Is Quantum Logic a Logic?

Thirty-seven years ago, Richard Greechie and Stanley Gudder wrote a paper entitled Is a Quantum Logic a Logic? [Greechie and Gudder, 1971] in which they strengthen a previous negative result of Josef Jauch and Constantin Piron. [Jauch and Piron, 1970] “Jauch and Piron have considered a possibility that a quantum propositional system is an infinite valued logic. . . and shown that standard propositional systems (that is, ones that are isomorphic to the lattice of all closed subspaces of a Hilbert space) are not conditional and thus cannot be logic in the usual sense.” [Greechie and Gudder, 1971] A conditional lattice is defined as follows. We define a valuation v[a] as a mapping from an element a of the lattice to the interval [0, 1]. We say that two elements a, b are conditional if there exists a unique c such that v[c] = min{1, 1 − v[a] + v[b]}. We call c the conditional of a and b and write c = a → b. We say that the lattice is conditional if every pair a, b is conditional. Greechie and Gudder then proved that a lattice is conditional if and only if it contains only two elements 0 and 1. This implies that [0,1] reduces to {0, 1} and that the lattice reduces to a two-valued Boolean algebra. In effect, this result shows that one cannot apply the same kind of valuation to both quantum and classical logics. It became obvious that if we wanted to arrive at a proper quantum logic, we should take an axiomatically defined set of propositions closed under substitutions and some rules of inference, and apply a model-theoretic approach to obtain valuations of every axiom and theorem of the logic. So, a valuation should not be a mapping to [0,1] or {0, 1} but to the elements of a model. For classical logic, a model for logic was a complemented distributive lattice, i.e., a Boolean algebra. For quantum logics the most natural candidate for a model was the orthomodular lattice, while the logics themselves were still to be formulated. Here we come to the question of what logic is. We take that logic is about propositions and inferences between them, so as to form an axiomatic deductive system. The system always has some algebras as models, and we always define valuations that map its propositions to elements of the algebra — we say, the system always has its semantics — but our definition stops short of taking semantics to be a part of the