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The well-known logic first degree entailment logic (FDE), introduced by Belnap and Dunn, is defined with |$\wedge $|⁠ , |$\vee $| and |$\sim $| as the sole primitive connectives. The aim of this paper is to establish the lattice formed by the class of all 4-valued C-extending implicative expansions of FDE verifying the axioms and rules of Routley and Meyer's basic logic B and its useful disjunctive extension B |$^{\textrm {d}}$|⁠. It is to be noted that Boolean negation (so, classical propositional logic) is definable in the strongest element in the said class. [ABSTRACT FROM AUTHOR]

The logic E4 is related to Brady's BN4 in a similar way to which Anderson and Belnap's logic of entailment E is related to their logic of the relevant implication R. In 'A companion to Brady's 4-valued relevant logic: the 4-valued logic of entailment E4', quoted in this paper, three alternatives to BN4 and another three to E4 are summarily introduced in a couple of pages as the only alternatives containing Routley and Meyer's basic logic B, provided some conditions are fulfilled. The aim of this note is to prove (i) BN4 and its three alternatives are the same logic up to some point, since they are functionally equivalent to each other, as it is the case with E4 and its three alternatives, which are also functionally equivalent to each other; (ii) to some extent, E4 is superior to BN4, since the latter is functionally included in the former, but not conversely. [ABSTRACT FROM AUTHOR]

La lógica Bp+ es B+ más el axioma de “prefijación”. Entendemos “negación mínima” en el sentido de Johansson. Nuestro objetivo es probar que que la negación mínima puede introducirse en Bp+; el resultado es el sistema Bpmcr. Mostramos también cómo extender Bpmcr a cualquier sistema incluido en la lógica intuicionista mínima

G3 is Gödelian 3-valued logic, G3Ł≤ is its paraconsistent counterpart and G3Ł≤ is a strong extension of G3Ł≤. The aim of this paper is to endow each one of the logics just mentioned with a 2 set-up binary Routley semantics. [ABSTRACT FROM AUTHOR]

G3 is Gödelian 3-valued logic, G3Ł≤ is its paraconsistent counterpart and G3Ł≤ is a strong extension of G3Ł≤. The aim of this paper is to endow each one of the logics just mentioned with a 2 set-up binary Routley semantics. [ABSTRACT FROM AUTHOR]

A classical result by Słupecki states that a logic L is functionally complete for the 3-element set of truth-values THREE if, in addition to functionally including Łukasiewicz's 3-valued logic Ł3, what he names the ' |$T$| -function' is definable in L. By leaning upon this classical result, we prove a general theorem for defining binary expansions (i.e. expansions with a binary connective) of Kleene's strong logic that are functionally complete for THREE. [ABSTRACT FROM AUTHOR]

Let MK3 I and MK3 I I be Kleene's strong 3-valued matrix with only one and two designated values, respectively. Next, let MK3 G (resp., MK3 d G ) be defined exactly as MK3 I (resp., MK3 I I ), except that the characteristic Łukasiewicz-type negation of Kleene's strong 3-valued matrix is replaced by a 'Gödel-type' negation (resp., 'dual Gödel-type' negation). The aim of this paper is to axiomatize the logics determined by all natural implicative expansions of MK3 G and MK3 d G . The axiomatic formulations are defined by using a 'two-valued' Belnap-Dunn semantics. [ABSTRACT FROM AUTHOR]

Equivalent overdetermined and underdetermined bivalent Belnap–Dunn type semantics for the logics determined by all natural implicative expansions of Kleene's strong 3-valued matrix with only one designated value are provided. [ABSTRACT FROM AUTHOR]

This paper is a sequel to 'Belnap-Dunn semantics for natural implicative expansions of Kleene's strong three-valued matrix with two designated values', where a 'bivalent' Belnap-Dunn semantics is provided for all the expansions referred to in its title. The aim of the present paper is to carry out a parallel investigation for all natural implicative expansions of Kleene's strong 3-valued matrix now with only one designated value. [ABSTRACT FROM AUTHOR]

A conditional is natural if it fulfils the three following conditions. (1) It coincides with the classical conditional when restricted to the classical values T and F; (2) it satisfies the Modus Ponens; and (3) it is assigned a designated value whenever the value assigned to its antecedent is less than or equal to the value assigned to its consequent. The aim of this paper is to provide a 'bivalent' Belnap-Dunn semantics for all natural implicative expansions of Kleene's strong 3-valued matrix with two designated elements. (We understand the notion 'natural conditional' according to N. Tomova, 'A lattice of implicative extensions of regular Kleene's logics', Reports on Mathematical Logic, 47, 173–182, 2012.) [ABSTRACT FROM AUTHOR]

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]

The aim of this paper is to define the logical system Sm4 characterised by the degree of truth-preserving consequence relation defined on the ordered set of values of Smiley’s four-element matrix MSm4. The matrix MSm4 has been of considerable importance in the development of relevant logics and it is at the origin of bilattice logics. It will be shown that Sm4 is a most interesting paraconsistent logic which encloses a sound theory of logical necessity similar to that of Anderson and Belnap’s logic of entailment E. Intuitively, Sm4 can be described as a four-valued expansion of the positive fragment of Lewis’ S5 or, alternatively, as a four-valued version of S5. [ABSTRACT FROM PUBLISHER]