Reflections on Brady's Logic of Meaning Containment (to appear in The Australasian Journal of Logic)
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 con- cepts 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 increase 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.
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