Changes

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 '''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'''.

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''.

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'''.

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:

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:

:{| cellpadding="4"

:{| cellpadding="4"

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

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

|-

|-

| align="center" | '''S''' || = || Sign Domain

| align="center" | ''S'' || = || Sign Domain

|-

|-

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

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

|}

|}

:{| cellpadding="4"

:{| cellpadding="4"

| align="center" | '''O'''

| align="center" | ''O''

| =

| =

| {Ann, Bob}

| {Ann, Bob}

| =

| =

| {A, B}

| {''A'', ''B''}

|-

|-

| align="center" | '''S'''

| align="center" | ''S''

| =

| =

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

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

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

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

|-

|-

| align="center" | '''I'''

| align="center" | ''I''

| =

| =

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

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

|}

|}

In the present Example, '''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 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.

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'', ''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.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"

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

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

|- style="background:paleturquoise"

|- style="background:paleturquoise"

! style="width:20%" | Object

! style="width:20%" | Object

! style="width:20%" | Interpretant

! style="width:20%" | Interpretant

|-

|-

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

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

|-

|-

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

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

|-

|-

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

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

|-

|-

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

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

|-

|-

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

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

|-

|-

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

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

|-

|-

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

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

|-

|-

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

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

|}

|}

<br>

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"

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

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

|- style="background:paleturquoise"

|- style="background:paleturquoise"

! style="width:20%" | Object

! style="width:20%" | Object

! style="width:20%" | Interpretant

! style="width:20%" | Interpretant

|-

|-

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

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

|-

|-

| '''A''' || '''"A"''' || '''"u"'''

| ''A'' || "A" || "u"

|-

|-

| '''A''' || '''"u"''' || '''"A"'''

| ''A'' || "u" || "A"

|-

|-

| '''A''' || '''"u"''' || '''"u"'''

| ''A'' || "u" || "u"

|-

|-

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

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

|-

|-

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

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

|-

|-

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

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

|-

|-

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

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

|}

|}

<br>

<br>

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 '''L''' is known as ''Den''('''L''').  Information about the denotative component of semantics can be derived from '''L''' 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, ''Proj''_'''OS''''''L''', '''L'''_'''OS''', or '''L'''₁₂, and defined as follows:

The dyadic relation that constitutes the ''denotative component'' of a sign relation ''L'' is known as ''Den''(''L'').  Information about the denotative component of semantics can be derived from ''L'' 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, ''Proj''_''OS''&nbsp;''L'', ''L''_''OS''&nbsp;, or ''L''₁₂&nbsp;, and defined as follows:

: ''Den''('''L''') = ''Proj''_'''OS''''''L''' = '''L'''_'''OS''' = {‹''o'', ''s''› &isin; '''O''' &times; '''S''' : ‹''o'', ''s'', ''i''› &isin; '''L''' for some ''i'' &isin; '''I'''}.

: ''Den''(''L'') = ''Proj''_''OS''&nbsp;''L'' = ''L''_''OS'' = {‹''o'', ''s''› &isin; ''O'' &times; ''S'' : ‹''o'', ''s'', ''i''› &isin; ''L'' for some ''i'' &isin; ''I''}.

Looking to the denotative aspects of the present example, various rows of the Tables specify that ''A'' uses "i" to denote ''A'' and "u" to denote ''B'', whereas ''B'' uses "i" to denote ''B'' and "u" to denote ''A''.  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 ''A'' uses "i" to denote ''A'' and "u" to denote ''B'', whereas ''B'' uses "i" to denote ''B'' and "u" to denote ''A''.  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 principal concern of this project is not with every conceivable sign relation but chiefly with those that are capable of supporting inquiry processes.  In these, the relationship between the connotational and the denotational aspects of meaning is not wholly arbitrary.  Instead, this relationship must be naturally constrained or deliberately designed in such a way that it:

The principal concern of this project is not with every conceivable sign relation but chiefly with those that are capable of supporting inquiry processes.  In these, the relationship between the connotational and the denotational aspects of meaning is not wholly arbitrary.  Instead, this relationship must be naturally constrained or deliberately designed in such a way that it:

1. Represents the embodiment of significant properties that have objective reality  

# Represents the embodiment of significant properties that have objective reality  

in the agent's domain.

in the agent's domain.

 

# Supports the achievement of particular purposes that have intentional value  

2. Supports the achievement of particular purposes that have intentional value  

for the agent.

for the agent.