Syntax====
{| align="center" cellpadding="0" cellspacing="0" width="90%"
{| align="center" cellpadding="0" cellspacing="0" width="90%"
1. &
1. &
\operatorname{If} &
\operatorname{If} &
^{\backprime\backprime}\operatorname{A}^{\prime\prime} &
{}^{\backprime\backprime}\operatorname{A}{}^{\prime\prime} &
\rightarrow &
\rightarrow &
\operatorname{Ann}, \\
\operatorname{Ann}, \\
&
&
\operatorname{that~is}, &
\operatorname{that~is}, &
^{\backprime\backprime}\operatorname{A}^{\prime\prime} &
{}^{\backprime\backprime}\operatorname{A}{}^{\prime\prime} &
\operatorname{denotes} &
\operatorname{denotes} &
\operatorname{Ann}, \\
\operatorname{Ann}, \\
&
&
\operatorname{Thus} &
\operatorname{Thus} &
^{\backprime\backprime}\operatorname{Ann}^{\prime\prime} &
{}^{\backprime\backprime}\operatorname{Ann}{}^{\prime\prime} &
\rightarrow &
\rightarrow &
\operatorname{A}, \\
\operatorname{A}, \\
&
&
\operatorname{that~is}, &
\operatorname{that~is}, &
^{\backprime\backprime}\operatorname{Ann}^{\prime\prime} &
{}^{\backprime\backprime}\operatorname{Ann}{}^{\prime\prime} &
\operatorname{denotes} &
\operatorname{denotes} &
\operatorname{A}. \\
\operatorname{A}. \\
\end{array}</math>
\end{array}\!</math>
|}
|}
\operatorname{Bob} &
\operatorname{Bob} &
\leftarrow &
\leftarrow &
^{\backprime\backprime}\operatorname{B}^{\prime\prime}, \\
{}^{\backprime\backprime}\operatorname{B}{}^{\prime\prime}, \\
&
&
\operatorname{that~is}, &
\operatorname{that~is}, &
\operatorname{Bob} &
\operatorname{Bob} &
\operatorname{is~denoted~by} &
\operatorname{is~denoted~by} &
^{\backprime\backprime}\operatorname{B}^{\prime\prime}, \\
{}^{\backprime\backprime}\operatorname{B}{}^{\prime\prime}, \\
&
&
\operatorname{then} &
\operatorname{then} &
\operatorname{B} &
\operatorname{B} &
\leftarrow &
\leftarrow &
^{\backprime\backprime}\operatorname{Bob}^{\prime\prime}, \\
{}^{\backprime\backprime}\operatorname{Bob}{}^{\prime\prime}, \\
&
&
\operatorname{that~is}, &
\operatorname{that~is}, &
\operatorname{B} &
\operatorname{B} &
\operatorname{is~denoted~by} &
\operatorname{is~denoted~by} &
^{\backprime\backprime}\operatorname{Bob}^{\prime\prime}. \\
{}^{\backprime\backprime}\operatorname{Bob}{}^{\prime\prime}. \\
\end{array}</math>
\end{array}</math>
|}
|}
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
^{\backprime\backprime}\operatorname{~}^{\prime\prime} &
{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime} &
\leftarrow &
\leftarrow &
^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\
{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime} \\
\\
\\
^{\backprime\backprime}\operatorname{blank}^{\prime\prime} &
{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime} &
\rightarrow &
\rightarrow &
^{\backprime\backprime}\operatorname{~}^{\prime\prime} \\
{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime} \\
\end{array}</math>
\end{array}</math>
|}
|}
|
|
<math>\begin{array}{lllll}
<math>\begin{array}{lllll}
^{\backprime\backprime}\operatorname{~}^{\prime\prime}
{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime}
& \leftarrow &
& \leftarrow &
^{\backprime\backprime}\operatorname{blank}^{\prime\prime}
{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime}
& = &
& = &
^{\backprime\backprime}\operatorname{b}^{\prime\prime} \, \cdot \,
{}^{\backprime\backprime}\operatorname{b}{}^{\prime\prime} \, \cdot \,
^{\backprime\backprime}\operatorname{l}^{\prime\prime} \, \cdot \,
{}^{\backprime\backprime}\operatorname{l}{}^{\prime\prime} \, \cdot \,
^{\backprime\backprime}\operatorname{a}^{\prime\prime} \, \cdot \,
{}^{\backprime\backprime}\operatorname{a}{}^{\prime\prime} \, \cdot \,
^{\backprime\backprime}\operatorname{n}^{\prime\prime} \, \cdot \,
{}^{\backprime\backprime}\operatorname{n}{}^{\prime\prime} \, \cdot \,
^{\backprime\backprime}\operatorname{k}^{\prime\prime} \\
{}^{\backprime\backprime}\operatorname{k}{}^{\prime\prime} \\
\end{array}</math>
\end{array}</math>
|}
|}
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
^{\backprime\backprime}\operatorname{~}^{\prime\prime} & = & \operatorname{blank} \\
{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime} & = & \operatorname{blank} \\
\end{array}</math>
\end{array}</math>
|}
|}
|
|
<math>\begin{array}{lclcl}
<math>\begin{array}{lclcl}
^{\backprime\backprime}\operatorname{~~}^{\prime\prime}
{}^{\backprime\backprime}\operatorname{~~}{}^{\prime\prime}
& = &
& = &
^{\backprime\backprime}\operatorname{~}^{\prime\prime} \, \cdot \,
{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime} \, \cdot \,
^{\backprime\backprime}\operatorname{~}^{\prime\prime}
{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime}
& = &
& = &
\operatorname{blank} \, \cdot \, \operatorname{blank} \\
\operatorname{blank} \, \cdot \, \operatorname{blank} \\
\\
\\
^{\backprime\backprime}\operatorname{~blank}^{\prime\prime}
{}^{\backprime\backprime}\operatorname{~blank}{}^{\prime\prime}
& = &
& = &
^{\backprime\backprime}\operatorname{~}^{\prime\prime} \, \cdot \,
{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime} \, \cdot \,
^{\backprime\backprime}\operatorname{blank}^{\prime\prime}
{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime}
& = &
& = &
\operatorname{blank} \, \cdot \,
\operatorname{blank} \, \cdot \,
^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\
{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime} \\
\\
\\
^{\backprime\backprime}\operatorname{blank~}^{\prime\prime}
{}^{\backprime\backprime}\operatorname{blank~}{}^{\prime\prime}
& = &
& = &
^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \, \cdot \,
{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime} \, \cdot \,
^{\backprime\backprime}\operatorname{~}^{\prime\prime}
{}^{\backprime\backprime}\operatorname{~}{}^{\prime\prime}
& = &
& = &
^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \, \cdot \,
{}^{\backprime\backprime}\operatorname{blank}{}^{\prime\prime} \, \cdot \,
\operatorname{blank}
\operatorname{blank}
\end{array}</math>
\end{array}</math>
{| align="center" cellpadding="4" style="text-align:center" width="90%"
{| align="center" cellpadding="4" style="text-align:center" width="90%"
|-
|-
| <math>\varepsilon</math>
| <math>\varepsilon\!</math>
| =
| =
| <math>^{\backprime\backprime\prime\prime}</math>
| <math>{}^{\backprime\backprime\prime\prime}\!</math>
| =
| =
| align="left" | the empty string.
| align="left" | the empty string.
|-
|-
| <math>\underline\varepsilon</math>
| <math>\underline\varepsilon\!</math>
| =
| =
| <math>\{ \varepsilon \}</math>
| <math>\{ \varepsilon \}\!</math>
| =
| =
| align="left" | the language consisting of a single empty string.
| align="left" | the language consisting of a single empty string.
& = &
& = &
\{ &
\{ &
^{\backprime\backprime} \, \operatorname{~} \, ^{\prime\prime} & , &
{}^{\backprime\backprime} \, \operatorname{~} \, {}^{\prime\prime} & , &
^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} & , &
{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} & , &
^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} & , &
{}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} & , &
^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} &
{}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime} &
\} \\
\} \\
& = &
& = &
</ol>
</ol>
The easiest way to define the language <math>\mathfrak{C}(\mathfrak{P})</math> 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 <math>\mathfrak{A}^*</math> that are called ''syntactic connectives''. If the strings on which they operate are exclusively sentences of <math>\mathfrak{C}(\mathfrak{P}),</math> 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''.
The easiest way to define the language <math>\mathfrak{C}(\mathfrak{P})\!</math> 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 <math>\mathfrak{A}^*\!</math> that are called ''syntactic connectives''. If the strings on which they operate are exclusively sentences of <math>\mathfrak{C}(\mathfrak{P}),\!</math> 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''.
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.
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.
The first step is to define two sets of basic operations on strings of <math>\mathfrak{A}^*.</math>
The first step is to define two sets of basic operations on strings of <math>\mathfrak{A}^*.\!</math>
<ol style="list-style-type:decimal">
<ol style="list-style-type:decimal">
<p>The ''concatenation'' of one string <math>s_1\!</math> is just the string <math>s_1.\!</math></p>
<p>The ''concatenation'' of one string <math>s_1\!</math> is just the string <math>s_1.\!</math></p>
<p>The ''concatenation'' of two strings <math>s_1, s_2\!</math> is the string <math>s_1 \cdot s_2.\!</math></p>
<p>The ''concatenation'' of two strings <math>{s_1, s_2}\!</math> is the string <math>{s_1 \cdot s_2}.\!</math></p>
<p>The ''concatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is the string of the form <math>s_1 \cdot \ldots \cdot s_k.\!</math></p></li>
<p>The ''concatenation'' of the <math>k\!</math> strings <math>{(s_j)_{j = 1}^k}\!</math> is the string of the form <math>{s_1 \cdot \ldots \cdot s_k}.\!</math></p></li>
<li>
<li>
<p>The ''surcatenation'' of one string <math>s_1\!</math> is the string <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p>
<p>The ''surcatenation'' of one string <math>s_1\!</math> is the string <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, s_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.\!</math></p>
<p>The ''surcatenation'' of two strings <math>s_1, s_2\!</math> is <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p>
<p>The ''surcatenation'' of two strings <math>{s_1, s_2}\!</math> is <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, s_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_2 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.\!</math></p>
<p>The ''surcatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is the string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p></li>
<p>The ''surcatenation'' of the <math>k\!</math> strings <math>{(s_j)_{j = 1}^k}\!</math> is the string of the form <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, s_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, \ldots \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_k \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.\!</math></p></li>
</ol>
</ol>
<ol style="list-style-type:lower-alpha">
<ol style="list-style-type:lower-alpha">
<li><math>\operatorname{Surc}_{j=1}^1 s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>
<li><math>\operatorname{Surc}_{j=1}^1 s_j \ = \ {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, s_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math></li>
<li>
<li>
<p>For <math>\ell > 1,\!</math></p>
<p>For <math>\ell > 1,\!</math></p>
<p><math>\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Surc}_{j=1}^{\ell - 1} s_j \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p></li>
<p><math>\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Surc}_{j=1}^{\ell - 1} s_j \, \cdot \, ( \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime} \, )^{-1} \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math></p></li>
</ol></ol>
</ol></ol>
<p>The conception of the <math>k\!</math>-place surcatenation operation can be extended to include its natural "prequel":</p>
<p>The conception of the <math>k\!</math>-place surcatenation operation can be extended to include its natural "prequel":</p>
<p><math>\operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math></p>
<p><math>\operatorname{Surc}^0 \ = \ {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}.</math></p>
<p>Finally, the construction of the <math>k\!</math>-place surcatenation can be broken into stages by means of the following conceptions:</p>
<p>Finally, the construction of the <math>k\!</math>-place surcatenation can be broken into stages by means of the following conceptions:</p>
<li>
<li>
<p>A ''subclause'' in <math>\mathfrak{A}^*</math> is a string that ends with a <math>^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p></li>
<p>A ''subclause'' in <math>\mathfrak{A}^*</math> is a string that ends with a <math>{}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math></p></li>
<li>
<li>
<p>The ''subcatenation'' <math>\operatorname{Subc} (s_1, s_2)</math> of a subclause <math>s_1\!</math> by a string <math>s_2\!</math> is the string that is defined as follows:</p>
<p>The ''subcatenation'' <math>\operatorname{Subc} (s_1, s_2)</math> of a subclause <math>s_1\!</math> by a string <math>s_2\!</math> is the string that is defined as follows:</p>
<p><math>\operatorname{Subc} (s_1, s_2) \ = \ s_1 \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p>
<p><math>\operatorname{Subc} (s_1, s_2) \ = \ s_1 \, \cdot \, ( \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime} \, )^{-1} \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_2 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math></p>
<li>
<li>
<p>The ''surcatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> can now be defined as an iterated subcatenation over the sequence of <math>k+1\!</math> strings that starts with the string <math>s_0 \ = \ \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}</math> and then continues on through the other <math>k\!</math> strings:</p></li>
<p>The ''surcatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> can now be defined as an iterated subcatenation over the sequence of <math>k+1\!</math> strings that starts with the string <math>s_0 \ = \ \operatorname{Surc}^0 \ = \ {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}</math> and then continues on through the other <math>k\!</math> strings:</p></li>
<ol style="list-style-type:lower-roman">
<ol style="list-style-type:lower-roman">
<li>
<li>
<p><math>\operatorname{Surc}_{j=0}^0 s_j \ = \ \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math></p></li>
<p><math>\operatorname{Surc}_{j=0}^0 s_j \ = \ \operatorname{Surc}^0 \ = \ {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}.</math></p></li>
<li>
<li>
Having defined the basic operations of concatenation and surcatenation on arbitrary strings, in effect, giving them operational meaning for the all-inclusive language <math>\mathfrak{L} = \mathfrak{A}^*,</math> it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.
Having defined the basic operations of concatenation and surcatenation on arbitrary strings, in effect, giving them operational meaning for the all-inclusive language <math>\mathfrak{L} = \mathfrak{A}^*,</math> it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.
