[FOM] How much of math is logic?
rgheck at brown.edu
Sun Feb 25 13:21:15 EST 2007
> Logicism seems to be considered passe nowadays, but I have not found the arguments against it convincing.
> Treating "logical validity" as an undefined term which I would like to understand better, I request examples of the folllowing:
> 1) A theorem of Peano Arithmetic which is not equivalent to a logical validity
The obvious answer would be, say: ~(Ex)(0 = Sx). That's not logically
valid and is therefore not equivalent to any logical validity, given
what is usually meant by "logically valid". Or (to channel for George
Boolos): the conjunction of ~(Ex)(0 = Sx) and (x)(y)(Sx = Sy --> x=y)
has no finite models and so is not logically valid on any understanding
of "logically valid" presently available to us.
> 2) A theorem of ZF without the axiom of Infinity which is not equivalent to a logical validity
And of course similar answers can be given here.
So, obviously, you mean something else by "logically valid", but then
> 4) An argument against second-order logic with standard semantics that does not simply amount to "a logic should have a well-behaved proof theory or a complete deductive calculus" or "since second-order logical validity depends on which set-theoretical assumptions are true it is not really logical validity".
There's an interesting paper by Peter Koellner that offers such an
argument. I'm not really qualified to evaluate it. Feferman has some
things to say about this in "Logic, Logics, and Logicism". Two of the
three considerations he offers are ones in which you're not interested,
but there's also a general approach in terms of a broadly Tarskian
attempt to characterize logical notions in terms of homomorphism that
leads to a restricted notion of what's logical.
To be sure, what one would really like to know is what the modifier
"logical" means and so what actually hangs on the question whether
arithmetic (say) is "logical" in character. Boolos, who was as
interested in the status of second-order logic as anyone, more or less
abandoned the claim that second-order logic was "really logic" late in
his life, not because he became convinced of the contrary view---though
the essentially G"odelian worry that second-order validity encodes too
much mathematics bothered him a lot---but rather because he ceased, he
said, to understand the notion of "logic" at issue. Is it supposed to be
epistemological? metaphysical? or what? I don't know that anyone really
has a good view about this.
Richard G Heck, Jr
Professor of Philosophy
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