## Sign Relations • Semiotic Equivalence Relations 1

A semiotic equivalence relation (SER) is a special type of equivalence relation arising in the analysis of sign relations.  Generally speaking, any equivalence relation induces a partition of the underlying set of elements, known as the domain or space of the relation, into a family of equivalence classes.  In the case of a SER the equivalence classes are called semiotic equivalence classes (SECs) and the partition is called a semiotic partition (SEP).

The sign relations $L_\mathrm{A}$ and $L_\mathrm{B}$ have many interesting properties over and above those possessed by sign relations in general.  Some of those properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of $L_\mathrm{A}$ and $L_\mathrm{B}$ on the $SI$-plane, notated as $\mathrm{proj}_{SI} L_\mathrm{A}$ and $\mathrm{proj}_{SI} L_\mathrm{B},$ respectively.  The dyadic relations on $S \times I$ induced by those projections are also referred to as the connotative components of the corresponding sign relations, notated as $\mathrm{Con}(L_\mathrm{A})$ and $\mathrm{Con}(L_\mathrm{B}),$ respectively.  Tables 6a and 6b show the corresponding connotative components. A nice property of the sign relations $L_\mathrm{A}$ and $L_\mathrm{B}$ is that their connotative components $\mathrm{Con}(L_\mathrm{A})$ and $\mathrm{Con}(L_\mathrm{B})$ form a pair of equivalence relations on their common syntactic domain $S = I.$  This type of equivalence relation is called a semiotic equivalence relation (SER) because it equates signs having the same meaning to some interpreter.

Each of the semiotic equivalence relations, $\mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I \cong S \times S$ partitions the collection of signs into semiotic equivalence classes.  This constitutes a strong form of representation in that the structure of the interpreters’ common object domain $\{ \mathrm{A}, \mathrm{B} \}$ is reflected or reconstructed, part for part, in the structure of each one’s semiotic partition of the syntactic domain $\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.$

It’s important to observe the semiotic partitions for interpreters $\mathrm{A}$ and $\mathrm{B}$ are not identical, indeed, they are orthogonal to each other.  Thus we may regard the form of the partitions as corresponding to an objective structure or invariant reality, but not the literal sets of signs themselves, independent of the individual interpreter’s point of view.

Information about the contrasting patterns of semiotic equivalence corresponding to the interpreters $\mathrm{A}$ and $\mathrm{B}$ is summarized in Tables 7a and 7b.  The form of the Tables serves to explain what is meant by saying the SEPs for $\mathrm{A}$ and $\mathrm{B}$ are orthogonal to each other. ### References

• Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75), in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.
• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  ArchiveJournal.  Online (doc) (pdf).

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