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Logic and Logical Philosophy

On Definability of Connectives and Modal Logics over FDE
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On Definability of Connectives and Modal Logics over FDE

Authors

  • Sergei P. Odintsov Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
  • Daniel Skurt Ruhr-Universität Bochum, Department of Philosophy I
  • Heinrich Wansing Ruhr-Universität Bochum, Department of Philosophy I http://orcid.org/0000-0002-0749-8847

DOI:

https://doi.org/10.12775/LLP.2019.010

Keywords

definability of connectives, first-degree entailment logic, modal logic, modal bilattice logic, functional completeness, translations between logics, weak definitional equivalence, definitional equivalence

Abstract

The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, model-theoretic proofs of the definability of connectives along the lines of [McCullough, “Logical connectives for intuitionistic propositional logic”, Journal of Symbolic Logic 36, 1 (1971): 15–20. DOI: 10.2307/2271511] and [[17] Wansing, “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: 10.1007/s11229-005-5518-5] are given for various FDE-based modal logics. On the other hand, building on [Odintsov and Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: 10.1007/s11225-017-9753-9], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK□×BK□, which underlines the central role played by BK among FDE-based modal logics.

References

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Odintsov, S.P., and H. Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: http://dx.doi.org/10.1007/s11225-017-9753-9

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Wansing, H., “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: http://dx.doi.org/10.1007/s11229-005-5518-5

Logic and Logical Philosophy

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Published

2019-04-11

How to Cite

1.
ODINTSOV, Sergei P., SKURT, Daniel & WANSING, Heinrich. On Definability of Connectives and Modal Logics over FDE. Logic and Logical Philosophy [online]. 11 April 2019, T. 28, nr 4, s. 631–659. [accessed 25.3.2023]. DOI 10.12775/LLP.2019.010.
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