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| | =====1.3.12.1. Syntactic Transformation Rules===== | | =====1.3.12.1. Syntactic Transformation Rules===== |
| − |
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| − | <pre>
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| − | Logical Translation Rule 1
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| − |
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| − | If S is a sentence
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| − |
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| − | about things in the universe U
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| − |
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| − | and P is a proposition : U -> B, such that:
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| − |
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| − | L1a. [S] = P,
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| − |
| |
| − | then the following equations hold:
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| − |
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| − | L1b00. [False] = () = 0 : U->B.
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| − |
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| − | L1b01. [Not S] = ([S]) = (P) : U->B.
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| − |
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| − | L1b10. [S] = [S] = P : U->B.
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| − |
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| − | L1b11. [True] = (()) = 1 : U->B.
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| − | </pre>
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| − |
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| − | <pre>
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| − | Geometric Translation Rule 1
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| − |
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| − | If X c U
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| − |
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| − | and P : U -> B, such that:
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| − |
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| − | G1a. {X} = P,
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| − |
| |
| − | then the following equations hold:
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| − |
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| − | G1b00. {{}} = () = 0 : U->B.
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| − |
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| − | G1b10. {~X} = ({X}) = (P) : U->B.
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| − |
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| − | G1b01. {X} = {X} = P : U->B.
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| − |
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| − | G1b11. {U} = (()) = 1 : U->B.
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| − | </pre>
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| − |
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| − | <pre>
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| − | Logical Translation Rule 2
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| − |
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| − | If S, T are sentences
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| − |
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| − | about things in the universe U
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| − |
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| − | and P, Q are propositions: U -> B, such that:
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| − |
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| − | L2a. [S] = P and [T] = Q,
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| − |
| |
| − | then the following equations hold:
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| − |
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| − | L2b00. [False] = () = 0 : U->B.
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| − |
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| − | L2b01. [Neither S nor T] = ([S])([T]) = (P)(Q).
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| − |
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| − | L2b02. [Not S, but T] = ([S])[T] = (P) Q.
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| − |
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| − | L2b03. [Not S] = ([S]) = (P).
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| − |
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| − | L2b04. [S and not T] = [S]([T]) = P (Q).
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| − |
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| − | L2b05. [Not T] = ([T]) = (Q).
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| − |
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| − | L2b06. [S or T, not both] = ([S], [T]) = (P, Q).
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| − |
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| − | L2b07. [Not both S and T] = ([S].[T]) = (P Q).
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| − |
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| − | L2b08. [S and T] = [S].[T] = P.Q.
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| − |
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| − | L2b09. [S <=> T] = (([S], [T])) = ((P, Q)).
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| − |
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| − | L2b10. [T] = [T] = Q.
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| − |
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| − | L2b11. [S => T] = ([S]([T])) = (P (Q)).
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| − |
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| − | L2b12. [S] = [S] = P.
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| − |
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| − | L2b13. [S <= T] = (([S]) [T]) = ((P) Q).
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| − |
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| − | L2b14. [S or T] = (([S])([T])) = ((P)(Q)).
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| − |
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| − | L2b15. [True] = (()) = 1 : U->B.
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| − | </pre>
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| − |
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| − | <pre>
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| − | Geometric Translation Rule 2
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| − |
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| − | If X, Y c U
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| − |
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| − | and P, Q U -> B, such that:
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| − |
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| − | G2a. {X} = P and {Y} = Q,
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| − |
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| − | then the following equations hold:
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| − |
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| − | G2b00. {{}} = () = 0 : U->B.
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| − |
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| − | G2b01. {~X n ~Y} = ({X})({Y}) = (P)(Q).
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| − |
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| − | G2b02. {~X n Y} = ({X}){Y} = (P) Q.
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| − |
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| − | G2b03. {~X} = ({X}) = (P).
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| − |
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| − | G2b04. {X n ~Y} = {X}({Y}) = P (Q).
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| − |
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| − | G2b05. {~Y} = ({Y}) = (Q).
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| − |
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| − | G2b06. {X + Y} = ({X}, {Y}) = (P, Q).
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| − |
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| − | G2b07. {~(X n Y)} = ({X}.{Y}) = (P Q).
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| − |
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| − | G2b08. {X n Y} = {X}.{Y} = P.Q.
