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Directory talk:Jon Awbrey/Papers/Syntactic Transformations (view source)

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=====1.3.12.1.  Syntactic Transformation Rules=====
 
=====1.3.12.1.  Syntactic Transformation Rules=====
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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.
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As another example of a ROST, consider the following logical equivalence, that holds for any <math>X \subseteq U\!</math> and for all <math>u \in U.</math>
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: -{X}-(u)  <=>  -{X}-(u) = 1.
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In practice, this logical equivalence is used to exchange an expression of the form "-{X}-(u)" with a sentence of the form "-{X}-(u) = 1" in any context where one has a relatively fixed X c U in mind and where one is conceiving u in U to vary over its whole domain, namely, the universe U.  This leads to the ROST that is given in Rule 2.
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<pre>
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o-------------------------------------------------o
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| Rule 2                                          |
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o-------------------------------------------------o
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|                                                |
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| If f : U -> %B%                                |
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|                                                |
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| and u in U,                                    |
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|                                                |
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| then the following are equivalent:              |
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|                                                |
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o-------------------------------------------------o
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|                                                |
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| R2a.  f(u).                                    |
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|                                                |
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| R2b.  f(u) = 1.                                |
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|                                                |
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o-------------------------------------------------o
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</pre>
      
Rules like these can be chained together to establish extended rules, just so long as their antecedent conditions are compatible.  For example, Rules 1 and 2 combine to give the equivalents that are listed in Rule 3.  This follows from a recognition that the function -{X}- : U -> %B% that is introduced in Rule 1 is an instance of the function f : U -> %B% that is mentioned in Rule 2.  By the time one arrives in the "consequence box" of either Rule, then, one has in mind a comparatively fixed X c U, a proposition f or -{X}- about things in U, and a variable argument u in U.
 
Rules like these can be chained together to establish extended rules, just so long as their antecedent conditions are compatible.  For example, Rules 1 and 2 combine to give the equivalents that are listed in Rule 3.  This follows from a recognition that the function -{X}- : U -> %B% that is introduced in Rule 1 is an instance of the function f : U -> %B% that is mentioned in Rule 2.  By the time one arrives in the "consequence box" of either Rule, then, one has in mind a comparatively fixed X c U, a proposition f or -{X}- about things in U, and a variable argument u in U.
Jon Awbrey
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