Completeness of relevant quantification theories.
Jon Michael Dunn, Hugues Leblanc, Robert K. Meyer · Notre Dame Journal of Formal Logic · 1974
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.