Relevance Logic as a Conservative Extension of Classical Logic

Part of the Outstanding Contributions to Logic book series (OCTR, volume 3)


Relevance logic is ordinarily seen as a subsystem of classical logic under the translation that replaces arrows by horseshoes. If, however, we consider the arrow as an additional connective alongside the horseshoe, then another perspective emerges: the theses of relevance logic, specifically the system R, may also be seen as the output of a conservative extension of the relation of classical consequence. We describe two ways in which this may be done. One is by defining a suitable closure relation out of the set of theses of relevance logic; the other is by adding to the usual natural deduction system for it further rules with ‘projective constraints’, whose application restricts the subsequent application of other rules. The significance of the two constructions is also discussed.


Relevance logic System R Consequence relations  Closure relations Natural deduction Projective constraints 



The author is very much indebted to Lloyd Humberstone, whose comments on a badly crashed early text helped the author to start again, and whose remarks on later drafts were also valuable. Thanks also to Diderik Batens, Michael Dunn, Jim Hawthorne, João Marcos and Peter Verdée for helpful discussions, as well as two anonymous referees for their suggestions.


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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of PhilosophyLogic and Scientific Method London School of EconomicsLondonUK

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