The Russell-kaplan Paradox and Other Modal Paradoxes: a New Solution

The most successful semantics for modal logic in the narrow sense, the logic of possibility, necessity, impossibility and contingency, has been possible worlds semantics. And though various kinds of algebraic semantics (as in Bealer 1982) are emerging as noteworthy rivals to possible worlds semantics in the area of general intensional logic (such as the logic of propositional attitudes, etc.), possible worlds semantics is still important even in that more general area. However, there is one big problem in the foundations of possible world semantics: paradoxes such as the so-called Russell-Kaplan paradox, the Forrest-Armstrong paradox, etc. These paradoxes are very serious obstacles for possible worlds semantics, especially if we want to interpret possible worlds realistically and thus have not only a modal logic, but also a possible worlds ontology. Some of these paradoxes also threaten algebraic semantics for intensional logic and thus the very possibility of intensional logic generally. I want to propose that at least most of the paradoxes can be avoided if we use as the background theory of possible worlds semantics the set theory NFU instead of ZF. Ironically, a set theory deriving from the work of Quine, the most famous opponent of modal logic, can be used to solve the greatest problem lurking in the foundations of modal logic!