# Directory talk:Jon Awbrey/Papers/Syntactic Transformations

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#### 1.3.12. Syntactic Transformations ✔

##### 1.3.12.1. Syntactic Transformation Rules

Value Rule 1 If v, w C B then "v = w" is a sentence about <v, w> C B2, [v = w] is a proposition : B2 -> B, and the following are identical values in B: V1a. [ v = w ](v, w) V1b. [ v <=> w ](v, w) V1c. ((v , w))

Value Rule 1 If v, w C B, then the following are equivalent: V1a. v = w. V1b. v <=> w. V1c. (( v , w )).

A rule that allows one to turn equivalent sentences into identical propositions:

- (S <=> T) <=> ([S] = [T])

Consider [ v = w ](v, w) and [ v(u) = w(u) ](u)

Value Rule 1 If v, w C B, then the following are identical values in B: V1a. [ v = w ] V1b. [ v <=> w ] V1c. (( v , w ))

Value Rule 1 If f, g : U -> B, and u C U then the following are identical values in B: V1a. [ f(u) = g(u) ] V1b. [ f(u) <=> g(u) ] V1c. (( f(u) , g(u) ))

Value Rule 1 If f, g : U -> B, then the following are identical propositions on U: V1a. [ f = g ] V1b. [ f <=> g ] V1c. (( f , g ))$

Evaluation Rule 1 If f, g : U -> B and u C U, then the following are equivalent: E1a. f(u) = g(u). :V1a :: E1b. f(u) <=> g(u). :V1b :: E1c. (( f(u) , g(u) )). :V1c :$1a :: E1d. (( f , g ))$(u). :$1b

Evaluation Rule 1 If S, T are sentences about things in the universe U, f, g are propositions: U -> B, and u C U, then the following are equivalent: E1a. f(u) = g(u). :V1a :: E1b. f(u) <=> g(u). :V1b :: E1c. (( f(u) , g(u) )). :V1c :$1a :: E1d. (( f , g ))$(u). :$1b