Negation in Relevant Logics (How I Stopped Worrying and Learned to Love the Routley Star)

Negation raises three thorny problems for anyone seeking to interpret relevant logics. the frame semantics for negation in relevant logics involves a ‘point shift’ operator *. Problem number one is the interpretation of this operator. Relevant logics commonly interpreted take the inference from A and ~ A ⋁ B to B to be invalid, because the corresponding relevant conditional A ⋀ (~A ⋁ B) → B is not a theorem. Yet we often make the inference from A and ~ A ⋁ B to B, and we seem to be reasoning validly when we do so. Problem number two is explaining what is really going on here. Finally, we can add an operation which Meyer has called Boolean negation to our logic, which is evaluated in the traditional way: x ⊨ −A if and only if x ⊭ A. Problem number three involves deciding which is the ‘real’ negation. How can we decide between orthodox negation and the new, ‘Boolean’ negation? In this paper, I present a new interpretation of the frame semantics for relevant logics which will allow us to give principled answers to each of these questions.

[1]  Greg Restall,et al.  Truthmakers, entailment and necessity , 1996 .

[2]  Greg Restall,et al.  On logics without contraction , 1994 .

[3]  Kit Fine,et al.  Semantics for quantified relevance logic , 1988, J. Philos. Log..

[4]  S. Read,et al.  Truthmakers, entailment and necessity , 1996 .

[5]  J. Bregetzer [The liar]. , 1989, Revue de l'infirmiere.

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

[7]  Richard Routley,et al.  The Semantics of First Degree Entailment , 1972 .

[8]  R. Routley,et al.  Relevantism, Material Detachment, and the Disjunctive Syllogism Argument , 1984, Canadian Journal of Philosophy.

[9]  John Slaney,et al.  A metacompleteness theorem for contraction-free relevant logics , 1984 .

[10]  James E. Tomberlin,et al.  On the Plurality of Worlds. , 1989 .

[11]  Greg Restall,et al.  Information Flow and Relevant Logics , 1996 .

[12]  N. Belnap,et al.  Entailment. The Logic of Relevance and Necessity. Volume I , 1978 .

[13]  N. Cocchiarella,et al.  Situations and Attitudes. , 1986 .

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

[15]  Nuel D. Belnap,et al.  A Useful Four-Valued Logic , 1977 .

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

[17]  P. Grim The Incomplete Universe: Totality, Knowledge, and Truth , 1994 .

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

[19]  B. Jack Copeland,et al.  On when a semantics is not a semantics: Some reasons for disliking the Routley-Meyer semantics for relevance logic , 1979, J. Philos. Log..

[20]  Chris Mortensen,et al.  Anything is possible , 1989 .

[21]  J. M. Dunn,et al.  Modern Uses of Multiple-Valued Logic , 1977 .

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

[23]  Nuel D. Belnap,et al.  Entailment and the disjunctive syllogism , 1981 .

[24]  Robert K. Meyer,et al.  Logic on the Australian plan , 1986, J. Philos. Log..

[25]  J. Dunn,et al.  Star and Perp: Two Treatments of Negation1 , 1993 .

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