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| − |
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| − | G2b09. {~(X + Y)} = (({X}, {Y})) = ((P, Q)).
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| − |
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| − | G2b10. {Y} = {Y} = Q.
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| − |
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| − | G2b11. {~(X n ~Y)} = ({X}({Y})) = (P (Q)).
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| − |
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| − | G2b12. {X} = {X} = P.
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| − |
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| − | G2b13. {~(~X n Y)} = (({X}) {Y}) = ((P) Q).
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| − |
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| − | G2b14. {X u Y} = (({X})({Y})) = ((P)(Q)).
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| − |
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| − | G2b15. {U} = (()) = 1 : U->B.
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| − | </pre>
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| | | | |
| | <pre> | | <pre> |
| Line 277: |
Line 145: |
| | </pre> | | </pre> |
| | | | |
| − | <pre>
| + | =====1.3.12.2. Derived Equivalence Relations <big>✔</big>===== |
| − | Definition 2
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| − | | |
| − | If X, Y c U,
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | D2a. X = Y.
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| − | | |
| − | D2b. u C X <=> u C Y, for all u C U.
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| − | </pre>
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| − | | |
| − | <pre>
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| − | Definition 3
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| − | | |
| − | If f, g : U -> V,
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | D3a. f = g.
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| − | | |
| − | D3b. f(u) = g(u), for all u C U.
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| − | </pre>
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| − | | |
| − | <pre>
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| − | Definition 4
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| − | | |
| − | If X c U,
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| − | | |
| − | then the following are identical subsets of UxB:
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| − | | |
| − | D4a. {X}
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| − | | |
| − | D4b. {< u, v> C UxB : v = [u C X]}
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| − | </pre>
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| − | | |
| − | <pre>
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| − | Definition 5
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| − | | |
| − | If X c U,
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| − | | |
| − | then the following are identical propositions:
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| − | | |
| − | D5a. {X}.
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| − | | |
| − | D5b. f : U -> B
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| − | | |
| − | : f(u) = [u C X], for all u C U.
| |
| − | </pre>
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| − | | |
| − | Given an indexed set of sentences, Sj for j C J, it is possible to consider the logical conjunction of the corresponding propositions. Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6.
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| − | | |
| − | <pre>
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| − | Definition 6
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| − | | |
| − | If Sj is a sentence
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| − | | |
| − | about things in the universe U,
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| − | | |
| − | for all j C J,
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | D6a. Sj, for all j C J.
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| − | | |
| − | D6b. For all j C J, Sj.
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| − | | |
| − | D6c. Conj(j C J) Sj.
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| − | | |
| − | D6d. ConjJ,j Sj.
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| − | | |
| − | D6e. ConjJj Sj.
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| − | </pre>
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| − | | |
| − | <pre>
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| − | Definition 7
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| − | | |
| − | If S, T are sentences
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| − | | |
| − | about things in the universe U,
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | D7a. S <=> T.
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| − | | |
| − | D7b. [S] = [T].
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| − | </pre>
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| − | | |
| − | <pre>
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| − | Rule 5
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| − | | |
| − | If X, Y c U,
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | R5a. X = Y. :D2a
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| − | | |
| − | ::
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| − | | |
| − | R5b. u C X <=> u C Y, for all u C U. :D2b
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| − | | |
| − | :D7a
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| − | | |
| − | ::
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| − | | |
| − | R5c. [u C X] = [u C Y], for all u C U. :D7b
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| − | | |
| − | :???
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| − | | |
| − | ::
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| − | | |
| − | R5d. {< u, v> C UxB : v = [u C X]}
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| − | | |
| − | =
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| − | | |
| − | {< u, v> C UxB : v = [u C Y]}. :???
