Reflections on Brady's Logic of Meaning Containment

Abstract

Abstract: This paper is a series of reflections on Ross Brady’s favourite substructural logic, the logic MC of meaning containment. In the first section, I describe some of the distinctive features of MC, including depth relevance, and its principled rejection of some concepts that have been found useful in many substructural logics, namely intensional or multiplicative conjunction (sometimes known as ‘fusion’), the Church constants (⊤ and ⊥), and the Ackermann constants (t and f). A further distinctive feature of the axiomatic formulation of MC is its meta-rule, which is a unique feature of MC Hilbert proofs. This meta-rule gives rise to one further special property of MC, in that the logic is distributive in one sense, and non-distributive in another. The distribution of additive conjunction over disjunction (the step from p∧(q∨r) to (p∧q)∨(p∧r)) holds in MC as a rule, but not as a provable conditional, and in this way, MC is distinctive among popular substructural logics. (Anderson and Belnap’s favourite logics R and E are distributive in both senses, while Girard’s linear logic is distributive in neither.)

In this paper, I aim to enhance our understanding of each of these distinctive features of MC, giving an account of what it might take for a propositional logic to meet these constraints. I will start with a presentation of Hilbert proofs for MC, and then showing how Brady Lattices (a natural class of algebraic models for MC) can help us understand each of these special features of Brady’s logic of meaning containment.