FOM: Re: The logical, the set-theoretical, and the mathematical

[Jeffrey Ketland]

Dear Ed

>I don't think logical truths are, in general, devoid of ontic commitments.
>(x)(x=x) is a logical truth, but it implies (Ex)(x=x), which is usually
>interpreted as saying that something exists. Maybe you want to use a
>universally free logic?

That's an old chestnut that always crops up here. It was occasionally
discussed by Quine. You can fix up a deductive system (universally free
logic) so that no logically true statement implies Ex(x = x). Then the set
of validities under consideration is the set of sentences true in all
models, including empty ones. This set is still r.e. and has a positive
proof procedure. It's just inconvenient. I think Neil Tennant has a
discussion of this in his "Natural Logic" (1978, pp. 167-175).
In usual FOL, we count Ex(x=x) as logically true but harmless in its
assumption that something exists, for the sake of convenience, for getting a
nice deductive system.
I don't think anything important hangs on this point.
We could say: the condition is that we don't want any logical truths to
imply the existence of anything "spooky" or "abstract" - like properties or
sets, whose existence is doubted by nominalists. As long as we bear in mind
the caveat above, I think it's OK to count Ex(x=x) as logically true, but if
someone were very insistent, then sure, we have to move to some kind of free
logic.

Best - Jeff

~~~~~~~~~~~ Jeffrey Ketland ~~~~~~~~~
Dept of Philosophy, University of Nottingham
Nottingham NG7 2RD United Kingdom
Tel: 0115 951 5843
Home: 0115 922 3978
E-mail: jeffrey.ketland at nottingham.ac.uk
Home: ketland at ketland.fsnet.co.uk
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