Thematics Of Order

[Jon Awbrey]

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TOO.  Note 3

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Re: TOO. http://stderr.org/pipermail/inquiry/2005-April/thread.html#2541

Before we stray too far from the triple point, let's return to
the idea of reflective order closure, whereby particular types
of statements about a given order are associated with elements
of that order.  As a rule, we would like the resulting element
of the order to give a measure of "how true" the corresponding
statement about the order happens to be.

There is of course a very simple example of reflective order closure
within the additive realm.  A basic type of statement about the order
takes the form "x < y", while the difference y - x provides a measure
of just how true the order statement actually is.

But what we are seeking here is a connection between the additive order,
at its simplest expressed by statements like "x < y", and the qualities
of the numbers x and y in multiplicative, or primes factorization terms.

The situation in propositional logic appears to be at least partly analogous
to the additive case, since a basic type of statement about the implicational
order takes the form "x => y", while a logical equivalent of this implication
is "~x or y", in this way comparable to the difference.  In the logical realm,
however, we have nice dualities between the additive and multiplicative forms --
for instance, "x => y" is also equivalent to "~(x & ~y)" -- dualities that we
lack in the case of natural number arithmetic.

Will have to leave it at that for today ...

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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