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| − | | |
| − | :D5b
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| − | | |
| − | ::
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| − | | |
| − | R5e. {X} = {Y}. :D5a
| |
| − | </pre>
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| − | | |
| − | <pre>
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| − | Rule 6
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| − | | |
| − | If f, g : U -> V,
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | R6a. f = g. :D3a
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| − | | |
| − | ::
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| − | | |
| − | R6b. f(u) = g(u), for all u C U. :D3b
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| − | | |
| − | :D6a
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| − | | |
| − | ::
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| − | | |
| − | R6c. ConjUu (f(u) = g(u)). :D6e
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| − | </pre>
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| − | | |
| − | <pre>
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| − | Rule 7
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| − | | |
| − | If P, Q : U -> B,
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | R7a. P = Q. :R6a
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| − | | |
| − | ::
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| − | | |
| − | R7b. P(u) = Q(u), for all u C U. :R6b
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| − | | |
| − | ::
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| − | | |
| − | R7c. ConjUu (P(u) = Q(u)). :R6c
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| − | | |
| − | :P1a
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| − | | |
| − | ::
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| − | | |
| − | R7d. ConjUu (P(u) <=> Q(u)). :P1b
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| − | | |
| − | ::
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| − | | |
| − | R7e. ConjUu (( P(u) , Q(u) )). :P1c
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| − | | |
| − | :$1a
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| − | | |
| − | ::
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| − | | |
| − | R7f. ConjUu (( P , Q ))$(u). :$1b
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| − | </pre>
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| − | | |
| − | <pre>
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| − | Rule 8
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| − | | |
| − | If S, T are sentences
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| − | | |
| − | about things in the universe U,
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | R8a. S <=> T. :D7a
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| − | | |
| − | ::
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| − | | |
| − | R8b. [S] = [T]. :D7b
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| − | | |
| − | :R7a
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| − | | |
| − | ::
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| − | | |
| − | R8c. [S](u) = [T](u), for all u C U. :R7b
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| − | | |
| − | ::
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| − | | |
| − | R8d. ConjUu ( [S](u) = [T](u) ). :R7c
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| − | | |
| − | ::
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| − | | |
| − | R8e. ConjUu ( [S](u) <=> [T](u) ). :R7d
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| − | | |
| − | ::
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| − | | |
| − | R8f. ConjUu (( [S](u) , [T](u) )). :R7e
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| − | | |
| − | ::
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| − | | |
| − | R8g. ConjUu (( [S] , [T] ))$(u). :R7f
| |
| − | </pre>
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| − | | |
| − | For instance, the observation that expresses the equality of sets in terms of their indicator functions can be formalized according to the pattern in Rule 9, namely, at lines (a, b, c), and these components of Rule 9 can be cited in future uses as "R9a", "R9b", "R9c", respectively. Using Rule 7, annotated as "R7", to adduce a few properties of indicator functions to the account, it is possible to extend Rule 9 by another few steps, referenced as "R9d", "R9e", "R9f", "R9g".
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| − | | |
| − | <pre>
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| − | Rule 9
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| − | | |
| − | If X, Y c U,
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | R9a. X = Y. :R5a
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| − | | |
| − | ::
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| − | | |
| − | R9b. {X} = {Y}. :R5e
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| − | | |
| − | :R7a
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| − | | |
| − | ::
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| − | | |
| − | R9c. {X}(u) = {Y}(u), for all u C U. :R7b
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| − | | |
| − | ::
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| − | | |
| − | R9d. ConjUu ( {X}(u) = {Y}(u) ). :R7c
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| − | | |
| − | ::
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| − | | |
| − | R9e. ConjUu ( {X}(u) <=> {Y}(u) ). :R7d
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| − | | |
| − | ::
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| − | | |
| − | R9f. ConjUu (( {X}(u) , {Y}(u) )). :R7e
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| − | | |
| − | ::
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| − | | |
| − | R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
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| − | </pre>
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| − | | |
| − | <pre>
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| − | Rule 10
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| − | | |
| − | If X, Y c U,
