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Six interesting variants of the logics BN4 and E4--which can be considered as the 4-valued logics of the relevant conditional and (relevant) entailment, respectively--were previously developed in the literature. All these systems are related to the family of relevant logics and contain Routley and Meyer's basic logic B, which is well-known to be specifically associated with the ternary relational semantics. The aim of this paper is to develop reduced general Routley-Meyer semantics for them. Strong soundness and completeness theorems are proved for each one of the logics. [ABSTRACT FROM AUTHOR]

This paper introduces the logics of super-strict implications, where a super-strict implication is a strengthening of C.I. Lewis' strict implication that avoids not only the paradoxes of material implication but also those of strict implication. The semantics of super-strict implications is obtained by strengthening the (normal) relational semantics for strict implication. We consider all logics of super-strict implications that are based on relational frames for modal logics in the modal cube. it is shown that all logics of super-strict implications are connexive logics in that they validate Aristotle's Theses and (weak) Boethius's Theses. A proof-theoretic characterisation of logics of super-strict implications is given by means of G3-style labelled calculi, and it is proved that the structural rules of inference are admissible in these calculi. It is also shown that validity in the S5-based logic of super-strict implications is equivalent to validity in G. Priest's negation-as-cancellation-based logic. Hence, we also give a cut-free calculus for Priest's logic. [ABSTRACT FROM AUTHOR]

We investigate the relationship between M. De's empirical negation in Kripke and Beth Semantics. It turns out empirical negation, as well as co-negation, corresponds to different logics under different semantics. We then establish the relationship between logics related to these negations under unified syntax and semantics based on R. Sylvan's CCw. [ABSTRACT FROM AUTHOR]

We define Kripke semantics for propositional intuitionistic logic with Suszko's identity (ISCI). We propose sequent calculus for ISCI along with cut-elimination theorem. We sketch a constructive interpretation of Suszko's propositional iden- tity connective. [ABSTRACT FROM AUTHOR]

Orthologic (OL) is non-classical logic and has been studied as a part of quantum logic. OL is based on an ortholattice and is also called minimal quantum logic. Sequent calculus is used as a tool for proof in logic and has been examined for several decades. Although there are many studies on sequent calculus for OL, these sequent calculi have some problems. In particular, they do not include implication connective and they are mostly incompatible with the cut-elimination theorem. In this paper, we introduce new labeled sequent calculus called LGOI, and show that this sequent calculus solve the above problems. It is already known that OL is decidable. We prove that decidability is preserved when the implication connective is added to OL. [ABSTRACT FROM AUTHOR]

We discuss relationships among the existence property, the disjunction property, and their weak variants in the setting of intermediate predicate logics. We deal with the weak and sentential existence properties, and the Z-normality, which is a weak variant of the disjunction property. These weak variants were presented in the author's previous paper [16]. In the present paper, the Kripke sheaf semantics is used. [ABSTRACT FROM AUTHOR]

The Craig interpolation property and interpolation property for deducibility are considered for special kind of normal extensions of the Brouwer logic. [ABSTRACT FROM AUTHOR]