Completeness of relevant quantification theories.

Abstract

In [20], Meyer and Dunn answered affirmatively for the relevant sentential logics E and R the question, "Is the rule y, 'From v-A and f-Av5, to infer B,' admissible?"This result, which confirmed an old conjecture of Anderson and Belnap, establishes the weak completeness of these and a number of related logics.In the present paper, some of whose principal results were announced without proof in [21], we shall extend the methods of past papers to prove both the admissibility of γ and, in a reasonable sense, weak completeness for the first-order extension RQ of R. In doing so, we replace the intuitively uninformative R-matrices of [20] with the theory of DeMorgan monoids, which furnishes a surprisingly smooth and natural algebraic semantics for R and, by extension, for RQ.1. Furnishing RQ with a viable algebraic semantics and a proof of γ is no unimportant task.In the first place, the Anderson-Belnap system R of relevant implication is at the sentential level the most stable and interesting of the relevant logics.R contains in exact and well-motivated ways both the intuitionistic and the classical sentential calculi. 1 Rj, the implicational fragment of R, 2 is the oldest of the relevant logics, having been independently investigated twenty years ago by Moh-Shaw-Kwei and by Church in important papers, which provide interesting deductive-methodological motivation (A relevantly implies B only if A is used in some deduction of B). 3 1.An exact translation of the Curry system HD into R, and hence of the intuitionistic sentential calculus, is presented in [18]; cf.[4] and [17].In &, v, -, R contains all classical tautologies; cf.[6] and also [1]. 2. In unpublished work Meyer has proved that R is a conservative extension of Rj when the latter is axiomatized as by Church in [10].This settles an open question for R of the sort raised by Anderson for E and Ej in [2].Cf. also Prawitz's [24].3. Cf. [10] and [22].