Changes

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 ''I''&nbsp;&sube;&nbsp;''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''&nbsp;&cup;&nbsp;''S''&nbsp;&cup;&nbsp;''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 <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:

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

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

:{| cellpadding="4"

{| align="center" cellspacing="6" width="90%"

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

|

| =

| {Ann, Bob}

O

| =

& = &

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

\{ \text{Ann}, \text{Bob} \} & = & \{ \text{A}, \text{B} \}

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

S

| =

& = &

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

\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}

| =

& = &

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

\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}

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

I

| =

& = &

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

\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}

| =

& = &

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

\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}

|}

|}

In the present example, ''S'' = ''I'' = syntactic domain.

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

The sign relation associated with a given interpreter ''J'' is denoted ''L''_''J''&nbsp; or ''L''(''J'').  Tables&nbsp;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'',&nbsp;''s'',&nbsp;''i''› that make up the corresponding sign relations, ''L''_''A''&nbsp;,&nbsp;''L''_''B''&nbsp;&nbsp;&sube;&nbsp;''O''&nbsp;&times;&nbsp;''S''&nbsp;&times;&nbsp;''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.

The sign relation associated with a given interpreter ''J'' is denoted ''L''_''J''&nbsp; or ''L''(''J'').  Tables&nbsp;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'',&nbsp;''s'',&nbsp;''i''› that make up the corresponding sign relations, ''L''_''A''&nbsp;,&nbsp;''L''_''B''&nbsp;&nbsp;&sube;&nbsp;''O''&nbsp;&times;&nbsp;''S''&nbsp;&times;&nbsp;''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.