UC Berkeley

We develop a probabilistic semantics for modal logic, which was introduced in recent years by Dana Scott. This semantics is intimately related to an older, topological semantics for modal logic developed by Tarski in the 1940's. Instead of interpreting modal languages in topological spaces, as Tarski did, we interpret them in the Lebesgue measure algebra, or algebra of measurable subsets of the real interval, [0, 1], modulo sets of measure zero. In the probabilistic semantics, each formula is assigned to some element of the algebra, and acquires a corresponding probability (or measure) value. A formula is satisfied in a model over the algebra if it is assigned to the top element in the algebra---or, equivalently, has probability 1.

The dissertation focuses on questions of completeness. We show that the propositional modal logic, S4, is sound and complete for the probabilistic semantics (formally, S4 is sound and complete for the Lebesgue measure algebra). We then show that we can extend this semantics to more complex, multi-modal languages. In particular, we prove that the dynamic topological logic, S4C, is sound and complete for the probabilistic semantics (formally, S4C is sound and complete for the Lebesgue measure algebra with O-operators). The connection with Tarski's topological semantics is developed throughout the text, and the first substantive chapter is devoted to a new and simplified proof of Tarski's completeness result via well-known fractal curves.

This work may be applied in the many formal areas of philosophy that exploit probability theory for philosophical purposes. One interesting application in metaphysics, or mereology, is developed in the introductory chapter. We argue, against orthodoxy, that on a 'gunky' conception of space---a conception of space according to which each region of space has a proper subregion---we can still introduce many of the usual topological notions that we have for ordinary, 'pointy' space.