R and Relevance Principle Revisited

This paper first shows that some versions of the logic R of Relevance do not satisfy the relevance principle introduced by Anderson and Belnap, the principle of which is generally accepted as the principle for relevance. After considering several possible (but defective) improvements of the relevance principle, this paper presents a new relevance principle for (three versions of) R, and explains why this principle is better than the original and others.

[1]  Greg Restall,et al.  An Introduction to Substructural Logics , 2000 .

[2]  Steve Giambrone,et al.  Where gamma fails , 1984 .

[3]  Franco Montagna,et al.  Substructural fuzzy logics , 2007, Journal of Symbolic Logic.

[4]  J. Michael Dunn,et al.  Relevance Logic and Entailment , 1986 .

[5]  Nuel Belnap,et al.  The pure calculus of entailment , 1962, Journal of Symbolic Logic.

[6]  Dov M. Gabbay,et al.  Handbook of Philosophical Logic , 2002 .

[7]  Richard Sylvan,et al.  Relevant Logics and Their Rivals, Volume II , 2005 .

[8]  Edwin D. Mares,et al.  “Four-Valued” Semantics for the Relevant Logic R , 2004, J. Philos. Log..

[9]  Robert K. Meyer,et al.  New axiomatics for relevant logics, I , 1974, J. Philos. Log..

[10]  J. Michael Dunn,et al.  Partiality and Its Dual , 2000, Stud Logica.

[11]  Soren Hallden,et al.  On the semantic non-completeness of certain Lewis calculi , 1951, Journal of Symbolic Logic.

[12]  R. Meyer,et al.  Intuitionism, Entailment, Negation , 1973 .

[13]  Nuel D. Belnap,et al.  Entailment : the logic of relevance and necessity , 1975 .

[14]  Hiroakira Ono,et al.  Proof-Theoretic Methods in Nonclassical Logic --an Introduction , 1998 .

[15]  Peter Jipsen,et al.  Residuated lattices: An algebraic glimpse at sub-structural logics , 2007 .

[16]  R. Meyer,et al.  The semantics of entailment — III , 1973 .

[17]  Richard Routley,et al.  Classical relevant logics II , 1973 .

[18]  Norihiro Kamide,et al.  Substructural Logics with Mingle , 2002, J. Log. Lang. Inf..

[19]  J. Michael Dunn.  Algebraic Completeness Results for R-Mingle and Its Extensions , 1970, J. Symb. Log..

[20]  Richard Routley,et al.  Relevant logics and their rivals , 1982 .

[21]  Richard Sylvan,et al.  The semantics of entailment—II , 1972, Journal of Philosophical Logic.