Solving Multimodal Paradoxes

Abstract

Recently, it has been observed that the usual type-theoretic restrictions are not enough to block certain paradoxes involving two or more predicates. In particular, when we have a self-referential language containing modal predicates, new paradoxes might appear even if there are type restrictions for the principles governing those predicates. In this article we consider two type-theoretic solutions to multimodal paradoxes. The first one adds types for each of the modal predicates. We argue that there are a number of problems with most versions of this approach. The second one, which we favour, represents modal notions by using the truth predicate together with the corresponding modal operator. This way of doing things is not only useful because it avoids multimodal paradoxes, but also because it preserves the expressive capacity of the language. As an example of the sort of theory we have in mind, we provide a type-theoretic axiomatization that combines truth with necessity and knowledge.