The paper argues, varying Cantor’s well-known set-theoretic reasoning, that the principle “(It is possible that p) iff (in some possible world, p)” leads to a paradoxical conclusion regarding how many possible worlds there are. The argument justifies some scepticism about whether there are possible worlds in any of the philosophical senses of the term (as opposed to the sense in which we talk about possible worlds in modal logic and which is minimally laden from a metaphysical viewpoint).

This paper shows that some plausible assumptions about any object that has a name lead to the conclusion that the object could not have failed to exist. Considering the conclusion wrong, I argue that the source of the error is the principle “If something is the case, it could not have been impossible”, which occurs in some systems of modal logic; if an object did not exist, it would not be possible for it to exist.

Take a formula of first-order logic which is a logical consequence of some other formulae according to model theory, and in all those formulae replace schematic letters with English expressions. Is the argument resulting from the replacement valid in the sense that the premisses could not have been true without the conclusion also being true? Can we reason from the model-theoretic concept of logical consequence to the modal concept of validity? Yes, if the model theory is the standard one for sentential logic; no, if it is the standard one for the predicate calculus; and yes, if it is a certain model theory for free logic. These conclusions rely inter alia on some assumptions about possible worlds, which are mapped into the models of model theory. Plural quantification is used in the last section, while part of the reasoning is relegated to an appendix that includes a proof of completeness for a version of free logic.