A Fresh Look at Relevant Number Theory

Abstract

Underlying numerical reasoning is the formal theory of arithmetic, and if the reasoning is to be carried out in a nonclassical logic then this will be a correspondingly nonclassical arithmetic. The relevant arithmetic R# was proposed around 50 years ago and is one of the few theories based on substructural logic to have been investigated in much detail. This paper surveys some old results concerning R# and recent attempts to extend it to deal with the rational numbers as well as the naturals. While the formal results here are not new, it is worthwhile to put them together and to present the topic as one of interest for contemporary research into nonclassical reasoning. Show more