Dynamic measure logic

Annals of Pure and Applied Logic 163 (12):1719-1737 (2012)
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Abstract

This paper brings together Dana Scottʼs 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 S4C. S4C is interpreted in ‘dynamic topological systems,’ or topological spaces together with a continuous function acting on the space. We extend Scottʼs 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

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Tamar Lando
Columbia University

Citations of this work

First order S4 and its measure-theoretic semantics.Tamar Lando - 2015 - Annals of Pure and Applied Logic 166 (2):187-218.

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References found in this work

Dynamic topological logic.Philip Kremer & Grigori Mints - 2005 - Annals of Pure and Applied Logic 131 (1-3):133-158.
Dynamic topological logic.Philip Kremer & Giorgi Mints - 2005 - Annals of Pure and Applied Logic 131 (1-3):133-158.
Completeness of S4 with respect to the real line: revisited.Guram Bezhanishvili & Mai Gehrke - 2004 - Annals of Pure and Applied Logic 131 (1-3):287-301.
Completeness of S4 for the Lebesgue Measure Algebra.Tamar Lando - 2012 - Journal of Philosophical Logic 41 (2):287-316.
A proof of topological completeness for S4 in.Grigori Mints & Ting Zhang - 2005 - Annals of Pure and Applied Logic 133 (1-3):231-245.

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