If <math>\mathfrak{L}</math> is an arbitrary formal language over an alphabet of the sort that
If <math>\mathfrak{L}\!</math> is an arbitrary formal language over an alphabet of the sort that
we are talking about, that is, an alphabet of the form <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P},</math> then there are a number of basic structural relations that can be defined on the strings of <math>\mathfrak{L}.</math>
we are talking about, that is, an alphabet of the form <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P},\!</math> then there are a number of basic structural relations that can be defined on the strings of <math>\mathfrak{L}.\!</math>
{| align="center" cellpadding="4" width="90%"
{| align="center" cellpadding="4" width="90%"
| 4. || <math>s\!</math> is a ''subclause'' of <math>\mathfrak{L}</math> if and only if
| 4. || <math>s\!</math> is a ''subclause'' of <math>\mathfrak{L}</math> if and only if
|-
|-
| || <math>s\!</math> is a sentence of <math>\mathfrak{L}</math> and <math>s\!</math> ends with a <math>^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>
| || <math>s\!</math> is a sentence of <math>\mathfrak{L}</math> and <math>s\!</math> ends with a <math>{}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math>
|-
|-
| 5. || <math>s\!</math> is the ''subcatenation'' of <math>s_1\!</math> by <math>s_2\!</math> if and only if
| 5. || <math>s\!</math> is the ''subcatenation'' of <math>s_1\!</math> by <math>s_2\!</math> if and only if
| || <math>s_1\!</math> is a subclause of <math>\mathfrak{L},</math> <math>s_2\!</math> is a sentence of <math>\mathfrak{L},</math> and
| || <math>s_1\!</math> is a subclause of <math>\mathfrak{L},</math> <math>s_2\!</math> is a sentence of <math>\mathfrak{L},</math> and
|-
|-
| || <math>s = s_1 \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>
| || <math>s = s_1 \, \cdot \, ( \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime} \, )^{-1} \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_2 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math>
|-
|-
| 6. || <math>s\!</math> is the ''surcatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> in <math>\mathfrak{L},</math>
| 6. || <math>s\!</math> is the ''surcatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> in <math>\mathfrak{L},</math>
|-
|-
| || if and only if <math>s_j\!</math> is a sentence of <math>\mathfrak{L},</math> for all <math>j = 1 \ldots k,\!</math> and
| || if and only if <math>s_j\!</math> is a sentence of <math>\mathfrak{L},</math> for all <math>{j = 1 \ldots k},\!</math> and
|-
|-
| || <math>s \ = \ \operatorname{Surc}_{j=1}^k s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>
| || <math>s \ = \ \operatorname{Surc}_{j=1}^k s_j \ = \ {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, s_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, \ldots \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, s_k \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math>
|}
|}
|}
|}
As usual, saying that <math>s\!</math> is a sentence is just a conventional way of stating that the string <math>s\!</math> belongs to the relevant formal language <math>\mathfrak{L}.</math> An individual sentence of <math>\mathfrak{C} (\mathfrak{P}),</math> for any palette <math>\mathfrak{P},</math> is referred to as a ''painted and rooted cactus expression'' (PARCE) on the palette <math>\mathfrak{P},</math> 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 <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> is also described as the set <math>\operatorname{PARCE} (\mathfrak{P})</math> of PARCE's on the palette <math>\mathfrak{P},</math> more generically, as the PARCE's that constitute the language <math>\operatorname{PARCE}.</math>
As usual, saying that <math>s\!</math> is a sentence is just a conventional way of stating that the string <math>s\!</math> belongs to the relevant formal language <math>\mathfrak{L}.\!</math> An individual sentence of <math>\mathfrak{C} (\mathfrak{P}),\!</math> for any palette <math>\mathfrak{P},</math> is referred to as a ''painted and rooted cactus expression'' (PARCE) on the palette <math>\mathfrak{P},</math> 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 <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> is also described as the set <math>\operatorname{PARCE} (\mathfrak{P})</math> of PARCE's on the palette <math>\mathfrak{P},</math> more generically, as the PARCE's that constitute the language <math>\operatorname{PARCE}.</math>
A ''bare'' PARCE, a bit loosely referred to as a ''bare cactus expression'', is a PARCE on the empty palette <math>\mathfrak{P} = \varnothing.</math> A bare PARCE is a sentence in the ''bare cactus language'', <math>\mathfrak{C}^0 = \mathfrak{C} (\varnothing) = \operatorname{PARCE}^0 = \operatorname{PARCE} (\varnothing).</math> This set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language <math>\mathfrak{C} (\mathfrak{P}).</math> 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 ''bare'' PARCE, a bit loosely referred to as a ''bare cactus expression'', is a PARCE on the empty palette <math>\mathfrak{P} = \varnothing.</math> A bare PARCE is a sentence in the ''bare cactus language'', <math>\mathfrak{C}^0 = \mathfrak{C} (\varnothing) = \operatorname{PARCE}^0 = \operatorname{PARCE} (\varnothing).</math> This set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language <math>\mathfrak{C} (\mathfrak{P}).</math> 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.
# To specify the intension or to signify the intention that every string that fits the conditions of the abstract type <math>T\!</math> must also fall under the grammatical heading of a sentence, as indicated by the type <math>S,\!</math> all within the target language <math>\mathfrak{L}.</math>
# To specify the intension or to signify the intention that every string that fits the conditions of the abstract type <math>T\!</math> must also fall under the grammatical heading of a sentence, as indicated by the type <math>S,\!</math> all within the target language <math>\mathfrak{L}.</math>
In these types of situation the letter <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> that signifies the type of 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 <math>^{\backprime\backprime} T \, ^{\prime\prime}</math> 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 these types of situation the letter <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> that signifies the type of 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 <math>{}^{\backprime\backprime} T \, {}^{\prime\prime}</math> 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 <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \}</math> whose sole member is the initial symbol with the set <math>\mathfrak{Q}</math> that assembles together all of the intermediate symbols results in the set <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q}</math> of ''non-terminal symbols''. Completing the package, the alphabet <math>\mathfrak{A}</math> of the language is also known as the set of ''terminal symbols''. In this discussion, I will adopt the convention that <math>\mathfrak{Q}</math> is the set of ''intermediate symbols'', but I will often use <math>q\!</math> as a typical variable that ranges over all of the non-terminal symbols, <math>q \in \{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q}.</math> Finally, it is convenient to refer to all of the symbols in <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> as the ''augmented alphabet'' of the prospective grammar for the language, and accordingly to describe the strings in <math>( \{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> 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 <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> from all other sorts of augmented strings. In these situations the strings in the disjoint union <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup (\mathfrak{Q} \cup \mathfrak{A} )^*</math> are known as the ''sentential forms'' of the associated grammar.
Combining the singleton set <math>\{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \}\!</math> whose sole member is the initial symbol with the set <math>\mathfrak{Q}\!</math> that assembles together all of the intermediate symbols results in the set <math>\{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \} \cup \mathfrak{Q}\!</math> of ''non-terminal symbols''. Completing the package, the alphabet <math>\mathfrak{A}</math> of the language is also known as the set of ''terminal symbols''. In this discussion, I will adopt the convention that <math>\mathfrak{Q}</math> is the set of ''intermediate symbols'', but I will often use <math>q\!</math> as a typical variable that ranges over all of the non-terminal symbols, <math>q \in \{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \} \cup \mathfrak{Q}.</math> Finally, it is convenient to refer to all of the symbols in <math>\{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> as the ''augmented alphabet'' of the prospective grammar for the language, and accordingly to describe the strings in <math>( \{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> 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 <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> from all other sorts of augmented strings. In these situations the strings in the disjoint union <math>\{ {}^{\backprime\backprime} S \, {}^{\prime\prime} \} \cup (\mathfrak{Q} \cup \mathfrak{A} )^*</math> are known as the ''sentential forms'' of the associated grammar.
In forming a grammar for a language statements of the form <math>W :> W',\!</math> where <math>W\!</math> and <math>W'\!</math> are augmented strings or sentential forms of specified 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 <math>\mathfrak{K}</math> that serves to complete the definition of the formal grammar in question.
In forming a grammar for a language statements of the form <math>W :> W',\!</math> where <math>W\!</math> and <math>W'\!</math> are augmented strings or sentential forms of specified 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 <math>\mathfrak{K}</math> that serves to complete the definition of the formal grammar in question.
Employing the notion of a covering relation it becomes possible to redescribe the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> in the following ways.
Employing the notion of a covering relation it becomes possible to redescribe the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> in the following ways.
=====2.3.1.1. Grammar 1=====
=====2.3.1.1. Grammar 1=====
Grammar 1 is something of a misnomer. It is nowhere near exemplifying 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 <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}</math> that comes with the territory of the cactus language <math>\mathfrak{C} (\mathfrak{P}),</math> it specifies <math>\mathfrak{Q} = \varnothing,</math> in other words, it employs no intermediate symbols, and it embodies the ''covering set'' <math>\mathfrak{K}</math> as listed in the following display.
Grammar 1 is something of a misnomer. It is nowhere near exemplifying 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 <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}\!</math> that comes with the territory of the cactus language <math>\mathfrak{C} (\mathfrak{P}),\!</math> it specifies <math>\mathfrak{Q} = \varnothing,\!</math> in other words, it employs no intermediate symbols, and it embodies the ''covering set'' <math>\mathfrak{K}\!</math> as listed in the following display.
<br>
<br>
& S
& S
& :>
& :>
& m_1 \ = \ ^{\backprime\backprime} \operatorname{~} ^{\prime\prime}
& m_1 \ = \ {}^{\backprime\backprime} \operatorname{~} {}^{\prime\prime}
\\
\\
2.
2.
& S
& S
& :>
& :>
& \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}
& \operatorname{Surc}^0 \ = \ {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}
\\
\\
5.
5.
& S
& S
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S \, \cdot \, ( \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \, )^* \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, S \, \cdot \, ( \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S \, )^* \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}
\\
\\
\end{array}</math>
\end{array}</math>
<li value="5"> The concept of a sentence in <math>\mathfrak{L}</math> covers any concatenation of sentences in <math>\mathfrak{L},</math> in effect, any number of freely chosen sentences that are available to be concatenated one after another.</li>
<li value="5"> The concept of a sentence in <math>\mathfrak{L}</math> covers any concatenation of sentences in <math>\mathfrak{L},</math> in effect, any number of freely chosen sentences that are available to be concatenated one after another.</li>
<li value="6"> The concept of a sentence in <math>\mathfrak{L}</math> covers any surcatenation of sentences in <math>\mathfrak{L},</math> in effect, any string that opens with a <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime},</math> continues with a sentence, possibly empty, follows with a finite number of phrases of the form <math>^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,</math> and closes with a <math>^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>
<li value="6"> The concept of a sentence in <math>\mathfrak{L}</math> covers any surcatenation of sentences in <math>\mathfrak{L},</math> in effect, any string that opens with a <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime},</math> continues with a sentence, possibly empty, follows with a finite number of phrases of the form <math>{}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S,</math> and closes with a <math>{}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}.</math></li>
</ol>
</ol>
|}
|}
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 <math>S :> S^*\!</math> is not a very productive piece of information, in the sense of telling very much about the language that falls under the type of a sentence <math>S.\!</math> In particular, since it implies that <math>S :> \underline\varepsilon,</math> and since <math>\underline\varepsilon \cdot \mathfrak{L} \, = \, \mathfrak{L} \cdot \underline\varepsilon \, = \, \mathfrak{L},</math> for any formal language <math>\mathfrak{L},</math> the empty string <math>\varepsilon</math> is counted over and over in every term of the union, and every non-empty sentence under <math>S\!</math> appears again and again in every term of the union that follows the initial appearance of <math>S.\!</math> 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.
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 <math>S :> S^*\!</math> is not a very productive piece of information, in the sense of telling very much about the language that falls under the type of a sentence <math>S.\!</math> In particular, since it implies that <math>S :> \underline\varepsilon,\!</math> and since <math>\underline\varepsilon \cdot \mathfrak{L} \, = \, \mathfrak{L} \cdot \underline\varepsilon \, = \, \mathfrak{L},\!</math> for any formal language <math>\mathfrak{L},\!</math> the empty string <math>\varepsilon\!</math> is counted over and over in every term of the union, and every non-empty sentence under <math>S\!</math> appears again and again in every term of the union that follows the initial appearance of <math>S.\!</math> 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.
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:
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:
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.
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.
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 <math>\mathfrak{C} (\mathfrak{P}),</math> one has the following definitions of concatenation as iterated precatenation and of surcatenation as iterated subcatenation, respectively:
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 <math>\mathfrak{C} (\mathfrak{P}),\!</math> one has the following definitions of concatenation as iterated precatenation and of surcatenation as iterated subcatenation, respectively:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
& \operatorname{Surc}_{j=1}^0
& \operatorname{Surc}_{j=1}^0
& =
& =
& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}
\\ \\
\\ \\
& \operatorname{Surc}_{j=1}^k S_j
& \operatorname{Surc}_{j=1}^k S_j
& \operatorname{Surc}
& \operatorname{Surc}
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}
\\ \\
\\ \\
& \operatorname{Surc}
& \operatorname{Surc}
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, S \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}
\\ \\
\\ \\
& \operatorname{Surc}
& \operatorname{Surc}
& :>
& :>
& \operatorname{Surc} \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
& \operatorname{Surc} \, \cdot \, ( \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime} \, )^{-1} \, \cdot \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}
\end{array}</math>
\end{array}</math>
|}
|}
# 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.
# 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.
=====2.3.1.2. Grammar 2=====
=====2.3.1.2. Grammar 2=====
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:
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:
A ''tract'' is a concatenation of a finite sequence of sentences, with a literal comma <math>^{\backprime\backprime} \operatorname{,} ^{\prime\prime}</math> interpolated between each pair of adjacent sentences. Thus, a typical tract <math>T\!</math> takes the form:
A ''tract'' is a concatenation of a finite sequence of sentences, with a literal comma <math>{}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}</math> interpolated between each pair of adjacent sentences. Thus, a typical tract <math>T\!</math> takes the form:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
& S_1
& S_1
& \cdot
& \cdot
& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}
& {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}
& \cdot
& \cdot
& \ldots
& \ldots
& \cdot
& \cdot
& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}
& {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}
& \cdot
& \cdot
& S_k
& S_k
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 2}\!</math>
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 2}\!</math>
| align="right" style="border-right:1px solid black;" width="50%" |
| align="right" style="border-right:1px solid black;" width="50%" |
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>
<math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \}</math>
|-
|-
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
& S
& S
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}
\\
\\
6.
6.
& T
& T
& :>
& :>
& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S
& T \, \cdot \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S
\\
\\
\end{array}</math>
\end{array}</math>
<br>
<br>
In this rendition, a string of type <math>T\!</math> 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 <math>T\!</math> gets off the ground, that is, to detect its minimal strings and to discover how its ensuing generations get started from these, it is useful to observe that the covering rule <math>T :> S\!</math> means that <math>T\!</math> ''inherits'' all of the initial conditions of <math>S,\!</math> namely, <math>T \, :> \, \varepsilon, m_1, p_j.</math> In accord with these simple beginnings it comes to parse that the rule <math>T \, :> \, T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,</math> with the substitutions <math>T = \varepsilon</math> and <math>S = \varepsilon</math> on the covered side of the rule, bears the germinal implication that <math>T \, :> \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}.</math>
In this rendition, a string of type <math>T\!</math> 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 <math>T\!</math> gets off the ground, that is, to detect its minimal strings and to discover how its ensuing generations get started from these, it is useful to observe that the covering rule <math>T :> S\!</math> means that <math>T\!</math> ''inherits'' all of the initial conditions of <math>S,\!</math> namely, <math>T \, :> \, \varepsilon, m_1, p_j.\!</math> In accord with these simple beginnings it comes to parse that the rule <math>T \, :> \, T \, \cdot \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S,\!</math> with the substitutions <math>T = \varepsilon\!</math> and <math>S = \varepsilon\!</math> on the covered side of the rule, bears the germinal implication that <math>T \, :> \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}.~\!</math>
Grammar 2 achieves a portion of its success through a higher degree of intermediate organization. Roughly speaking, the level of organization can be seen as reflected in the cardinality of the intermediate alphabet <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math> 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.
Grammar 2 achieves a portion of its success through a higher degree of intermediate organization. Roughly speaking, the level of organization can be seen as reflected in the cardinality of the intermediate alphabet <math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \}</math> 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.
