A Two-Dimensional Approach to Modal Logics Containing an Actuality Operator

Abstract

It has long been agreed that the standard language of (quantified) modal logic is insufficient to express such natural language sentences as "It might have been that some people, who are actually princes, were paupers". The solution is also relatively non-contentious: add an actuality operator to the modal language. At this stage, the consensus ceases. How should one axiomatize this new logic? What are the correct rules of inference? How, if at all, should we modify our semantics? Following Hodes and Davies and Humberstone , this dissertation suggests that the best (by virtue of naturalness and generality) way to formalise such actual talk is by means of a two-dimensional semantics. That is, instead of evaluating the truth of a formula with respect to a single world, or state, we ought to consider formulas with respect to pairs of worlds. We suggest that by doing so, not only do we obtain an elegant, and intuitively appealing, solution to the original problem, we are also able to see more clearly the logical structure and consequences of the various points of disagreement surrounding this topic. From a more formal perspective, we demonstrate that our proposed approach is technically satisfactory, admitting soundness and completeness proofs. In addition, we explore various expansions as well as relationships with other logics.