Completeness of S4 for the Lebesgue Measure Algebra
- Tamar Lando
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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, \(\Box\) and \(\Diamond\) . 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, \(\mathcal M\) , and we add to this a non-trivial interior operator constructed from the frame of ‘open’ elements—elements in \(\mathcal M\) with an open representative. We prove completeness of the modal logic S4 for the algebra \(\mathcal M\) . 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 \(\mathcal M\) .
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- Completeness of S4 for the Lebesgue Measure Algebra
Journal of Philosophical Logic
Volume 41, Issue 2 , pp 287-316
- Cover Date
- Print ISSN
- Online ISSN
- Springer Netherlands
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- Measure algebra
- Topological modal logic
- Topological semantics
- Modal logic
- Probabilistic semantics
- Tamar Lando (1)
- Author Affiliations
- 1. Department of Philosophy, University of California, Berkeley, 314 Moses Hall #2390, Berkeley, CA, 94720-2390, USA