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | R10a. X = Y. :D2a
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| − | | |
| − | ::
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| − | | |
| − | R10b. u C X <=> u C Y, for all u C U. :D2b
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| − | | |
| − | :R8a
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| − | | |
| − | ::
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| − | | |
| − | R10c. [u C X] = [u C Y]. :R8b
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| − | | |
| − | ::
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| − | | |
| − | R10d. For all u C U,
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| − | | |
| − | [u C X](u) = [u C Y](u). :R8c
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| − | | |
| − | ::
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| − | | |
| − | R10e. ConjUu ( [u C X](u) = [u C Y](u) ). :R8d
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| − | | |
| − | ::
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| − | | |
| − | R10f. ConjUu ( [u C X](u) <=> [u C Y](u) ). :R8e
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| − | | |
| − | ::
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| − | | |
| − | R10g. ConjUu (( [u C X](u) , [u C Y](u) )). :R8f
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| − | | |
| − | ::
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| − | | |
| − | R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
| |
| − | </pre>
| |
| − | | |
| − | <pre>
| |
| − | Rule 11
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| − | | |
| − | If X c U
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | R11a. X = {u C U : S}. :R5a
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| − | | |
| − | ::
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| − | | |
| − | R11b. {X} = { {u C U : S} }. :R5e
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| − | | |
| − | ::
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| − | | |
| − | R11c. {X} c UxB
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| − | | |
| − | : {X} = {< u, v> C UxB : v = [S](u)}. :R
| |
| − | | |
| − | ::
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| − | | |
| − | R11d. {X} : U -> B
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| − | | |
| − | : {X}(u) = [S](u), for all u C U. :R
| |
| − | | |
| − | ::
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| − | | |
| − | R11e. {X} = [S]. :R
| |
| − | </pre>
| |
| − | | |
| − | An application of Rule 11 involves the recognition of an antecedent condition as a case under the Rule, that is, as a condition that matches one of the sentences in the Rule's chain of equivalents, and it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that has to be checked on input for whether it fits the antecedent condition, and there is the choice of three types of output that are generated as a consequence, only one of which is generally needed at any given time. More often than not, though, a rule is applied in only a few of its possible ways. The usual antecedent and the usual consequents for Rule 11 can be distinguished in form and specialized in practice as follows:
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| − | | |
| − | a. R11a marks the usual starting place for an application of the Rule, that is, the standard form of antecedent condition that is likely to lead to an invocation of the Rule.
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| − | | |
| − | b. R11b records the trivial consequence of applying the spiny braces to both sides of the initial equation.
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| − | | |
| − | c. R11c gives a version of the indicator function with {X} c UxB, called its "extensional form".
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| − | | |
| − | d. R11d gives a version of the indicator function with {X} : U->B, called its "functional form".
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| − | | |
| − | Applying Rule 9, Rule 8, and the Logical Rules to the special case where S <=> (X = Y), one obtains the following general fact.
| |
| − | | |
| − | <pre>
| |
| − | Fact 1
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| − | | |
| − | If X,Y c U,
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| − | | |
| − | then the following are equivalent:
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| − | | |
| − | F1a. S <=> X = Y. :R9a
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| − | | |
| − | ::
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| − | | |
| − | F1b. S <=> {X} = {Y}. :R9b
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| − | | |
| − | ::
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| − | | |
| − | F1c. S <=> {X}(u) = {Y}(u), for all u C U. :R9c
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| − | | |
| − | ::
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| − | | |
| − | F1d. S <=> ConjUu ( {X}(u) = {Y}(u) ). :R9d
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| − | | |
| − | :R8a
| |
| − | | |
| − | ::
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| − | | |
| − | F1e. [S] = [ ConjUu ( {X}(u) = {Y}(u) ) ]. :R8b
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| − | | |
| − | :???
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| − | | |
| − | ::
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| − | | |
| − | F1f. [S] = ConjUu [ {X}(u) = {Y}(u) ]. :???
| |
| − | | |
| − | ::
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| − | | |
| − | F1g. [S] = ConjUu (( {X}(u) , {Y}(u) )). :$1a
| |
| − | | |
| − | ::
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| − | | |
| − | F1h. [S] = ConjUu (( {X} , {Y} ))$(u). :$1b
| |
| − | | |
| − | ///
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| − | | |
| − | {u C U : (f, g)$(u)}
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| − | | |
| − | = {u C U : (f(u), g(u))}
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| − | | |
| − | = {u C
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| − | | |
| − | ///
| |
| − | </pre>
| |
| − | | |
| − | =====1.3.12.2. Derived Equivalence Relations===== | |
| − | | |
| − | One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.