=====2.3.1.3. Grammar 3=====
=====2.3.1.3. Grammar 3=====
Although it is not strictly necessary to do so, it is possible to organize the materials of our developing 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:
Although it is not strictly necessary to do so, it is possible to organize the materials of our developing 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:
|}
|}
When there is no possibility of confusion, the letter <math>^{\backprime\backprime} R \, ^{\prime\prime}</math> can be used either as a string variable that ranges over the set of runes or else as a type name for the class of runes. The latter reading amounts to the enlistment of a fresh intermediate symbol, <math>^{\backprime\backprime} R \, ^{\prime\prime} \in \mathfrak{Q},</math> as a part of a new grammar for <math>\mathfrak{C} (\mathfrak{P}).</math> In effect, <math>^{\backprime\backprime} R \, ^{\prime\prime}</math> affords a grammatical recognition for any rune that forms a part of a sentence in <math>\mathfrak{C} (\mathfrak{P}).</math> In situations where these variant usages are likely to be confused, the types of strings can be indicated by means of expressions like <math>r <: R\!</math> and <math>W <: R.\!</math>
When there is no possibility of confusion, the letter <math>{}^{\backprime\backprime} R \, {}^{\prime\prime}</math> can be used either as a string variable that ranges over the set of runes or else as a type name for the class of runes. The latter reading amounts to the enlistment of a fresh intermediate symbol, <math>{}^{\backprime\backprime} R \, {}^{\prime\prime} \in \mathfrak{Q},</math> as a part of a new grammar for <math>\mathfrak{C} (\mathfrak{P}).</math> In effect, <math>{}^{\backprime\backprime} R \, {}^{\prime\prime}</math> affords a grammatical recognition for any rune that forms a part of a sentence in <math>\mathfrak{C} (\mathfrak{P}).</math> In situations where these variant usages are likely to be confused, the types of strings can be indicated by means of expressions like <math>r <: R\!</math> and <math>W <: R.\!</math>
A ''foil'' is a string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},</math> where <math>T\!</math> is a tract. Thus, a typical foil <math>F\!</math> has the form:
A ''foil'' is a string of the form <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime},</math> where <math>T\!</math> is a tract. Thus, a typical foil <math>F\!</math> has the form:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
F
F
& =
& =
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime}
& \cdot
& \cdot
& S_1
& S_1
& \cdot
& \cdot
& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}
& {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}
& \cdot
& \cdot
& \ldots
& \ldots
& \cdot
& \cdot
& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}
& {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime}
& \cdot
& \cdot
& S_k
& S_k
& \cdot
& \cdot
& ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}
\\
\\
\end{array}</math>
\end{array}</math>
|}
|}
This is just the surcatenation of the sentences <math>S_1, \ldots, S_k.\!</math> Given the possibility that this sequence of sentences is empty, and thus that the tract <math>T\!</math> is the empty string, the minimum foil <math>F\!</math> is the expression <math>^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math> Explicitly marking each foil <math>F\!</math> that is embodied in a cactus expression is tantamount to recognizing another intermediate symbol, <math>^{\backprime\backprime} F \, ^{\prime\prime} \in \mathfrak{Q},</math> further articulating the structures of sentences and expanding the grammar for the language
This is just the surcatenation of the sentences <math>S_1, \ldots, S_k.\!</math> Given the possibility that this sequence of sentences is empty, and thus that the tract <math>T\!</math> is the empty string, the minimum foil <math>F\!</math> is the expression <math>{}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}.</math> Explicitly marking each foil <math>F\!</math> that is embodied in a cactus expression is tantamount to recognizing another intermediate symbol, <math>{}^{\backprime\backprime} F \, {}^{\prime\prime} \in \mathfrak{Q},</math> further articulating the structures of sentences and expanding the grammar for the language
<math>\mathfrak{C} (\mathfrak{P}).</math> 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 <math>^{\backprime\backprime} F \, ^{\prime\prime}.</math>
<math>\mathfrak{C} (\mathfrak{P}).</math> 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 <math>{}^{\backprime\backprime} F \, {}^{\prime\prime}.</math>
<br>
<br>
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 3}\!</math>
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 3}\!</math>
| align="right" style="border-right:1px solid black;" width="50%" |
| align="right" style="border-right:1px solid black;" width="50%" |
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>
<math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} F \, {}^{\prime\prime}, \, {}^{\backprime\backprime} R \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \}</math>
|-
|-
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
& F
& F
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}
\\
\\
9.
9.
& T
& T
& :>
& :>
& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S
& T \, \cdot \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S
\\
\\
\end{array}</math>
\end{array}</math>
<br>
<br>
In Grammar 3, the first three Rules say that a sentence (a string of type <math>S\!</math>), is a rune (a string of type <math>R\!</math>), a foil (a string of type <math>F\!</math>), or an arbitrary concatenation of strings of these two types. Rules 4 through 7 specify that a rune <math>R\!</math> is an empty string <math>\varepsilon,</math> a blank symbol <math>m_1,\!</math> a paint <math>p_j,\!</math> or any concatenation of strings of these three types. Rule 8 characterizes a foil <math>F\!</math> as a string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},</math> where <math>T\!</math> is a tract. The last two Rules say that a tract <math>T\!</math> is either a sentence <math>S\!</math> or else the concatenation of a tract, a comma, and a sentence, in that order.
In Grammar 3, the first three Rules say that a sentence (a string of type <math>S\!</math>), is a rune (a string of type <math>R\!</math>), a foil (a string of type <math>F\!</math>), or an arbitrary concatenation of strings of these two types. Rules 4 through 7 specify that a rune <math>R\!</math> is an empty string <math>\varepsilon,</math> a blank symbol <math>m_1,\!</math> a paint <math>p_j,\!</math> or any concatenation of strings of these three types. Rule 8 characterizes a foil <math>F\!</math> as a string of the form <math>{}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime},</math> where <math>T\!</math> is a tract. The last two Rules say that a tract <math>T\!</math> is either a sentence <math>S\!</math> or else the concatenation of a tract, a comma, and a sentence, in that order.
At this point in the succession of grammars for <math>\mathfrak{C} (\mathfrak{P}),</math> the explicit uses of indefinite iterations, like the kleene star operator, are now completely reduced to finite forms of concatenation, but the problems that some styles of analysis have with allowing non-terminal symbols to cover both themselves and the empty string are still present.
At this point in the succession of grammars for <math>\mathfrak{C} (\mathfrak{P}),\!</math> the explicit uses of indefinite iterations, like the kleene star operator, are now completely reduced to finite forms of concatenation, but the problems 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: 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.
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.
Returning to the case of the cactus language <math>\mathfrak{C} (\mathfrak{P}),</math> in other words, the formal language <math>\operatorname{PARCE}</math> of ''painted and rooted cactus expressions'', it serves the purpose of efficient accounting to partition the language into the following couple of sublanguages:
Returning to the case of the cactus language <math>\mathfrak{C} (\mathfrak{P}),\!</math> in other words, the formal language <math>\operatorname{PARCE}\!</math> of ''painted and rooted cactus expressions'', it serves the purpose of efficient accounting to partition the language into the following couple of sublanguages:
<ol style="list-style-type:decimal">
<ol style="list-style-type:decimal">
For brevity in the present case, and to serve as a generic device in any similar array of situations, let <math>S\!</math> be the type of an arbitrary sentence, possibly empty, and let <math>S'\!</math> be the type of a specifically non-empty sentence. In addition, let <math>\underline\varepsilon</math> be the type of the empty sentence, in effect, the language
For brevity in the present case, and to serve as a generic device in any similar array of situations, let <math>S\!</math> be the type of an arbitrary sentence, possibly empty, and let <math>S'\!</math> be the type of a specifically non-empty sentence. In addition, let <math>\underline\varepsilon</math> be the type of the empty sentence, in effect, the language
<math>\underline\varepsilon = \{ \varepsilon \}</math> that contains a single empty string, and let a plus sign <math>^{\backprime\backprime} + ^{\prime\prime}</math> signify a disjoint union of types. In the most general type of situation, where the type <math>S\!</math> is permitted to include the empty string, one notes the following relation among types:
<math>\underline\varepsilon = \{ \varepsilon \}</math> that contains a single empty string, and let a plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> signify a disjoint union of types. In the most general type of situation, where the type <math>S\!</math> is permitted to include the empty string, one notes the following relation among types:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
With the distinction between empty and significant expressions in mind, I return to the grasp of the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) = \operatorname{PARCE} (\mathfrak{P})</math> 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.
With the distinction between empty and significant expressions in mind, I return to the grasp of the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) = \operatorname{PARCE} (\mathfrak{P})</math> 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.
=====2.3.1.4. Grammar 4=====
=====2.3.1.4. Grammar 4=====
If one imposes the distinction between empty and significant types on each non-terminal symbol in Grammar 2, then the non-terminal symbols <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> and <math>^{\backprime\backprime} T \, ^{\prime\prime}</math> give rise to the expanded set of non-terminal symbols <math>^{\backprime\backprime} S \, ^{\prime\prime}, \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime},</math> leaving the last three of these to form the new intermediate alphabet. Grammar 4 has the intermediate alphabet <math>\mathfrak{Q} \, = \, \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime} \, \},</math> with the set <math>\mathfrak{K}</math> of covering rules as listed in the next display.
If one imposes the distinction between empty and significant types on each non-terminal symbol in Grammar 2, then the non-terminal symbols <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} T \, {}^{\prime\prime}</math> give rise to the expanded set of non-terminal symbols <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}, \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T' \, {}^{\prime\prime},</math> leaving the last three of these to form the new intermediate alphabet. Grammar 4 has the intermediate alphabet <math>\mathfrak{Q} \, = \, \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T' \, {}^{\prime\prime} \, \},</math> with the set <math>\mathfrak{K}</math> of covering rules as listed in the next display.
<br>
<br>
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 4}\!</math>
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 4}\!</math>
| align="right" style="border-right:1px solid black;" width="50%" |
| align="right" style="border-right:1px solid black;" width="50%" |
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime} \, \}</math>
<math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T' \, {}^{\prime\prime} \, \}</math>
|-
|-
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
& S'
& S'
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}
\\
\\
6.
6.
& T'
& T'
& :>
& :>
& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S
& T \, \cdot \, {}^{\backprime\backprime} \operatorname{,} {}^{\prime\prime} \, \cdot \, S
\\
\\
\end{array}</math>
\end{array}</math>
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.
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.
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 <math>q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q}</math> and sentential forms <math>W \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*.</math>
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 <math>q \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q}</math> and sentential forms <math>W \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*.</math>
<br>
<br>
& q
& q
& =
& =
& ^{\backprime\backprime} S \, ^{\prime\prime}
& {}^{\backprime\backprime} S \, {}^{\prime\prime}
\\
\\
\end{array}</math>
\end{array}</math>
<br>
<br>
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, <math>^{\backprime\backprime}\!< \, ^{\prime\prime}.</math>
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, <math>{}^{\backprime\backprime}\!< \, {}^{\prime\prime}.</math>
# The ordering <math>^{\backprime\backprime}\!< \, ^{\prime\prime}</math> on the set of non-terminal symbols, <math>q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q},</math> ordains the initial symbol <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> to be strictly prior to every intermediate symbol. This is tantamount to the axiom that <math>^{\backprime\backprime} S \, ^{\prime\prime} < q,</math> for all <math>q \in \mathfrak{Q}.</math>
# The ordering <math>{}^{\backprime\backprime}\!< \, {}^{\prime\prime}</math> on the set of non-terminal symbols, <math>q \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q},</math> ordains the initial symbol <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> to be strictly prior to every intermediate symbol. This is tantamount to the axiom that <math>{}^{\backprime\backprime} S \, {}^{\prime\prime} < q,</math> for all <math>q \in \mathfrak{Q}.</math>
# The ordering <math>^{\backprime\backprime}\!< \, ^{\prime\prime}</math> on the collection of sentential forms, <math>W \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*,</math> ordains the empty string to be strictly minor to every other sentential form. This is stipulated in the axiom that <math>\varepsilon < W,</math> for every non-empty sentential form <math>W.\!</math>
# The ordering <math>{}^{\backprime\backprime}\!< \, {}^{\prime\prime}</math> on the collection of sentential forms, <math>W \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*,</math> ordains the empty string to be strictly minor to every other sentential form. This is stipulated in the axiom that <math>\varepsilon < W,</math> for every non-empty sentential form <math>W.\!</math>
Given these two orderings, the constraint in question on intermediate significance can be stated as follows:
Given these two orderings, the constraint in question on intermediate significance can be stated as follows:
& q
& q
& >
& >
& ^{\backprime\backprime} S \, ^{\prime\prime}
& {}^{\backprime\backprime} S \, {}^{\prime\prime}
\\
\\
\text{then}
\text{then}
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.
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.
=====2.3.1.5. Grammar 5=====
=====2.3.1.5. Grammar 5=====
With the foregoing array of considerations in mind, one is gradually led to a grammar for <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> in which all of the covering productions have either one of the following two forms:
With the foregoing array of considerations in mind, one is gradually led to a grammar for <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> in which all of the covering productions have either one of the following two forms:
& :>
& :>
& W,
& W,
& \text{with} \ q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \ \text{and} \ W \in (\mathfrak{Q} \cup \mathfrak{A})^+
& \text{with} \ q \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q} \ \text{and} \ W \in (\mathfrak{Q} \cup \mathfrak{A})^+
\\
\\
\end{array}</math>
\end{array}</math>
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'', while the second type of rewrite rule is called an ''ordinary production''. An ''ordinary derivation'' is one that employs only ordinary productions. In ordinary productions, those that have the form <math>q :> W,\!</math> the replacement string <math>W\!</math> 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 <math>S :> \varepsilon,</math> 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.
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'', while the second type of rewrite rule is called an ''ordinary production''. An ''ordinary derivation'' is one that employs only ordinary productions. In ordinary productions, those that have the form <math>q :> W,\!</math> the replacement string <math>W\!</math> 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 <math>S :> \varepsilon,</math> 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 that uses <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},</math> with <math>\mathfrak{K}</math> as listed in the next display.
Grammar 5 is a context-free grammar for the painted cactus language that uses <math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \},</math> with <math>\mathfrak{K}</math> as listed in the next display.
<br>
<br>
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 5}\!</math>
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 5}\!</math>
| align="right" style="border-right:1px solid black;" width="50%" |
| align="right" style="border-right:1px solid black;" width="50%" |
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>
<math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \}</math>
|-
|-
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
& S'
& S'
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}
\\
\\
7.
7.
& S'
& S'
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}
\\
\\
8.
8.
& T
& T
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime}
\\
\\
9.
9.
& T
& T
& :>
& :>
& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}
& T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime}
\\
\\
11.
11.
& T
& T
& :>
& :>
& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S'
& T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, S'
\\
\\
\end{array}</math>
\end{array}</math>
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.
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.
=====2.3.1.6. Grammar 6=====
=====2.3.1.6. Grammar 6=====
Grammar 6 has the intermediate alphabet <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},</math> with the production set <math>\mathfrak{K}</math> as listed in the next display.
Grammar 6 has the intermediate alphabet <math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} F \, {}^{\prime\prime}, \, {}^{\backprime\backprime} R \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \},</math> with the production set <math>\mathfrak{K}</math> as listed in the next display.
