Basic Intuitionistic Conditional Logic

Abstract

Conditional logics have traditionally been intended to formalize various intuitively correct modes of reasoning involving (counterfactual) conditional expressions in natural language. Although conditional logics have by now been thoroughly studied in a classical context, they have yet to be systematically examined in an intuitionistic context, despite compelling philosophical and technical reasons to do so. This paper addresses this gap by thoroughly examining the basic intuitionistic conditional logic ICK, the intuitionistic counterpart of Chellas’ important classical system CK. I give ICK both worlds semantics and algebraic semantics, and prove that these are equivalent. I give a Godel-type embedding of ICK into CK (augmented with an S4 box connective) and a Glivenko-type embedding of CK into ICK. I axiomatize ICK and prove soundness, completeness, and decidability results. Finally, I discuss extending ICK.