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===6.29. Projects of Representation===

===6.29. Projects of Representation===

'''Note.'''  This section is very rough and will need to be rewritten even more than most.

'''Note.'''  This section is very rough and will need to be rewritten.

There are numerous modalities of description and representation that are involved in linking the extensions and intensions of terms and concepts.  To facilitate the building of a suitable analytic and synthetic framework for this task, and to abbreviate future references to the categories of modalities that come into play, I will employ a set of technical notions, along with their aliases and acronyms, to be indicated next.

There are numerous modalities of description and representation that are involved in linking the extensions and intensions of terms and concepts.  To facilitate the building of a suitable analytic and synthetic framework for this task, and to abbreviate future references to the categories of modalities that come into play, I will employ a set of technical notions, along with their aliases and acronyms, to be indicated next.

For the rest of this section I restrict the discussion to sign relations of the type <math>L \subseteq O \times S \times I\!</math> and elementary sign relations of the form <math>(o, s, i) \in L.\!</math>

For the rest of this section I restrict the discussion to sign relations of the type <math>L \subseteq O \times S \times I\!</math> and elementary sign relations of the form <math>(o, s, i) \in L.\!</math>

A ''discussion of concrete examples'', intended to serve as a preparatory treatment for approaching a significantly more complex area, is necessarily limited in its focus to isolated cases, in effect, to those that remain simple enough to be instructive in a preliminary approach to the topic.  This means that the observable properties of the initial examples, with respect to the class they are aimed to exemplify, will sort themselves into two kinds:  (1) their essential, generic, or genuine properties, and (2) their accidental, factitious, or spurious properties.

A "discussion of concrete examples" (DOCE), intended to serve as a preparatory treatment for approaching a significantly more complex area, is necessarily limited in its focus to isolated cases, in effect, to those that remain simple enough to be instructive in a preliminary approach to the topic.  This means that the observable properties of the initial examples, with respect to the class they are aimed to exemplify, will sort themselves into two kinds:  (1) their essential, generic, or genuine properties, and (2) their accidental, factitious, or spurious properties.

But the present discussion of sign relations cannot illustrate the properties of even these elementary examples in an adequate way without considering extended multitudes of other relations, both those that share the same properties and those that do not.  Consequently, by way of getting the comparative study of sign relations started on a casual basis, an end that is served in addition by placing sign relations within the broader setting of n place relations, I will exploit a few devices of taxonomic nomenclature, intending them to be applied for the moment in a purely informal way.

But the present discussion of sign relations cannot illustrate the properties of even these elementary examples in an adequate way without considering extended multitudes of other relations, both those that share the same properties and those that do not.  Consequently, by way of getting the comparative study of sign relations started on a casual basis, an end that is served in addition by placing sign relations within the broader setting of <math>n\!</math>-place relations, I will exploit a few devices of taxonomic nomenclature, intending them to be applied for the moment in a purely informal way.

An "order", "genus", or "species" of relations is a class or set of relations that obey a particular collection of axioms, or that satisfy a certain combination of operational constraints and axiomatic properties.  These respective terms, given in order of increasing specificity, are not intended to be applied too systematically, but only roughly to indicate how many axioms are listed in the specification of a class of relations and thus how narrowly the indicated class is pinned down relative to other classes within the context of a particular discussion.

An ''order'', ''genus'', or ''species'' of relations is a class or set of relations that obey a particular collection of axioms, or that satisfy a certain combination of operational constraints and axiomatic properties.  These respective terms, given in order of increasing specificity, are not intended to be applied too systematically, but only roughly to indicate how many axioms are listed in the specification of a class of relations and thus how narrowly the indicated class is pinned down relative to other classes within the context of a particular discussion.

For example, this terminology allows me to indicate a "general order" G of sign relations R, each of whose connotative components RSI is an equivalence relation, and then it allows me to extend this investigation by pursuing the prospective existence of a "generalized order" G' of sign relations R, each of which has many properties analogous to the sign relations in G, with the exception or extension that G' is more broadly formulated in certain designated respects.

For example, this terminology allows me to indicate a ''general order'' <math>G\!</math> of sign relations <math>L,\!</math> each of whose connotative components <math>L_{SI}\!</math> is an equivalence relation, and then it allows me to extend this investigation by pursuing the prospective existence of a generalized order <math>G'\!</math> of sign relations <math>L,\!</math> each of which has many properties analogous to the sign relations in <math>G,\!</math> with the exception or extension that <math>G'\!</math> is more broadly formulated in certain designated respects.