<br>
<br>
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
| align="left" style="border-left:1px solid black;" width="50%" |
| align="left" style="border-left:1px solid black;" width="50%" |
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 6}\!</math>
<math>{\mathfrak{C} (\mathfrak{P}) : \text{Grammar 6}}\!</math>
| align="right" style="border-right:1px solid black;" width="50%" |
| align="right" style="border-right:1px solid black;" width="50%" |
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>
<math>\mathfrak{Q} = \{ \, {}^{\backprime\backprime} S' \, {}^{\prime\prime}, \, {}^{\backprime\backprime} F \, {}^{\prime\prime}, \, {}^{\backprime\backprime} R \, {}^{\prime\prime}, \, {}^{\backprime\backprime} T \, {}^{\prime\prime} \, \}\!</math>
|-
|-
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
& F
& F
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{()} \, {}^{\prime\prime}
\\
\\
10.
10.
& F
& F
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}
\\
\\
11.
11.
& T
& T
& :>
& :>
& ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}
& {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime}
\\
\\
12.
12.
& T
& T
& :>
& :>
& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}
& T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime}
\\
\\
14.
14.
& T
& T
& :>
& :>
& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S'
& T \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, S'
\\
\\
\end{array}</math>
\end{array}</math>
|-
|-
|
|
| <math>\operatorname{Surc}_{j=1}^k S_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}</math>
| <math>\operatorname{Surc}_{j=1}^k S_j \ = \ {}^{\backprime\backprime} \, \operatorname{(} \, {}^{\prime\prime} \, \cdot \, S_1 \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, \ldots \, \cdot \, {}^{\backprime\backprime} \, \operatorname{,} \, {}^{\prime\prime} \, \cdot \, S_k \, \cdot \, {}^{\backprime\backprime} \, \operatorname{)} \, {}^{\prime\prime}</math>
|-
|-
|
|
|}
|}
====2.3.2. Generalities About Formal Grammars====
====2.3.2. Generalities About Formal Grammars====
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).
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 ''formal grammar'' <math>\mathfrak{G}</math> is given by a four-tuple <math>\mathfrak{G} = ( \, ^{\backprime\backprime} S \, ^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )</math> that takes the following form of description:
A ''formal grammar'' <math>\mathfrak{G}</math> is given by a four-tuple <math>\mathfrak{G} = ( \, {}^{\backprime\backprime} S \, {}^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )</math> that takes the following form of description:
<ol style="list-style-type:decimal">
<ol style="list-style-type:decimal">
<li><math>^{\backprime\backprime} S \, ^{\prime\prime}</math> is the ''initial'', ''special'', ''start'', or ''sentence'' symbol. Since the letter <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> 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.</li>
<li><math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> is the ''initial'', ''special'', ''start'', or ''sentence'' symbol. Since the letter <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> 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.</li>
<li><math>\mathfrak{Q} = \{ q_1, \ldots, q_m \}</math> is a finite set of ''intermediate symbols'', all distinct from <math>^{\backprime\backprime} S \, ^{\prime\prime}.</math></li>
<li><math>\mathfrak{Q} = \{ q_1, \ldots, q_m \}</math> is a finite set of ''intermediate symbols'', all distinct from <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}.</math></li>
<li><math>\mathfrak{A} = \{ a_1, \dots, a_n \}</math> is a finite set of ''terminal symbols'', also known as the ''alphabet'' of <math>\mathfrak{G},</math> all distinct from <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> and disjoint from <math>\mathfrak{Q}.</math> Depending on the particular conception of the language <math>\mathfrak{L}</math> that is ''covered'', ''generated'', ''governed'', or ''ruled'' by the grammar <math>\mathfrak{G},</math> that is, whether <math>\mathfrak{L}</math> is conceived to be a set of words, sentences, paragraphs, or more extended structures of discourse, it is usual to describe <math>\mathfrak{A}</math> as the ''alphabet'', ''lexicon'', ''vocabulary'', ''liturgy'', or ''phrase book'' of both the grammar <math>\mathfrak{G}</math> and the language <math>\mathfrak{L}</math> that it regulates.</li>
<li><math>\mathfrak{A} = \{ a_1, \dots, a_n \}</math> is a finite set of ''terminal symbols'', also known as the ''alphabet'' of <math>\mathfrak{G},</math> all distinct from <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> and disjoint from <math>\mathfrak{Q}.</math> Depending on the particular conception of the language <math>\mathfrak{L}</math> that is ''covered'', ''generated'', ''governed'', or ''ruled'' by the grammar <math>\mathfrak{G},</math> that is, whether <math>\mathfrak{L}</math> is conceived to be a set of words, sentences, paragraphs, or more extended structures of discourse, it is usual to describe <math>\mathfrak{A}</math> as the ''alphabet'', ''lexicon'', ''vocabulary'', ''liturgy'', or ''phrase book'' of both the grammar <math>\mathfrak{G}</math> and the language <math>\mathfrak{L}</math> that it regulates.</li>
<li><math>\mathfrak{K}</math> 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''.</li>
<li><math>\mathfrak{K}</math> 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''.</li>
<ol style="list-style-type:lower-latin">
<ol style="list-style-type:lower-latin">
<li>The symbols in <math>\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> form the ''augmented alphabet'' of <math>\mathfrak{G}.</math></li>
<li>The symbols in <math>\{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> form the ''augmented alphabet'' of <math>\mathfrak{G}.</math></li>
<li>The symbols in <math>\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q}</math> are the ''non-terminal symbols'' of <math>\mathfrak{G}.</math></li>
<li>The symbols in <math>\{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q}</math> are the ''non-terminal symbols'' of <math>\mathfrak{G}.</math></li>
<li>The symbols in <math>\mathfrak{Q} \cup \mathfrak{A}</math> are the ''non-initial symbols'' of <math>\mathfrak{G}.</math></li>
<li>The symbols in <math>\mathfrak{Q} \cup \mathfrak{A}</math> are the ''non-initial symbols'' of <math>\mathfrak{G}.</math></li>
<li>The strings in <math>( \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> are the ''augmented strings'' for <math>\mathfrak{G}.</math></li>
<li>The strings in <math>( \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> are the ''augmented strings'' for <math>\mathfrak{G}.</math></li>
<li>The strings in <math>\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*</math> are the ''sentential forms'' for <math>\mathfrak{G}.</math></li>
<li>The strings in <math>\{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*</math> are the ''sentential forms'' for <math>\mathfrak{G}.</math></li>
</ol>
</ol>
In this scheme, <math>S_1\!</math> and <math>S_2\!</math> are members of the augmented strings for <math>\mathfrak{G},</math> more precisely, <math>S_1\!</math> is a non-empty string and a sentential form over <math>\mathfrak{G},</math> while <math>S_2\!</math> is a possibly empty string and also a sentential form over <math>\mathfrak{G}.</math>
In this scheme, <math>S_1\!</math> and <math>S_2\!</math> are members of the augmented strings for <math>\mathfrak{G},</math> more precisely, <math>S_1\!</math> is a non-empty string and a sentential form over <math>\mathfrak{G},</math> while <math>S_2\!</math> is a possibly empty string and also a sentential form over <math>\mathfrak{G}.</math>
Here also, <math>q\!</math> is a non-terminal symbol, that is, <math>q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q},</math> while <math>Q_1, Q_2,\!</math> and <math>W\!</math> are possibly empty strings of non-initial symbols, a fact that can be expressed in the form, <math>Q_1, Q_2, W \in (\mathfrak{Q} \cup \mathfrak{A})^*.</math>
Here also, <math>q\!</math> is a non-terminal symbol, that is, <math>q \in \{ \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, \} \cup \mathfrak{Q},</math> while <math>Q_1, Q_2,\!</math> and <math>W\!</math> are possibly empty strings of non-initial symbols, a fact that can be expressed in the form, <math>Q_1, Q_2, W \in (\mathfrak{Q} \cup \mathfrak{A})^*.</math>
In practice, the couplets in <math>\mathfrak{K}</math> are used to ''derive'', to ''generate'', or to ''produce'' sentences of the corresponding language <math>\mathfrak{L} = \mathfrak{L} (\mathfrak{G}).</math> The language <math>\mathfrak{L}</math> is then said to be ''governed'', ''licensed'', or ''regulated'' by the grammar <math>\mathfrak{G},</math> a circumstance that is expressed in the form <math>\mathfrak{L} = \langle \mathfrak{G} \rangle.</math> In order to facilitate this active employment of the grammar, it is conventional to write the abstract characterization <math>(S_1, S_2)\!</math> and the specific characterization <math>(Q_1 \cdot q \cdot Q_2, \ Q_1 \cdot W \cdot Q_2)</math> in the following forms, respectively:
In practice, the couplets in <math>\mathfrak{K}</math> are used to ''derive'', to ''generate'', or to ''produce'' sentences of the corresponding language <math>\mathfrak{L} = \mathfrak{L} (\mathfrak{G}).</math> The language <math>\mathfrak{L}</math> is then said to be ''governed'', ''licensed'', or ''regulated'' by the grammar <math>\mathfrak{G},</math> a circumstance that is expressed in the form <math>\mathfrak{L} = \langle \mathfrak{G} \rangle.</math> In order to facilitate this active employment of the grammar, it is conventional to write the abstract characterization <math>(S_1, S_2)\!</math> and the specific characterization <math>(Q_1 \cdot q \cdot Q_2, \ Q_1 \cdot W \cdot Q_2)</math> in the following forms, respectively:
In this usage, the characterization <math>S_1 :> S_2\!</math> is tantamount to a grammatical license to transform a string of the form <math>Q_1 \cdot q \cdot Q_2</math> into a string of the form <math>Q1 \cdot W \cdot Q2,</math> in effect, replacing the non-terminal symbol <math>q\!</math> with the non-initial string <math>W\!</math> in any selected, preserved, and closely adjoining context of the form <math>Q1 \cdot \underline{~~~} \cdot Q2.</math> In this application the notation <math>S_1 :> S_2\!</math> can be read to say that <math>S_1\!</math> ''produces'' <math>S_2\!</math> or that <math>S_1\!</math> ''transforms into'' <math>S_2.\!</math>
In this usage, the characterization <math>S_1 :> S_2\!</math> is tantamount to a grammatical license to transform a string of the form <math>Q_1 \cdot q \cdot Q_2</math> into a string of the form <math>Q1 \cdot W \cdot Q2,</math> in effect, replacing the non-terminal symbol <math>q\!</math> with the non-initial string <math>W\!</math> in any selected, preserved, and closely adjoining context of the form <math>Q1 \cdot \underline{~~~} \cdot Q2.</math> In this application the notation <math>S_1 :> S_2\!</math> can be read to say that <math>S_1\!</math> ''produces'' <math>S_2\!</math> or that <math>S_1\!</math> ''transforms into'' <math>S_2.\!</math>
An ''immediate derivation'' in <math>\mathfrak{G}</math> is an ordered pair <math>(W, W')\!</math> of sentential forms in <math>\mathfrak{G}</math> such that:
An ''immediate derivation'' in <math>\mathfrak{G}\!</math> is an ordered pair <math>(W, W^\prime)\!</math> of sentential forms in <math>\mathfrak{G}\!</math> such that:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|}
|}
The language <math>\mathfrak{L} = \mathfrak{L} (\mathfrak{G}) = \langle \mathfrak{G} \rangle</math> that is ''generated'' by the formal grammar <math>\mathfrak{G} = ( \, ^{\backprime\backprime} S \, ^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )</math> is the set of strings over the terminal alphabet <math>\mathfrak{A}</math> that are derivable from the initial symbol <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> by way of the intermediate symbols in <math>\mathfrak{Q}</math> according to the characterizations in <math>\mathfrak{K}.</math> In sum:
The language <math>\mathfrak{L} = \mathfrak{L} (\mathfrak{G}) = \langle \mathfrak{G} \rangle</math> that is ''generated'' by the formal grammar <math>\mathfrak{G} = ( \, {}^{\backprime\backprime} S \, {}^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )</math> is the set of strings over the terminal alphabet <math>\mathfrak{A}</math> that are derivable from the initial symbol <math>{}^{\backprime\backprime} S \, {}^{\prime\prime}</math> by way of the intermediate symbols in <math>\mathfrak{Q}</math> according to the characterizations in <math>\mathfrak{K}.</math> In sum:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
| <math>\mathfrak{L} (\mathfrak{G}) \ = \ \langle \mathfrak{G} \rangle \ = \ \{ \, W \in \mathfrak{A}^* \, : \, ^{\backprime\backprime} S \, ^{\prime\prime} \, :\!*\!:> \, W \, \}.</math>
| <math>\mathfrak{L} (\mathfrak{G}) \ = \ \langle \mathfrak{G} \rangle \ = \ \{ \, W \in \mathfrak{A}^* \, : \, {}^{\backprime\backprime} S \, {}^{\prime\prime} \, :\!*\!:> \, W \, \}.</math>
|}
|}
Finally, a string <math>W\!</math> is called a ''word'', a ''sentence'', or so on, of the language generated by <math>\mathfrak{G}</math> if and only if <math>W\!</math> is in <math>\mathfrak{L} (\mathfrak{G}).</math>
Finally, a string <math>W\!</math> is called a ''word'', a ''sentence'', or so on, of the language generated by <math>\mathfrak{G}</math> if and only if <math>W\!</math> is in <math>\mathfrak{L} (\mathfrak{G}).</math>
====2.3.3. The Cactus Language : Stylistics====
====2.3.3. The Cactus Language : Stylistics====
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{| align="center" cellpadding="0" cellspacing="0" width="90%"
Any style of declarative programming, also called ''logic programming'', depends on a capacity, as embodied in a programming language or other 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.
Any style of declarative programming, also called ''logic programming'', depends on a capacity, as embodied in a programming language or other 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.
This patch of the ongoing discussion is concerned with describing a particular variety of formal languages, whose typical representative is the painted cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}).</math> 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?
This patch of the ongoing discussion is concerned with describing a particular variety of formal languages, whose typical representative is the painted cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}).\!</math> 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?
In order to speak to these questions, I have to express what a grammar says about a language in terms of what a language can say on its own. In effect, it is necessary to analyze the kinds of meaningful statements that grammars are capable of making about languages in general and to relate them to the kinds of meaningful statements that the syntactic ''sentences'' of the cactus language might be interpreted as making about the very same topics. So far 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.
In order to speak to these questions, I have to express what a grammar says about a language in terms of what a language can say on its own. In effect, it is necessary to analyze the kinds of meaningful statements that grammars are capable of making about languages in general and to relate them to the kinds of meaningful statements that the syntactic ''sentences'' of the cactus language might be interpreted as making about the very same topics. So far 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.
Having broached the distinction between propositions and sentences, one 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.
Having broached the distinction between propositions and sentences, one 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: Do not worry too much about what roles the empty string <math>\varepsilon \, = \, ^{\backprime\backprime\prime\prime}</math> and the blank symbol <math>m_1 \, = \, ^{\backprime\backprime} \operatorname{~} ^{\prime\prime}</math> 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.
The application of these considerations in the immediate setting is this: Do not worry too much about what roles the empty string <math>\varepsilon \, = \, ^{\backprime\backprime\prime\prime}</math> and the blank symbol <math>m_1 \, = \, {}^{\backprime\backprime} \operatorname{~} {}^{\prime\prime}</math> 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. 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.
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.
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It is useful to examine the relationship between the grammatical covering or production relation <math>(:>\!)</math> and the logical relation of implication <math>(\Rightarrow),</math> with one eye to what they have in common and one eye to how they differ. The production <math>q :> W\!</math> says that the appearance of the symbol <math>q\!</math> in a sentential form implies the possibility of exchanging it for <math>W.\!</math> Although this sounds like a ''possible implication'', to the extent that ''<math>q\!</math> implies a possible <math>W\!</math>'' or that ''<math>q\!</math> possibly implies <math>W,\!</math>'' 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 <math>^{\backprime\backprime} \, q \Leftarrow W \, ^{\prime\prime}</math> as the best analogue for the sense of the production.
