Difference between revisions of "Directory talk:Jon Awbrey/Papers/Syntactic Transformations"

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==Alternate Version : Needs To Be Reconciled==
 
==Alternate Version : Needs To Be Reconciled==
  
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====1.3.12.  Syntactic Transformations====
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====1.3.12.  Syntactic Transformations <big>&#10004;</big>====
  
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We have been examining several distinct but closely related notions of indication.  To discuss the import of these ideas in greater depth, it serves to establish a number of logical relations and set-theoretic identities that can be found to hold among their roughly parallel arrays of conceptions and constructions.  Facilitating this task requires in turn a number of auxiliary concepts and notations.
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=====1.3.12.1.  Syntactic Transformation Rules=====
 +
 
 +
<pre>
 +
Value Rule 1
 +
 
 +
If v, w C B
  
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The diverse notions of indication presently under discussion are expressed in a variety of different notations, enumerated as follows:
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then "v = w" is a sentence about <v, w> C B2,
  
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# The functional language of propositions
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[v = w] is a proposition : B2 -> B,
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# The logical language of sentences
 
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# The geometric language of sets
 
  
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Correspondingly, one way to explain the relationships that exist among the various notions of indication is to describe the translations that they induce among the associated families of notation.
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and the following are identical values in B:
  
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=====1.3.12.1.  Syntactic Transformation Rules=====
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V1a. [ v = w ](v, w)
  
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A good way to summarize the necessary translations between different styles of indication, and along the way to organize their use in practice, is by means of the ''rules of syntactic transformation'' (ROSTs) that partially formalize the translations in question.
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V1b. [ v <=> w ](v, w)
  
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Rudimentary examples of ROSTs are readily mined from the raw materials that are already available in this area of discussion. To begin as near the beginning as possible, let the definition of an indicator function be recorded in the following form:
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V1c. ((v , w))
 +
</pre>
  
 
<pre>
 
<pre>
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o-------------------------------------------------o
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Value Rule 1
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| Definition 1.  Indicator Function              |
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o-------------------------------------------------o
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If v, w C B,
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|                                                |
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| If     Q  c X,                               |
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then the following are equivalent:
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|                                                |
+
 
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| then -{Q}- : X -> %B%                          |
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V1a. v = w.
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|                                                |
+
 
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| such that, for all x in X:                      |
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V1b. v <=> w.
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|                                                |
+
 
−
o-------------------------------------------------o
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V1c. (( v , w )).
−
|                                                |
 
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| D1a. -{Q}-(x)  <=> x in Q.                   |
 
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|                                                |
 
−
o-------------------------------------------------o
 
 
</pre>
 
</pre>
  
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In practice, a definition like this is commonly used to substitute one of two logically equivalent expressions or sentences for the other in a context where the conditions of using the definition in this way are satisfied and where the change is perceived as potentially advancing a proof.  The employment of a definition in this way can be expressed in the form of a ROST that allows one to exchange two expressions of logically equivalent forms for one another in every context where their logical values are the only consideration.  To be specific, the ''logical value'' of an expression is the value in the boolean domain %B% = {%0%, %1%} that the expression represents to its context or that it stands for in its context.
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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 ]
  
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In the case of Definition 1, the corresponding ROST permits one to exchange a sentence of the form "x in Q" with an expression of the form "-{Q}-(x)" in any context that satisfies the conditions of its use, namely, the conditions of the definition that lead up to the stated equivalence.  The relevant ROST is recorded in Rule 1.  By way of convention, I list the items that fall under a rule in rough order of their ascending conceptual subtlety or their increasing syntactic complexity, without regard for the normal or the typical orders of their exchange, since this can vary from widely from case to case.
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V1c. (( v , w ))
 +
</pre>
  
 
<pre>
 
<pre>
−
o-------------------------------------------------o
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Value Rule 1
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| Rule 1                                         |
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o-------------------------------------------------o
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If f, g : U -> B,
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|                                                |
+
 
−
| If     Q  c X,                               |
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and u C U
−
|                                                |
+
 
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| then  -{Q}- : X -> %B%,                         |
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then the following are identical values in B:
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|                                                |
+
 
−
| and if  x  in X,                                |
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V1a. [ f(u) = g(u) ]
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|                                                |
+
 
−
| then the following are equivalent:             |
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V1b. [ f(u) <=> g(u) ]
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|                                                |
+
 
−
o-------------------------------------------------o
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V1c. (( f(u) , g(u) ))
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|                                                |
 
−
| R1a.   x in Q.                                  |
 
−
|                                                |
 
−
| R1b. -{Q}-(x).                                |
 
−
|                                                |
 
−
o-------------------------------------------------o
 
 
</pre>
 
</pre>
  
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Conversely, any rule of this sort, properly qualified by the conditions under which it applies, can be turned back into a summary statement of the logical equivalence that is involved in its application. This mode of conversion between a static principle and a transformational rule, in other words, between a statement of equivalence and an equivalence of statements, is so automatic that it is usually not necessary to make a separate note of the "horizontal" versus the "vertical" versions of what amounts to the same abstract principle.
+
<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>
  
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=====1.3.12.2.  Derived Equivalence Relations=====
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=====1.3.12.2.  Derived Equivalence Relations <big>&#10004;</big>=====
  
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=====1.3.12.3.  Digression on Derived Relations=====
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=====1.3.12.3.  Digression on Derived Relations <big>&#10004;</big>=====

Latest revision as of 14:58, 12 September 2010

Alternate Version : Needs To Be Reconciled

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
1.3.12.2. Derived Equivalence Relations ✔
1.3.12.3. Digression on Derived Relations ✔
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