Directory talk:Jon Awbrey/Papers/Syntactic Transformations

Alternate Version : Needs To Be Reconciled

1.3.12. Syntactic Transformations

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.

The diverse notions of indication presently under discussion are expressed in a variety of different notations, enumerated as follows:

1. The functional language of propositions
2. The logical language of sentences
3. The geometric language of sets

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.

1.3.12.1. Syntactic Transformation Rules

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.

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:

o-------------------------------------------------o
| Definition 1.  Indicator Function               |
o-------------------------------------------------o
|                                                 |
| If      Q   c X,                                |
|                                                 |
| then  -{Q}- : X -> %B%                          |
|                                                 |
| such that, for all x in X:                      |
|                                                 |
o-------------------------------------------------o
|                                                 |
| D1a.  -{Q}-(x)  <=>  x in Q.                    |
|                                                 |
o-------------------------------------------------o

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.

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.

o-------------------------------------------------o
| Rule 1                                          |
o-------------------------------------------------o
|                                                 |
| If      Q   c X,                                |
|                                                 |
| then  -{Q}- : X -> %B%,                         |
|                                                 |
| and if  x  in X,                                |
|                                                 |
| then the following are equivalent:              |
|                                                 |
o-------------------------------------------------o
|                                                 |
| R1a.   x in Q.                                  |
|                                                 |
| R1b.  -{Q}-(x).                                 |
|                                                 |
o-------------------------------------------------o

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.