Notre Dame Journal of Formal Logic

Algebraic Study of Two Deductive Systems of Relevance Logic

Josep Maria Font and Gonzalo Rodríguez


In this paper two deductive systems (i.e., two consequence relations) associated with relevance logic are studied from an algebraic point of view. One is defined by the familiar, Hilbert-style, formalization of R; the other one is a weak version of it, called WR, which appears as the semantic entailment of the Meyer-Routley-Fine semantics, and which has already been suggested by Wójcicki for other reasons. This weaker consequence is first defined indirectly, using R, but we prove that the first one turns out to be an axiomatic extension of WR. Moreover we provide WR with a natural Gentzen calculus (of a classical kind). It is proved that both deductive systems have the same associated class of algebras but different classes of models on these algebras. The notion of model used here is an abstract logic, that is, a closure operator on an abstract algebra; the abstract logics obtained in the case of WR are also the models, in a natural sense, of the given Gentzen calculus.

Article information

Notre Dame J. Formal Logic, Volume 35, Number 3 (1994), 369-397.

First available in Project Euclid: 21 December 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B46
Secondary: 03G25: Other algebras related to logic [See also 03F45, 06D20, 06E25, 06F35] 06D30: De Morgan algebras, Lukasiewicz algebras [See also 03G20]


Font, Josep Maria; Rodríguez, Gonzalo. Algebraic Study of Two Deductive Systems of Relevance Logic. Notre Dame J. Formal Logic 35 (1994), no. 3, 369--397. doi:10.1305/ndjfl/1040511344.

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