Quantum Logic and Meaning
Sebastian Horvat, Iulian D. Toader
TL;DR
The paper formalizes quantum logic (QL) within the Dunn–Hardegree algebraic framework to enable a rigorous, platform-agnostic comparison with classical logic (CL). It analyzes viable bivalent semantics via two valuation notions, $H$-classes and $H$-klasses, showing that while conjunction can be truth-functional, disjunction and negation are not in the standard framework, and that no single class makes all connectives TF. The work argues that a realist interpretation of quantum phenomena is not ruled out by bivalence, provided one allows non-truth-functional connectives and carefully reframes arguments about validity and meaning. It further critiques Hellman’s meaning-variance claim, offering an enhanced argument that specifies which QL connectives may differ in meaning from their CL counterparts and highlighting the dependence on the chosen semantic apparatus (classes vs. klasses). Overall, the paper lays a foundational semantic framework for reconciling QL with realism and for re-evaluating the alleged semantic divergence between classical and quantum connectives.
Abstract
This paper gives a formulation of quantum logic in the abstract algebraic setting laid out by Dunn and Hardegree (2001). On this basis, it provides a comparative analysis of viable quantum logical bivalent semantics and their classical counterparts, thereby showing that the truth-functional status of classical and quantum connectives is not as different as usually thought. Then it points out that bivalent semantics for quantum logic -- compatible with realism about quantum mechanics -- can be maintained, albeit at the price of truth-functionality. Finally, the paper critically addresses Geoffrey Hellman's argument (1980) that this lack of truth-functionality entails a change of meaning between classical and quantum connectives.