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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 {Ann, Bob}.

:* 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 {"Ann",&nbsp;"Bob",&nbsp;"I",&nbsp;"You"}.

:* 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 ''L'' is a ''[[subset]]'' of a ''[[cartesian product]]'' ''O''&nbsp;&times;&nbsp;''S''&nbsp;&times;&nbsp;''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''&nbsp;&sube;&nbsp;''O''&nbsp;&times;&nbsp;''S''&nbsp;&times;&nbsp;''I''.

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 ''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''.