Definition: An interpretation of quantum mechanics developed by John von Neumann in the late 1930′s. Quantum logic says that everyday logic cannot be applied to the quantum world. Contrary to Boolean logic, quantum logic says that and and either do not have the same meaning in the quantum world.
More about the term: This interpretation isn’t known to be deterministic or indeterministic. That is still up for debate. However, there is no collapse of the wave function. Also, local causation is uncertain as well. There is little debate on the issue but the majority understanding of this is that it has a unique history (contrary to other non-collapse interpretations like Many Worlds). When using Boolean logic to assess quantum logic it may seem that quantum logic is self-contradictory; however, if quantum logic is assessed internally it is indeed consistent. The major problem for this system is extrapolating applied logic. Boolean logic is certainly valid in everyday life but is invalid in the quantum world. One way or another it seems that contradictions may arise somewhere along the way. For more information see John Gribbin’s Q is for Quantum.
Example of use: Consider the double slit experiment where a photon is shot at a wall with two slits and the photon goes through either one (or both). So, because there is no wave collapse the photon actually goes through both slits. There’s a different logical significance in this experiment.