It is useful to examine the relationship between the grammatical covering or production relation <math>(:>\!)</math> and the logical relation of implication <math>(\Rightarrow),</math> with one eye to what they have in common and one eye to how they differ. The production <math>q :> W\!</math> says that the appearance of the symbol <math>q\!</math> in a sentential form implies the possibility of exchanging it for <math>W.\!</math> Although this sounds like a ''possible implication'', to the extent that ''<math>q\!</math> implies a possible <math>W\!</math>'' or that ''<math>q\!</math> possibly implies <math>W,\!</math>'' 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 <math>{}^{\backprime\backprime} \, q \Leftarrow W \, {}^{\prime\prime}</math> 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 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 <math>W,\!</math> then it is an admissible hypothesis, according to the theory of the language that is constituted by the formal grammar, that this piece not only fits the type <math>q\!</math> but even comes to be generated under the auspices of the non-terminal symbol <math>^{\backprime\backprime} q ^{\prime\prime}.</math>
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 <math>W,\!</math> then it is an admissible hypothesis, according to the theory of the language that is constituted by the formal grammar, that this piece not only fits the type <math>q\!</math> but even comes to be generated under the auspices of the non-terminal symbol <math>{}^{\backprime\backprime} q {}^{\prime\prime}.</math>
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: "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 to pose it.
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: "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 to pose it.
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The concatenation <math>\mathfrak{L}_1 \cdot \mathfrak{L}_2</math> of the formal languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> is just the cartesian product of sets <math>\mathfrak{L}_1 \times \mathfrak{L}_2</math> without the extra <math>\times</math>'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 is to focus on the information that is needed to specify each construction, and thus to reflect on the signs that are used to carry this information. As a first approach to the topic of information, according to a strategy that seeks to be as elementary and as informal as possible, I introduce the following set of ideas, intended to be taken in a very provisional way.
The concatenation <math>\mathfrak{L}_1 \cdot \mathfrak{L}_2\!</math> of the formal languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> is just the cartesian product of sets <math>\mathfrak{L}_1 \times \mathfrak{L}_2\!</math> without the extra <math>\times\!</math>’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 is to focus on the information that is needed to specify each construction, and thus to reflect on the signs that are used to carry this information. As a first approach to the topic of information, according to a strategy that seeks to be as elementary and as informal as possible, I introduce the following set of ideas, intended to be taken in a very provisional way.
A ''stricture'' is a specification of a certain set in a certain place, relative to a number of other sets, yet to be specified. It is assumed that one knows enough 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 proper concern of the definition, no matter how convenient they are found to be for a particular discussion. As a schematic form of illustration, a stricture can be pictured in the following shape:
A ''stricture'' is a specification of a certain set in a certain place, relative to a number of other sets, yet to be specified. It is assumed that one knows enough 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 proper concern of the definition, no matter how convenient they are found to be for a particular discussion. As a schematic form of illustration, a stricture can be pictured in the following shape:
:{| cellpadding="8"
:{| cellpadding="8"
| <math>^{\backprime\backprime}</math>
| <math>{}^{\backprime\backprime}</math>
| <math>\ldots \times X \times Q \times X \times \ldots</math>
| <math>\ldots \times X \times Q \times X \times \ldots</math>
| <math>^{\prime\prime}</math>
| <math>{}^{\prime\prime}</math>
|}
|}
|
|
<math>\begin{array}{ccccccc}
<math>\begin{array}{ccccccc}
^{\backprime\backprime} Q ^{\prime\prime}
{}^{\backprime\backprime} Q {}^{\prime\prime}
& , &
& , &
^{\backprime\backprime} X \times Q \times X ^{\prime\prime}
{}^{\backprime\backprime} X \times Q \times X {}^{\prime\prime}
& , &
& , &
^{\backprime\backprime} X \times X \times Q \times X \times X ^{\prime\prime}
{}^{\backprime\backprime} X \times X \times Q \times X \times X {}^{\prime\prime}
& , &
& , &
\ldots
\ldots
|
|
<math>\begin{matrix}
<math>\begin{matrix}
^{\backprime\backprime} X ^{\prime\prime} &
{}^{\backprime\backprime} X {}^{\prime\prime} &
^{\backprime\backprime} P ^{\prime\prime} &
{}^{\backprime\backprime} P {}^{\prime\prime} &
^{\backprime\backprime} Q ^{\prime\prime} \\
{}^{\backprime\backprime} Q {}^{\prime\prime} \\
\\
\\
^{\backprime\backprime} X \times X ^{\prime\prime} &
{}^{\backprime\backprime} X \times X {}^{\prime\prime} &
^{\backprime\backprime} X \times P ^{\prime\prime} &
{}^{\backprime\backprime} X \times P {}^{\prime\prime} &
^{\backprime\backprime} X \times Q ^{\prime\prime} \\
{}^{\backprime\backprime} X \times Q {}^{\prime\prime} \\
\\
\\
^{\backprime\backprime} P \times X ^{\prime\prime} &
{}^{\backprime\backprime} P \times X {}^{\prime\prime} &
^{\backprime\backprime} P \times P ^{\prime\prime} &
{}^{\backprime\backprime} P \times P {}^{\prime\prime} &
^{\backprime\backprime} P \times Q ^{\prime\prime} \\
{}^{\backprime\backprime} P \times Q {}^{\prime\prime} \\
\\
\\
^{\backprime\backprime} Q \times X ^{\prime\prime} &
{}^{\backprime\backprime} Q \times X {}^{\prime\prime} &
^{\backprime\backprime} Q \times P ^{\prime\prime} &
{}^{\backprime\backprime} Q \times P {}^{\prime\prime} &
^{\backprime\backprime} Q \times Q ^{\prime\prime} \\
{}^{\backprime\backprime} Q \times Q {}^{\prime\prime} \\
\end{matrix}</math>
\end{matrix}</math>
|}
|}
<math>\begin{array}{lcccc}
<math>\begin{array}{lcccc}
\text{High:}
\text{High:}
& ^{\backprime\backprime} P \times P ^{\prime\prime}
& {}^{\backprime\backprime} P \times P {}^{\prime\prime}
& ^{\backprime\backprime} P \times Q ^{\prime\prime}
& {}^{\backprime\backprime} P \times Q {}^{\prime\prime}
& ^{\backprime\backprime} Q \times P ^{\prime\prime}
& {}^{\backprime\backprime} Q \times P {}^{\prime\prime}
& ^{\backprime\backprime} Q \times Q ^{\prime\prime}
& {}^{\backprime\backprime} Q \times Q {}^{\prime\prime}
\\
\\
\\
\\
\text{Med:}
\text{Med:}
& ^{\backprime\backprime} P ^{\prime\prime}
& {}^{\backprime\backprime} P {}^{\prime\prime}
& ^{\backprime\backprime} X \times P ^{\prime\prime}
& {}^{\backprime\backprime} X \times P {}^{\prime\prime}
& ^{\backprime\backprime} P \times X ^{\prime\prime}
& {}^{\backprime\backprime} P \times X {}^{\prime\prime}
\\
\\
\\
\\
\text{Med:}
\text{Med:}
& ^{\backprime\backprime} Q ^{\prime\prime}
& {}^{\backprime\backprime} Q {}^{\prime\prime}
& ^{\backprime\backprime} X \times Q ^{\prime\prime}
& {}^{\backprime\backprime} X \times Q {}^{\prime\prime}
& ^{\backprime\backprime} Q \times X ^{\prime\prime}
& {}^{\backprime\backprime} Q \times X {}^{\prime\prime}
\\
\\
\\
\\
\text{Low:}
\text{Low:}
& ^{\backprime\backprime} X ^{\prime\prime}
& {}^{\backprime\backprime} X {}^{\prime\prime}
& ^{\backprime\backprime} X \times X ^{\prime\prime}
& {}^{\backprime\backprime} X \times X {}^{\prime\prime}
\\
\\
\end{array}</math>
\end{array}</math>
# <math>q_{[2]}\!</math> says that <math>q\!</math> is in the second place of the product element under construction.
# <math>q_{[2]}\!</math> says that <math>q\!</math> is in the second place of the product element under construction.
Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> in this way, one shifts the level of the active construction from the tupling of the elements in <math>P\!</math> and <math>Q\!</math> or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> 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, <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{[1]}</math> and <math>(\mathfrak{L}_2)_{[2]},</math> 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 <math>^{\backprime\backprime} [1] ^{\prime\prime}</math> and <math>^{\backprime\backprime} [2] ^{\prime\prime}</math> can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external ''mark-up'' language.
Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> in this way, one shifts the level of the active construction from the tupling of the elements in <math>P\!</math> and <math>Q\!</math> or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> 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, <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{[1]}\!</math> and <math>(\mathfrak{L}_2)_{[2]},\!</math> 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 <math>{}^{\backprime\backprime} [1] {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} [2] {}^{\prime\prime}\!</math> 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 relations that strictures and straits placed at higher levels of complexity, constraint, information, and organization have with those that are placed at the associated lower levels, I introduce the following pair of definitions:
In order to systematize the relations that strictures and straits placed at higher levels of complexity, constraint, information, and organization have with those that are placed at the associated lower levels, I introduce the following pair of definitions:
The <math>j^\text{th}\!</math> ''excerpt'' of a stricture of the form <math>^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime},</math> regarded within a frame of discussion where the number of places is limited to <math>k,\!</math> is the stricture of the form <math>^{\backprime\backprime} \, X \times \ldots \times S_j \times \ldots \times X \, ^{\prime\prime}.</math> In the proper context, this can be written more succinctly as the stricture <math>^{\backprime\backprime} \, (S_j)_{[j]} \, ^{\prime\prime},</math> an assertion that places the <math>j^\text{th}\!</math> set in the <math>j^\text{th}\!</math> place of the product.
The <math>j^\text{th}\!</math> ''excerpt'' of a stricture of the form <math>{}^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, {}^{\prime\prime},</math> regarded within a frame of discussion where the number of places is limited to <math>k,\!</math> is the stricture of the form <math>{}^{\backprime\backprime} \, X \times \ldots \times S_j \times \ldots \times X \, {}^{\prime\prime}.</math> In the proper context, this can be written more succinctly as the stricture <math>{}^{\backprime\backprime} \, (S_j)_{[j]} \, {}^{\prime\prime},</math> an assertion that places the <math>j^\text{th}\!</math> set in the <math>j^\text{th}\!</math> place of the product.
The <math>j^\text{th}\!</math> ''extract'' of a strait of the form <math>S_1 \times \ldots \times S_k,\!</math> constrained to a frame of discussion where the number of places is restricted to <math>k,\!</math> is the strait of the form <math>X \times \ldots \times S_j \times \ldots \times X.</math> In the appropriate context, this can be denoted more succinctly by the stricture <math>^{\backprime\backprime} \, (S_j)_{[j]} \, ^{\prime\prime},</math> an assertion that places the <math>j^\text{th}\!</math> set in the <math>j^\text{th}\!</math> place of the product.
The <math>j^\text{th}\!</math> ''extract'' of a strait of the form <math>S_1 \times \ldots \times S_k,\!</math> constrained to a frame of discussion where the number of places is restricted to <math>k,\!</math> is the strait of the form <math>X \times \ldots \times S_j \times \ldots \times X.</math> In the appropriate context, this can be denoted more succinctly by the stricture <math>{}^{\backprime\backprime} \, (S_j)_{[j]} \, {}^{\prime\prime},</math> an assertion that places the <math>j^\text{th}\!</math> set in the <math>j^\text{th}\!</math> place of the product.
In these terms, a stricture of the form <math>^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime}</math> can be expressed in terms of simpler strictures, to wit, as a conjunction of its <math>k\!</math> excerpts:
In these terms, a stricture of the form <math>{}^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, {}^{\prime\prime}</math> can be expressed in terms of simpler strictures, to wit, as a conjunction of its <math>k\!</math> excerpts:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|
|
<math>\begin{array}{lll}
<math>\begin{array}{lll}
^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime}
{}^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, {}^{\prime\prime}
& = &
& = &
^{\backprime\backprime} \, (S_1)_{[1]} \, ^{\prime\prime}
{}^{\backprime\backprime} \, (S_1)_{[1]} \, {}^{\prime\prime}
\, \land \, \ldots \, \land \,
\, \land \, \ldots \, \land \,
^{\backprime\backprime} \, (S_k)_{[k]} \, ^{\prime\prime}.
{}^{\backprime\backprime} \, (S_k)_{[k]} \, {}^{\prime\prime}.
\end{array}</math>
\end{array}</math>
|}
|}
There is a measure of ambiguity that remains in this formulation, but it is the best that I can do in the present informal context.
There is a measure of ambiguity that remains in this formulation, but it is the best that I can do in the present informal context.
====2.3.4. The Cactus Language : Mechanics====
====2.3.4. The Cactus Language : Mechanics====
{| align="center" cellpadding="0" cellspacing="0" width="90%"
{| align="center" cellpadding="0" cellspacing="0" width="90%"
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.
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.
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 <math>\{ \, ^{\backprime\backprime} \operatorname{~} ^{\prime\prime} \, \} \cup \mathfrak{P} = \{ m_1 \} \cup \{ p_1, \ldots, p_k \}.</math>
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 <math>\{ \, {}^{\backprime\backprime} \operatorname{~} {}^{\prime\prime} \, \} \cup \mathfrak{P} = \{ m_1 \} \cup \{ p_1, \ldots, p_k \}.</math>
It is conceivable, from a purely graph-theoretical point of view, to have 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).
