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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 <math>\{ \text{Ann}, \text{Bob} \}</math>.

:* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}\!</math>.

:* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}</math>.

:* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}\!</math>.

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.

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 theory|set-theoretic]] [[extension (logic)|extension]]'', a sign relation <math>L\!</math> is a ''[[subset]]'' of a ''[[cartesian product]]'' <math>O \times S \times I</math>.  Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I</math>.

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

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 <math>I \subseteq S</math>.  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, <math>S\!</math> and <math>I\!</math> 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 <math>O\!</math>, <math>S\!</math>, <math>I\!</math> for a given sign relation <math>L\!</math>, one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I</math>.

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 <math>I \subseteq S\!</math>.  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, <math>S\!</math> and <math>I\!</math> 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 <math>O\!</math>, <math>S\!</math>, <math>I\!</math> for a given sign relation <math>L\!</math>, one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I\!</math>.

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:

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:

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In the present example, <math>S = I = \text{Syntactic Domain}</math>.

In the present example, <math>S = I = \text{Syntactic Domain}\!</math>.

The sign relation associated with a given interpreter <math>J\!</math> is denoted <math>L_J</math> or <math>L(J)</math>.  Tables&nbsp;1 and 2 give the sign relations associated with the interpreters <math>\text{A}</math> and <math>\text{B}</math>, respectively, putting them in the form of ''[[relational database]]s''.  Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)</math> that make up the corresponding sign relations, <math>L_\text{A}, L_\text{B} \subseteq</math><math>O \times S \times I</math>.  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.

The sign relation associated with a given interpreter <math>J\!</math> is denoted <math>L_J\!</math> or <math>L(J)\!</math>.  Tables&nbsp;1 and 2 give the sign relations associated with the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math>, respectively, putting them in the form of ''[[relational database]]s''.  Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\text{A}, L_\text{B} \subseteq O \times S \times I\!</math>.  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.

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|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Sign Relation of Interpreter A}</math>

|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Sign Relation of Interpreter A}\!</math>

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|- style="height:40px; background:#f0f0ff"

| <math>\text{Object}</math>

| <math>\text{Object}\!</math>

| <math>\text{Sign}</math>

| <math>\text{Sign}\!</math>

| <math>\text{Interpretant}</math>

| <math>\text{Interpretant}\!</math>

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|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Sign Relation of Interpreter B}</math>

|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Sign Relation of Interpreter B}\!</math>

|- style="height:40px; background:#f0f0ff"

|- style="height:40px; background:#f0f0ff"

| <math>\text{Object}</math>

| <math>\text{Object}\!</math>

| <math>\text{Sign}</math>

| <math>\text{Sign}\!</math>

| <math>\text{Interpretant}</math>

| <math>\text{Interpretant}\!</math>

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These Tables codify a rudimentary level of interpretive practice for the agents <math>\text{A}</math> and <math>\text{B}</math> 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 <math>(o, s, i)</math> that is called an ''elementary relation'', that is, one element of the relation's set-theoretic extension.

These Tables codify a rudimentary level of interpretive practice for the agents <math>\text{A}\!</math> and <math>\text{B}\!</math> 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 <math>(o, s, i)\!</math> that is called an ''elementary relation'', that is, one element of the relation's set-theoretic extension.

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.

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.

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.

The dyadic relation that constitutes the ''denotative component'' of a sign relation <math>L</math> is denoted <math>\operatorname{Den}(L)</math>.  Information about the denotative component of semantics can be derived from <math>L</math> by taking its ''dyadic projection'' on the plane that is generated by the object domain and the sign domain, indicated by any one of the equivalent forms, <math>\operatorname{proj}_{OS} L</math>, <math>L_{OS}</math>, or <math>L_{12}</math>, and defined as follows:

The dyadic relation that constitutes the ''denotative component'' of a sign relation <math>L\!</math> is denoted <math>\operatorname{Den}(L)\!</math>.  Information about the denotative component of semantics can be derived from <math>L\!</math> by taking its ''dyadic projection'' on the plane that is generated by the object domain and the sign domain, indicated by any one of the equivalent forms, <math>\operatorname{proj}_{OS} L\!</math>, <math>L_{OS}\!</math>, or <math>L_{12}\!</math>, and defined as follows:

: <math>\operatorname{Den}(L) = \operatorname{proj}_{OS} 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) = \operatorname{proj}_{OS} L = L_{OS} = \{ (o, s) \in O \times S : (o, s, i) \in L ~\text{for some}~ i \in I \}\!</math>.

