Completeness of S4 for the Lebesgue Measure Algebra

Journal of Philosophical Logic 41 (2):287-316 (2012)
Abstract
We prove completeness of the propositional modal logic S 4 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, , and we add to this a non-trivial interior operator constructed from the frame of ‘open’ elements—elements in with an open representative. We prove completeness of the modal logic S 4 for the algebra . A corollary to the main result is that non-theorems of S 4 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
Keywords Measure algebra  Topological modal logic  Topological semantics  Completeness  Modal logic  Probabilistic semantics
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References found in this work BETA
Philip Kremer & Grigori Mints (2005). Dynamic Topological Logic. Annals of Pure and Applied Logic 131 (1-3):133-158.
Citations of this work BETA
Tamar Lando (2012). Dynamic Measure Logic. Annals of Pure and Applied Logic 163 (12):1719-1737.
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