It is conceivable, from a purely graph-theoretical point of view, to have 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).
|}
|}
Working from a structural description of the cactus language, or any suitable formal grammar for <math>\mathfrak{C} (\mathfrak{P}),</math> it is possible to give a recursive definition of the function called <math>\operatorname{Parse}</math> that maps each sentence in <math>\operatorname{PARCE} (\mathfrak{P})</math> to the corresponding graph in <math>\operatorname{PARC} (\mathfrak{P}).</math> One way to do this proceeds as follows:
Working from a structural description of the cactus language, or any suitable formal grammar for <math>\mathfrak{C} (\mathfrak{P}),\!</math> it is possible to give a recursive definition of the function called <math>\operatorname{Parse}\!</math> that maps each sentence in <math>\operatorname{PARCE} (\mathfrak{P})\!</math> to the corresponding graph in <math>\operatorname{PARC} (\mathfrak{P}).\!</math> One way to do this proceeds as follows:
<ol style="list-style-type:decimal">
<ol style="list-style-type:decimal">
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" cellpadding="10" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"
|+ '''Table 13. Algorithmic Translation Rules'''
|+ style="height:30px" | <math>\text{Table 13.} ~~ \text{Algorithmic Translation Rules}\!</math>
|- style="background:whitesmoke"
|- style="height:50px; background:ghostwhite"
| width="33%" | <math>\mathrm{Sentence~in~PARCE}\!</math>
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
| width="33%" | <math>\text{Sentence in PARCE}\!</math>
| width="33%" | <math>\mathrm{Graph~in~PARC}\!</math>
| align="center" | <math>\xrightarrow{\operatorname{Parse}}</math>
| width="33%" | <math>\text{Graph in PARC}\!</math>
|-
|-
|
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Conc}^0
\\[8pt]
\mathrm{Conc}_{j=1}^k s_j
\end{matrix}</math>
| style="border-top:1px solid black" |
| align="center" | <math>\xrightarrow{\operatorname{Parse}}</math>
<math>\begin{matrix}
| width="33%" | <math>\operatorname{Node}_{j=1}^k \operatorname{Parse} (s_j)</math>
\xrightarrow{\mathrm{Parse}}
\\[8pt]
\xrightarrow{\mathrm{Parse}}
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Node}^0
\\[8pt]
\mathrm{Node}_{j=1}^k \mathrm{Parse}(s_j)
\end{matrix}</math>
|-
|-
|
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Surc}^0
\\[8pt]
\mathrm{Surc}_{j=1}^k s_j
\end{matrix}</math>
| style="border-top:1px solid black" |
| align="center" | <math>\xrightarrow{\operatorname{Parse}}</math>
<math>\begin{matrix}
| width="33%" | <math>\operatorname{Lobe}_{j=1}^k \operatorname{Parse} (s_j)</math>
\xrightarrow{\mathrm{Parse}}
\\[8pt]
\xrightarrow{\mathrm{Parse}}
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Lobe}^0
\\[8pt]
\mathrm{Lobe}_{j=1}^k \mathrm{Parse}(s_j)
\end{matrix}</math>
|}
|}
A ''substructure'' of a PARC is defined recursively as follows. Starting at the root node of the cactus <math>C,\!</math> any attachment is a substructure of <math>C.\!</math> If a substructure is a blank or a paint, then it constitutes a minimal substructure, meaning that no further substructures of <math>C\!</math> arise from it. If a substructure is a lobe, then each one of its accoutrements is also a substructure of <math>C,\!</math> and has to be examined for further substructures.
A ''substructure'' of a PARC is defined recursively as follows. Starting at the root node of the cactus <math>C,\!</math> any attachment is a substructure of <math>C.\!</math> If a substructure is a blank or a paint, then it constitutes a minimal substructure, meaning that no further substructures of <math>C\!</math> arise from it. If a substructure is a lobe, then each one of its accoutrements is also a substructure of <math>C,\!</math> and has to be examined for further substructures.
The concept of substructure can be used to define varieties of deletion and erasure operations that respect the structure of the abstract graph. For the purposes of this depiction, a blank symbol <math>^{\backprime\backprime} ~ ^{\prime\prime}</math> is treated as a ''primer'', in other words, as a ''clear paint'' or a ''neutral tint''. In effect, one is letting <math>m_1 = p_0.\!</math> In this frame of discussion, it is useful to make the following distinction:
The concept of substructure can be used to define varieties of deletion and erasure operations that respect the structure of the abstract graph. For the purposes of this depiction, a blank symbol <math>{}^{\backprime\backprime} ~ {}^{\prime\prime}</math> is treated as a ''primer'', in other words, as a ''clear paint'' or a ''neutral tint''. In effect, one is letting <math>m_1 = p_0.\!</math> In this frame of discussion, it is useful to make the following distinction:
# To ''delete'' a substructure is to replace it with an empty node, in effect, to reduce the whole structure to a trivial point.
# To ''delete'' a substructure is to replace it with an empty node, in effect, to reduce the whole structure to a trivial point.
</ol>
</ol>
====2.3.5. The Cactus Language : Semantics====
====2.3.5. The Cactus Language : Semantics====
{| align="center" cellpadding="0" cellspacing="0" width="90%"
{| align="center" cellpadding="0" cellspacing="0" width="90%"
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" cellpadding="10" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"
|+ '''Table 14. Semantic Translation : Functional Form'''
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Semantic Translation : Functional Form}\!</math>
|- style="background:whitesmoke"
|- style="height:50px; background:ghostwhite"
| width="20%" | <math>\mathrm{Sentence}\!</math>
| width="20%" | <math>\xrightarrow[~~~~~~~~~~]{\mathrm{Parse}}\!</math>
| width="20%" | <math>\operatorname{Sentence}</math>
| width="20%" | <math>\mathrm{Graph}\!</math>
| width="20%" | <math>\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Parse}}</math>
| width="20%" | <math>\xrightarrow[~~~~~~~~~~]{\mathrm{Denotation}}\!</math>
| width="20%" | <math>\operatorname{Graph}</math>
| width="20%" | <math>\mathrm{Proposition}\!</math>
| width="20%" | <math>\xrightarrow[\operatorname{~~~~~~~~~~}]{\operatorname{Denotation}}</math>
| width="20%" | <math>\operatorname{Proposition}</math>
|-
|-
|
| style="border-top:1px solid black" | <math>s_j\!</math>
| style="border-top:1px solid black" | <math>\xrightarrow{~~~~~~~~~~}</math>
| style="border-top:1px solid black" | <math>C_j\!</math>
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
| style="border-top:1px solid black" | <math>\xrightarrow{~~~~~~~~~~}</math>
| width="20%" | <math>C_j\!</math>
| style="border-top:1px solid black" | <math>q_j\!</math>
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
| width="20%" | <math>q_j\!</math>
|-
|-
|
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Conc}^0
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
\\[8pt]
| width="20%" | <math>\operatorname{Node}^0</math>
\mathrm{Conc}_{j=1}^k s_j
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
\end{matrix}</math>
| width="20%" | <math>\underline{1}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\xrightarrow{~~~~~~~~~~}
\\[8pt]
\xrightarrow{~~~~~~~~~~}
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Node}^0
\\[8pt]
\mathrm{Node}_{j=1}^k C_j
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\xrightarrow{~~~~~~~~~~}
\\[8pt]
\xrightarrow{~~~~~~~~~~}
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\underline{1}
\\[8pt]
\mathrm{Conj}_{j=1}^k q_j
\end{matrix}</math>
|-
|-
| width="20%" | <math>\operatorname{Conc}^k_j s_j</math>
| style="border-top:1px solid black" |
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
<math>\begin{matrix}
\mathrm{Surc}^0
\\[8pt]
| width="20%" | <math>\operatorname{Conj}^k_j q_j</math>
\mathrm{Surc}_{j=1}^k s_j
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\xrightarrow{~~~~~~~~~~}
\\[8pt]
\xrightarrow{~~~~~~~~~~}
\end{matrix}</math>
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\mathrm{Lobe}^0
\\[8pt]
\mathrm{Lobe}_{j=1}^k C_j
\end{matrix}</math>
| width="20%" | <math>\xrightarrow{\operatorname{~~~~~~~~~~}}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\xrightarrow{~~~~~~~~~~}
\\[8pt]
\xrightarrow{~~~~~~~~~~}
\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\underline{0}
\\[8pt]
\mathrm{Surj}_{j=1}^k q_j
\end{matrix}</math>
|}
|}
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" cellpadding="10" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"
|+ '''Table 15. Semantic Translation : Equational Form'''
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Semantic Translation : Equational Form}\!</math>
|- style="background:whitesmoke"
|- style="height:50px; background:ghostwhite"
| width="20%" | <math>\downharpoonleft \mathrm{Sentence} \downharpoonright\!</math>
| width="20%" | <math>\stackrel{\mathrm{Parse}}{=}\!</math>
| width="20%" | <math>\downharpoonleft \operatorname{Sentence} \downharpoonright</math>
| width="20%" | <math>\downharpoonleft \mathrm{Graph} \downharpoonright\!</math>
| width="20%" | <math>\stackrel{\operatorname{Parse}}{=}</math>
| width="20%" | <math>\stackrel{\mathrm{Denotation}}{=}\!</math>
| width="20%" | <math>\downharpoonleft \operatorname{Graph} \downharpoonright</math>
| width="20%" | <math>\mathrm{Proposition}\!</math>
| width="20%" | <math>\stackrel{\operatorname{Denotation}}{=}</math>
| width="20%" | <math>\operatorname{Proposition}</math>
|-
|-
|
| style="border-top:1px solid black" | <math>\downharpoonleft s_j \downharpoonright\!</math>
| style="border-top:1px solid black" | <math>=\!</math>
| style="border-top:1px solid black" | <math>\downharpoonleft C_j \downharpoonright\!</math>
| width="20%" | <math>=\!</math>
| style="border-top:1px solid black" | <math>=\!</math>
| width="20%" | <math>\downharpoonleft C_j \downharpoonright</math>
| style="border-top:1px solid black" | <math>q_j\!</math>
| width="20%" | <math>=\!</math>
| width="20%" | <math>q_j\!</math>
|-
|-
|
| style="border-top:1px solid black" |
<math>\begin{matrix}
\downharpoonleft \mathrm{Conc}^0 \downharpoonright
| width="20%" | <math>=\!</math>
\\[8pt]
| width="20%" | <math>\downharpoonleft \operatorname{Node}^0 \downharpoonright</math>
\downharpoonleft \mathrm{Conc}_{j=1}^k s_j \downharpoonright
| width="20%" | <math>=\!</math>
\end{matrix}</math>
| width="20%" | <math>\underline{1}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}=\\[8pt]=\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\downharpoonleft \mathrm{Node}^0 \downharpoonright
\\[8pt]
\downharpoonleft \mathrm{Node}_{j=1}^k C_j \downharpoonright
\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}=\\[8pt]=\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\underline{1}
\\[8pt]
\mathrm{Conj}_{j=1}^k q_j
\end{matrix}</math>
|-
|-
| width="20%" | <math>\downharpoonleft \operatorname{Conc}^k_j s_j \downharpoonright</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\downharpoonleft \mathrm{Surc}^0 \downharpoonright
| width="20%" | <math>=\!</math>
\\[8pt]
\downharpoonleft \mathrm{Surc}_{j=1}^k s_j \downharpoonright
|}
\end{matrix}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}=\\[8pt]=\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
\downharpoonleft \mathrm{Lobe}^0 \downharpoonright
\\[8pt]
\downharpoonleft \mathrm{Lobe}_{j=1}^k C_j \downharpoonright
\end{matrix}</math>
| width="20%" | <math>\underline{0}</math>
| style="border-top:1px solid black" | <math>\begin{matrix}=\\[8pt]=\end{matrix}</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
| width="20%" | <math>=\!</math>
\underline{0}
\\[8pt]
\mathrm{Surj}_{j=1}^k q_j
\end{matrix}</math>
|}
|}
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
|+ '''Table 16. Boolean Functions on Zero Variables'''
|+ style="height:30px" | <math>\text{Table 16.} ~~ \text{Boolean Functions on Zero Variables}\!</math>
|- style="background:whitesmoke"
|- style="height:40px; background:ghostwhite"
| width="14%" | <math>F\!</math>
| width="14%" | <math>F\!</math>
| width="14%" | <math>F\!</math>
| width="14%" | <math>F\!</math>
| width="24%" | <math>F\!</math>
| width="24%" | <math>F\!</math>
|-
|-
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>F_0^{(0)}\!</math>
| <math>F_0^{(0)}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>(~)</math>
| <math>\texttt{(~)}\!</math>
|-
|-
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>F_1^{(0)}\!</math>
| <math>F_1^{(0)}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>((~))</math>
| <math>\texttt{((~))}\!</math>
|}
|}
Column 3 lists the functional values for each boolean function, or possibly a boolean element appearing in the guise of a function, for each combination of its domain values.
Column 3 lists the functional values for each boolean function, or possibly 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, conforming to the practice of omitting the underlines in display formats. Here I illustrate also the convention of using the expression <math>^{\backprime\backprime} ((~)) ^{\prime\prime}</math> as a visible stand-in for the expression of the logical value <math>\operatorname{true},</math> a value that is minimally represented by a blank expression that tends to elude our giving it much notice in the context of more demonstrative texts.
Column 4 shows the usual expressions of these elements in the cactus language, conforming to the practice of omitting the underlines in display formats. Here I illustrate also the convention of using the expression <math>{}^{\backprime\backprime} ((~)) {}^{\prime\prime}</math> as a visible stand-in for the expression of the logical value <math>\operatorname{true},</math> a value that is minimally represented by a blank expression that tends to elude our giving it much notice in the context of more demonstrative texts.
Table 17 presents the boolean functions on one variable, <math>F^{(1)} : \underline\mathbb{B} \to \underline\mathbb{B},</math> of which there are precisely four.
Table 17 presents the boolean functions on one variable, <math>F^{(1)} : \underline\mathbb{B} \to \underline\mathbb{B},</math> of which there are precisely four.
