Relevant and substructural logics

This essay is structured around the bifurcation between proofs and models: The first section discusses Proof Theory of relevant and substructural logics, and the second covers the Model Theory of these logics. This order is a natural one for a history of relevant and substructural logics, because much of the initial work — especially in the Anderson–Belnap tradition of relevant logics — started by developing proof theory. The model theory of relevant logic came some time later. As we will see, Dunn’s algebraic models [76, 77] Urquhart’s operational semantics [267, 268] and Routley and Meyer’s relational semantics [239, 240, 241] arrived decades after the initial burst of activity from Alan Anderson and Nuel Belnap. The same goes for work on the Lambek calculus: although inspired by a very particular application in linguistic typing, it was developed first proof-theoretically, and only later did model theory come to the fore. Girard’s linear logic is a different story: it was discovered though considerations of the categorical models of coherence..
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