Itamar Pitowsky on Logic and Probability in Modern Physics: The Case of Quantum Mechanics

What is the basic structure of the world responsible for quantum "weirdness"? We shall review the attempts, which began with the mathematicians Birkhoff and von Neumann, to derive the basic structure underlying quantum mechanics (the Hilbert space) from simple logical axioms; and the arguments in support of the view that this is the quantum analogue of the logical (Boolean) structure of classical physics. It is a remarkable feature of the quantum formalism that it can be obtained from "almost nothing".

Another remarkable feature is that quantum logic dictates Born's rule, that is, the quantum algorithm for calculating probabilities (this is a theorem by Gleason). We shall see how this fact provides for a natural and quite minimal semantics for quantum logic, which is in an important sense complete.

Finally we shall see how this semantics accounts for the usual features of quantum mechanics-and thus for quantum "weirdness"- in a straightforward way: for example, the uncertainty relations, the violation of Bell's inequalities, the Kochen and Specker theorem, and the apparently classical behavior of macroscopic objects.