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: f(u) = [u C X], for all u C U. | : f(u) = [u C X], for all u C U. | ||
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======Facts====== | ======Facts====== | ||
Revision as of 02:42, 11 September 2010
Alternate Version : Needs To Be Reconciled
1.3.12. Syntactic Transformations ✔
1.3.12.1. Syntactic Transformation Rules
Value 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
Definitions
Definition 5
If X c U,
then the following are identical propositions:
D5a. {X}.
D5b. f : U -> B
: f(u) = [u C X], for all u C U.
Facts
Applying Rule 9, Rule 8, and the Logical Rules to the special case where S <=> (X = Y), one obtains the following general fact.
Fact 1
If X,Y c U,
then the following are equivalent:
F1a. S <=> X = Y. :R9a
::
F1b. S <=> {X} = {Y}. :R9b
::
F1c. S <=> {X}(u) = {Y}(u), for all u C U. :R9c
::
F1d. S <=> ConjUu ( {X}(u) = {Y}(u) ). :R9d
:R8a
::
F1e. [S] = [ ConjUu ( {X}(u) = {Y}(u) ) ]. :R8b
:???
::
F1f. [S] = ConjUu [ {X}(u) = {Y}(u) ]. :???
::
F1g. [S] = ConjUu (( {X}(u) , {Y}(u) )). :$1a
::
F1h. [S] = ConjUu (( {X} , {Y} ))$(u). :$1b
///
{u C U : (f, g)$(u)}
= {u C U : (f(u), g(u))}
= {u C
///
