Belnap-Dunn Semantics for the Variants of BN4 and E4 which Contain Routley and Meyer’s Logic B

Abstract

The logics BN4 and E4 can be considered as the 4-valued logics of the relevant conditional and (relevant) entailment, respectively. The logic BN4 was developed by Brady in 1982 and the logic E4 by Robles and Méndez in 2016. The aim of this paper is to investigate the implicative variants (of both systems) which contain Routley and Meyer's logic B and endow them with a Belnap-Dunn type bivalent semantics.

Ternary Relational Semantics for the Variants of BN4 and E4 which Contain Routley and Meyer's Logic B

“…Then, we have (3) B ∧ E ∈ b and also (4 3 and 5. This is, ( 7 (8) [((A∨D)∨C) → (B ∧E)] ∈ c. Similarly as in 4, we have ( 9) (A∨D)∨C ∈ d by A4 and T3 given 2 (D ∈ d). Then, (10) B ∧ E ∈ e by 8 and 9 given Rcde in 1.…”

“…The variants of BN4 and E4 which contain Routley and Meyer's logic B were developed in [6] as possible alternatives to the systems BN4 and E4. The Lti-logics are clearly related to the family of relevant logics since they enjoy the quasi relevance property characteristic of logics such as R-Mingle 6 .…”

“…The logic FDE is a well-known core non-classical system among many-valued and relevant logics (about its importance, cf. [8] and references therein) and is equivalent to Belnap and Dunn's logic B4 [2]. It is worth to mention here that, according to Slaney [12, p. 289], BN4 has the truth-functional implication most naturally associated the logic FDE.…”

“…Although E4 was presented as a companion to BN4 worthy of consideration, Robles and Méndez asserted that E4 might not be the only alternative to BN4 and set out six different variations of the conditional function of MBN4 which could turn out to be possibly interesting 4-valued logics in the family of relevant logics. Some research on the logics built upon these tables has recently been conducted in [6]. In particular, it has been proved that they are the only variants of MBN4 and ME4 which verify Routley and Meyer's logic B.…”

“…Suppose there are a, b, c ∈ K C and wffs A, B, C, D, E, F , G, H such that (1) Rabc but (2) A → B ∈ c * , A ∈ a, B / ∈ b * , (3) C → D ∈ c * , C ∈ b, D / ∈ a * , (4) E → F ∈ c * , E ∈ a, F / ∈ a * and (5) G → H ∈ c * , G ∈ b, H / ∈ b * .Given 2 and 4, we have A ∧ E ∈ a and by A4,(6) (A ∧ E) ∨ (C ∧ G) ∈ a.Similarly, given 2 and 3, we have D ∨ F / ∈ a * (i.e., ¬(D ∨ F ) ∈ a) and by A4, (7)¬(D ∨ F ) ∨ ¬(B ∨ H) ∈ a, this is, (8) ¬[(D ∨ F ) ∧ (B ∨ H)] ∈ a by applying T6. Next, we have (9) [(A ∧ E) ∨ (C ∧ G)] ∧ ¬[(D ∨ F ) ∧ (B ∨ H)] ∈ a by A24 in the form (A ∨ B) → [(A ∨ B) ∧ ¬[C → (A ∨ B)]] → C, we get (4) [(A ∨ B) ∧ ¬[C → (A ∨ B)]] → C ∈ a.Given 1, 2 (C / ∈ c) and 4, we get (5) (A ∨ B) ∧ ¬[C → (A ∨ B)] / ∈ b.…”