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===6.25. Analytic Intensional Representations===

===6.25. Analytic Intensional Representations===

In this section the ERs of <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are translated into a variety of different IRs that actually accomplish some measure of analytic work.  These are referred to as ''analytic intensional representations'' (AIRs).  This strategy of representation is referred to as a ''structural coding'' or a ''sensitive coding'', because it pays attention to the structure of its object domain and attends to the nuances of each sign's interpretation to fashion its code.  It may also be characterized as a ''<math>\log(n)\!</math> coding'', because it uses roughly <math>\log_2(n)\!</math> binary features to represent a domain of <math>n\!</math> elements.

In this section the ERs of <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are translated into a variety of different IRs that actually accomplish some measure of analytic work.  These are referred to as ''analytic intensional representations'' (AIRs).  This strategy of representation is referred to as a ''structural coding'' or a ''sensitive coding'', because it pays attention to the structure of its object domain and attends to the nuances of each sign's interpretation to fashion its code.  It may also be characterized as a <math>\log(n)\!</math> coding, because it uses roughly <math>\log_2(n)\!</math> binary features to represent a domain of <math>n\!</math> elements.

For the domain <math>O = \{ \text{A}, \text{B} \}\!</math> of two elements one needs to use a single logical feature.  It is often convenient to use an object feature that is relative to the interpreter using it, for instance, telling whether the object described is the self or the other.

For the domain O = {A, B} of two elements one needs to use a single logical feature.  It is often convenient to use an object feature that is relative to the interpreter using it, for instance, telling whether the object described is the self or the other.  

For the domain S = I = {"A", "B", "i", "u"} of four elements one needs to use two logical features.  One possibility is to classify each element:  according to its syntactic category, as being a noun or a pronoun, and according to its semantic category, as denoting the self or the other.

For the domain <math>S = I = \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> of four elements one needs to use two logical features.  One possibility is to classify each element according to its syntactic category, as being a noun or a pronoun, and according to its semantic category, as denoting the self or the other.

Tables 62.1 through 62.3 show several ways of representing these categories in terms of feature value pairs and propositional codes.  In each Table, Column 1 describes the category in question, Column 2 gives the mnemonic form of a propositional expression for that category, and Column 3 gives the abbreviated form of that expression, using a notation for propositional calculus where parentheses circumscribing a term or expression are interpreted as forming its logical negation.  

Tables 62.1 through 62.3 show several ways of representing these categories in terms of feature value pairs and propositional codes.  In each Table, Column 1 describes the category in question, Column 2 gives the mnemonic form of a propositional expression for that category, and Column 3 gives the abbreviated form of that expression, using a notation for propositional calculus where parentheses circumscribing a term or expression are interpreted as forming its logical negation.