Sub-sub-intuitionistic logic

TL;DR

This paper introduces sub-sub-intuitionistic logic, a weakened form of intuitionistic logic, providing semantics, categorical duality, and completeness results, along with exploring its extensions with additional axioms.

Contribution

It offers a new logical system with semantics and duality theory, expanding the understanding of weakened intuitionistic logics.

Findings

Semantics via semilattices with selection functions

Categorical duality for algebraic semantics

Completeness results for the logic

Abstract

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 the extension of sub-sub-intuitionistic logic with a variety of axioms.