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In contrast to what has become known as standard deontic logic, Anderson and Kanger independently proposed alethic modal logics with a deontic constant, in which an obligation operator can be defined. It can be proven that several deontic logics are fragments of these alethic modal logics. This has been called "one of the main mathematical results on propositional monadic deontic logic". We show that a similar translation theorem can be proven for a number of predicative deontic logics, known as term-modal deontic logics (TMDLs). In this paper we define several TMDLs and several first-order (alethic) modal logics (FOMLs) with predicative constants, as well as a translation from the TMDLs to the FOMLs. We prove that a number of the TMDLs are fragments of the FOMLs. This is interesting not only from a conceptual, but also from a technical perspective. It shows that (some) term-modal logics are actually reducible to orthodox (i.e. not term-modal) FOMLs. We also present a few negative results, showing that some TMDLs cannot straightforwardly be shown to be fragments of FOMLs. Finally, we briefly discuss the possible repercussions of both the positive and negative results for other reductions, in particular for the reduction of propositional multi-modal logics to propositional uni-modal logics. [ABSTRACT FROM AUTHOR]

In the paper, we consider a particular way of obtaining modal extensions of the discussive logic D 2 (Jaśkowski 1948; 1949). The proposed framework of defining modal discussive logics is a novelty compared to the one presented in Mruczek-Nasieniewska et al. (2019Logic Journal of IGPL, 27(4), 451–477). In this case, different discussants may have different views about what is necessary (resp. possible) in a given world. We enrich the intuitive model of discussion given by Jaśkowski. This is obtained by considering two-sorted frames. The first sort consists of discussants and the second of situations. We have a universal accessibility relation between the discussants, i.e., we use the standard semantic formulation of S 5 , which is characteristic of the original variant of the discussive logic D 2 . For a frame, a valuation assigns to each discussant a set of atomic sentences they accept in a given situation. Although different discussants may have different views about what occurs in a given world, their views should be within the same "domain of possibilities". Therefore, we have a common set of possible situations and an accessibility relation for all discussants. The introduction of modal operators allows for specifying strict discussive implications, which—as it seems—correspond to colloquial discussive conditionals better than Jaśkowski's discussive implication. We also analyze various semantic and syntactic (including axiomatizations) aspects of modal discussive logics resulting from the proposed framework. [ABSTRACT FROM AUTHOR]

The need for more flexible and robust models to reason about systems in the presence of conflicting information is becoming more and more relevant in different contexts. This has prompted the introduction of paraconsistent transition systems, where transitions are characterized by two pairs of weights: one representing the evidence that the transition effectively occurs and the other its absence. Such a pair of weights can express scenarios of vagueness and inconsistency. This paper establishes a foundation for a compositional and structured specification approach of paraconsistent transition systems, framed as paraconsistent institution. The proposed methodology follows the stepwise implementation process outlined by Sannella and Tarlecki. [ABSTRACT FROM AUTHOR]

Three new Gentzen-style sequent calculi GS41, GS42, and GS43 for normal modal logic S4 are introduced. These calculi are obtained from a Gentzen-style sequent calculus GMA4 for a modal extension MA4 of Avron's self-extensional paradefinite four-valued logic by adding some special inference rules and/or some special initial sequents. The cut-elimination theorems for GS41, GS42, GS43, and GMA4 are proved. An alternative Kripke semantics for S4 is introduced and the completeness theorems with respect to this semantics are proved for GS41 and GS42. An extended Kripke semantics is introduced for an extended S4 with an auxiliary negation connective and the completeness theorem with respect to this semantics is proved for GS43. Finally, a Gödel--McKinsey--Tarski theorem for embedding Gurevich logic into S4 is proved. [ABSTRACT FROM AUTHOR]