The algebra of conditional logic

Abstract

Conditional logic is the non-commutative regular extension of Boolean logic to 3 truth values; the third truth value stands for “undefined” or “non-terminating evaluation”. In conditional logic, expressions are evaluated from left to right and evaluation stops as soon as the answer may be obtained. For example, “x and y” is false wheneverx is false, undefined wheneverx is undefined and takes the value ofy wheneverx is true. In this paper we study the variety generated by the 3-element algebra associated with conditional logic. We obtain the following results for this variety: a finite complete set of laws; a detailed description of free algebras as sets of ordered, rooted, labelled, binary trees; a representation theorem for the algebras in this variety, analogous to that for Boolean algebras and a recursive formula and an asymptotic approximation for the orders of the finitely-generated free algebras.