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{{DISPLAYTITLE:Inquiry Driven Systems : Part 2}}

{{DISPLAYTITLE:Inquiry Driven Systems : Part 2}}

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<div class="nonumtoc">__TOC__</div>

====1.3.9.  Reconnaissance====

==2.==

 

===2.1.  Reconnaissance===

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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.

=====1.3.9.1.  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 &mdash; those between the formal arenas, bowers, courts and the informal context that surrounds them all &mdash; 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 &mdash; those between the formal arenas, bowers, courts and the informal context that surrounds them all &mdash; 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.

=====1.3.9.2.  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.

=====1.3.9.3.  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.

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====1.3.10.  Recurring Themes====

===2.2.  Recurring Themes===

The overall purpose of the next several Sections is threefold:

The overall purpose of the next several Sections is threefold:

# 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.

# 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.

=====1.3.10.1.  Preliminary Notions=====

====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 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.

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.

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.

=====1.3.10.2.  Intermediary Notions=====

====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.

A number of additional definitions are relevant to sign relations whose connotative components constitute equivalence relations, if only in part.

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.

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.

=====1.3.10.3.  Propositions and Sentences=====

====2.2.3.  Propositions and Sentences====

The concept of a sign relation is typically extended as a set <math>\mathcal{L} \subseteq \mathcal{O} \times \mathcal{S} \times \mathcal{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 that it is likely to be encountered in, a good amount of set-theoretic machinery is necessary to carry out a reasonably detailed analysis of sign relations in general.

The concept of a sign relation is typically extended as a set <math>\mathcal{L} \subseteq \mathcal{O} \times \mathcal{S} \times \mathcal{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 that it is likely to be encountered in, a good amount of set-theoretic machinery is necessary to carry out a reasonably detailed analysis of sign relations in general.

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.

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.

=====1.3.10.4.  Empirical Types and Rational Types=====

====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.

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.

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.

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.

=====1.3.10.5.  Articulate Sentences=====

====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.

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 ''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''.

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''.

=====1.3.10.6.  Stretching Principles=====

====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:

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:

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.

=====1.3.10.7.  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.

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>

====1.3.11.  The Cactus Patch====

===2.3.  The Cactus Patch===

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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.

=====1.3.11.1.  The Cactus Language : Syntax=====

====2.3.1.  The Cactus Language : Syntax====

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=====1.3.11.2.  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&ndash;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&ndash;61).

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>

=====1.3.11.3.  The Cactus Language : Stylistics=====

====2.3.3.  The Cactus Language : Stylistics====

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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.

=====1.3.11.4.  The Cactus Language : Mechanics=====

====2.3.4.  The Cactus Language : Mechanics====

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</ol>

</ol>

=====1.3.11.5.  The Cactus Language : Semantics=====

====2.3.5.  The Cactus Language : Semantics====

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Let me now address one last question that may have occurred to some.  What has happened, in this suggested scheme of functional reasoning, to the distinction that is quite pointedly made by careful logicians between (1) the connectives called ''conditionals'' and symbolized by the signs <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called ''implications'' and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, and, in a related question:  What has happened to the distinction that is equally insistently made between (3) the connective called the ''biconditional'' and signified by the sign <math>(\leftrightarrow)</math> and (4) the assertion that is called an ''equivalence'' and signified by the sign <math>(\Leftrightarrow)</math>?  My answer is this:  For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.

Let me now address one last question that may have occurred to some.  What has happened, in this suggested scheme of functional reasoning, to the distinction that is quite pointedly made by careful logicians between (1) the connectives called ''conditionals'' and symbolized by the signs <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called ''implications'' and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, and, in a related question:  What has happened to the distinction that is equally insistently made between (3) the connective called the ''biconditional'' and signified by the sign <math>(\leftrightarrow)</math> and (4) the assertion that is called an ''equivalence'' and signified by the sign <math>(\Leftrightarrow)</math>?  My answer is this:  For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.

=====1.3.11.6. Stretching Exercises=====

====2.3.6. Stretching Exercises====

Taking up the preceding arrays of particular connections, namely, the boolean functions on up to two variables, <math>F^{(k)} : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>k\!</math> in <math>\{ 0, 1, 2 \},\!</math> it is possible to illustrate the use of the stretch operation in a variety of concrete cases.

Taking up the preceding arrays of particular connections, namely, the boolean functions on up to two variables, <math>F^{(k)} : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>k\!</math> in <math>\{ 0, 1, 2 \},\!</math> it is possible to illustrate the use of the stretch operation in a variety of concrete cases.

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====1.3.12.  Syntactic Transformations====

===2.4.  Syntactic Transformations===

To discuss the import of the above definitions in greater depth, it serves to establish a number of logical relations and set-theoretic identities that can be found to hold among this array of conceptions and constructions.  Facilitating this task requires in turn a number of auxiliary concepts and notations.

To discuss the import of the above definitions in greater depth, it serves to establish a number of logical relations and set-theoretic identities that can be found to hold among this array of conceptions and constructions.  Facilitating this task requires in turn a number of auxiliary concepts and notations.

The diverse notions of ''indication'' under discussion are expressed in a variety of different notations, in particular, the logical language of sentences, the functional language of propositions, and the geometric language of sets.  Thus, one way to explain the relationships that exist among these concepts is to describe the ''translations'' that they induce among the allied families of notation.

The diverse notions of ''indication'' under discussion are expressed in a variety of different notations, in particular, the logical language of sentences, the functional language of propositions, and the geometric language of sets.  Thus, one way to explain the relationships that exist among these concepts is to describe the ''translations'' that they induce among the allied families of notation.

=====1.3.12.1.  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.  A rudimentary example of a STR is 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.  A rudimentary example of a STR is 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:

<br>

<br>

=====1.3.12.2.  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.

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=====1.3.12.3.  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.