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Semiproducts, Products, and Modal Predicate Logics: Some Examples

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Abstract

We study two kinds of combined modal logics, semiproducts and products with \({\mathbf{S5}}\), and their connection with modal predicate logics. We present examples of propositional modal logics, for which semiproducts or products with \({\mathbf{S5}}\) are axiomatized in the minimal way (they are called semiproduct- or product-matching with \({\mathbf{S5}}\)) and also present counterexamples for these properties. The finite model property for (semi)products, together with (semi)product-matching, allow us to obtain decidability of corresponding 1-variable modal predicate logics.

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Fig. 2.

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Notes

  1. Usually, \({\mathbf{QK}}\), \({\mathbf{Q}\boldsymbol\Lambda }\), and \({\mathbf{Q}\boldsymbol\Lambda \mathbf{C}}\) denote modal logics in the language with predicate letters of every arity, but in this paper we use the same notation for the monadic fragments of those logics.

  2. These are probably the largest known decidable fragments of modal predicate logics; most of 2-variable fragments, even in signatures with a single monadic predicate letter, are undecidable [8].

  3. This fact can also be inferred from Fig. 2.

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ACKNOWLEDGMENTS

We thank an anonymous reviewer for the comments that helped to improve the paper.

Funding

The work of the first author was carried out at Steklov Mathematical Institute and supported by the Russian Scientific Foundation, project no. 21-11-00318.

The work on Sections 1, 2, and 5 has been carried out by the first author; the work on Sections 2, 4, and 6 by the second author.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    V. B. Shehtman

  2. University of the Witwatersrand, Johannesburg, South Africa

    D. Shkatov

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  1. V. B. Shehtman
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  2. D. Shkatov
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Correspondence to V. B. Shehtman.

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Shehtman, V.B., Shkatov, D. Semiproducts, Products, and Modal Predicate Logics: Some Examples. Dokl. Math. 108, 411–418 (2023). https://doi.org/10.1134/S1064562423701296

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  • DOI: https://doi.org/10.1134/S1064562423701296

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