Changes

In the initial slice of semantics presented for the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> the sign domain <math>S\!</math> is identical to the interpretant domain <math>I,\!</math> and this set is disjoint from the object domain <math>O.\!</math>  In order for this discussion to develop more interesting examples of sign relations these constraints will need to be generalized.  As a start in this direction, one can preserve the identification of the syntactic domain as <math>S = I\!</math> and contemplate ways of varying the pattern of intersection between <math>S\!</math> and <math>O.\!</math>

In the initial slice of semantics presented for the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> the sign domain <math>S\!</math> is identical to the interpretant domain <math>I,\!</math> and this set is disjoint from the object domain <math>O.\!</math>  In order for this discussion to develop more interesting examples of sign relations these constraints will need to be generalized.  As a start in this direction, one can preserve the identification of the syntactic domain as <math>S = I\!</math> and contemplate ways of varying the pattern of intersection between <math>S\!</math> and <math>O.\!</math>

One direction of generalization is motivated by the desire to give interpreters a measure of &ldquo;reflective capacity&rdquo;.  This is a property of sign relations that can be associated with the overlap of <math>O\!</math> and <math>S\!</math> and gauged by the extent to which <math>S\!</math> is contained in <math>O.\!</math> In intuitive terms, interpreters are said to have a reflective capacity to the extent that they can refer to their own signs independently of their denotations.  An interpretive system with a sufficient amount of reflective capacity can support the maintenance and manipulation of textual objects like expressions and programs without necessarily having to evaluate the expressions or execute the programs.

One direction of generalization is motivated by the desire to give interpreters a measure of "reflective capacity".  This is a property of sign relations that can be associated with the overlap of O and S and gauged by the extent to which S is contained in O.  In intuitive terms, interpreters are said to have a "reflective capacity" to the extent that they can refer to their own signs independently of their denotations.  An interpretive system with a sufficient amount of reflective capacity can support the maintainence and manipulation of textual objects like expressions and programs without necessarily having to evaluate the expressions or execute the programs.

In ordinary discourse HA signs are usually generated by means of linguistic devices for quoting pieces of text.  In computational frameworks these quoting mechanisms are implemented as functions that map syntactic arguments into numerical or syntactic values.  A quoting function, given a sign or expression as its single argument, needs to accomplish two things:  first, to defer the reference of that sign, in other words, to inhibit, delay, or prevent the evaluation of its argument expression, and then, to exhibit or produce another sign whose object is precisely that argument expression.

In ordinary discourse HA signs are usually generated by means of linguistic devices for quoting pieces of text.  In computational frameworks these quoting mechanisms are implemented as functions that map syntactic arguments into numerical or syntactic values.  A quoting function, given a sign or expression as its single argument, needs to accomplish two things:  first, to defer the reference of that sign, in other words, to inhibit, delay, or prevent the evaluation of its argument expression, and then, to exhibit or produce another sign whose object is precisely that argument expression.

The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects.  In each case, these extensions are assumed to begin with sign relations like A and B that have disjoint sets of objects and signs and thus have no reflective capacity at the outset.  The status of A and B as the "reflective origins" of a "reflective development" is recalled by saying that A and B themselves are the "zeroth order reflective extensions" of A and B, in symbols, A = Ref0(A) and B = Ref0(B).

The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects.  In each case, these extensions are assumed to begin with sign relations like A and B that have disjoint sets of objects and signs and thus have no reflective capacity at the outset.  The status of A and B as the "reflective origins" of a "reflective development" is recalled by saying that A and B themselves are the "zeroth order reflective extensions" of A and B, in symbols, A = Ref0(A) and B = Ref0(B).