Sun, 25 Oct 1998 17:12:06 +0100
> I was wondering if anybody can give me a
> sensible explanation of the reason for basing most logics on several
> certain operators.
Simplicity? See for instance how these operators arise simply
in a category-theory reconstruction of intuitionnistic logic/lambda-calculus.
For a more thorough discussion of the notion of simplicity/complexity,
see "An Introduction to Kolmogorov Complexity and its Applications"
by Li & Vitanyi.
> It seems that this operator-class constitutes some sort of dirty
> reflectivity with a limited (unary) scope.
Unarity is usually chosen because it leads
to a simple finitary presentation of the logic,
which allows for easier reflection.
Now, there are *lots* of variants of various calculi
with varying arity of operators.
For instance, people involved in Pi-calculi usually prefer polyadic variants.
> "For all variables x, A(...x...) is true" is equivalent to "There does
> not exist a variable x such that A(...x...) is not true"
This only holds in classical logic, but not in most intuitionnistic logic.
> these operators are declarations about symbols, not intensions.
As soon as you've got internal notions (be it in a meta-language),
you're dealing with extensions. Intension is an ever fleeing concept.
Whenever you closely look at intensions, they become extensions.
It only becomes back an intension when you don't look at it.
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