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Representability in second-order propositional poly-modal logic

Published online by Cambridge University Press:  12 March 2014

G. Aldo Antonelli
Affiliation:
Department of Logic and Philosophy of Science, University of California, Irvine, CA 92697-5100, USA, E-mail: aldo@uci.edu
Richmond H. Thomason
Affiliation:
Department of Philosophy, University of Michigan, Ann Arbor, MI 48109-1003, USA, E-mail: rich@thomason.org

Abstract

A propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.

In this paper we generalize this framework by allowing multiple modalities. While this does not affect the undecidability of K, B, T, K4 and S4, poly-modal second-order S5 is dramatically more expressive than its mono-modal counterpart. As an example, we establish the definability of the transitive closure of finitely many modal operators. We also take up the decidability issue, and, using a novel encoding of sets of unordered pairs by partitions of the leaves of certain graphs, we show that the second-order propositional logic of two S5 modalitities is also equivalent to full second-order logic.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2002

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