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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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| − | If S is a sentence
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| − | about things in the universe U
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| − | and P is a proposition : U -> B, such that:
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| − | L1a. [S] = P,
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| − | then the following equations hold:
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| − | L1b00. [False] = () = 0 : U->B.
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| − | L1b01. [Not S] = ([S]) = (P) : U->B.
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| − | L1b10. [S] = [S] = P : U->B.
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| − | L1b11. [True] = (()) = 1 : U->B.
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| − | </pre>
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| − | <pre>
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| − | Geometric Translation Rule 1
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| − | If X c U
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| − | and P : U -> B, such that:
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| − | G1a. {X} = P,
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| − |
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| − | then the following equations hold:
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| − | G1b00. {{}} = () = 0 : U->B.
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| − | G1b10. {~X} = ({X}) = (P) : U->B.
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| − | G1b01. {X} = {X} = P : U->B.
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| − | G1b11. {U} = (()) = 1 : U->B.
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| − | </pre>
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| − | <pre>
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| − | Logical Translation Rule 2
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| − | If S, T are sentences
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| − | about things in the universe U
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| − | and P, Q are propositions: U -> B, such that:
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| − | L2a. [S] = P and [T] = Q,
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| − |
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| − | then the following equations hold:
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| − | L2b00. [False] = () = 0 : U->B.
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| − | L2b01. [Neither S nor T] = ([S])([T]) = (P)(Q).
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| − | L2b02. [Not S, but T] = ([S])[T] = (P) Q.
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| − | L2b03. [Not S] = ([S]) = (P).
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| − | L2b04. [S and not T] = [S]([T]) = P (Q).
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| − | L2b05. [Not T] = ([T]) = (Q).
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| − | L2b06. [S or T, not both] = ([S], [T]) = (P, Q).
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| − | L2b07. [Not both S and T] = ([S].[T]) = (P Q).
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| − | L2b08. [S and T] = [S].[T] = P.Q.
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| − | L2b09. [S <=> T] = (([S], [T])) = ((P, Q)).
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| − | L2b10. [T] = [T] = Q.
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| − | L2b11. [S => T] = ([S]([T])) = (P (Q)).
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| − | L2b12. [S] = [S] = P.
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| − | L2b13. [S <= T] = (([S]) [T]) = ((P) Q).
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| − | L2b14. [S or T] = (([S])([T])) = ((P)(Q)).
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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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| − | If X, Y c U
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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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| − | G2b00. {{}} = () = 0 : U->B.
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| − | G2b01. {~X n ~Y} = ({X})({Y}) = (P)(Q).
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| − | G2b02. {~X n Y} = ({X}){Y} = (P) Q.
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| − | G2b03. {~X} = ({X}) = (P).
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| − | G2b04. {X n ~Y} = {X}({Y}) = P (Q).
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| − | G2b05. {~Y} = ({Y}) = (Q).
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| − | G2b06. {X + Y} = ({X}, {Y}) = (P, Q).
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| − | G2b07. {~(X n Y)} = ({X}.{Y}) = (P Q).
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| − | G2b08. {X n Y} = {X}.{Y} = P.Q.
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| − | G2b09. {~(X + Y)} = (({X}, {Y})) = ((P, Q)).
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| − | G2b10. {Y} = {Y} = Q.
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| − | G2b11. {~(X n ~Y)} = ({X}({Y})) = (P (Q)).
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| − | G2b12. {X} = {X} = P.
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| − | G2b13. {~(~X n Y)} = (({X}) {Y}) = ((P) Q).
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| − | G2b14. {X u Y} = (({X})({Y})) = ((P)(Q)).
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| − | G2b15. {U} = (()) = 1 : U->B.
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| − | </pre>
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| | <pre> | | <pre> |
