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The rest of this section discusses the relationship between higher order signs and a concept called the ''reflective extension'' of a sign relation.  Reflective extensions will be subjected to a more detailed study in a later part of this work.  For now, just to see how the process works, the sign relations <math>L(A)\!</math> and <math>L(B)\!</math> are taken as starting points to illustrate the more common forms of reflective development.

The rest of this section discusses the relationship between HO signs and a concept called the "reflective extension" of a sign relation.  Reflective extensions will be subjected to a more detailed study in a later part of this work.  For now, just to see how the process works, the sign relations A and B are taken as starting points to illustrate the more common forms of reflective development.

In the most typical scenario, HO sign relations come into being as the "reflective extensions" of simpler, possibly unreflective sign relations.  Conversely, the incorporation of HO signs within a sign relation leads to a larger sign relation that constitutes one of its "reflective extensions".  In general, there are many different ways that a reflective extension can get started and many different structures that can result.

In the most typical scenario, higher order sign relations come into being as the reflective extensions of simpler, possibly unreflective sign relations.  Conversely, the incorporation of higher order signs within a sign relation leads to a larger sign relation that constitutes one of its reflective extensions.  In general, there are many different ways that a reflective extension can get started and many different structures that can result.

In the initial slice of semantics presented for the dialogue of A and B, the sign domain S is identical to the interpretant domain I, and this set is disjoint from the object domain O.  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 S = I and contemplate ways of varying the pattern of intersection between S and O.

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 "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.

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.