The Relevance of Relevance to Relevance Logic - Logic and Its Applications - page 11

The Relevance of Relevance to Relevance Logic

J. Michael Dunn

School of Informatics and Computing, and Department of Philosophy,
Indiana University - Bloomington

Abstract. I explore the question of whether the concept of relevance
is relevant to the study of what Anderson and Belnap call “relevance
logic.” The answer should be “Of course!” But there are some twists and
turns, as is shown by the fact that it has taken over 50 years to get here.
Despite protests by R. K. Meyer that the concept of relevance is not
part of what he calls “relevant logic,” I suggest and defend interpreting
the Routley-Meyer ternary accessibility relation using information states
a, b, c ,so Rabc means “in the context a , b is relevant to c .” Motivations
are provided from Sperber and Wilson's work in linguistics on relevance.

1 Introduction

As the title suggests I will be looking at the question of whether the concept
of relevance is relevant to the study of what Anderson and Belnap have called
“relevance logic.” The reader might think that a short abstract of my paper, if
not the paper itself, would be “Of course!” After all Anderson and Belnap titled
their magnum opus Entailment: The Logic of Relevance and Necessity ,andthis
topic (vol. I, p. xxii) opens with the claim that Wilhelm Ackermann's system
of strenge Implikation “give us for the first time a mathematically satisfactory
way of grasping the elusive notion of relevance of antecedent to consequent in “if
. . . then —” propositions; such is the topic of this topic.” Ackermann's system
with a few important modifications became Anderson and Belnap's system E of
entailment, which Anderson and Belnap promote as their system that captures
both relevance and necessity. It is at one and the same time both what they
term a relevance logic, and also a modal logic. They also present the system
R of relevant implication, which was intended to be E stripped of modality.
The system R has taken on a life of its own and in many ways has become the
focus of relevance logic, and of course, one can always add necessity. Maksimova
(1973) showed that one cannot get E back from R, defining entailment as a
necessary relevant implication by adding what would seem to be the appropriate
S4 type necessity operator. And when one sees how this breaks down, this can