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Brady's logic BN4 can be viewed as the 4-valued logic of the relevant conditional. The aim of this article is to introduce the logic E4, which can be regarded as the 4-valued logic of (relevant) entailment on the basis of the two following facts: (1) as the logic R-mingle, E4 has the 'quasi-relevance property'; (2) as Anderson and Belnap's logic of entailment E, E4 encloses a sound theory of logical necessity when the necessity operator is introduced via the definition ♥A=df(A → A) → A. [ABSTRACT FROM AUTHOR]

Łukasiewicz 3-valued logic Ł3 is often understood as the set of all valid formulas according to Łukasiewicz 3-valued matrices MŁ3. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: ‘truth-preserving’ Ł3a and ‘well-determined’ Ł3b defined by two different consequence relations on the 3-valued matrices MŁ3. The aim of this article is to provide a Routley–Meyer ternary semantics for each one of these three versions of Łukasiewicz 3-valued logic: Ł3, Ł3a and Ł3b. [ABSTRACT FROM AUTHOR]

In our paper "A general characterization of the variable-sharing property by means of logical matrices", a general class of so-called "Relevant logical matrices", RMLs, is defined. The aim of this paper is to define a class of simpler Relevant logical matrices RMLs' serving the same purpose that RMLs, to wit: any logic verified by an RML' has the variable-sharing property and related properties predicable of the logic of entailment E and of the logic of relevance R. [ABSTRACT FROM AUTHOR]

A restriction of R-Mingle with the variable-sharing property and the Ackermann properties is defined. From an intuitive semantical point of view, this restriction is an alternative to Anderson and Belnap's logic of entailment E. [ABSTRACT FROM AUTHOR]