POSSIBLE WORLDS SEMANTICS AND THE LIAR Reflections on a Problem Posed by Kaplan

In this paper I discuss a paradox, due to David Kaplan, that in his view threatens the use of possible worlds semantics as a model-theoretic framework for intensional logic.1 Kaplan’s paradox starts out from an intuitively reasonable principle that I refer to as the Principle of Plenitude. From this principle he derives a contradiction in what he calls Naive Possible World Theory. Kaplan’s metatheoretic argument can be restated in the modal object language as an intensional version of the Liar paradox. To dissolve the paradox, Kaplan favors a ramified theory of propositions, along the lines of Russell’s ramified theory of types. According to this approach: (i) the propositions are divided into orders 0, 1,. . . ; (ii) a propositional quantifier never ranges over the totality of all propositions but only over the propositions of some given order; and (iii) a proposition that involves quantification over the propositions of a given order will itself be of higher order than the propositions in the domain of quantification. Accordingly, a sentence can never express a proposition that is itself within the range of one of the propositional quantifiers in the sentence. This approach avoids the intensional Liar paradox, but its implementation involves severe complications both in the syntax and in the model theory of intensional logic.2 Here I shall attempt an alternative, less drastic, modification of the standard possible worlds methodology than the one favored by Kaplan. The idea is to regard sentences that involve propositional quantifiers, like the sentence