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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] more...

In this paper, we provide an axiom system for the relevant logic of equivalence relation frames and prove completeness for it. This provides a partial answer to the longstanding open problem of axiomatizing frames for relevant modal logics where the modal accessibility relation is symmetric. Following this, we show that the logic enjoys Halldén completeness and that a related logic enjoys the disjunction property. [ABSTRACT FROM AUTHOR] more...