Cf: Sign Relations • Semiotic Equivalence Relations 2
http://inquiryintoinquiry.com/2020/07/03/sign-relations-%e2%80%a2-semiotic-equivalence-relations-2/
A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in
particular. (NB. The notations will be much more readable on the blog page linked above.)
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
Thus we have the following definitions.
• [x]_E = {y ∈ X : (x, y) ∈ E}
• x =_E y ⇔ (x, y) ∈ E
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. That is, let [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
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. These modifications are 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-components-con-la-con-lb.png
The semiotic equivalence relation for interpreter A yields the following
semiotic equations.
• [“A”]_{L_A} = [“i”]_{L_A}
• [“B”]_{L_A} = [“u”]_{L_A}
or
• “A” =_{L_A} “i”
• “B” =_{L_A} “u”
Thus it induces the semiotic partition:
• {{“A”, “i”}, {“B”, “u”}}
The semiotic equivalence relation for interpreter B yields the following
semiotic equations.
• [“A”]_{L_B} = [“u”]_{L_B}
• [“B”]_{L_B} = [“i”]_{L_B}
or
• “A” =_{L_B} “u”
• “B” =_{L_B} “i”
Thus it induces the semiotic partition:
• {{“A”, “u”}, {“B”, “i”}}
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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