1992/03/01 by Edwin D. Mares, Edwin Mares, Robert K. Meyer · 3 citations
Mathematics · Computer Science · #Analytic Number Theory Research #Mathematical Approximation and Integration #Polynomial and algebraic computation
paper · doi:10.1305/ndjfl/1093636096
The logic NR of Meyer's "Entailment and relevant implication" is extended to include the axiom scheme Π (A vB)-> (OA v ΠB) to create the logic R4, so named because it is a conservative extension of S4.It has been an open problem since the writing of "Entailment and relevant implication" whether Ackermann's rule y is admissible in R4.In this paper, we close this problem by proving that y is admissible in this system.In [4], Meyer formulates the system NR, which combines the axioms governing implication from Anderson and Belnap's system R with the axioms governing the behavior of necessity from S4. NR, however, does not contain S4 on a direct translation.For example, NR does not contain the scheme ~Π\(AvB)v (OA v ΠB).To overcome this deficiency, Belnap and Meyer have suggested adding the postulateto the axioms of NR (see Routley and Meyer [6], p. 70).We call the system that results from the addition of this new axiom scheme R4 (to signify the fact that it contains all of S4).R4 has not yet been adopted as the system of modality and relevance, at least to a large extent, because it has not been shown to be complete over the semantics suggested in [6], and Ackermann's rule y (from \-~A vB and \-A infer h#) had not been shown to be admissible in it.The purpose of this paper is to remove the latter difficulty.That is, we show that y is admissible in R4.Our argument uses a modified version of Meyer's method of metavaluations (following, e.g., Meyer [5]).We build a structure of regular, prime R4 theories that mimics, for the most part, a Kripke model for S4.We impose a binary accessibility relation on this structure.We then show, using a version of the method of metavaluations, that each of the theories in our structure can be reduced to a theory that is prime, regular, and consistent, while retaining the same accessibility relation between these reduced theories.As a corollary of this construc-