[CG] Re: Sign Relations

Cf: Sign Relations • Semiotic Equivalence Relations 2 https://inquiryintoinquiry.com/2022/07/08/sign-relations-semiotic-equivalen… All, A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular. In general, if E is an equivalence relation on a set X then every element x of X belongs to a unique equivalence class under E called “the equivalence class of x under E”. Convention provides the “square bracket notation” for denoting such equivalence classes, in either the form [x]_E or the simpler form [x] when the subscript E is understood. A statement that the elements x and y are equivalent under E is called an “equation” or an “equivalence” and may be expressed in any of the following ways. • (x, y) ∈ E • x ∈ [y]_E • y ∈ [x]_E • [x]_E = [y]_E • x =_E y Display 1 https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-1.png Thus we have the following definitions. • [x]_E = {y ∈ X : (x, y) ∈ E} • x =_E y ⇔ (x, y) ∈ E Display 2 https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-2.png In the application to sign relations it is useful to extend the square bracket notation in the following ways. If L is a sign relation whose connotative component L_SI is an equivalence relation on S = I, let [s]_L be the equivalence class of s under L_SI. In short, [s]_L = [s]_{L_{SI}}. A statement that the signs x and y belong to the same equivalence class under a semiotic equivalence relation L_SI is called a “semiotic equation” (SEQ) and may be written in either of the following forms. • [x]_L = [y]_L • x =_L y Display 3 https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-3.png In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple (o, s, i) in a sign relation L, it is permissible to let [o]_L be defined as [s]_L. This modifications is designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions. Applying the array of equivalence notations to the sign relations for A and B will serve to illustrate their use and utility. Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B) https://inquiryintoinquiry.files.wordpress.com/2020/06/connotative-componen… The semiotic equivalence relation for interpreter A yields the following semiotic equations. Display 4 https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-4.png or Display 5 https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-5.png Thus it induces the semiotic partition: • {{“A”, “i”}, {“B”, “u”}}. Display 6 https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-6.png The semiotic equivalence relation for interpreter B yields the following semiotic equations. Display 7 https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-7.png or Display 8 https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-8.png Thus it induces the semiotic partition: • {{“A”, “u”}, {“B”, “i”}}. Display 9 https://inquiryintoinquiry.files.wordpress.com/2022/07/ser-2-display-9.png Tables 7a and 7b. Semiotic Partitions for Interpreters A and B https://inquiryintoinquiry.files.wordpress.com/2020/06/semiotic-partitions-… Regards, Jon