Abstract

In classical and intuitionistic arithmetics, any formula implies a true equation, and a false equation implies anything. In weaker logics fewer implications hold. In this paper we rehearse known results about the relevant arithmetic R, and we show that in linear arithmetic LL by contrast false equations never imply true ones. As a result, linear arithmetic is desecsed. A formula A which entails 0 = 0 is a secondary equation; one entailed by 0 6= 0 is a secondary unequation. A system of formal arithmetic is secsed if every extensional formula is either a secondary equation or a secondary unequation. We are indebted to the program MaGIC for the simple countermodel SZ7, on which 0 = 1 is not a secondary formula. This is a small but signi cant success for automated reasoning.