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Re: intuitionism
Posted:
Oct 25, 2009 2:02 AM
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On Oct 24, 11:01 am, nukeymusic <nukeymu...@gmail.com> wrote:
|Is the following a correct interpretation of what Brouwer said
|concerning the principle of the excluded third?:
|there exist p : (p or not(p)) is not equal to 1
Brouwer was deeply suspicious of the possibility of
formalism becoming an end in itself. So we should be
careful to know what you mean to say by this.
There is a sense in which for a statement to be "equal"
to 1 or T (verum) just means that it is true. Being not
equal could well mean being false or absurd. But Brouwer
did not assert the absurdity of "p or not p" for any p.
In formal terms, ~(p or ~p) is unacceptable in
intuitionistic logic; one can prove in fact for each p
that ~~(p or ~p).
There are formalizations of intuitionism in which
"~(for all p)(p or ~p)" is a theorem in fact, even
though "(for all p)~~(p or ~p)" is also a theorem.
I think it's a much better description of Brouwer's
point of view to say that it's not justified to assert
"p or not p" for every p. If you can find his collected
works try an early paper titled, ""The untrustworthiness
of the principles of logic" and get it just the way
that he put it himself.
Keith Ramsay
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