16 Feb
2024
16 Feb
'24
7:08 p.m.
Sign Relations • Semiotic Equivalence Relations 2
•
https://inquiryintoinquiry.com/2024/02/16/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
In this way it induces the following 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
In this way it induces the following 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-…
Resources —
Sign Relations
• https://oeis.org/wiki/Sign_relation
Semiotic Equivalence Relations 2
• https://oeis.org/wiki/Sign_relation#Semiotic_Equivalence_Relations_2
Document History
• https://oeis.org/wiki/Sign_relation#Document_history
Regards,
Jon
cc: https://www.academia.edu/community/VvO8Je
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