Quantum Logic and Its Role in Interpreting Quantum Theory

Foundational studies in quantum theory have consisted in large part of attempts to avoid paradoxical situations in describing properties of a physical system. Quantum logic addresses some of the perplexing problems that plague interpretations of quantum theory by invoking a non-classical propositional calculus for the set of propositions of a quantum system. Historically, quantum logic derives from von Neumann’s observation that the set of projection operators on a Hilbert space constitute a ‘logic’ of experimental propositions. More than two decades passed from the pioneering work of Birkhoff and von Neumann before interest in quantum logic was re-ignited by Mackey’s probabilistic analysis of quantum theory. Piron’s axiomatization provided a significant extension of Mackey’s formulation, leading to further developments focused primarily on establishing concrete operational foundations to quantum logic, most notably, the empirical framework introduced by Foulis and Randall in the 1980s. Recent advances in the past two decades feature the employment of powerful techniques from pure mathematics, in particular, various algebraic notions taken from category theory and computational logic. Aside from establishing the formal structures characterizing the syntax of quantum logic, there have also been concerted efforts to attribute realist or empiricist readings to the semantic content embodied in the projection lattice of Hilbert space. Some prominent quantum-logical interpretations, at least those critically evaluated in this essay, include Reichenbach’s trivalent logic, Putnam’s realist quantum logic, and Finkelstein’s operational quantum logic.