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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 22:00, 27 January 2009

138 bytes removed ,  22:00, 27 January 2009
→‎Work Area
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==Work Area==
 
==Work Area==
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===Logical Translation Rule 1===
    
<pre>
 
<pre>
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<br>
 
<br>
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===Geometric Translation Rule 1===
    
<pre>
 
<pre>
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<br>
 
<br>
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===Logical Translation Rule 2===
    
<pre>
 
<pre>
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<br>
 
<br>
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===Geometric Translation Rule 2===
    
<pre>
 
<pre>
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<br>
 
<br>
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===Value Rule 1===
    
<pre>
 
<pre>
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<br>
 
<br>
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===Value Rule 1===
    
<pre>
 
<pre>
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V1c. (( v , w )).
 
V1c. (( v , w )).
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</pre>
     −
<pre>
   
A rule that allows one to turn equivalent sentences into identical propositions:
 
A rule that allows one to turn equivalent sentences into identical propositions:
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V1c. (( f , g ))$
 
V1c. (( f , g ))$
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</pre>
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===Evaluation Rule 1===
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<pre>
 
Evaluation Rule 1
 
Evaluation Rule 1
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::
 
::
 
E1d. (( f , g ))$(u). :$1b
 
E1d. (( f , g ))$(u). :$1b
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</pre>
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===Definition 2===
    +
<pre>
 
Definition 2
 
Definition 2
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D2b. u C X  <=>  u C Y, for all u C U.
 
D2b. u C X  <=>  u C Y, for all u C U.
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</pre>
    +
===Definition 3===
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<pre>
 
Definition 3
 
Definition 3
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D3b. f(u) = g(u), for all u C U.
 
D3b. f(u) = g(u), for all u C U.
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</pre>
    +
===Definition 4===
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<pre>
 
Definition 4
 
Definition 4
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D4b. {<u, v> C UxB : v = [u C X]}
 
D4b. {<u, v> C UxB : v = [u C X]}
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</pre>
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===Definition 5===
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<pre>
 
Definition 5
 
Definition 5
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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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</pre>
   −
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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===Definition 6===
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<pre>
 
Definition 6
 
Definition 6
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D6e. ConjJj Sj.
 
D6e. ConjJj Sj.
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</pre>
    +
===Definition 7===
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<pre>
 
Definition 7
 
Definition 7
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D7b. [S] = [T].
 
D7b. [S] = [T].
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</pre>
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===Rule 5===
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<pre>
 
Rule 5
 
Rule 5
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If X, Y c U,
 
If X, Y c U,
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::
 
::
 
R5e. {X} = {Y}. :D5a
 
R5e. {X} = {Y}. :D5a
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</pre>
    +
===Rule 6===
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 +
<pre>
 
Rule 6
 
Rule 6
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::
 
::
 
R6c. ConjUu (f(u) = g(u)). :D6e
 
R6c. ConjUu (f(u) = g(u)). :D6e
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</pre>
    +
===Rule 7===
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<pre>
 
Rule 7
 
Rule 7
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::
 
::
 
R7f. ConjUu (( P , Q ))$(u). :$1b
 
R7f. ConjUu (( P , Q ))$(u). :$1b
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</pre>
    +
===Rule 8===
    +
<pre>
 
Rule 8
 
Rule 8
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::
 
::
 
R8g. ConjUu (( [S] , [T] ))$(u). :R7f
 
R8g. ConjUu (( [S] , [T] ))$(u). :R7f
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</pre>
   −
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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===Rule 9===
    +
<pre>
 
Rule 9
 
Rule 9
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::
 
::
 
R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
 
R9g. ConjUu (( {X} , {Y} ))$(u). :R7f
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</pre>
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===Rule 10===
    +
<pre>
 
Rule 10
 
Rule 10
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::
 
::
 
R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
 
R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
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</pre>
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===Rule 11===
    +
<pre>
 
Rule 11
 
Rule 11
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R11e. {X} = [S]. :R
 
R11e. {X} = [S]. :R
 
</pre>
 
</pre>
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===Fact 1===
    
<pre>
 
<pre>
Jon Awbrey
12,080

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