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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]

A logic L has the 'variable-sharing property' (VSP) if in all L-theorems of the form |$A\rightarrow B$|⁠ , |$A$| and |$B$| share at least a propositional variable. A logic L has the 'Ackermann property' (AP) if in all L-theorems of the form |$A\rightarrow (B\rightarrow C)$|⁠ , |$A$| contains at least a conditional connective (⁠|$\rightarrow $|⁠). Anderson and Belnap consider the VSP a necessary property of any relevant logic, and both the VSP and the AP necessary properties of any (relevant) entailment logic. Now, among relevant logicians, Brady's logic BN4 is widely viewed as the adequate 4-valued implicative logic. But BN4 lacks the VSP and the AP. The aim of this paper is to define the logics BN4 |$^{\text {VSP}}$| and BN4 |$^{\text {AP}}$|⁠. The former one has the VSP, whereas the latter one has the VSP and the AP. Moreover, both logics have some properties that do not support their consideration as mere artificial constructs. [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]

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]

This document is a correction to a previous article titled "A note on functional relations in a certain class of implicative expansions of FDE related to Brady's 4-valued logic BN4." The correction addresses an error in the definition used in the proof of Proposition 3.2. The corrected definition states that A→B is equal to (A k3 B) ∨ (A→B), rather than (A k11 B) ∨ (A→B) as previously stated. The correction is reported by Gemma Robles and José M Méndez. [Extracted from the article]