| |
| − | | |
| − | A classic way of showing that two sets are equal is to show that every element of the first belongs to the second and that every element of the second belongs to the first. The problem with this strategy is that one can exhaust a considerable amount of time trying to prove that two sets are equal before it occurs to one to look for a counterexample, that is, an element of the first that does not belong to the second or an element of the second that does not belong to the first, in cases where that is precisely what one ought to be seeking. It would be nice if there were a more balanced, impartial, neutral, or nonchalant way to go about this task, one that did not require such an undue commitment to either side, a technique that helps to pinpoint the counterexamples when they exist, and a method that keeps in mind the original relation of "proving that" and "showing that" to probing, testing, and seeing "whether".
| |
| − | | |
| − | A different way of seeing that two sets are equal, or of seeing whether two sets are equal, is based on the following observation:
| |
| − | | |
| − | <pre> | |
| − | Two sets are equal as sets
| |
| − | | |
| − | <=> the indicator functions of these sets are equal as functions
| |
| − | | |
| − | <=> the values of these functions are equal on all domain elements.
| |
| − | </pre>
| |
| − | | |
| − | It is important to notice the hidden quantifier, of a universal kind, that lurks in all three equivalent statements but is only revealed in the last.
| |
| − | | |
| − | In making the next set of definitions and in using the corresponding terminology it is taken for granted that all of the references of signs are relative to a particular sign relation R c OxSxI that either remains to be specified or is already understood. Further, I continue to assume that S = I, in which case this set is called the "syntactic domain" of R.
| |
| − | | |
| − | In the following definitions let R c OxSxI, let S = I, and let x, y C S.
| |
| − | | |
| − | Recall the definition of Con(R), the connotative component of R, in the following form:
| |
| − | | |
| − | : Con(R) = RSI = {< s, i> C SxI : <o, s, i> C R for some o C O}.
| |
| − | | |
| − | Equivalent expressions for this concept are recorded in Definition 8.
| |
| − | | |
| − | <pre>
| |
| − | Definition 8
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | then the following are identical subsets of SxI:
| |
| − | | |
| − | D8a. RSI
| |
| − | | |
| − | D8b. ConR
| |
| − | | |
| − | D8c. Con(R)
| |
| − | | |
| − | D8d. PrSI(R)
| |
| − | | |
| − | D8e. {< s, i> C SxI : <o, s, i> C R for some o C O}
| |
| − | </pre>
| |
| − | | |
| − | The dyadic relation RIS that constitutes the converse of the connotative relation RSI can be defined directly in the following fashion:
| |
| − | | |
| − | : Con(R)^ = RIS = {< i, s> C IxS : <o, s, i> C R for some o C O}.
| |
| − | | |
| − | A few of the many different expressions for this concept are recorded in Definition 9.
| |
| − | | |
| − | <pre>
| |
| − | Definition 9
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | then the following are identical subsets of IxS:
| |
| − | | |
| − | D9a. RIS
| |
| − | | |
| − | D9b. RSI^
| |
| − | | |
| − | D9c. ConR^
| |
| − | | |
| − | D9d. Con(R)^
| |
| − | | |
| − | D9e. PrIS(R)
| |
| − | | |
| − | D9f. Conv(Con(R))
| |
| − | | |
| − | D9g. {< i, s> C IxS : <o, s, i> C R for some o C O}
| |
| − | </pre>
| |
| − | | |
| − | Recall the definition of Den(R), the denotative component of R, in the following form:
| |
| − | | |
| − | : Den(R) = ROS = {<o, s> C OxS : <o, s, i> C R for some i C I}.
| |
| − | | |
| − | Equivalent expressions for this concept are recorded in Definition 10.
| |
| − | | |
| − | <pre>
| |
| − | Definition 10
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | then the following are identical subsets of OxS:
| |
| − | | |
| − | D10a. ROS
| |
| − | | |
| − | D10b. DenR
| |
| − | | |
| − | D10c. Den(R)
| |
| − | | |
| − | D10d. PrOS(R)
| |
| − | | |
| − | D10e. {<o, s> C OxS : <o, s, i> C R for some i C I}
| |
| − | </pre>
| |
| − | | |
| − | The dyadic relation RSO that constitutes the converse of the denotative relation ROS can be defined directly in the following fashion:
| |
| − | | |
| − | : Den(R)^ = RSO = {< s, o> C SxO : <o, s, i> C R for some i C I}.
| |
| − | | |
| − | A few of the many different expressions for this concept are recorded in Definition 11.
