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Directory talk:Jon Awbrey/Papers/Syntactic Transformations (view source)
Revision as of 14:58, 12 September 2010
, 14:58, 12 September 2010→1.3.12.1. Syntactic Transformation Rules: del merged text
==Bump==
<div class="nonumtoc">__TOC__</div>
==Alternate Version : Needs To Be Reconciled==
====1.3.12. Syntactic Transformations <big>✔</big>====
=====1.3.12.1. Syntactic Transformation Rules=====
<pre>
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))
</pre>
<pre>
Value Rule 1
If v, w C B,
then the following are equivalent:
V1a. v = w.
V1b. v <=> w.
V1c. (( v , w )).
</pre>
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)
<pre>
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 ))
</pre>
<pre>
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) ))
</pre>
<pre>
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 ))$
</pre>
<pre>
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
</pre>
<pre>
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
</pre>
=====1.3.12.2. Derived Equivalence Relations <big>✔</big>=====
=====1.3.12.3. Digression on Derived Relations <big>✔</big>=====