Single Axioms and Axiom-Pairs for the Implicational Fragments of R, R-Mingle, and Some Related Systems

Abstract

Various axiom sets for the implicational fragments R→and RM→of R and of R-Mingle have appeared in the literature over the last six-and-a-half decades, some of them in other guises well before the full systems with ∼, &, and ∨ were even introduced. Most such sets are comprised of three or four axioms. For other logics of pure implication, the historical progression has typically been from longer axiom sets to the discovery of deductively equivalent two- and one-axiom bases. This paper continues in that pattern, presenting such bases for R→and RM→. Along the way, new axiom pairs and new single axioms are given for a number of other implicational logics as well, some in the paper itself and many in the Appendix attached to it. Prominent among these is C.A. Meredith’s system BCI. Though single axioms for BCI are of independent interest, one of them in particular also plays an invaluable role in the construction of those provided here for implicational R and R-Mingle.