Relevant Logics Obeying Component Homogeneity

Abstract

This paper discusses three relevant logics (S^fde; dS^  fde; crossS^fde) that obey Component Homogeneity|a principle that Goddard and Routley [29] introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity|that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S^fde; dS  ^fde; crossS^fde. Second, the paper establishes complete sequent calculi for S^fde; dS^fde; crossS^fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar [5], Hallden [30], Deutsch [16] and Daniels [13], we provide a general recipe to define (a given family of) containment logics, we explore the single-premise/single-conclusion of S^fde; dS^fde; crossS^fde and the connections between crossS^fde and the logic Eq of equality by [22]. Also, we present S^fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley [29]. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues