Notre Dame Journal of Formal Logic 35 (3):369-397 (1994)

In this paper two deductive systems 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. 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.
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DOI 10.1305/ndjfl/1040511344
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References found in this work BETA

Protoalgebraic Logics.W. J. Blok & Don Pigozzi - 1986 - Studia Logica 45 (4):337 - 369.
The Independence of Connectives.Timothy Smiley - 1962 - Journal of Symbolic Logic 27 (4):426-436.
Note on Algebraic Models for Relevance Logic.Josep M. Font & Gonzalo Rodríguez - 1990 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (6):535-540.
Reduced Models for Relevant Logics Without ${\rm WI}$.John K. Slaney - 1987 - Notre Dame Journal of Formal Logic 28 (3):395-407.

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Selfextensional Logics with a Conjunction.Ramon Jansana - 2006 - Studia Logica 84 (1):63-104.
Taking Degrees of Truth Seriously.Josep Maria Font - 2009 - Studia Logica 91 (3):383-406.
Fregean Logics.J. Czelakowski & D. Pigozzi - 2004 - Annals of Pure and Applied Logic 127 (1-3):17-76.
The Suszko Operator. Part I.Janusz Czelakowski - 2003 - Studia Logica 74 (1-2):181 - 231.

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