TY - JOUR
AU - Odintsov, Sergei P.
AU - Skurt, Daniel
AU - Wansing, Heinrich
PY - 2019/04/11
Y2 - 2021/10/25
TI - On Definability of Connectives and Modal Logics over FDE
JF - Logic and Logical Philosophy
JA - LLP
VL - 28
IS - 4
SE - Articles
DO - 10.12775/LLP.2019.010
UR - https://apcz.umk.pl/LLP/article/view/LLP.2019.010
SP - 631-659
AB - <p>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<sup>□</sup>×BK<sup>□</sup>, which underlines the central role played by BK among FDE-based modal logics.</p>
ER -