[FOM] intuitionism and the liar paradox
Panu Raatikainen
panu.raatikainen at helsinki.fi
Mon Apr 19 00:22:04 EDT 2010
"Nik Weaver" responds to my challenge to explain what, more exactly,
is the intuitionistic understanding of truth, with a reference to my
paper:
http://www.mv.helsinki.fi/home/praatika/conceptions%20of%20truth%20in%20intuitionism.pdf
> This criticism isn't really specific to my paper. It doesn't address
> my central insight that there is a problem with the inference "A is
> true implies A" under the proof interpretation of the logical symbols.
Fair enough. I should certainly think more about Weaver's specific
ideas. I find the "central insight" at least very interestning.
However, isn't that inference needed in the quite standard
intuitionistic reasoning, which derives the Law of the Excluded Middle
and the Principle of Bivalence from each other?
> Rather, the paper Panu cites is a broad critique of the semantic
> notion of a valid proof.
I am afraid I did not understand why this would be the case.
> Turning to his "challenge", I have looked at his paper and I don't
> see the circularity. His argument at the bottom of page 140 seems to
> presume that proof validity is decidable, which I don't accept. If I
> am mistaken that decidability is assumed here, perhaps he can clarify.
Very interesting, again. But now we need to explain why it has been so
common among intuitionists to insist it must be decidable, and why
exactly they are mistaken.
My reflection about decidability was just a further worry.
My central worry about circularity is this: if mathematical truth is
explicated as provability, which means just that it would be possible
to prove the relevant statement, one can then ask exactly what kind of
notion of possibility is assumed here; and the plausible answers seem,
one way or other, to lean on the notion of mathematical truth. And
that would certainly be circular...
Perhaps this worry can be circumvented - but so far, I haven't seen
any convincing answers...
All the Best
Panu
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