While studying quantum mechanics, I've seen two contradictory opinions about quantum logic. Some people say that it indeed defies classical logic (see Emilio Santos (1986). The Bell inequalities as tests of classical logic and Samson Abramsky. Classical logic, classical probability, and quantum mechanics).

A violation of Boole’s conditions of possible experience cannot be encountered when all the frequencies concerned have been measured on a single sample. Such a violation simply entails a logical contradiction; ‘observing’ it would be like ‘observing’ a round square. We expect Boole’s conditions to hold even when the frequencies are measured on distinct large random samples. But they are systematically violated, and there is no easy way out (see below). We thus live ‘on the edge of a logical contradiction’. An interpretation of quantum mechanics, an attempt to answer the WHY question, is thus an effort to save logic.

Itamar Pitowsky. George Boole’s “Conditions of Possible Experience” and the Quantum Puzzle.

Other people say that quantum logic is completely irrelevant, such as the eminent philosopher of science Tim Maudlin.

the horse of quantum logic has been so thrashed, whipped and pummeled, and is so thoroughly deceased that...the question is not whether the horse will rise again, it is: how in the world did this horse get here in the first place? The tale of quantum logic is not the tale of a promising idea gone bad, it is rather the tale of the unrelenting pursuit of a bad idea.

Tim Maudlin. The Tale of Quantum Logic

I would like to know how does Bell's Inequality relate with the failure of the distributive law of classical logic. Is classical logic really insufficient to explain quantum phenomena?

  • 1
    $\begingroup$ Non-paywalled Pitowsky and Maudlin $\endgroup$
    – benrg
    Mar 23, 2021 at 2:23
  • $\begingroup$ Hopefully somebody would provide more details, but as far as I have seen quantum logic is no more than the idea that probabilities must be changed to probability amplitudes. Not to be confused with quantum computing difference between qubits and bits. $\endgroup$
    – Mauricio
    Sep 9, 2022 at 17:15

1 Answer 1


The essence of Bell's theorem, from this point of view, is related to the fact that the lattice of quantum elementary propositions is not Boolean. This is not the only ingredient however. Quantum logic is however a disjoint issue. The lattice is the one of orthoprojectors in a Hilbert space product of two Hilbert spaces. The existence of incompatible propositions, which also implies failure of distributivity of the two connectives, plays a crucial role in fact. But this is a physical evidence of quantum systems, which possibly may be treated from the viewpoint of quantum logic. However nothing forces to force us to adopt that point of view.


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