FOM: Intuitionism (Tait)

[Dean Buckner]

William Tait gives a neat argument (23 May 2002) that the proposition that p
is the same as the proposition that it is provable that p.

"What would prove "P is provable" other than a proof of P".

He means (I believe), that the proof that p can be proved = the proof that
p, and thus (by substitution), p = p can be proved.

I am unable to see quite what is wrong with the argument, but the conclusion
hardly seems correct.  E.g. if Bill says that grass is green, does he also
say that it is provable that grass is green?

A proof that p is a set of symbols, a picture, a form of words or even a
fact (such as grass being green) that expresses the proposition in such a
way that makes it clear or evident or obvious that p.  It is the manner
or mode of expression of the proposition, rather than anything in the
proposition itself, that constitutes the proof.  If all propositions
themselves contained this clarity and self-evidence, we wouldn't need to
prove them.  So

    that p can be proved = that there is a proof-like way M of showing that
p.

If there are ways (and therefore different ways) of showing the same
proposition, it follows that the proposition, and the showing of it, cannot
be the same thing.  Whereas showing something, and showing that you are
showing something, are more closely related.  Just as to know, is to know
that you know (Sartre / Alain).





Dean Buckner
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