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The four-valued semantics of Belnap–Dunn logic, consisting of the truth values True, False, Neither, and Both, gives rise to several nonclassical logics depending on which feature of propositions we wish to preserve: truth, nonfalsity, or exact truth (truth and nonfalsity). Interpreting equality of truth values in this semantics as material equivalence of propositions, we can moreover see the equational consequence relation of this four-element algebra as a logic of material equivalence. In this paper, we axiomatize all combinations of these four-valued logics, for example, the logic of truth and exact truth or the logic of truth and material equivalence. These combined systems are consequence relations which allow us to express implications involving more than one of these features of propositions.

This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and significance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the nonalgebraizable fragments of relevance logic are considered.

A term $\mathit{td}(p,q,r)$ is called a ternary deductive (TD) term for a variety of algebras $\mathcal{V}$ if the identity $\mathit{td}(p,p,r)\approxr$ holds in $\mathcal{V}$ and $(\mathsf{c},\mathsf{d})\in\theta(\mathsf{a},\mathsf{b})$ yields $\mathit{td}(\mathsf{a},\mathsf{b},\mathsf{c})\approx\mathit{td}(\mathsf{a},\mathsf{b},\mathsf{d})$ for any $\mathscr{A}\in\mathcal{V}$ and any principal congruence $\theta$ on $\mathscr{A}$ . A connective $f(p_{1},\dots,p_{n})$ is called $\mathit{td}$ -distributive if $\mathit{td}(p,q,f(r_{1},\dots,r_{n}))\approx$ $f(\mathit{td}(p,q,r_{1}),\dots,\mathit{td}(p,q,r_{n}))$ . If $\mathsf{L}$ is a propositional logic and $\mathcal{V}$ is a corresponding variety (algebraic semantic) that has a TD term $\mathit{td}$ , then any admissible in $\mathsf{L}$ rule, the premises of which contain only $\mathit{td}$ -distributive operations, is derivable, and the substitution $r\mapsto\mathit{td}(p,q,r)$ is a projective $\mathsf{L}$ -unifier for any formula containing only $\mathit{td}$ -distributive connectives. The above substitution is a generalization of the substitution introduced by T. Prucnal to prove structural completeness of the implication fragment of intuitionistic propositional logic.

In this paper, we give a possible characterisation of the distributivity on bi-approximation semantics. To this end, we introduce new notions of special elements on polarities and show the distributivity is first-order definable on bi-approximation semantics. In addition, we shall investigate the dual representation of those structures and compare them with biapproximation semantics for intuitionistic logic. We shall also discuss that two different methods to validate the distributivity by the splitters and by the adjointness can be explicated with help of the Axiom of Choice as well.

Finitely algebraizable deductive systems were introduced by Blok and Pigozzi to capture the essential properties of those deductive systems that are very tightly connected to quasivarieties of universal algebras. They include the equivalential logics of Czelakowski. Based on Blok and Pigozzi’s work, Herrmann defined algebraizable deductive systems. These are the equivalential deductive systems that are also truth-equational, in the sense that the truth predicate of the class of their reduced matrix models is explicitly definable by some set of unary equations. Raftery undertook the task of characterizing the property of truth-equationality for arbitrary deductive systems. In this paper, following Raftery, we extend the notion of truth-equationality for logics formalized as $\pi$ -institutions and abstract several of the results that hold for deductive systems in this more general categorical context.