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Re: Whitehead & Russell on Reductio A.A. and when Constructivism is
mainstream #200; 2nd ed; Correcting Math
Posted:
Oct 12, 2009 1:51 AM
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On Oct 11, 12:07 pm, Marshall <marshall.spi...@gmail.com> wrote:
|OK. I understood that. This is a redefinition of the symbols
|from their more usual meaning, though, isn't it? Or perhaps
|an overlay on top of and in addition to the usual meaning.
|It seems a confusing terminological approach, but every
|human endeavor is full of such.
Well, "more usual" seems to mean essentially "most popular".
Do we want to make it a popularity contest? The intuitionist
interpretation of the logical connectives and quantifiers is
much more natural than it seems at first glance if you're
only familiar with classical logic (the "usual").
|Perhaps I could re-ask my question in a slightly different
|way? Is it possible for a mathematical structure
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|http://en.wikipedia.org/wiki/Structure_(mathematical_logic)
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|to exist in which (?x)~Fx is false and (?x)Fx is also false?
|It seems to me that it is clearly not possible.
It's intuitionistically valid to reason that if (Ex)Fx is
false then (Ax)~Fx is true. So indeed, they cannot both
be false.
Intuitionism distinguishes between "they cannot both be
false" and "one of them is true". If the two statements
are A and B, then "they cannot both be false" is
~(~A & ~B) while "one of them is true is A v B. The
former is equivalent to ~~(A v B). The latter implies
that there's a way to find one of them that is true.
|Sure, but does it then follow that there is no useful
|distinction to be had between knowing how to find
|a value x such that Fx and knowing that such a value
|exists? It seems to me that such a distinction is a
|useful one, in which case the constructivist stricture
|seems dubious, unreasonable, anti-pragmatic.
The distinction is still there. You could for example
be told by a reliable person that something exists
without being told how to find it.
Intuitionism and constructive mathematics generally
is more systematic about making distinctions, not
less. When you "know that a value exists" classically,
you are making some kind of discovery, which you have
arranged to describe using "existential" language.
If (as is typically the case) what has happened is that
you've found a contradiction arises from assuming that
no such value exists, then might it not be more
illuminating to say ~(Ax) ~Fx rather than Ex Fx?
That strikes me as being a very direct way of stating
what you know, as opposed to "one exists but I don't
know how to find it", which could be the case if
someone else has found out how to find it but only
told you that it is possible. The apparent advantage
of using classical reasoning is just that you get to
"simplify" ~(Ax)~Fx down to (Ex)Fx, but then have this
side explanation for why (Ex)Fx has a weaker meaning
for you than constructive mathematics gives it. And
then you have this additional way of boosting up the
meaning of the quantifier by saying, "and I know how
to find it". I don't find it obviously easier to go
through all those maneuvers just to say either that
~(Ax)~Fx or (Ex)Fx.
Keith Ramsay
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