Solutions to Russell’s paradox of propositions and to Kaplan’s paradox are proposed based on an extension of von Neumann’s method of avoiding paradox. It is shown that Russell’s ‘anti-Cantorian’ mappings can be preserved using this method, but Kaplan’s mapping cannot. In addition, several versions of the Epimenides paradox are discussed in light of von Neumann’s method.

The principle of universal instantiation plays a pivotal role both in the derivation of intensional paradoxes such as Prior’s paradox and Kaplan’s paradox and the debate between necessitism and contingentism. We outline a distinctively free logical approach to the intensional paradoxes and note how the free logical outlook allows one to distinguish two different, though allied themes in higher-order necessitism. We examine the costs of this solution and compare it with the more familiar ramificationist approaches to higher-order logic. Our assessment (...) of both approaches is largely pessimistic, and we remain reluctantly inclined to take Prior’s and Kaplan’s derivations at face value. (shrink)