| |
| − | | |
| − | <pre>
| |
| − | Definition 11
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | then the following are identical subsets of SxO:
| |
| − | | |
| − | D11a. RSO
| |
| − | | |
| − | D11b. ROS^
| |
| − | | |
| − | D11c. DenR^
| |
| − | | |
| − | D11d. Den(R)^
| |
| − | | |
| − | D11e. PrSO(R)
| |
| − | | |
| − | D11f. Conv(Den(R))
| |
| − | | |
| − | D11g. {< s, o> C SxO : <o, s, i> C R for some i C I}
| |
| − | </pre>
| |
| − | | |
| − | The "denotation of x in R", written "Den(R, x)", is defined as follows:
| |
| − | | |
| − | : Den(R, x) = {o C O : <o, x> C Den(R)}.
| |
| − | | |
| − | In other words:
| |
| − | | |
| − | : Den(R, x) = {o C O : <o, x, i> C R for some i C I}.
| |
| − | | |
| − | Equivalent expressions for this concept are recorded in Definition 12.
| |
| − | | |
| − | <pre>
| |
| − | Definition 12
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | and x C S,
| |
| − | | |
| − | then the following are identical subsets of O:
| |
| − | | |
| − | D12a. ROS.x
| |
| − | | |
| − | D12b. DenR.x
| |
| − | | |
| − | D12c. DenR|x
| |
| − | | |
| − | D12d. DenR(, x)
| |
| − | | |
| − | D12e. Den(R, x)
| |
| − | | |
| − | D12f. Den(R).x
| |
| − | | |
| − | D12g. {o C O : <o, x> C Den(R)}
| |
| − | | |
| − | D12h. {o C O : <o, x, i> C R for some i C I}
| |
| − | </pre>
| |
| − | | |
| − | Signs are "equiferent" if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be "denotatively equivalent" or "referentially equivalent", but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumping to the conclusions that are implied by these latter terms.
| |
| − | | |
| − | To define the "equiference" of signs in terms of their denotations, one says that "x is equiferent to y under R", and writes "x =R y", to mean that Den(R, x) = Den(R, y). Taken in extension, this notion of a relation between signs induces an "equiference relation" on the syntactic domain.
| |
| − | | |
| − | For each sign relation R, this yields a binary relation Der(R) c SxI that is defined as follows:
| |
| − | | |
| − | : Der(R) = DerR = {<x, y> C SxI : Den(R, x) = Den(R, y)}.
| |
| − | | |
| − | These definitions and notations are recorded in the following display.
| |
| − | | |
| − | <pre>
| |
| − | Definition 13
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | then the following are identical subsets of SxI:
| |
| − | | |
| − | D13a. DerR
| |
| − | | |
| − | D13b. Der(R)
| |
| − | | |
| − | D13c. {<x,y> C SxI : DenR|x = DenR|y}
| |
| − | | |
| − | D13d. {<x,y> C SxI : Den(R, x) = Den(R, y)}
| |
| − | </pre>
| |
| − | | |
| − | The relation Der(R) is defined and the notation "x =R y" is meaningful in every situation where Den(-,-) makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation.
| |
| − | | |
| − | # Reflexive property. Is it true that x =R x for every x C S = I? By definition, x =R x if and only if Den(R, x) = Den(R, x). Thus, the reflexive property holds in any setting where the denotations Den(R, x) are defined for all signs x in the syntactic domain of R.
| |
| − | # Symmetric property. Does x =R y => y =R x for all x, y C S? In effect, does Den(R, x) = Den(R, y) imply Den(R, y) = Den(R, x) for all signs x and y in the syntactic domain S? Yes, so long as the sets Den(R, x) and Den(R, y) are well-defined, a fact which is already being assumed.
| |
| − | # Transitive property. Does x =R y & y =R z => x =R z for all x, y, z C S? To belabor the point, does Den(R, x) = Den(R, y) and Den(R, y) = Den(R, z) imply Den(R, x) = Den(R, z) for all x, y, z in S? Yes, again, under the stated conditions. | |
| − | | |
| − | It should be clear at this point that any question about the equiference of signs reduces to a question about the equality of sets, specifically, the sets that are indexed by these signs. As a result, so long as these sets are well-defined, the issue of whether equiference relations induce equivalence relations on their syntactic domains is almost as trivial as it initially appears.
