Matrix Semantics for Classical Logic: The Case of the Lattice O6

Abstract

It is well established that classical propositional logic is Boolean. However, this view has recently been challenged. In their paper Non-Orthomodular Models for Both Standard Quantum Logic and Standard Classical Logic: Repercussions for Quantum Computers, Mladen Pavic̆ić and Norman Megill present a non-distributive, non-orthomodular model for both classical and quantum logic based on lattice O6, and argue that classical propositional logic is non-distributive.

In this paper, we examine this claim. Pavic̆ić and Megill’s model is formulated within unital matrix semantics rather than as an algebraic model in the sense of Abstract Algebraic Logic. An analysis of the lattice O6 in the framework of matrix semantics reveals that the matrix (O6,{1,a,b}) is adequate for CL, but not reduced, and induces the same consequence relation as the two-element Boolean matrix B2. Similarly, the unital matrix (O6,{1}) is adequate for CL through reduction to the four-element Boolean matrix B4. Furthermore, we present two lattice constructions that yield matrix models for CL lacking nontrivial lattice-theoretic properties.

These results show that the adequacy of O6 is not intrinsic to its algebraic structure, but is inherited from its reducibility to Boolean matrices, and more generally that classical logic admits models with highly unconstrained lattice structure. Consequently, the existence of such non-distributive models does not undermine the distributive character of classical propositional logic.