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 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, Hallden, Deutsch and Daniels, we provide a general recipe to define (a given family of) containment logics, we explore the single-premise/single-conclusion fragment of S^*_fde , dS^*_fde , crossS^*_fde and the connections between crossS^*_fde and the logic Eq of equality by Epstein. Also, we present S^*_fde  as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues.