Logic with truth values in A linearly ordered heyting algebra

Abstract

It is known that the theorems of the intuitionist predicate calculus are exactly those formulas which are valid in every Heyting algebra (that is, pseudo-Boolean algebra). The simplest kind of Heyting algebra is a linearly ordered set. This paper concerns the question of determining all formulas which are valid in every linearly ordered Heyting algebra. The question is of interest because it is a particularly simple case intermediate between the intuitionist and classical logics. Also the interpretation of implication is such that in general there exists no nondiscrete Hausdorff topology for which this operation is continuous.