The problem of inconsistency in relevant connexive logics

The article addresses the issue of inconsistency in relevant connexive logic CR. It demonstrates that inconsistency can be eliminated by rejecting the semantic postulate of idempotence. This modification yields a new logical theory, CRW, which retains the properties of relevance and connexivity while ensuring consistency. The work is grounded in the broader context of combining relevant and connexive logics, which aim to address classical logical paradoxes and better align with natural reasoning patterns. The study builds upon the foundational system of relevant logic R, developed by Anderson and Belnap, as a framework for analysis. While the initial attempts to integrate Aristotle’s thesis into R led to trivialization, recent efforts to extend certain subsystems of R with connexive principles have avoided this problem. The CR system proposed by Francez serves as the basis for further exploration. Although CR satisfies relevance and connexivity, it exhibits inconsistency - a property the study aims to resolve. This inconsistency is addressed through both syntactic adjustments and semantic innovations. The article introduces CRW, a variant of CR that excludes the contraction axiom (A → B) → (A → (A → B)). This modification is supported by an alternative semantics based on commutative monoids. The CRW system is axiomatically defined and satisfies key properties of relevance, connexivity, and consistency. The article rigorously demonstrates these properties through semantic models and logical proofs, including Aristotle’s and Boethius’ theses, and highlights CRW’s non-trivial nature. The results suggest that CRW offers a promising framework for further exploration of relevant and connexive logics. Future research directions include proving the completeness of CRW’s semantics, developing natural deduction or Gentzen-style systems for CRW, and investigating extensions incorporating additional logical connectives. These efforts aim to refine CRW’s theoretical underpinnings and enhance its applicability in modeling natural reasoning processes. The author declares no conflicts of interests.

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