Cf: Sign Relations • Semiotic Equivalence Relations 1
http://inquiryintoinquiry.com/2020/07/01/sign-relations-%e2%80%a2-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 is associated with a family of equivalence classes which
partition the underlying set of elements, known as the “domain” or “space” of the relation. 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_A and L_B have many interesting properties over and above those possessed by sign relations in
general. Some of these properties have to do with the relation between signs and their interpretant signs, as reflected
in the projections of L_A and L_B on the SI-plane, notated as proj_SI L_A and proj_SI L_B, respectively. The dyadic
relations on S × I induced by these projections are also referred to as the connotative components of the corresponding
sign relations, notated as Con(L_A) and Con(L_B), respectively. Tables 6a and 6b show the corresponding connotative
components.
Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B)
https://inquiryintoinquiry.files.wordpress.com/2020/06/connotative-components-con-la-con-lb.png
A nice property of the sign relations L_A and L_B is that their connotative components Con(L_A) and Con(L_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, Con(L_A), Con(L_B) ⊆ S × I ≅ S × S partitions the collection of signs into
semiotic equivalence classes. This makes for a strong form of representation in that the structure of the interpreters'
common object domain {A, B} is reflected or reconstructed, part for part, in the structure of each one's semiotic
partition of the syntactic domain {“A”, “B”, “i”, “u”}. But it needs to be observed that the semiotic partitions for
interpreters A and B are not identical, indeed, they are orthogonal to each other. This allows us to regard the “form”
of these 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 A and B is
summarized in Tables 7a and 7b. The form of these Tables serves to explain what is meant by saying the SEPs for A and B
are orthogonal to each other.
Tables 7a and 7b. Semiotic Partitions for Interpreters A and B
https://inquiryintoinquiry.files.wordpress.com/2020/06/semiotic-partitions-for-interpreters-a-b.png
Regards,
Jon
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