| |
| − | | |
| − | Taken in its set-theoretic extension, a relation of equiference induces a "denotative equivalence relation" (DER) on its syntactic domain S = I. This leads to the formation of "denotative equivalence classes" (DEC's), "denotative partitions" (DEP's), and "denotative equations" (DEQ's) on the syntactic domain. But what does it mean for signs to be equiferent?
| |
| − | | |
| − | Notice that this is not the same thing as being "semiotically equivalent", in the sense of belonging to a single "semiotic equivalence class" (SEC), falling into the same part of a "semiotic partition" (SEP), or having a "semiotic equation" (SEQ) between them. It is only when very felicitous conditions obtain, establishing a concord between the denotative and the connotative components of a sign relation, that these two ideas coalesce.
| |
| − | | |
| − | In general, there is no necessity that the equiference of signs, that is, their denotational equivalence or their referential equivalence, induces the same equivalence relation on the syntactic domain as that defined by their semiotic equivalence, even though this state of accord seems like an especially desirable situation. This makes it necessary to find a distinctive nomenclature for these structures, for which I adopt the term "denotative equivalence relations" (DER's). In their train they bring the allied structures of "denotative equivalence classes" (DEC's) and "denotative partitions" (DEP's), while the corresponding statements of "denotative equations" (DEQ's) are expressible in the form "x =R y".
| |
| − | | |
| − | The uses of the equal sign for denoting equations or equivalences are recalled and extended in the following ways:
| |
| − | | |
| − | 1. If E is an arbitrary equivalence relation,
| |
| − | | |
| − | then the equation "x =E y" means that <x, y> C E.
| |
| − | | |
| − | 2. If R is a sign relation such that RSI is a SER on S = I,
| |
| − | | |
| − | then the semiotic equation "x =R y" means that <x, y> C RSI.
| |
| − | | |
| − | 3. If R is a sign relation such that F is its DER on S = I,
| |
| − | | |
| − | then the denotative equation "x =R y" means that <x, y> C F,
| |
| − | | |
| − | in other words, that Den(R, x) = Den(R, y).
| |
| − | | |
| − | The uses of square brackets for denoting equivalence classes are recalled and extended in the following ways:
| |
| − | | |
| − | 1. If E is an arbitrary equivalence relation,
| |
| − | | |
| − | then "[x]E" denotes the equivalence class of x under E.
| |
| − | | |
| − | 2. If R is a sign relation such that Con(R) is a SER on S = I,
| |
| − | | |
| − | then "[x]R" denotes the SEC of x under Con(R).
| |
| − | | |
| − | 3. If R is a sign relation such that Der(R) is a DER on S = I,
| |
| − | | |
| − | then "[x]R" denotes the DEC of x under Der(R).
| |
| − | | |
| − | By applying the form of Fact 1 to the special case where X = Den(R, x) and Y = Den(R, y), one obtains the following facts.
| |
| − | | |
| − | <pre>
| |
| − | Fact 2.1
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | then the following are identical subsets of SxI:
| |
| − | | |
| − | F2.1a. DerR :D13a
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1b. Der(R) :D13b
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1c. {<x, y> C SxI :
| |
| − | | |
| − | Den(R, x) = Den(R, y)
| |
| − | | |
| − | } :D13c
| |
| − | | |
| − | :R9a
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1d. {<x, y> C SxI :
| |
| − | | |
| − | {Den(R, x)} = {Den(R, y)}
| |
| − | | |
| − | } :R9b
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1e. {<x, y> C SxI :
| |
| − | | |
| − | for all o C O
| |
| − | | |
| − | {Den(R, x)}(o) = {Den(R, y)}(o)
| |
| − | | |
| − | } :R9c
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1f. {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) = {Den(R, y)}(o)
| |
| − | | |
| − | } :R9d
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1g. {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | (( {Den(R, x)}(o) , {Den(R, y)}(o) ))
| |
| − | | |
| − | } :R9e
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1h. {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | (( {Den(R, x)} , {Den(R, y)} ))$(o)