Looking to the denotative aspects of the present example, various rows of the Tables specify that <math>\text{A}</math> uses <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}</math> to denote <math>\text{A}</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}</math> to denote <math>\text{B}</math>, whereas <math>\text{B}</math> uses <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}</math> to denote <math>\text{B}</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}</math> to denote <math>\text{A}</math>.  It is utterly amazing that even these impoverished remnants of natural language use have properties that quickly bring the usual prospects of formal semantics to a screeching halt.

Looking to the denotative aspects of the present example, various rows of the Tables specify that <math>\text{A}\!</math> uses <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math> to denote <math>\text{A}\!</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math> to denote <math>\text{B}\!</math>, whereas <math>\text{B}\!</math> uses <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math> to denote <math>\text{B}\!</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math> to denote <math>\text{A}\!</math>.  It is utterly amazing that even these impoverished remnants of natural language use have properties that quickly bring the usual prospects of formal semantics to a screeching halt.

The other dyadic aspects of semantics that might be considered concern the reference that a sign has to its interpretant and the reference that an interpretant has to its object.  As before, either type of reference can be multiple, unique, or empty in its collection of terminal points, and both can be formalized as different types of dyadic relations that are obtained as planar projections of the triadic sign relations.

The other dyadic aspects of semantics that might be considered concern the reference that a sign has to its interpretant and the reference that an interpretant has to its object.  As before, either type of reference can be multiple, unique, or empty in its collection of terminal points, and both can be formalized as different types of dyadic relations that are obtained as planar projections of the triadic sign relations.

The connection that a sign makes to an interpretant is called its ''connotation''.  In the general theory of sign relations, this aspect of semantics includes the references that a sign has to affects, concepts, impressions, intentions, mental ideas, and to the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct.  This complex ecosystem of references is unlikely ever to be mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the ''connotative'' import of language.  Given a particular sign relation <math>L</math>, the dyadic relation that constitutes the ''connotative component'' of <math>L</math> is denoted <math>\operatorname{Con}(L)</math>.

The connection that a sign makes to an interpretant is called its ''connotation''.  In the general theory of sign relations, this aspect of semantics includes the references that a sign has to affects, concepts, impressions, intentions, mental ideas, and to the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct.  This complex ecosystem of references is unlikely ever to be mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the ''connotative'' import of language.  Given a particular sign relation <math>L\!</math>, the dyadic relation that constitutes the ''connotative component'' of <math>L\!</math> is denoted <math>\operatorname{Con}(L)\!</math>.

The bearing that an interpretant has toward a common object of its sign and itself has no standard name.  If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this omits the mediational character of the whole transaction.

The bearing that an interpretant has toward a common object of its sign and itself has no standard name.  If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this omits the mediational character of the whole transaction.

Given the service that interpretants supply in furnishing a locus for critical, reflective, and explanatory glosses on objective scenes and their descriptive texts, it is easy to regard them as ''annotations'' both of objects and of signs, but this function points in the opposite direction to what is needed in this connection.  What does one call the inverse of the annotation function?  More generally asked, what is the converse of the annotation relation?

Given the service that interpretants supply in furnishing a locus for critical, reflective, and explanatory glosses on objective scenes and their descriptive texts, it is easy to regard them as ''annotations'' both of objects and of signs, but this function points in the opposite direction to what is needed in this connection.  What does one call the inverse of the annotation function?  More generally asked, what is the converse of the annotation relation?

