Completeness of S4 with Respect to the Lebesgue Measure Algebra Based on the Unit Interval

We prove completeness of the propositional modal logic S4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connectives and necessity and possibility operators, and ♦. Propositional modal formulae are assigned to Lebesgue-measurable subsets of the real interval [0, 1], modulo sets of measure zero. Equivalence classes of Lebesgue-measurable subsets form a measure algebra, M, and we add to this a non-trivial interior operator constructed from the frame of ‘open’ elements—elements in M with an open representative. We prove completeness of the modal logic S4 for the algebra M. A corollary to the main result is that non-theorems of S4 can be falsified at each point in a subset of the real interval [0, 1] of measure arbitrarily close to 1. A second corollary is that Intuitionistic propositional logic (IPC) is complete for the frame of open elements in M.