[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D4

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Philosophy of Notation:  Formal Terms and Flexible Types

| Where number is irrelevant, regimented mathematical technique has hitherto
| tended to be lacking.  Thus it is that the progress of natural science has
| depended so largely upon the discernment of measurable quantity of one sort
| or another.
|
| W.V. Quine, 'Mathematical Logic', [Qui, 7]

For much of our discussion propositions and boolean functions are treated as the
same formal objects, or as different interpretations of the same formal calculus.
This rule of has exceptions, though.  There is a distinctively logical interest
in the use of propositional calculus that is not exhausted by its functional
interpretation.  It is part of our task in this study to deal with these
uniquely logical characteristics as they present themselves both in our
subject matter and in our formal calculus.  Just to provide a hint of
what's at stake:  In logic, as opposed to the more imaginative realms
of mathematics, we consider it a good thing to always know what we are
talking about.  Where mathematics encourages tolerance for uninterpreted
symbols as intermediate terms, logic exerts a keener effort to interpret
directly each oblique carrier of meaning, no matter how slight, and to
unfold the complicities of every indirection in the flow of information.
Translated into functional terms, this means that we want to maintain a
continual, immediate, and persistent sense of both the converse relation
f^(-1) c B x B^n, or what is the same thing, f^(-1) : B -> Pow(B^n), and
the "fibers" or inverse images f^(-1)(0) and f^(-1)(1), associated with
each boolean function f : B^n -> B that we use.  In practical terms, the
desired implementation of a propositional interpreter should incorporate
our intuitive recognition that the induced partition of the functional
domain into level sets f^(-1)(b), for b in B, is part and parcel of
understanding the denotative uses of each propositional function f.

Jon Awbrey

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