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Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter

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We prove that the positive fragment of first-order intuitionistic logic in the language with two individual variables and a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals \([\mathbf{QBL}, \mathbf{QKC}]\) and \([\mathbf{QBL}, \mathbf{QFL}]\), where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser’s basic and formal logics, respectively. We also show that, for most “natural” first-order modal logics, the two-variable fragment with a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz—among them, QK, QT, QKB, QD, QK4, and QS4.

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We are grateful to the anonymous referees for comments that helped to significantly improve the presentation of the paper.


This work has been supported by Russian Foundation for Basic Research, Projects 17-03-00818 and 18-011-00869.

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Correspondence to Dmitry Shkatov.

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Presented by Yde Venema

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Rybakov, M., Shkatov, D. Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter. Stud Logica 107, 695–717 (2019).

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