🤖 AI Summary
This study addresses the unification of implication structures in intuitionistic logic and orthomodular logic while preserving both constructivity and quantum logical features. To this end, it introduces Sasaki implication as the central connective and constructs a novel logical system, termed iEx-logic, which is axiomatized for the first time. Employing methods from algebraic logic—integrating lattice theory, orthomodular algebras, and intermediate logics—the paper demonstrates that the lattice of extensions of iEx-logic decomposes as the direct product of the lattice of intermediate logics and the lattice of orthomodular logics. This work not only achieves a refined integration of these two logical frameworks but also reveals the robustness and decomposability of their underlying algebraic structure.
📝 Abstract
We investigate logics that generalize both intuitionistic logic and quantum logic. In earlier work, we introduced Ex-logic, an extension of Holliday's fundamental logic that coincides with the intersection of orthologic and the implication-free fragment of intuitionistic logic. In this paper, we add an implication connective to Ex-logic and axiomatize iEx-logic, the intersection of full intuitionistic logic and orthomodular logic with the implication connective interpreted as the Sasaki hook. As a consequence, we obtain a characterization of the lattice of logics extending iEx-logic as the product of the lattice of intermediate logics and the lattice of orthomodular logics. We also explore the robustness of our algebraic approach by briefly discussing extensions of iEx-logic with modal operators.