The paper provides a new semantics for positive modal logic using Kripke frames having a quasi ordering on the set of possible worlds and an accessibility relation connected to the quasi ordering by the conditions (1) that the composition of with is included in the composition of with and (2) the analogous for the inverse of and . This semantics has an advantage over the one used by Dunn in "Positive modal logic," Studia Logica (1995) and works fine for extensions (...) of the minimal system of normal positive modal logic. (shrink)

In [12] the study of Positive Modal Logic is initiated using standard Kripke semantics and the positive modal algebras are introduced. The minimum system of Positive Modal Logic is the -fragment of the local consequence relation defined by the class of all Kripke models. It can be axiomatized by a sequent calculus and extensions of it can be obtained by adding sequents as new axioms. In [6] a new semantics for PML is proposed to overcome some frame incompleteness problems discussed (...) in [12]. The frames of this semantics consists of a set of indexes, a quasi-order on them and an accessibility relation. The models are obtained by using increasing valuations relatively to the quasi-order of the frame. This semantics is coherent with the dual structures obtained by developing the Priestley duality for positive modal algebras, one of the topics or the present paper, and can be seen also as arising from the Kripke semantics for a suitable intuitionistic modal logic. The present paper is devoted to the study of the mentioned duality as well as to proving some d-persistency results as well as a Sahlqist Theorem for sequents and the semantics proposed in [6]. Also a Goldblatt-Thomason theorem that characterizes the elementary classes of frames of that semantics that are definable by sets of sequents is proved. (shrink)

The Logics of Deontic (In)Consistency (LDI's) can be considered as the deontic counterpart of the paraconsistent logics known as Logics of Formal (In)Consistency. This paper introduces and studies new LDI's and other paraconsistent deontic logics with different properties: systems tolerant to contradictory obligations; systems in which contradictory obligations trivialize; and a bimodal paraconsistent deontic logic combining the features of previous systems. These logics are used to analyze the well-known Chisholm's paradox, taking profit of the fact that, besides contradictory obligations do (...) not trivialize in LDI's, several logical dependencies of classical logic are blocked in the context of LDI's, allowing to dissolve the paradox. (shrink)