<br>
<br>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:70%"
|+ '''Table 17. Boolean Functions on One Variable'''
|+ style="height:30px" | <math>\text{Table 17.} ~~ \text{Boolean Functions on One Variable}\!</math>
|- style="background:whitesmoke"
|- style="height:40px; background:ghostwhite"
| width="14%" | <math>F\!</math>
| width="14%" | <math>F\!</math>
| width="14%" | <math>F\!</math>
| width="14%" | <math>F\!</math>
| colspan="2" | <math>F(x)\!</math>
| colspan="2" | <math>F(x)\!</math>
| width="24%" | <math>F\!</math>
| width="24%" | <math>F\!</math>
|- style="background:whitesmoke"
|- style="background:ghostwhite"
| width="14%" |
| width="14%" |
| width="14%" |
| width="14%" |
| width="24%" | <math>F(\underline{1})</math>
| width="24%" | <math>F(\underline{1})~\!</math>
| width="24%" | <math>F(\underline{0})</math>
| width="24%" | <math>F(\underline{0})~\!</math>
| width="24%" |
| width="24%" |
|-
|-
| <math>F_0^{(1)}\!</math>
| <math>{F_0^{(1)}}\!</math>
| <math>F_{00}^{(1)}\!</math>
| <math>{F_{00}^{(1)}}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>(~)</math>
| <math>\texttt{(~)}\!</math>
|-
|-
| <math>F_1^{(1)}\!</math>
| <math>{F_1^{(1)}}\!</math>
| <math>F_{01}^{(1)}\!</math>
| <math>{F_{01}^{(1)}}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>(x)\!</math>
| <math>\texttt{(} x \texttt{)}\!</math>
|-
|-
| <math>F_2^{(1)}\!</math>
| <math>{F_2^{(1)}}\!</math>
| <math>F_{10}^{(1)}\!</math>
| <math>{F_{10}^{(1)}}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>x\!</math>
| <math>x\!</math>
|-
|-
| <math>F_3^{(1)}\!</math>
| <math>{F_3^{(1)}}\!</math>
| <math>F_{11}^{(1)}\!</math>
| <math>{F_{11}^{(1)}}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>((~))</math>
| <math>\texttt{((~))}\!</math>
|}
|}
<br>
<br>
<br>
<br>
{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:90%"
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:70%"
|+ '''Table 18. Boolean Functions on Two Variables'''
|+ style="height:30px" | <math>\text{Table 18.} ~~ \text{Boolean Functions on Two Variables}\!</math>
|- style="background:whitesmoke"
|- style="height:40px; background:ghostwhite"
| width="14%" | <math>F\!</math>
| width="14%" | <math>F\!</math>
| width="14%" | <math>F\!</math>
| width="14%" | <math>F\!</math>
| colspan="4" | <math>F(x, y)\!</math>
| colspan="4" | <math>F(x, y)\!</math>
| width="24%" | <math>F\!</math>
| width="24%" | <math>F\!</math>
|- style="background:whitesmoke"
|- style="background:ghostwhite"
| width="14%" |
| width="14%" |
| width="14%" |
| width="14%" |
|-
|-
| <math>F_{0}^{(2)}\!</math>
| <math>F_{0}^{(2)}\!</math>
| <math>F_{0000}^{(2)}\!</math>
| <math>F_{0000}^{(2)}~\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>(~)</math>
| <math>\texttt{(~)}</math>
|-
|-
| <math>F_{1}^{(2)}\!</math>
| <math>F_{1}^{(2)}</math>
| <math>F_{0001}^{(2)}\!</math>
| <math>F_{0001}^{(2)}~\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>(x)(y)\!</math>
| <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math>
|-
|-
| <math>F_{2}^{(2)}\!</math>
| <math>F_{2}^{(2)}\!</math>
| <math>F_{0010}^{(2)}\!</math>
| <math>F_{0010}^{(2)}~\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>(x) y\!</math>
| <math>\texttt{(} x \texttt{)} y\!</math>
|-
|-
| <math>F_{3}^{(2)}\!</math>
| <math>F_{3}^{(2)}\!</math>
| <math>F_{0011}^{(2)}\!</math>
| <math>F_{0011}^{(2)}~\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>(x)\!</math>
| <math>\texttt{(} x \texttt{)}\!</math>
|-
|-
| <math>F_{4}^{(2)}\!</math>
| <math>F_{4}^{(2)}\!</math>
| <math>F_{0100}^{(2)}\!</math>
| <math>F_{0100}^{(2)}~\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>x (y)\!</math>
| <math>x \texttt{(} y \texttt{)}\!</math>
|-
|-
| <math>F_{5}^{(2)}\!</math>
| <math>F_{5}^{(2)}\!</math>
| <math>F_{0101}^{(2)}\!</math>
| <math>F_{0101}^{(2)}~\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>(y)\!</math>
| <math>\texttt{(} y \texttt{)}\!</math>
|-
|-
| <math>F_{6}^{(2)}\!</math>
| <math>F_{6}^{(2)}\!</math>
| <math>F_{0110}^{(2)}\!</math>
| <math>F_{0110}^{(2)}~\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>(x, y)\!</math>
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math>
|-
|-
| <math>F_{7}^{(2)}\!</math>
| <math>F_{7}^{(2)}\!</math>
| <math>F_{0111}^{(2)}\!</math>
| <math>F_{0111}^{(2)}~\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>(x y)\!</math>
| <math>\texttt{(} x y \texttt{)}\!</math>
|-
|-
| <math>F_{8}^{(2)}\!</math>
| <math>F_{8}^{(2)}\!</math>
| <math>F_{1000}^{(2)}\!</math>
| <math>F_{1000}^{(2)}~\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>x y\!</math>
| <math>x y\!</math>
|-
|-
| <math>F_{9}^{(2)}\!</math>
| <math>F_{9}^{(2)}\!</math>
| <math>F_{1001}^{(2)}\!</math>
| <math>F_{1001}^{(2)}~\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>((x, y))\!</math>
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math>
|-
|-
| <math>F_{10}^{(2)}\!</math>
| <math>F_{10}^{(2)}\!</math>
| <math>F_{1010}^{(2)}\!</math>
| <math>F_{1010}^{(2)}~\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>y\!</math>
| <math>y\!</math>
|-
|-
| <math>F_{11}^{(2)}\!</math>
| <math>F_{11}^{(2)}\!</math>
| <math>F_{1011}^{(2)}\!</math>
| <math>F_{1011}^{(2)}~\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>(x (y))\!</math>
| <math>\texttt{(} x \texttt{(} y \texttt{))}\!</math>
|-
|-
| <math>F_{12}^{(2)}\!</math>
| <math>F_{12}^{(2)}\!</math>
| <math>F_{1100}^{(2)}\!</math>
| <math>F_{1100}^{(2)}~\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>x\!</math>
| <math>x\!</math>
|-
|-
| <math>F_{13}^{(2)}\!</math>
| <math>F_{13}^{(2)}\!</math>
| <math>F_{1101}^{(2)}\!</math>
| <math>F_{1101}^{(2)}~\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>((x)y)\!</math>
| <math>\texttt{((} x \texttt{)} y \texttt{)}\!</math>
|-
|-
| <math>F_{14}^{(2)}\!</math>
| <math>F_{14}^{(2)}\!</math>
| <math>F_{1110}^{(2)}\!</math>
| <math>F_{1110}^{(2)}~\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{0}</math>
| <math>\underline{0}\!</math>
| <math>((x)(y))\!</math>
| <math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math>
|-
|-
| <math>F_{15}^{(2)}\!</math>
| <math>F_{15}^{(2)}\!</math>
| <math>F_{1111}^{(2)}\!</math>
| <math>F_{1111}^{(2)}~\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>\underline{1}</math>
| <math>\underline{1}\!</math>
| <math>((~))</math>
| <math>\texttt{((~))}</math>
|}
|}
For example, suppose that <math>F\!</math> is a connection of the form <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> that is, any one of the sixteen possibilities in Table 18, while <math>p\!</math> and <math>q\!</math> are propositions of the form <math>p, q : X \to \underline\mathbb{B},</math> that is, propositions about things in the universe <math>X,\!</math> or else the indicators of sets contained in <math>X.\!</math>
For example, suppose that <math>F\!</math> is a connection of the form <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> that is, any one of the sixteen possibilities in Table 18, while <math>p\!</math> and <math>q\!</math> are propositions of the form <math>p, q : X \to \underline\mathbb{B},</math> that is, propositions about things in the universe <math>X,\!</math> or else the indicators of sets contained in <math>X.\!</math>
Then one has the imagination <math>\underline{f} = (f_1, f_2) = (p, q) : (X \to \underline\mathbb{B})^2,</math> and the stretch of the connection <math>F\!</math> to <math>\underline{f}</math> on <math>X\!</math> amounts to a proposition <math>F^\$ (p, q) : X \to \underline\mathbb{B}</math> that may be read as the ''stretch of <math>F\!</math> to <math>p\!</math> and <math>q.\!</math>'' If one is concerned with many different propositions about things in <math>X,\!</math> or if one is abstractly indifferent to the particular choices for <math>p\!</math> and <math>q,\!</math> then one may detach the operator <math>F^\$ : (X \to \underline\mathbb{B}))^2 \to (X \to \underline\mathbb{B})),</math> called the ''stretch of <math>F\!</math> over <math>X,\!</math>'' and consider it in isolation from any concrete application.
Then one has the imagination <math>\underline{f} = (f_1, f_2) = (p, q) : (X \to \underline\mathbb{B})^2,\!</math> and the stretch of the connection <math>F\!</math> to <math>\underline{f}\!</math> on <math>X\!</math> amounts to a proposition <math>F^\$ (p, q) : X \to \underline\mathbb{B}\!</math> that may be read as the ''stretch of <math>F\!</math> to <math>p\!</math> and <math>q.\!</math>'' If one is concerned with many different propositions about things in <math>X,\!</math> or if one is abstractly indifferent to the particular choices for <math>p\!</math> and <math>q,\!</math> then one may detach the operator <math>F^\$ : (X \to \underline\mathbb{B}))^2 \to (X \to \underline\mathbb{B})),\!</math> called the ''stretch of <math>F\!</math> over <math>X,\!</math>'' and consider it in isolation from any concrete application.
When the cactus notation is used to represent boolean functions, a single <math>\$</math> sign at the end of the expression is enough to remind the reader that the connections are meant to be stretched to several propositions on a universe <math>X.\!</math>
When the cactus notation is used to represent boolean functions, a single <math>\$</math> sign at the end of the expression is enough to remind the reader that the connections are meant to be stretched to several propositions on a universe <math>X.\!</math>
For example, take the connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> such that:
For example, take the connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> such that:
: <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \underline{(}~x~,~y~\underline{)}</math>
: <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \underline{(}~x~,~y~\underline{)}\!</math>
The connection in question is a boolean function on the variables <math>x, y\!</math> that returns a value of <math>\underline{1}</math> just when just one of the pair <math>x, y\!</math> is not equal to <math>\underline{1},</math> or what amounts to the same thing, just when just one of the pair <math>x, y\!</math> is equal to <math>\underline{1}.</math> There is clearly an isomorphism between this connection, viewed as an operation on the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \},</math> and the dyadic operation on binary values <math>x, y \in \mathbb{B} = \operatorname{GF}(2)</math> that is otherwise known as <math>x + y\!.</math>
The connection in question is a boolean function on the variables <math>x, y\!</math> that returns a value of <math>\underline{1}\!</math> just when just one of the pair <math>x, y\!</math> is not equal to <math>\underline{1},\!</math> or what amounts to the same thing, just when just one of the pair <math>x, y\!</math> is equal to <math>\underline{1}.\!</math> There is clearly an isomorphism between this connection, viewed as an operation on the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \},\!</math> and the dyadic operation on binary values <math>x, y \in \mathbb{B} = \operatorname{GF}(2)\!</math> that is otherwise known as <math>x + y.\!</math>
The same connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math> If <math>s\!</math> is a sentence that denotes the proposition <math>F,\!</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}.</math> In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set:
The same connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math> If <math>s\!</math> is a sentence that denotes the proposition <math>F,\!</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>{}^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, {}^{\prime\prime}.</math> In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|}
|}
Notice the distinction, that I continue to maintain at this point, between the logical values <math>\{ \operatorname{falsehood}, \operatorname{truth} \}</math> and the algebraic values <math>\{ 0, 1 \}.\!</math> This makes it legitimate to write a sentence directly into the righthand side of a set-builder expression, for instance, weaving the sentence <math>s\!</math> or the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}</math> into the context <math>^{\backprime\backprime} \, \{ (x, y) \in \underline{B}^2 : \ldots \} \, ^{\prime\prime},</math> thereby obtaining the corresponding expressions listed above. It also allows us to assert the proposition <math>F(x, y)\!</math> in a more direct way, without detouring through the equation <math>F(x, y) = \underline{1},</math> since it already has a value in <math>\{ \operatorname{falsehood}, \operatorname{true} \},</math> and thus can be taken as tantamount to an actual sentence.
Notice the distinction, that I continue to maintain at this point, between the logical values <math>\{ \operatorname{falsehood}, \operatorname{truth} \}</math> and the algebraic values <math>\{ 0, 1 \}.\!</math> This makes it legitimate to write a sentence directly into the righthand side of a set-builder expression, for instance, weaving the sentence <math>s\!</math> or the sentence <math>{}^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, {}^{\prime\prime}</math> into the context <math>{}^{\backprime\backprime} \, \{ (x, y) \in \underline{B}^2 : \ldots \} \, {}^{\prime\prime},</math> thereby obtaining the corresponding expressions listed above. It also allows us to assert the proposition <math>F(x, y)\!</math> in a more direct way, without detouring through the equation <math>F(x, y) = \underline{1},</math> since it already has a value in <math>\{ \operatorname{falsehood}, \operatorname{true} \},</math> and thus can be taken as tantamount to an actual sentence.
If the appropriate safeguards can be kept in mind, avoiding all danger of 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.
If the appropriate safeguards can be kept in mind, avoiding all danger of 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.
As a result, the application of the proposition <math>F^\$ (p, q)</math> to each <math>x \in X</math> returns a logical value in <math>\underline\mathbb{B},</math> all in accord with the following equations:
As a result, the application of the proposition <math>F^\$ (p, q)</math> to each <math>x \in X</math> returns a logical value in <math>\underline\mathbb{B},</math> all in accord with the following equations:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
|
|
<math>\begin{matrix}
<math>\begin{matrix}
F^\$ (p, q)(x) & = & \underline{(}~p~,~q~\underline{)}^\$ (x) & \in & \underline\mathbb{B}
F^\$ (p, q)(x) & = & \underline{(}~p~,~q~\underline{)}^\$ (x) & \in & \underline\mathbb{B}
\\
\\
\\
\\
\Updownarrow & & \Updownarrow
\Updownarrow & & \Updownarrow
\\
\\
\\
\\
F(p(x), q(x)) & = & \underline{(}~p(x)~,~q(x)~\underline{)} & \in & \underline\mathbb{B}
F(p(x), q(x)) & = & \underline{(}~p(x)~,~q(x)~\underline{)} & \in & \underline\mathbb{B}
\\
\\
\end{matrix}</math>
\end{matrix}</math>
|}
For each choice of propositions <math>p\!</math> and <math>q\!</math> about things in <math>X,\!</math> the stretch of <math>F\!</math> to <math>p\!</math> and <math>q\!</math> on <math>X\!</math> is just another proposition about things in <math>X,\!</math> a simple proposition in its own right, no matter how complex its current expression or its present construction as <math>F^\$ (p, q) = \underline{(}~p~,~q~\underline{)}^\$</math> makes it appear in relation to <math>p\!</math> and <math>q.\!</math> Like any other proposition about things in <math>X,\!</math> it indicates a subset of <math>X,\!</math> namely, the fiber that is variously described in the following ways:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lll}
[| F^\$ (p, q) |]
& = & [| \underline{(}~p~,~q~\underline{)}^\$ |]
\\[6pt]
& = & (F^\$ (p, q))^{-1} (\underline{1})
\\[6pt]
& = & \{~ x \in X ~:~ F^\$ (p, q)(x) ~\}
\\[6pt]
& = & \{~ x \in X ~:~ \underline{(}~p~,~q~\underline{)}^\$ (x) ~\}
\\[6pt]
& = & \{~ x \in X ~:~ \underline{(}~p(x)~,~q(x)~\underline{)} ~\}
\\[6pt]
& = & \{~ x \in X ~:~ p(x) + q(x) ~\}
\\[6pt]
& = & \{~ x \in X ~:~ p(x) \neq q(x) ~\}
\\[6pt]
& = & \{~ x \in X ~:~ \upharpoonleft P \upharpoonright (x) ~\neq~ \upharpoonleft Q \upharpoonright (x) ~\}
\\[6pt]
& = & \{~ x \in X ~:~ x \in P ~\nLeftrightarrow~ x \in Q ~\}
\\[6pt]
& = & \{~ x \in X ~:~ x \in P\!-\!Q ~\operatorname{or}~ x \in Q\!-\!P ~\}
\\[6pt]
& = & \{~ x \in X ~:~ x \in P\!-\!Q ~\cup~ Q\!-\!P ~\}
\\[6pt]
& = & \{~ x \in X ~:~ x \in P + Q ~\}
\\[6pt]
& = & P + Q ~\subseteq~ X
\\[6pt]
& = & [|p|] + [|q|] ~\subseteq~ X
\end{array}</math>
|}
|}
===2.4. Syntactic Transformations===
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. The notions of indication in question are expressed in a variety of different notations, enumerated as follows:
# The functional language of propositions
# The logical language of sentences
# The geometric language of sets
Thus, one way to explain the relationships that hold among these concepts is to describe the ''translations'' that are induced among their allied families of notation.
