FOM: Existential commitments in logic
Vaughan Pratt
pratt at CS.Stanford.EDU
Sat Oct 16 16:18:55 EDT 1999
>if one treats the empty universe as one of the universes quantified over
>in the normal definition of validity (S is valid iff in every universe and
>under every interpretation in that universe, S is true) then the resulting
>notions of validity and logical consequence for free logic, under the
>Russellian assumption the truth of any atomic predication requires
>denotations for all the singular terms involved, call for the rules of
>inference that I mentioned.
I think that's what you said before. Where I'm stuck is in seeing how
free logic contradicts what I said. To help get me unstuck, which is
the first sentence in the following that is falsified by the assumptions
of free logic?
Under (a) the Russellian assumption (all singular terms must denote
something), (b) the existence of at least one singular term in the
language, and (c) empty universe, the set of all interpretations of
the terms of the language is empty. This in turn implies that when the
only universe is the empty one, every proposition is vacuously valid.
Therefore bringing in the empty universe cannot affect any "monotone"
logic (one that can only lose laws when universes are introduced)
because no sentence is thereby rendered invalid.
There simply *is* no call for more or different rules, all the old rules
continue to work as before and no new laws come into existence.
>I know that it seems terribly counterintuitive to consider an (the) empty
>universe; but some logicians (I am not one of them) insist that logic
>should be absolutely neutral on ontological matters. That is, logic should
>not even commit one to the claim that something exists.
One consequence of my point is that any logic of valid formulas *cannot*
commit one to the assumption of a nonempty universe. There are many
ontological premises that could divide logicians into classes, but the
nonemptiness of the universe is not among them. On this particular
ontological matter logic must be neutral, independently of the number
of logicians insisting that it be so.
Vaughan
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