[FOM] Kaplan's Sentence
hdeutsch at ilstu.edu
Fri May 9 16:06:06 EDT 2008
David Kaplan wrote a paper titled "A Problem for Possible Worlds
Semantics." It's in A volume of essays in honor of Ruth Barcan Marcus
(Modality, Morality, and Belief, Armstrong et al, eds. (I'm sorry I
don't have the name of the publisher at hand.) In this paper Kaplan
considers and extension of S5 which has variables, p, q, etc. ranging
over sets of possible worlds (which are regarded as propositions in
possible worlds semantics.
Kaplan points out that the following sentence is not satisfiable in
this "second order" extension of S5:
(A) For any proposition, p, it is possible that, for any proposition q,
(Qp < > p = q),
where Q is a sentential operator, and < > is material implication (in
my inadequate email notation!).
Kaplan argues that this sentence should be satisfiable--that it
represents a genuine "possibility" and hence that possible worlds
semantics faces the problem of not being able to underwrite this
But it seems to me that we should conclude only that the possibilities
are not what we might have thought--as often happens in logic.
In fact it seems to that Kaplan's (A) literally asserts the existence
of an onto mapping from worlds to propositions (sets of worlds), and
hence of course is not satisfiable.
But it is interesting that (if I am right in my interpretation) this
"Cantorian" proposition can be expressed in the language of modal
logic. And I am wondering if people agree with my interpretaation (I
am prepared to be wrong), and if so, are there any wider implications?
The following may be a little helpful: I've noticed that if Qp is
interpreted as meaning that p is empty (i.e. Qp is short for
"necessarily not-p") then the negation of A (with Q so interpreted)
asserts that there are always (in the domain of propositions (sets of
worlds)) at least two propositions: the empty one and at least one
other one (minimally a single singleton), which of course i true.
Thanks for any help.
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