Sub-Sub-Intuitionistic Logic
Jonte Deakin, Jim de Groot · Notre Dame Journal of Formal Logic · 2026
Sub-sub-intuitionistic logic is obtained from intuitionistic logic by weakening the implication and removing distributivity. It can alternatively be viewed as conditional weak positive logic. We provide semantics for sub-sub-intuitionistic logic by means of semilattices with a selection function, prove a categorical duality for the algebraic semantics of the logic, and use this to derive completeness. We then consider extensions of sub-sub-intuitionistic logic with a variety of axioms.