Quantum logic

In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account. This research area and its name originated in a 1936 paper by Garrett Birkhoff and John von Neumann, who were attempting to reconcile the apparent inconsistency of classical logic with the facts concerning the measurement of complementary variables in quantum mechanics, such as position and momentum.

Quantum logic can be formulated either as a modified version of propositional logic or as a noncommutative and non-associative many-valued (MV) logic.

Quantum logic has some properties that clearly distinguish it from classical logic, most notably, the failure of the distributive law of propositional logic:

where the symbols p, q and r are propositional variables. To illustrate why the distributive law fails, consider a particle moving on a line and let

(using some system of units where the reduced Planck's constant is 1) then we might observe that: