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.

In recent years, AI research has started to take seriously the role of context in developing models. This trend is heralded by a departure from logic-based views and their over-emphasis on the formal aspects of thought processes. The departure, however, has not been complete--- some fundamental assumptions of formal logic are still maintained in the new approaches. Similarly, "relevance" remains a major challenge for AI research. This paper outlines an alternative proposal that takes context and relevance as intertwined aspects of thought and intelligence. We argue for a pragmatic approach, which shows better promise than formal logic in dealing with such issues.

Abstract. Shortest possible axiomatizations for the implicational fragments of the modal logics S4 and S5 are reported. Among these axiomatizations is included a shortest single axiom for implicational S4—which to our knowledge is the first reported single axiom for that system—and several new shortest single axioms for implicational S5. A variety of automated reasoning strategies were essential to our discoveries.