Rejection and Truth-Value Gaps

A theorem due to Shoesmith and Smiley that axiomatizes twovalued multiple-conclusion logics is extended to partial logics. Rumfitt [1] extends Smiley’s [3] discussion of rejection by axiomatizing a calculus where truth values of sentences are given by truth tables that admit truth-value gaps. “The Smiley multiple-conclusion consequence relation” for the calculus is defined over assertions and rejections. Rumfitt gives a complex Henkin-style proof of completeness for this calculus. Our goal is to show that there is a simple procedure for axiomatizing calculi of the sort that he considers. We do this by imitating Shoesmith and Smiley’s [2] proof of a similar result (their Theorem 18.1) where truth tables do not admit truth-value gaps and the consequence relation is defined without using rejections. Let A1, A2, . . . be sentences. And let +p and ∗p be assertions and rejections, respectively, given that p is a sentence. Assertions and rejections are judgments. We let +J, . . . ,∗J, . . ., and J, . . . range over sets of assertions, sets of rejections, and sets of judgments, respectively. Let a valuation v be a function that maps sentences into {t, n, f } (true, neither true nor false, and false) and judgments into {c, i} (correct and incorrect), where v(+p) = c if and only if v(p) = t and v(∗p) = c if and only if v(p) = f . The Smiley multiple-conclusion consequence relation, |=, is defined as follows: J |= K if and only if, for every valuation v, v assigns i to a member of J or c to a member of K (so |= preserves correctness). Assume a language with connectives c1, . . . , cn where valuations are determined by truth tables for the connectives. To define J K (K is deducible from J) we use the following structural rules together with the truth-table rules: