Categorical Consequence for Paraconsistent Logic

Abstract

Consequence relations over sets of "judgments" are defined by using "overdetermined" as well as "underdetermined" valuations. Some of these relations are shown to be categorical. And generalized soundness and completeness results are given for both multiple and single conclusion categorical consequence relations. Rumfitt(1] discusses multiple-conclusion consequence relations defined over sets whose members, if any, are assertions or rejections. The consequence relations are generated by sets of valuations whose m~mbers include those that admit truth value gaps-there are sentences that are neither true nor false on some valuations. Rumfitt shows that the consequence relations are categorical-that is, the consequence relations generated by distinct sets of valuations are distinct. Given Rumfitt's work, it is natural to ask whether categoricity holds for multiple-conclusion consequence relations generated by sets of valuations whose members include those that admit truth value gluts -there are sentences that are both true and false on some valuations. We will show that the answer is Yes.Johnson [2] extends Rumfitt's work to obtain categoricity results for single conclusion consequence relations generated by valuations that allow truth-value gaps. We will extend his discussion by considering valuations that allow truth-value gluts.We also will extend Shoesmith and Smiley's (3] generalized soundness and completeness results for consequence relations generated by valuations that admit neither gaps nor gluts to those that do. 1 1 Our discussion is heavily influenced by Shoesmith and Smiley [3] and Smiley [4]. The latter gives credit to Carnap [5] and [6] for initiating discussions of rejection and categoricity, respectively. This remark by Smiley [4, p. 7] is especially noteworthy: 'If I am right, the practice of identifying a calculus with its consequence relation is only justified if specifying the consequence relation is sufficient to determine the entire sentential output of the semantics.' We endorse Smiley's view that the common practice of identifying calculi by using non-categorical consequence relations is a mistake. 141