Simple consequence relations

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 to characterize several known logics (including linear logic and non-monotonic logics) and for a general, semantics-independent classification of standard onnectives 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.

Authors

• Arnon Avron

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