This is the Editorial Introduction to “S.I.: Strong and Weak Kleene Logics”.

Paraorthomodular posets are bounded partially ordered sets with an antitone involution induced by quantum structures arising from the logico-algebraic approach to quantum mechanics. The aim of the present work is starting a systematic inquiry into paraorthomodular posets theory both from algebraic and order-theoretic perspectives. On the one hand, we show that paraorthomodular posets are amenable of an algebraic treatment by means of a smooth representation in terms of bounded directoids with antitone involution. On the other, we investigate their order-theoretical features (...) in terms of forbidden configurations. Moreover, sufficient and necessary conditions characterizing bounded posets with an antitone involution whose Dedekind–MacNeille completion is paraorthomodular are provided. (shrink)

This paper essentially originates from the notion of a block in an orthomodular lattice. What happens to orthomodularity when orthocomplementation is weakened? We will show that, under definitely smooth conditions, a great deal of the theory of orthomodular lattices carries over naturally.

This article is about Avicenna’s account of syllogisms comprising opposite premises. We examine the applications and the truth conditions of these syllogisms. Finally, we discuss the relation between these syllogisms and the principle of non-contradiction.

This paper offers a logico-algebraic investigation of AGM belief revision based on the logic of paradox ( \(\mathrm {LP}\) ). First, we define a concrete belief revision operator for \(\mathrm {LP}\), proving that it satisfies a generalised version of the traditional AGM postulates. Moreover, we investigate to what extent the Levi and Harper identities, in their classical formulation, can be applied to a paraconsistent account of revision. We show that a generalised Levi-type identity still yields paraconsistent-based revisions that are fully (...) compatible with the AGM postulates. The main outcome is that, once the classical AGM framework is lifted up to an appropriate level of generality, it still appears as a regulative ideal for treating of paraconsistent-based epistemic operators. (shrink)

Arising in the study of Quantum Logics, PBZ \(^{*}\) -_lattices_ are the paraorthomodular Brouwer–Zadeh lattices in which the pairs of elements with their Kleene complements satisfy the Strong De Morgan condition with respect to the Brouwer complement. They form a variety \(\mathbb {PBZL}^{*}\) which includes that of orthomodular lattices considered with an extended signature (by endowing them with a Brouwer complement coinciding with their Kleene complement), as well as antiortholattices (whose Brouwer complements are trivial). The former turn out to have (...) directly irreducible lattice reducts and, under distributivity, no nontrivial elements with bounded lattice complements, since the elements with bounded lattice complements coincide to the sharp elements in distributive PBZ \(^{*}\) -lattices. The variety \(\mathbb {DIST}\) of distributive PBZ \(^{*}\) -lattices has an infinite ascending chain of subvarieties, and the variety generated by orthomodular lattices and antiortholattices has an infinity of pairwise disjoint infinite ascending chains of subvarieties. (shrink)