Changes

=====1.3.10.14.  Syntactic Transformations=====

=====1.3.10.14.  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, in turn, requires a number of auxiliary concepts and notations.

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, in turn, requires

a number of auxiliary concepts and notations.

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.  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.

of sets, and the logical language of sentences.  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 asssociated families of notation.  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" (ROST's)

that partially formalize the translations in question.

Rudimentary examples of ROST's are readily mined from the

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:

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

o-------------------------------------------------o

| Definition 1.  Indicator Function              |

| Definition 1.  Indicator Function              |

|                                                |

|                                                |

o-------------------------------------------------o

o-------------------------------------------------o

In practice, a definition like this is commonly used to substitute

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.

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

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.

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

o-------------------------------------------------o

| Rule 1                                          |

| Rule 1                                          |

|                                                |

|                                                |

o-------------------------------------------------o

o-------------------------------------------------o

Conversely, any rule of this sort, properly qualified by the

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.

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.

==Where I Left Off In June 2004==

==Where I Left Off In June 2004==