[FOM] intuitionism and the liar paradox
nweaver at math.wustl.edu
Wed Apr 21 08:29:31 EDT 2010
In response to Panu Raatikainen's message (number 014566):
I may have misunderstood your position. Is it fair to say that your
objection is not to the semantic notion of proof, but rather to the
identification of mathematical truth with provability? And your
complaint is that explicating the notion of provability requires a
prior understanding of mathematical truth.
Well, this might not affect my account of the liar paradox. I don't
need to equate mathematical truth with provability (and I explicitly
don't do so; see the "I do not see anything wrong" paragraph in Section
2.8 of my paper). I just need to equate the heuristic self-applicative
truth concept appearing in the liar sentence with provability.
In any case, I still don't see the circularity you allege, assuming we
agree that proof validity might not be decidable. In the language of
your paper, we may postulate an "objective realm" of possible proofs
(say, all the grammatical expressions of some well-defined language)
but we deny that it must determinately be the case that each expression
is or is not a valid proof. Then we lack obviously lack bivalence,
with no need to appeal to a proof interpretation of "or".
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