Is Logic Empirical? - Quantum Logic

Quantum Logic

The formal laws of a physical theory are justified by a process of repeated controlled observations. This from a physicist's point of view is the meaning of the empirical nature of these laws.

The idea of a propositional logic with rules radically different from Boolean logic in itself was not new. Indeed a sort of analogy had been established in the mid-nineteen thirties by Garrett Birkhoff and John von Neumann between a non-classical propositional logic and some aspects of the measurement process in quantum mechanics. Putnam and the physicist David Finkelstein proposed that there was more to this correspondence than a loose analogy: that in fact there was a logical system whose semantics was given by a lattice of projection operators on a Hilbert space. This, actually, was the correct logic for reasoning about the microscopic world.

In this view, classical logic was merely a limiting case of this new logic. If this were the case, then our "preconceived" Boolean logic would have to be rejected by empirical evidence in the same way Euclidean geometry (taken as the correct geometry of physical space) was rejected on the basis of (the facts supporting the theory of) general relativity. This argument is in favour of the view that the rules of logic are empirical.

That logic came to be known as quantum logic. There are, however, few philosophers today who regard this logic as a replacement for classical logic; Putnam himself may no longer hold that view. Quantum logic is still used as a foundational formalism for quantum mechanics: but in a way in which primitive events are not interpreted as atomic sentences but rather in operational terms as possible outcomes of observations. As such, quantum logic provides a unified and consistent mathematical theory of physical observables and quantum measurement.