Meeting of the Association for Symbolic Logic

Abstract

S OF PAPERS 1427 rules treating the new connective exactly as the old rules treat the original connective, then the two connectives form (from the same components) equivalent compounds. Following Belnap (in his famous reply to A. N. Prior's Tonk paper-both conveniently accessible in [1]), we say in such a case that the rules uniquely characterize the connective in question. Examples: one can easily check that Gentzen's natural deduction rules uniquely characterize conjunction, disjunction and implication. The present paper, after introducing the topic in ?1, draws out, in ?2, a philosophical moral from certain examples where rules governing a connective are more than strong enough to uniquely characterize that connective. ?3 is devoted to clarifying the concept of unique characterization with examples and precise definitions: it pays here to be particularly careful about the notion of a rule, and the familiar admissible/derivable distinction is supplemented by a distinction between weakly and strongly derivable rules, the latter concept involving consideration of a range of languages instead of a single language. ?4 explores the extent of the uniqueness phenomenon with special attention to modal logic. Through the paper, logical systems are thought of as manipulating sequents by means of rules. By a logical framework we mean a conception of what sort of object a sequent is (e.g.: are multiple succedents allowed?): we shall see that connectives uniquely characterizable in one framework need not be so in another.