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There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many parallels and coincidences that it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, "thinking" in its best sense, "is a term denoting the various ways in which things acquire significance" (Dewey).  Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation.  Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view.

There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many parallels and coincidences that it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, "thinking" in its best sense, "is a term denoting the various ways in which things acquire significance" (Dewey).  Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation.  Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view.

Because the examples in this section have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations.  Still, these examples have subtleties of their own, and their careful treatment will serve to illustrate important issues in the general theory of signs.

Because the examples in this section have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations.  Still, these examples have subtleties of their own, and their careful treatment will serve to illustrate important issues in the general theory of signs.

Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns:  "Ann", "Bob", "I", "you".

Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns:  "Ann", "Bob", "I", "you".

The ''object domain'' of this discussion fragment is the set of two people {Ann, Bob}.

:* The ''object domain'' of this discussion fragment is the set of two people {Ann, Bob}.

The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs {"Ann", "Bob", "I", "You"}.

:* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs {"Ann", "Bob", "I", "You"}.

In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use.  The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual three-place relation called the ''[[sign relation]]'' of that interpreter.

In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use.  The ''system of interpretation'' (SOI) associated with each language user can be represented in the form of an individual [[triadic relation|three-place relation]] called the ''[[sign relation]]'' of that interpreter.

Understood in terms of its set-theoretic extension, a sign relation ''L'' is a subset of a cartesian product ''O''×''S''×''I''.  Here, ''O'', ''S'', and ''I'' are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation ''L'' ⊆ ''O''×''S''×''I''.  In general, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations that are contemplated in a computational framework are usually constrained to having ''I'' ⊆ ''S''.  In this case, interpretants are just a special variety of signs, and this makes it convenient to lump signs and interpretants together into a ''syntactic domain''.  In the forthcoming examples, ''S'' and ''I'' are identical as sets, so the very same elements manifest themselves in two distinct roles of the sign relations in question.  When it is necessary to refer to the whole set of objects and signs in the union of the domains ''O'', ''S'', and ''I'' for a given sign relation ''L'', one may call this the "world of ''L''" and write ''W'' = ''W''(''L'') = ''O'' ∪ ''S'' ∪ ''I''.

Understood in terms of its ''[[set theory|set-theoretic]] [[extension (logic)|extension]]'', a sign relation '''L''' is a ''[[subset]]'' of a ''[[cartesian product]]'' '''O''' × '''S''' × '''I'''.  Here, '''O''', '''S''', '''I''' are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation '''L''' ⊆ '''O''' × '''S''' × '''I'''.

To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible when the examples become more complicated, I introduce the following abbreviations:

In general, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations that are contemplated in a computational framework are usually constrained to having '''I''' ⊆ '''S'''.  In this case, interpretants are just a special variety of signs, and this makes it convenient to lump signs and interpretants together into a single class called the ''syntactic domain''.  In the forthcoming examples, '''S''' and '''I''' are identical as sets, so the very same elements manifest themselves in two different roles of the sign relations in question.  When it is necessary to refer to the whole set of objects and signs in the union of the domains '''O''', '''S''', '''I''' for a given sign relation '''L''', one may refer to this set as the ''world of '''L''''' and write '''W''' = '''W'''_'''L''' = '''O''' ∪ '''S''' ∪ '''I'''.

To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations:

O = Object Domain,

S = Sign Domain,

O = { Ann, Bob } = { A, B }.

| align="center" | '''O''' || = || Object Domain

| align="center" | '''I''' || = || Interpretant Domain

|}

S = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}.

Introducing a few abbreviations for use in considering the present Example, we have the following data:

I = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}.

| {"Ann", "Bob", "I", "You"}

| =

| {"Ann", "Bob", "I", "You"}

| =

| {"A", "B", "i", "u"}

In the present examples, ''S'' = ''I'' = Syntactic Domain.

In the present Example, '''S''' = '''I''' = Syntactic Domain.

Tables 1 and 2 give the sign relations associated with the interpreters ''A'' and ''B'', respectively, putting them in the form of relational databases.  Thus, the rows of each Table list the ordered triples of the form ‹''o'', ''s'', ''i''› that make up the corresponding sign relations:  ''A'', ''B'' ⊆ ''O''×''S''×''I''.  The issue of using the same names for objects and for relations involving these objects will be taken up later, after the less problematic features of these relations have been treated.

Tables 1 and 2 give the sign relations associated with the interpreters ''A'' and ''B'', respectively, putting them in the form of ''[[relational database]]s''.  Thus, the rows of each Table list the ordered triples of the form ‹''o'', ''s'', ''i''› that make up the corresponding sign relations, '''L'''_A and '''L'''_B ⊆ '''O''' × '''S''' × '''I'''.  It is often tempting to use the same names for objects and for relations involving these objects, but it is best to avoid this in a first approach, taking up the issues that this practice raises after the less problematic features of these relations have been treated.

These Tables codify a rudimentary level of interpretive practice for the agents ''A'' and ''B'', and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain.  Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form ‹''o'', ''s'', ''i''› that is called an ''elementary relation'', that is, one element of the relation's set-theoretic extension.

|+ '''L'''_A = Sign Relation of Interpreter A

| '''A''' || '''"A"''' || '''"i"'''

| '''B''' || '''"B"''' || '''"B"'''

|-

| '''B''' || '''"u"''' || '''"B"'''

|-

Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semantics''.  In the process of discussing these alternatives, I will introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.

|+ '''L'''_B = Sign Relation of Interpreter B

| '''B''' || '''"i"''' || '''"B"'''

These Tables codify a rudimentary level of interpretive practice for the agents ''A'' and ''B'', and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain.  Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form ‹''o'', ''s'', ''i''› that is called an ''elementary relation'', that is, one element of the relation's set-theoretic extension.

Table 1.  Sign Relation of Interpreter A

A "A" "i"

Already in this elementary context, there are several different meanings that might attach to the project of a ''formal semiotics'', or a formal theory of meaning for signs.  In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.

Table 2.  Sign Relation of Interpreter B

One aspect of semantics is concerned with the reference that a sign has to its object, which is called its ''denotation''.  For signs in general, neither the existence nor the uniqueness of a denotation is guaranteed.  Thus, the denotation of a sign can refer to a plural, a singular, or a vacuous number of objects.  In the pragmatic theory of signs, these references are formalized as certain types of dyadic relations that are obtained by projection from the triadic sign relations.

One aspect of semantics is concerned with the reference that a sign has to its object, which is called its ''denotation''.  For signs in general, neither the existence nor the uniqueness of a denotation is guaranteed.  Thus, the denotation of a sign can refer to a plural, a singular, or a vacuous number of objects.  In the pragmatic theory of signs, these references are formalized as certain types of dyadic relations that are obtained by projection from the triadic sign relations.