| |
| − | | |
| − | } :R9f
| |
| − | | |
| − | :D12e
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.1i. {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | (( {ROS.x} , {ROS.y} ))$(o)
| |
| − | | |
| − | } :D12a
| |
| − | </pre>
| |
| − | | |
| − | <pre>
| |
| − | Fact 2.2
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | then the following are equivalent:
| |
| − | | |
| − | F2.2a. DerR = {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | } :R11a
| |
| − | ::
| |
| − | | |
| − | F2.2b. {DerR} = { {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | }
| |
| − | | |
| − | } :R11b
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.2c. {DerR} c SxIxB
| |
| − | | |
| − | :
| |
| − | | |
| − | {DerR} = {<x, y, v> C SxIxB :
| |
| − | | |
| − | v =
| |
| − | | |
| − | [ Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | ]
| |
| − | | |
| − | } :R11c
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.2d. {DerR} = {<x, y, v> C SxIxB :
| |
| − | | |
| − | v =
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | [ {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | ]
| |
| − | | |
| − | } :Log
| |
| − | | |
| − | F2.2e. {DerR} = {<x, y, v> C SxIxB :
| |
| − | | |
| − | v =
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | (( {Den(R, x)}(o),
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | ))
| |
| − | | |
| − | } :Log
| |
| − | | |
| − | F2.2f. {DerR} = {<x, y, v> C SxIxB :
| |
| − | | |
| − | v =
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | (( {Den(R, x)},
| |
| − | | |
| − | {Den(R, y)}
| |
| − | | |
| − | ))$(o)
| |
| − | | |
| − | } :$
| |
| − | </pre>
| |
| − | | |
| − | <pre>
| |
| − | Fact 2.3
| |
| − | | |
| − | If R c OxSxI,
| |
| − | | |
| − | then the following are equivalent:
| |
| − | | |
| − | F2.3a. DerR = {<x, y> C SxI :
| |
| − | | |
| − | Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | } :R11a
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.3b. {DerR} : SxI -> B
| |
| − | | |
| − | :
| |
| − | | |
| − | {DerR}(x, y) = [ Conj(o C O)
| |
| − | | |
| − | {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | ] :R11d
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.3c. {DerR}(x, y) = Conj(o C O)
| |
| − | | |
| − | [ {Den(R, x)}(o) =
| |
| − | | |
| − | {Den(R, y)}(o)
| |
| − | | |
| − | ] :Log
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.3d. {DerR}(x, y) = Conj(o C O)
| |
| − | | |
| − | [ {DenR}(o, x) =
| |
| − | | |
| − | {DenR}(o, y)
| |
| − | | |
| − | ] :Def
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.3e. {DerR}(x, y) = Conj(o C O)
| |
| − | | |
| − | (( {DenR}(o, x),
| |
| − | | |
| − | {DenR}(o, y)
| |
| − | | |
| − | )) :Log
| |
| − | | |
| − | :D10b
| |
| − | | |
| − | ::
| |
| − | | |
| − | F2.3f. {DerR}(x, y) = Conj(o C O)
| |
| − | | |
| − | (( {ROS}(o, x),
| |
| − | | |
| − | {ROS}(o, y)
| |
| − | | |
| − | )) :D10a
| |
| − | </pre> | |
| − | | |
| − | =====1.3.12.3. Digression on Derived Relations===== | |
| − | | |
| − | A better understanding of derived equivalence relations (DER's) can be achieved by placing their constructions within a more general context, and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation R into a dyadic relation Der(R), with other types of operations on triadic relations. The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations.
| |
| − | | |
| − | To that end, let the derivation Der(R) be expressed in the following way:
| |
| − | | |
| − | : {DerR}(x, y) = Conj(o C O) (( {RSO}(x, o) , {ROS}(o, y) )).
| |
| − | | |
| − | From this abstract a form of composition, temporarily notated as "P#Q", where P c XxM and Q c MxY are otherwise arbitrary dyadic relations, and where P#Q c XxY is defined as follows:
| |
| − | | |
| − | : {P#Q}(x, y) = Conj(m C M) (( {P}(x, m) , {Q}(m, y) )).
| |
| − | | |
| − | Compare this with the usual form of composition, typically notated as "P.Q" and defined as follows:
| |
| | | | |
| − | : {P.Q}(x, y) = Disj(m C M) ( {P}(x, m) . {Q}(m, y) ).
| + | =====1.3.12.3. Digression on Derived Relations <big>✔</big>===== |
