Undecidability of First-Order Intuitionistic and Modal Logics with Two variables
Roman Kontchakov, Agi Kurucz, Michael Zakharyaschev · Bulletin of Symbolic Logic · 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 logicLwith expanding first-order domains is undecidable whenever there is a Kripke frame forLwith a point having infinitely many successors (such are, in particular, the first-order extensions of practically all standard modal logics likeK, 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.