In light of these considerations, I find myself still experimenting with terms to suit this last-mentioned dimension of semantics.  On a trial basis, I refer to it as the ''ideational'', the ''intentional'', or the ''canonical'' component of the sign relation, and I provisionally refer to the reference of an interpretant sign to its object as its ''ideation'', its ''intention'', or its ''conation''.  Given a particular sign relation <math>L</math>, the dyadic relation that constitutes the ''intentional component'' of <math>L</math> is denoted <math>\operatorname{Int}(L)</math>.

In light of these considerations, I find myself still experimenting with terms to suit this last-mentioned dimension of semantics.  On a trial basis, I refer to it as the ''ideational'', the ''intentional'', or the ''canonical'' component of the sign relation, and I provisionally refer to the reference of an interpretant sign to its object as its ''ideation'', its ''intention'', or its ''conation''.  Given a particular sign relation <math>L\!</math>, the dyadic relation that constitutes the ''intentional component'' of <math>L\!</math> is denoted <math>\operatorname{Int}(L)\!</math>.

A full consideration of the connotative and intentional aspects of semantics would force a return to difficult questions about the true nature of the interpretant sign in the general theory of sign relations.  It is best to defer these issues to a later discussion.  Fortunately, omission of this material does not interfere with understanding the purely formal aspects of the present example.

A full consideration of the connotative and intentional aspects of semantics would force a return to difficult questions about the true nature of the interpretant sign in the general theory of sign relations.  It is best to defer these issues to a later discussion.  Fortunately, omission of this material does not interfere with understanding the purely formal aspects of the present example.

The connotative component of a sign relation ''L'' can be formalized as its dyadic projection on the plane generated by the sign domain and the interpretant domain, defined as follows:

The connotative component of a sign relation ''L'' can be formalized as its dyadic projection on the plane generated by the sign domain and the interpretant domain, defined as follows:

: <math>\operatorname{Con}(L) = \operatorname{proj}_{SI} L = L_{SI} = \{ (s, i) \in S \times I : (o, s, i) \in L ~\text{for some}~ o \in O \}</math>.

: <math>\operatorname{Con}(L) = \operatorname{proj}_{SI} L = L_{SI} = \{ (s, i) \in S \times I : (o, s, i) \in L ~\text{for some}~ o \in O \}\!</math>.

The intentional component of semantics for a sign relation <math>L</math>, or its ''second moment of denotation'', is adequately captured by its dyadic projection on the plane generated by the object domain and interpretant domain, defined as follows:

The intentional component of semantics for a sign relation <math>L\!</math>, or its ''second moment of denotation'', is adequately captured by its dyadic projection on the plane generated by the object domain and interpretant domain, defined as follows:

: <math>\operatorname{Int}(L) = \operatorname{proj}_{OI} L = L_{OI} = \{ (o, i) \in O \times I : (o, s, i) \in L ~\text{for some}~ s \in S \}</math>.

: <math>\operatorname{Int}(L) = \operatorname{proj}_{OI} L = L_{OI} = \{ (o, i) \in O \times I : (o, s, i) \in L ~\text{for some}~ s \in S \}\!</math>.

As it happens, the sign relations <math>L_\text{A}</math> and <math>L_\text{B}</math> in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of <math>(L_\text{A})_{OS}</math> and <math>(L_\text{B})_{OS}</math> is merely echoed in <math>(L_\text{A})_{OI}</math> and <math>(L_\text{B})_{OI}</math>, respectively.

As it happens, the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}\!</math> in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of <math>(L_\text{A})_{OS}\!</math> and <math>(L_\text{B})_{OS}\!</math> is merely echoed in <math>(L_\text{A})_{OI}\!</math> and <math>(L_\text{B})_{OI}\!</math>, respectively.

'''Note on notation.'''  When there is only one sign relation <math>L_J = L(J)</math> associated with a given interpreter <math>J</math>, it is convenient to use the following forms of abbreviation:

'''Note on notation.'''  When there is only one sign relation <math>L_J = L(J)\!</math> associated with a given interpreter <math>J\!</math>, it is convenient to use the following forms of abbreviation:

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