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Representability in second-order propositional poly-modal logic

Published online by Cambridge University Press:  12 March 2014

G. Aldo Antonelli
Department of Logic and Philosophy of Science, University of California, Irvine, CA 92697-5100, USA, E-mail:
Richmond H. Thomason
Department of Philosophy, University of Michigan, Ann Arbor, MI 48109-1003, USA, E-mail:


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.

Research Article
Copyright © Association for Symbolic Logic 2002

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Ackermann, Wilhelm [1954], Solvable cases of the decision problem, North-Holland Publishing Co., Amsterdam.Google Scholar
Fagin, Ronald, Halpern, Joseph Y., Moses, Yoram, and Vardi, Moshe Y. [1995], Reasoning about knowledge, The MIT Press, Cambridge, Massachusetts.Google Scholar
Fine, Kit [1970], Propositional quantifiers in modal logic, Theoria, pp. 336346.Google Scholar
Kaplan, David [1970], S5 with quantifiable propositional variables, this Journal, vol. 35, no. 2, p. 355.Google Scholar
Kremer, Philip [1997], On the complexity of propositional quantification in intuitionistic logic, this Journal, vol. 62, no. 2, pp. 529544.Google Scholar
Nerode, Anil and Shore, Richard [1980], Second order logic and theories of reducibility orderings, The Kleene symposium (Amsterdam) (Barwise, Jon, Keisler, H. Jerome, and Kunen, Kenneth, editors), North-Holland, pp. 181200.CrossRefGoogle Scholar
Rabin, Michael O. [1965], A simple method for undecidability proofs and some applications, Logic, methodology and philosophy of science. Proceedings of the 1964 international congress (Amsterdam) (Bar-Hillel, Yehoshua, editor), North-Holland, pp. 5868.Google Scholar
Tarski, Alfred, Mostowski, Andrzej, and Robinson, Raphael M. [1953], Undecidable theories, North-Holland Publishing Co., Amsterdam.Google Scholar