Search

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]

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]

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]

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]

'Binary Routley-semantics' (bR-semantics) differs from Routley-Meyer semantics (RM-semantics) mainly in that the accessibility relation is binary instead of ternary. Intuitionistic bR-semantics is essentially defined when introducing the Routley operator (used for modelling negation) in Kripke models for positive intuitionistic logic. The aim of this article is to define the minimal logic in intuitionistic bR-semantics, HM, as well as a number of its extension. HM is minimal in intuitionistic bR-semantics in the same sense in which Sylvan and Plumwood's BM is minimal in RM-semantics with a set of designated points. [ABSTRACT FROM AUTHOR]

The aim of the article is to introduce a paraconsistent logic, G3Ł<, related to Gödel 3-valued logic G3. In order to adequately explain G3Ł<, the akin logics G3Ł(FD) and G3Ł1 are also investigated. A Henkin-style completeness proof is provided for each one of these logics by endowing them with dually equivalent under-determined and over-determined bivalent semantics of the type defined by Dunn. G3Ł(FD), G3Ł1 and G3Ł< are related to ‘quasi relevant’ logics such as R-Mingle or RM3, as well as to non-relevant logics such as Łukasiewicz Ł3 or some logics in the vicinity of intuitionistic logic. The logic G3Ł< belongs to the class of paraconsistent logics extending positive intuitionistic logic rather than positive classical logic. [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]

The logic TM is the result of adding the mingle axiom, M to Ticket Entailment logic, T. In the present study, it is proved that TM has the variable-sharing property (vsp). Ternary relational semantics for TM is provided. Finally, an interesting extension of TM with the vsp is briefly discussed. [ABSTRACT FROM PUBLISHER]

The logic BKc1 is the basic constructive logic in the ternary relational semantics (without a set of designated points) adequate to consistency understood as the absence of the negation of any theorem. Negation is introduced in BKc1 with a negation connective. The aim of this paper is to define the logic BKc1F. In this logic negation is introduced via a propositional falsity constant. We prove that BKc1 and BKc1F are definitionally equivalent. [ABSTRACT FROM PUBLISHER]