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  1. Neighbourhood Semantics for FDE-Based Modal Logics.S. Drobyshevich & D. Skurt - 2021 - Studia Logica 109 (6):1273-1309.
    We investigate some non-normal variants of well-studied paraconsistent and paracomplete modal logics that are based on N. Belnap’s and M. Dunn’s four-valued logic. Our basic non-normal modal logics are characterized by a weak extensionality rule, which reflects the four-valued nature of underlying logics. Aside from introducing our basic framework of bi-neighbourhood semantics, we develop a correspondence theory in order to prove completeness results with respect to our neighbourhood semantics for non-normal variants of \, \ and \.
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  • Modal and Intuitionistic Variants of Extended Belnap–Dunn Logic with Classical Negation.Norihiro Kamide - 2021 - Journal of Logic, Language and Information 30 (3):491-531.
    In this study, we introduce Gentzen-type sequent calculi BDm and BDi for a modal extension and an intuitionistic modification, respectively, of De and Omori’s extended Belnap–Dunn logic BD+ with classical negation. We prove theorems for syntactically and semantically embedding BDm and BDi into Gentzen-type sequent calculi S4 and LJ for normal modal logic and intuitionistic logic, respectively. The cut-elimination, decidability, and completeness theorems for BDm and BDi are obtained using these embedding theorems. Moreover, we prove the Glivenko theorem for embedding (...)
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  • The Modal Logics of Kripke–Feferman Truth.Carlo Nicolai & Johannes Stern - 2021 - Journal of Symbolic Logic 86 (1):362-396.
    We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model $\mathcal {M}$, or an axiomatization S thereof, we find a modal logic M such that a modal sentence $\varphi $ is a theorem of M if and only if the sentence $\varphi ^*$ obtained by translating the modal operator with the truth predicate is true in $\mathcal {M}$ or a theorem of S under all such translations. (...)
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  • Belnap–Dunn Modal Logic with Value Operators.Yuanlei Lin & Minghui Ma - 2020 - Studia Logica 109 (4):759-789.
    The language of Belnap–Dunn modal logic \ expands the language of Belnap–Dunn four-valued logic with the modal operator \. We introduce the polarity semantics for \ and its two expansions \ and \ with value operators. The local finitary consequence relation \ in the language \ with respect to the class of all frames is axiomatized by a sequent system \ where \. We prove by using translations between sequents and formulas that these languages under the polarity semantics have the (...)
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  • On a Multilattice Analogue of a Hypersequent S5 Calculus.Oleg Grigoriev & Yaroslav Petrukhin - forthcoming - Logic and Logical Philosophy:1.
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  • Belnap–Dunn Modal Logics: Truth Constants Vs. Truth Values.Sergei P. Odintsov & Stanislav O. Speranski - 2020 - Review of Symbolic Logic 13 (2):416-435.
    We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more (...)
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  • Propositional Dynamic Logic with Belnapian Truth Values.Igor Sedlár - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. CSLI Publications. pp. 503-519.
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  • Disentangling FDE -Based Paraconsistent Modal Logics.Sergei P. Odintsov & Heinrich Wansing - 2017 - Studia Logica 105 (6):1221-1254.
    The relationships between various modal logics based on Belnap and Dunn’s paraconsistent four-valued logic FDE are investigated. It is shown that the paraconsistent modal logic \, which lacks a primitive possibility operator \, is definitionally equivalent with the logic \, which has both \ and \ as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with \ without the absurdity constant. Moreover, (...)
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