Relevance, Relatedness and Restricted Set Theory

1. Introduction Relevance logic has become ontologically fertile. No longer is the idea of relevance restricted in its application to purely logical relations among propositions, for as Dunn has shown in his (1987), it is possible to extend the idea in such a way that we can distinguish also between relevant and irrelevant predications, as for example between " Reagan is tall " and " Reagan is such that Socrates is wise ". Dunn shows that we can exploit certain special properties of identity within the context of standard relevance logic in a way which allows us to discriminate further between relevant and irrelevant properties, as also between relevant and irrelevant relations. The idea yields a family of ontologically interesting results concerning the different ways in which attributes and objects may hang together. Because of certain notorious peculiarities of relevance logic, however, 1 Dunn's idea breaks down where the attempt is made to have it bear fruit in application to relations among entities which are of homogeneous type. Let us suppose that is red and is prime are relevant properties in the sense of Dunn. Then the formal machinery underlying his approach dictates that so also are: is either red or prime, is both red and prime, is if red then prime, and so on. 2 This consequence is surely both counterintuitive and in conflict with Dunn's stated aims. Still worse, however, are the implications in relation to what Dunn calls " relevant properties of pairs ". A pair has such a relevant property, Dunn tells us, 3 iff there are relevant monadic properties F and G—relevant monadic properties of any old sort—such that Fa ∨ Gb. But this seems to be a good first candidate definition of what one might better call an irrelevant property of a pair. Can we, then, provide a more adequate account—a means of sifting out the relevant pairs (Gilbert and Sullivan, Anderson and Belnap, Xanthippe and Socrates) from the irrelevant pairs (Napoleon and the moon, the number 2 and the Eiffel Tower, Quine's left foot and Chisholm's right ear)? Can we, more generally, find a means of sifting out the relevant sets from those irrelevant sets which, while constituting a formal unity, lack all material connectedness among their members. Examples of relevant sets would be: The Berlin Philharmonic Orchestra, the set of molecules in this nut, the natio hungarica. Examples of irrelevant …