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=====1.3.4.3.  Semiotic Equivalence Relations=====

=====1.3.4.3.  Semiotic Equivalence Relations=====

If one examines the sign relations ''L''_''A'' and ''L''_''B'' that are associated with the interpreters ''A'' and ''B'', respectively, one observes that they have many contingent properties that are not possessed by sign relations in general.  One nice property possessed by the sign relations ''L''_''A'' and ''L''_''B'' is that their connotative components ''A''_''SI''  and ''B''_''SI''  constitute a pair of [[equivalence relation]]s on their common syntactic domain ''S'' = ''I''.  It is convenient to refer to such structures as ''[[semiotic equivalence relation]]s'' (SER's) since they equate signs that mean the same thing to somebody.  Each of the SER's, ''A''_''SI'' , ''B''_''SI''  ⊆ ''S'' × ''I'' = ''S'' × ''S'' partitions the whole collection of signs into ''[[semiotic equivalence class]]es'' (SEC's).  This makes for a strong form of representation in that the structure of the participants' common object domain is reflected or reconstructed, part for part, in the structure of each of their ''[[semiotic partition]]s'' (SEP's) of the syntactic domain.

If one examines the sign relations <math>L_\text{A}</math> and <math>L_\text{B}</math> that are associated with the interpreters <math>\text{A}</math> and <math>\text{B}</math>, respectively, one observes that they have many contingent properties that are not possessed by sign relations in general.  One nice property possessed by the sign relations <math>L_\text{A}</math> and <math>L_\text{B}</math> is that their connotative components <math>\text{A}_{SI}</math> and <math>\text{B}_{SI}</math> constitute a pair of [[equivalence relation]]s on their common syntactic domain <math>S = I</math>.  It is convenient to refer to such structures as ''[[semiotic equivalence relation]]s'' (SERs) since they equate signs that mean the same thing to somebody.  Each of the SERs, <math>\text{A}_{SI}, \text{B}_{SI} \subseteq S \times I = S \times S</math>, partitions the whole collection of signs into ''[[semiotic equivalence class]]es'' (SECs).  This makes for a strong form of representation in that the structure of the participants' common object domain is reflected or reconstructed, part for part, in the structure of each of their ''[[semiotic partition]]s'' (SEPs) of the syntactic domain.

The main trouble with this notion of semantics in the present situation is that the two semiotic partitions for ''A'' and ''B'' are not the same, indeed, they are orthogonal to each other.  This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view (POV).

The main trouble with this notion of semantics in the present situation is that the two semiotic partitions for <math>\text{A}</math> and <math>\text{B}</math> are not the same, indeed, they are orthogonal to each other.  This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view.

Information about the different forms of semiotic equivalence induced by the interpreters ''A'' and ''B'' is summarized in Tables&nbsp;3 and 4.  The form of these Tables should suffice to explain what is meant by saying that the SEP's for ''A'' and ''B'' are orthogonal to each other.

Information about the different forms of semiotic equivalence induced by the interpreters <math>\text{A}</math> and <math>\text{B}</math> is summarized in Tables&nbsp;3 and 4.  The form of these Tables should suffice to explain what is meant by saying that the SEPs for <math>\text{A}</math> and <math>\text{B}</math> are orthogonal to each other.

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