Philip Kremer Relevant Predication : Grammatical

This paper reformulates and decides a certain conjecture in Dunn’s ‘Relevant Predication 1: The Formal Theory’ (Journal of Philosophical Logic 16, 347-381, 1987). This conjecture of Dunn’s relates his object-language characterisation of a property’s being relevant in a variable x to certain grammatical characterisations of relevance, analogous to some given by Helman, in ‘Relevant Implication and Relevant Functions’ (to appear in Entailment: The Logic of Relevance and Necessity, vol. 2, by Alan Ross Anderson, Nuel Belnap, and J. Michael Dunn et al.) In the course of the investigation this paper also investigates Kit Fine’s semantics for quantified relevance logics, which appears in his appropriately titled ‘Semantics for Quantified Relevance Logics’ (Journal of Phiiosophical Logic 17, 27-59, 1988). 1. BACKGROUND AND INTRODUCTION Dunn (1987) investigates the notion of relevant predication in the context of relevance logic; after suggesting an object-language definition specifying when a formula Ax is of a “kind that determines relevant properties (with respect to the variable x)“, he leaves us with a technical conjecture relating his object-language characterisation of relevance to various possible meta-linguistic characterisations in particular, to some characterisations given in Helman (to appear; see also Helman 1977). In this paper we restate this conjecture in various forms and show which forms of the conjecture are true and which are false, thereby linking Dunn’s and Helman’s research projects. Dunn’s project Dunn motivates his object-language definition of relevance by considering the following pair of statements: (1) Socrates is such that he is wise (2) Reagan is such that Socrates is wise Despite the surface similarities of these statements and the classical logician’s temptation to treat (2) as a kind of degenerate case of the Journal of Philosophical Logic 18: 349-382, 1989.