====2.4.1. Syntactic Transformation Rules====
====2.4.1. Syntactic Transformation Rules====
A good way to summarize these translations and to organize their use in practice is by means of the ''syntactic transformation rules'' (STRs) that partially formalize them. Rudimentary examples of STRs are readily mined from the raw materials that are already available in this area of discussion. To begin, let the definition of an indicator function be recorded in the following form:
A good way to summarize these translations and to organize their use in practice is by means of the ''syntactic transformation rules'' (STRs) that partially formalize them. Rudimentary examples of STRs are readily mined from the raw materials that are already available in this area of discussion. To begin, let the definition of an indicator function be recorded in the following form:
|- style="height:40px"
|- style="height:40px"
|
|
| <math>\text{such that:}\!</math>
| <math>\text{such that:}~\!</math>
|
|
|}
|}
| <math>\operatorname{R3c.}</math>
| <math>\operatorname{R3c.}</math>
| <math>\upharpoonleft Q \upharpoonright (x) ~=~ \underline{1}</math>
| <math>\upharpoonleft Q \upharpoonright (x) ~=~ \underline{1}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R3c~:~R2b}</math></p>
| style="border-left:1px solid black; text-align:center" | <p><math>\operatorname{R3c~:~R2b}</math></p>
|}
|}
|}
|}
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 hand margin of the Rule Box interweave the ''numerators'' and the ''denominators'' of the paradigm being employed, in other words, the alternating terms of comparison in a sequence of analogies. Taking the syntactic transformations marked in the Rule Box one at a time, each step is licensed by its formal analogy to a previously established rule.
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 hand margin of the Rule Box interweave the ''numerators'' and the ''denominators'' of the paradigm being employed, in other words, the alternating terms of comparison in a sequence of analogies. Taking the syntactic transformations marked in the Rule Box one at a time, each step is licensed by its formal analogy to a previously established rule.
For example, the annnotation <math>X_1 : A_1 :: X_2 : A_2\!</math> may be read to say that <math>X_1\!</math> is to <math>A_1\!</math> as <math>X_2\!</math> is to <math>A_2,\!</math> where the step from <math>A_1\!</math> to <math>A_2\!</math> is permitted by a previously accepted rule.
For example, the annotation <math>X_1 : A_1 :: X_2 : A_2\!</math> may be read to say that <math>X_1\!</math> is to <math>A_1\!</math> as <math>X_2\!</math> is to <math>A_2,\!</math> where the step from <math>A_1\!</math> to <math>A_2\!</math> is permitted by a previously accepted rule.
This can be illustrated by considering the derivation of Rule 3 in the augmented form that follows:
This can be illustrated by considering the derivation of Rule 3 in the augmented form that follows:
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 stating their equality in the appropriate form of equation.
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 stating their equality in the appropriate form of equation.
For example, extracting the expressions <math>\text{R3a}\!</math> and <math>\text{R3c}\!</math> that are given as equivalents in Rule 3 and explicitly stating their equivalence produces the equation recorded in Corollary 1.
For example, extracting the expressions <math>\text{R3a}~\!</math> and <math>\text{R3c}~\!</math> that are given as equivalents in Rule 3 and explicitly stating their equivalence produces the equation recorded in Corollary 1.
<br>
<br>
<br>
<br>
The first and last items on this list, namely, the sentence <math>\text{R4a}\!</math> stating <math>x \in Q</math> and the sentence <math>\text{R4e}\!</math> stating <math>\upharpoonleft Q \upharpoonright (x) = \underline{1},</math> are just the pair of sentences from Rule 3 whose equivalence for all <math>x \in X</math> is usually taken to define the idea of an indicator function <math>\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}.</math> 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 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 example, the expression <math>^{\backprime\backprime} \downharpoonleft x \in Q \downharpoonright \, ^{\prime\prime}</math> 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 at most be a sign of one.
The first and last items on this list, namely, the sentence <math>\text{R4a}\!</math> stating <math>x \in Q</math> and the sentence <math>\text{R4e}\!</math> stating <math>\upharpoonleft Q \upharpoonright (x) = \underline{1},</math> are just the pair of sentences from Rule 3 whose equivalence for all <math>x \in X</math> is usually taken to define the idea of an indicator function <math>\upharpoonleft Q \upharpoonright ~:~ X \to \underline\mathbb{B}.</math> 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 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 example, the expression <math>{}^{\backprime\backprime} \downharpoonleft x \in Q \downharpoonright \, {}^{\prime\prime}</math> 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 at most be a sign of one.
The use of the basic logical connectives can be expressed in the form of an STR as follows:
The use of the basic logical connectives can be expressed in the form of an STR as follows:
|- style="height:48px"
|- style="height:48px"
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| <math>\text{such that:}\!</math>
| <math>\text{such that:}~\!</math>
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|- style="height:48px"
|- style="height:48px"
<br>
<br>
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.
As a general rule, the application of an 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.
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.
|- style="height:40px; text-align:center"
|- style="height:40px; text-align:center"
| width="80%" |
| width="80%" |
| width="20%" | <math>\operatorname{Definition~7}</math>
| width="20%" | <math>\operatorname{Definition~7}\!</math>
|}
|}
|-
|-
| width="2%" style="border-top:1px solid black" |
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="80%" style="border-top:1px solid black" | <math>s, t ~\text{are sentences about things in the universe}~ X</math>
| width="80%" style="border-top:1px solid black" | <math>s, t ~\text{are sentences about things in the universe}~ X\!</math>
|- style="height:40px"
|- style="height:40px"
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|
|- style="height:40px; text-align:center"
|- style="height:40px; text-align:center"
| width="80%" |
| width="80%" |
| width="20%" style="border-left:1px solid black" | <math>\operatorname{Rule~7}</math>
| width="20%" style="border-left:1px solid black" | <math>\operatorname{Rule~7}\!</math>
|}
|}
|-
|-
| <math>\operatorname{R8f.}</math>
| <math>\operatorname{R8f.}</math>
| <math>\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft s \downharpoonright (x) ~,~ \downharpoonleft t \downharpoonright (x) ~\underline{))}</math>
| <math>\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft s \downharpoonright (x) ~,~ \downharpoonleft t \downharpoonright (x) ~\underline{))}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R8f~:~R7e}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R8f~:~R7e}\!</math>
|- style="height:20px"
|- style="height:20px"
|
|
|
|
| <math>\operatorname{R10c.}</math>
| <math>\operatorname{R10c.}</math>
| <math>\downharpoonleft x \in P \downharpoonright ~=~ \downharpoonleft x \in Q \downharpoonright</math>
| <math>\downharpoonleft x \in P \downharpoonright ~=~ \downharpoonleft x \in Q \downharpoonright\!</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10c~:~R8b}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10c~:~R8b}\!</math>
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|
|- style="height:40px"
|- style="height:40px"
|
|
| <math>\operatorname{R11f.}</math>
| <math>\operatorname{R11f.}\!</math>
| <math>\upharpoonleft Q \upharpoonright ~=~ \downharpoonleft s \downharpoonright</math>
| <math>\upharpoonleft Q \upharpoonright ~=~ \downharpoonleft s \downharpoonright\!</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R11f~:~\_\_?\_\_}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R11f~:~\_\_?\_\_}\!</math>
|}
|}
|}
|}
| style="width:2%; border-top:1px solid black" |
| style="width:2%; border-top:1px solid black" |
| style="width:14%; border-top:1px solid black" | <math>\operatorname{F1a.}</math>
| style="width:14%; border-top:1px solid black" | <math>\operatorname{F1a.}</math>
| style="width:64%; border-top:1px solid black" | <math>s \quad \Leftrightarrow \quad (P ~=~ Q)</math>
| style="width:64%; border-top:1px solid black" | <math>s \quad \Leftrightarrow \quad (P ~=~ Q)\!</math>
| style="width:20%; border-top:1px solid black; border-left:1px solid black; text-align:center" |
| style="width:20%; border-top:1px solid black; border-left:1px solid black; text-align:center" |
<math>\operatorname{F1a~:~R9a}</math>
<math>\operatorname{F1a~:~R9a}\!</math>
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<br>
<br>
====2.4.2. Derived Equivalence Relations====
====2.4.2. 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.
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.
|
|
| <math>\operatorname{D8e.}</math>
| <math>\operatorname{D8e.}</math>
| <math>\{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\operatorname{for~some}~ o \in O \}</math>
| <math>\{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\operatorname{for~some}~ o \in O \}\!</math>
|}
|}
|}
|}
<br>
<br>
Recall the definition of <math>\operatorname{Den} (L),</math> the denotative component of <math>L,\!</math> in the following form:
Recall the definition of <math>\operatorname{Den}(L),</math> the denotative component of <math>L,\!</math> in the following form:
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{| align="center" cellpadding="8" width="90%"
| <math>\operatorname{Den} (L) ~=~ L_{OS} ~=~ \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}.</math>
| <math>\operatorname{Den}(L) ~=~ L_{OS} ~=~ \{ (o, s) \in O \times S ~:~ (o, s, i) \in L ~\text{for some}~ i \in I \}.</math>
|}
|}
<br>
<br>
The dyadic relation <math>L_{SO}\!</math> that is the converse of the denotative relation <math>L_{OS}\!</math> can be defined directly in the following fashion:
The dyadic relation <math>{L_{SO}}\!</math> that is the converse of the denotative relation <math>L_{OS}\!</math> can be defined directly in the following fashion:
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
| width="2%" style="border-top:1px solid black" |
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D11a.}</math>
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D11a.}</math>
| width="80%" style="border-top:1px solid black" | <math>L_{SO}\!</math>
| width="80%" style="border-top:1px solid black" | <math>{L_{SO}}\!</math>
|- style="height:50px"
|- style="height:50px"
|
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| <math>\operatorname{D11b.}</math>
| <math>\operatorname{D11b.}</math>
| <math>\overset{\smile}{L_{OS}}</math>
| <math>\overset{\smile}{L_{OS}}\!</math>
|- style="height:50px"
|- style="height:50px"
|
|
| <math>\operatorname{D11c.}</math>
| <math>\operatorname{D11c.}</math>
| <math>\overset{\smile}{\operatorname{Den}^L}</math>
| <math>\overset{\smile}{\operatorname{Den}^L}\!</math>
|- style="height:50px"
|- style="height:50px"
|
|
| <math>\operatorname{D11d.}</math>
| <math>\operatorname{D11d.}</math>
| <math>\overset{\smile}{\operatorname{Den}(L)}</math>
| <math>\overset{\smile}{\operatorname{Den}(L)}\!</math>
|- style="height:50px"
|- style="height:50px"
|
|
| <math>\operatorname{D11e.}</math>
| <math>\operatorname{D11e.}</math>
| <math>\operatorname{proj}_{SO}(L)</math>
| <math>\operatorname{proj}_{SO}(L)\!</math>
|- style="height:50px"
|- style="height:50px"
|
|
| <math>\operatorname{D11f.}</math>
| <math>\operatorname{D11f.}</math>
| <math>\operatorname{Conv}(\operatorname{Den}(L))</math>
| <math>\operatorname{Conv}(\operatorname{Den}(L))\!</math>
|- style="height:50px"
|- style="height:50px"
|
|
| <math>\operatorname{D11g.}</math>
| <math>\operatorname{D11g.}</math>
| <math>\{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\operatorname{for~some}~ i \in I \}</math>
| <math>\{ (s, o) \in S \times O ~:~ (o, s, i) \in L ~\operatorname{for~some}~ i \in I \}\!</math>
|}
|}
|}
|}
| width="2%" style="border-top:1px solid black" |
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D12a.}</math>
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D12a.}</math>
| width="80%" style="border-top:1px solid black" | <math>L_{OS} \cdot x</math>
| width="80%" style="border-top:1px solid black" | <math>L_{OS} \cdot x\!</math>
|- style="height:40px"
|- style="height:40px"
|
|
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.
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.
To define the ''equiference'' of signs in terms of their denotations, one says that ''<math>x\!</math> is equiferent to <math>y\!</math> under <math>L,\!</math>'' and writes <math>x ~\overset{L}{=}~ y,\!</math> to mean that <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y).</math> Taken in extension, this notion of a relation between signs induces an ''equiference relation'' on the syntactic domain.
To define the ''equiference'' of signs in terms of their denotations, one says that ''<math>x\!</math> is equiferent to <math>y\!</math> under <math>L,\!</math>'' and writes <math>{x ~\overset{L}{=}~ y},\!</math> to mean that <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y).\!</math> Taken in extension, this notion of a relation between signs induces an ''equiference relation'' on the syntactic domain.
For each sign relation <math>L,\!</math> this yields a binary relation <math>\operatorname{Der}(L) \subseteq S \times I</math> that is defined as follows:
For each sign relation <math>L,\!</math> this yields a binary relation <math>\operatorname{Der}(L) \subseteq S \times I</math> that is defined as follows:
<p>'''Transitive property.'''</p>
<p>'''Transitive property.'''</p>
<p>Does <math>x ~\overset{L}{=}~ y</math> and <math>y ~\overset{L}{=}~ z</math> imply <math>x ~\overset{L}{=}~ z</math> for all <math>x, y, z \in S</math>?</p>
<p>Does <math>{x ~\overset{L}{=}~ y}\!</math> and <math>y ~\overset{L}{=}~ z</math> imply <math>{x ~\overset{L}{=}~ z}\!</math> for all <math>x, y, z \in S\!</math>?</p>
<p>To belabor the point, does <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y)</math> and <math>\operatorname{Den}(L, y) = \operatorname{Den}(L, z)</math> imply <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, z)</math> for all <math>x, y, z \in S</math>?</p>
<p>To belabor the point, does <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y)\!</math> and <math>\operatorname{Den}(L, y) = \operatorname{Den}(L, z)\!</math> imply <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, z)\!</math> for all <math>x, y, z \in S\!</math>?</p>
<p>Yes, once again, under the stated conditions.</p></li>
<p>Yes, once again, under the stated conditions.</p></li>
|- style="height:60px"
|- style="height:60px"
|
|
| valign="top" | <math>\operatorname{F2.1f.}</math>
| valign="top" | <math>\operatorname{F2.1f.\!}</math>
| valign="top" |
| valign="top" |
<math>\begin{array}{ll}
<math>\begin{array}{ll}
\end{array}</math>
\end{array}</math>
| style="border-left:1px solid black; text-align:center" |
| style="border-left:1px solid black; text-align:center" |
<math>\operatorname{F2.2c~:~R11c}</math></p>
<p><math>\operatorname{F2.2c~:~R11c}</math></p>
|- style="height:20px"
|- style="height:20px"
| colspan="3" |
| colspan="3" |
& & & \\
& & & \\
\end{array}</math>
\end{array}</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{F2.3c~:~Log}</math></p>
| style="border-left:1px solid black; text-align:center" | <p><math>\operatorname{F2.3c~:~Log}</math></p>
|- style="height:20px"
|- style="height:20px"
| colspan="3" |
| colspan="3" |
<br>
<br>
====2.4.3. Digression on Derived Relations====
====2.4.3. Digression on Derived Relations====
A better understanding of derived equivalence relations (DERs) 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 <math>L\!</math> into a dyadic relation <math>\operatorname{Der}(L),</math> 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.
A better understanding of derived equivalence relations (DERs) 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 <math>L\!</math> into a dyadic relation <math>\operatorname{Der}(L),</math> 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.
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<div align="center" ><big>
<div align="center">
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Contents]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Contents]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1|Part 1]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1|Part 1]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2|Part 2]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 2|Part 2]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 3|Part 3]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 3|Part 3]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 4|Part 4]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 5|Part 5]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6|Part 6]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 7|Part 7]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 8|Part 8]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Appendices|Appendices]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : References|References]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Document History|Document History]]
• [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Document History|Document History]]
•
•
</div>
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[[Category:Artificial Intelligence]]
[[Category:Artificial Intelligence]]
