Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1

We provide a general investigation of Logic in which the notion of a simple consequence relation is taken to be fundamental. Our notion is more general than the usual one since we give up monotonicity and use multisets rather than sets. We use our notion for characterizing several known logics (including Linear Logic and non-monotonic logics) and for a general, semantics-independent classification of standard connectives via equations on consequence relations (these include Girard's "multiplicatives" and "additives"). We next investigate the standard methods for uniformly representing consequence relations: Hilbert type, Natural Deduction and Gentzen type. The advantages and disadvantages of using each system and what should be taken as good representations in each case (especially from the implementation point of view) are explained. We end by briefly outlining (with examples) some methods for developing non-uniform, but still efficient, representations of consequence relations.

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Irrelevance reasoning refers to the process in which a system reasons about which parts of its knowledge are relevant (or irrelevant) to a specific query. Aside from its importance in speeding up inferences from large knowledge bases, relevance reasoning is crucial in advanced applications such as modeling complex physical devices and information gathering in distributed heterogeneous systems. This article presents a novel framework for studying the various kinds of irrelevance that arise in inference and efficient algorithms for relevance reasoning. We present a

We show that the rule that allows the inference of A from A\Omega B is admissible in many of the basic multiplicative (intensional) systems. By adding this rule to these systems we get, therefore, conservative extensions in which the tensor behaves as classical conjunction. Among the systems obtained in this way the one derived from RMIm (= multiplicative linear logic together with contraction and its converse) has a particular interest. We show that this system has a simple infinite-valued semantics, relative to which it is strongly complete, and a nice cut-free Gentzen-type formulation which employs hypersequents (= finite sequences of ordinary sequents). Moreover: classical logic has a simple, strong translation into this logic. This translation uses definable connectives and preserves the consequence relation of classical logic (not just the set of theorems). Similar results, but with a 3-valued semantics, obtain if instead of RMIm we use RMm (the purely multiplicative fragment ...