The purpose of these informal taxonomic distinctions is not to specify absolute levels of generality, something that could not be achieved in a global manner without splitting hierarchies of hereditary properties and the whole host of their successive heirs down to the ultimate pedigrees, but merely to organize properties relative to each other in comparative terms, in which case three levels of generality are usually enough to orient oneself locally in any ontology, no matter how wide or deep.  Thus, the main interest that the terms "order", "genus", and "species" will subserve in this connection is to indicate the taxonomic directions of generalization and specialization that a particular investigation is trying to achieve among classes of relations:  "generalizing" a class by abstracting features or removing constraints from its original definition, and "specializing" a class by concretizing features or adding constraints to its initial characterization.

The purpose of these informal taxonomic distinctions is not to specify absolute levels of generality, something that could not be achieved in a global manner without splitting hierarchies of hereditary properties and the whole host of their successive heirs down to the ultimate pedigrees, but merely to organize properties relative to each other in comparative terms, in which case three levels of generality are usually enough to orient oneself locally in any ontology, no matter how wide or deep.  Thus, the main interest that the terms ''order'', ''genus'', and ''species'' will subserve in this connection is to indicate the taxonomic directions of generalization and specialization that a particular investigation is trying to achieve among classes of relations:  ''generalizing'' a class by abstracting features or removing constraints from its original definition, and ''specializing'' a class by concretizing features or adding constraints to its initial characterization.

In order to talk and think about any sign relation at all, not to mention addressing the topic of a "generic order" (GO) of sign relations, one has to use signs to do it, and this requires one's taking part in what can be called a "higher order" (HO) of sign relations.  By way of definition, a sign relation is a HO sign relation if some of its signs refer to objects that are themselves sign relations or classes of sign relations.

In order to talk and think about any sign relation at all, not to mention addressing the topic of a ''generic order'' of sign relations, one has to use signs to do it, and this requires one's taking part in what can be called a ''higher order'' of sign relations.  By way of definition, a sign relation is a ''higher order sign relation'' if some of its signs refer to objects that are themselves sign relations or classes of sign relations.

So long as one expects to deal with only a few sign relations at a time, managing to use only a few conventional names to denote each of them, then one's concomitant participation in a HO sign relation hardly ever becomes too problematic, and it rarely needs to be formalized in order for one to cope with the duties of serving as its unofficial interpreter.  Once a reflective involvement with HO sign relations gets started, however, there will be difficulties that continue to lurk and grow just beneath the apparently conversant surface of their all too facile fluency.

So long as one expects to deal with only a few sign relations at a time, managing to use only a few conventional names to denote each of them, then one's participation in a higher order sign relation hardly ever becomes too problematic, and it rarely needs to be formalized in order for one to cope with the duties of serving as its unofficial interpreter.  Once a reflective involvement with higher order sign relations gets started, however, there will be difficulties that continue to grow and lurk just beneath the apparently conversant surface of their all too facile fluency.

By way of example, a singular sign that denotes an entire sign relation refers by extension to a class of elementary sign relations, or a set of "transaction triples" <o, s, i>.  So far, this is still not too much of a problem.  But when one begins to develop large numbers of conventional symbols and complicated formulas for referring to the same classes of sign transactions, then considerations of effective and efficient interpretation will demand that these symbols and formulas be organized into semantic equivalence classes with recognizable characters.  That is, one is forced to find computable types of similarity relations defined on pairs of symbols and formulas that tell whether they refer to the same class of sign transactions or not.  It is almost inevitable in such a situation that canonical representatives of these equivalence classes will have to be developed, and a means for transforming arbitrarily complex and obscure expressions into optimally simple and clear equivalents will also become necessary.

By way of example, a singular sign that denotes an entire sign relation refers by extension to a class of elementary sign relations, or a set of "transaction triples" <o, s, i>.  So far, this is still not too much of a problem.  But when one begins to develop large numbers of conventional symbols and complicated formulas for referring to the same classes of sign transactions, then considerations of effective and efficient interpretation will demand that these symbols and formulas be organized into semantic equivalence classes with recognizable characters.  That is, one is forced to find computable types of similarity relations defined on pairs of symbols and formulas that tell whether they refer to the same class of sign transactions or not.  It is almost inevitable in such a situation that canonical representatives of these equivalence classes will have to be developed, and a means for transforming arbitrarily complex and obscure expressions into optimally simple and clear equivalents will also become necessary.