Undecidability of first-order intuitionistic and modal logics with two variables

Bulletin of Symbolic Logic 11 (3):428-438 (2005)
Abstract
We prove that the two-variable fragment of first-order intuitionistic logic is undecidable, even without constants and equality. We also show that the two-variable fragment of a quantified modal logic L with expanding first-order domains is undecidable whenever there is a Kripke frame for L with a point having infinitely many successors (such are, in particular, the first-order extensions of practically all standard modal logics like K, K4, GL, S4, S5, K4.1, S4.2, GL.3, etc.). For many quantified modal logics, including those in the standard nomenclature above, even the monadic two-variable fragments turn out to be undecidable
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DOI 10.2178/bsl/1122038996
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References found in this work BETA

The Undecidability of Monadic Modal Quantification Theory.Saul A. Kripke - 1962 - Mathematical Logic Quarterly 8 (2):113-116.
A Note on the Entscheidungsproblem.Alonzo Church - 1936 - Journal of Symbolic Logic 1 (1):40-41.
Decidable Fragments of First-Order Temporal Logics.Ian Hodkinson, Frank Wolter & Michael Zakharyaschev - 2000 - Annals of Pure and Applied Logic 106 (1-3):85-134.
On Languages with Two Variables.Michael Mortimer - 1975 - Mathematical Logic Quarterly 21 (1):135-140.
Decidable Fragments of First-Order Modal Logics.Frank Wolter & Michael Zakharyaschev - 2001 - Journal of Symbolic Logic 66 (3):1415-1438.

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