Dynamic measure logic

This paper brings together Dana Scott’s measure-based semantics for the propositional modal logic S4, and recent work in Dynamic Topological Logic. In a series of recent talks, Scott showed that the language of S4 can be interpreted in the Lebesgue measure algebra, M, or algebra of Borel subsets of the real interval, [0, 1], modulo sets of measure zero. Conjunctions, disjunctions and negations are interpreted via the Boolean structure of the algebra, and we add an interior operator on M that interprets the -modality. In this paper we show how to extend this measure-based semantics to the bimodal logic of S4C. S4C is a dynamic topological logic that is interpreted in ‘dynamic topological systems,’ or topological spaces together with a continuous function acting on the space. We extend Scott’s measure based semantics to this bimodal logic by defining a class of operators on the algebra M, which we call O-operators and which take the place of continuous functions in the topological semantics for S4C. The main result of the paper is that S4C is complete for the Lebesgue measure algebra. A strengthening of this result, also proved here, is that there is a single measure-based model in which all non-theorems of S4C are refuted.