FOM: Existential commitments in logic
pratt at CS.Stanford.EDU
Fri Oct 15 15:24:17 EDT 1999
From: Jeff Ketland
> "The exclusion of the empty universe from predicate logic is purely
>a matter of efficiency. If you leave the empty universe out of account,
>you gain a lot of convenient laws that you can use in your deductions
>in applied logic and which would not hold if you included the empty
>universe. .... If you want to allow for the possibility that your
>universe of discourse might prove empty, you can still handle the empty
>case separately. It is trivial, in that all formulas there admit of an
>immediate decision procedure: just mark the existentials as false and the
>universals as true and resolve out truth-functionally. So there is
>nothing philosophical about the empty universe in predicate logic"
It seems to me that this is one of those moments when it is important
to distinguish truth from validity. Ordinarily one would not want
any falsehoods among one's theorems, but under the (very reasonable)
assumption that your language includes constants, all propositions
true false or contingent become vacuously valid in the empty universe
for lack of interpretations of those constants.
Hence if a law is understood to be a necessarily (i.e. universally)
true proposition rather than merely a true one, leaving out the empty
universe cannot gain you any new laws, contrary to Quine's claim.
This of course assumes a notion of interpretation that assigns to all
constants rather than just to the constants appearing in the proposition
in question, yet another fine distinction that needs to be drawn in
order to get one's story straight.
There can be no doubt that the residents of the empty universe are at
this moment all chuckling over the irony that Quine's evidence for their
world having nothing philosophical about it can be used to prove the
opposite as well.
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