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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?

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    $\begingroup$ Non-paywalled Pitowsky and Maudlin $\endgroup$
    – benrg
    Commented 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
    Commented Sep 9, 2022 at 17:15

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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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On Bell's theorem

Bell's theorem does in no way imply the failure of classical logic. This is clear, since there are examples of theories such as Bohmian mechanics, which agree with QM predictions and therefore violate Bell's inequality (in its generalized form), but are formulated just with ordinary logic.

The purpose of Bell's theorem is to show that the physical world is nonlocal. Einstein had pointed out in the famous EPR paper, that orthodox QM is either incomplete or nonlocal (since the wave collapse is instantenous). Einstein had hoped that locality could be rescued by introducing further physics not yet discovered. Bell showed that this is not possible (if the predictions of QM are right). He starts by formulating what it means to be a local theory of physics. Then he shows that certain inequalities cannot be violated by such a theory. See https://hal.science/jpa-00220688/document for a good exposure of the details.

Now QM predicts a violation of these inequalities. But more importantly the experiments violating Bell's inequality have been confirmed in experiments (see e.g https://arxiv.org/abs/1508.05949). So what ever the true theory of nature is, it must be nonlocal (even if QM turns out to be false).

About Quantum logic

Among other aspects of Quantum mechanics the nonlocality is a very unintuitive feature. So unintuitive, that people have put enormous effort to get around Bell's theorem. This might have connection to quantum logic.

If you are willing to abandon ordinary logic, Bell's theorem might not imply non-locality. If some reasoning turns out to be unlogical, you could call it quantum logical. On the other hand, more or less anything goes, if you give up logical reasoning. As Tim Maudlin put it:

There is no point in arguing with somebody how does not belive in logic.

As the mathematical physicist Roderich Tumulka pointed out. Quantum Logic can mean three different things

  1. A certain piece of mathematics that is rather pretty

  2. A certain analogy between two formalisms that is rather limited

  3. A certain philosophical idea that is rather wrongheaded

Quantum logic as piece of mathematics, namely an orthomodular lattice which has some similarities to Boolean algebras, which can be quite interesting.

The analogy to quantum mechanics is rather limited. For example for a Spin $\frac{1}{2}$ particle orthogonal complement to $\mathbb C |up \rangle$ is given by $\mathbb C |down \rangle $. But the negation to $\Psi \in \mathbb C |up \rangle $ is not spin down but $\Psi \in \mathscr H \setminus \mathbb C |up \rangle$. In other words the negation of spin up, is not spin down, but anything else then spin up.

The third point is, that logic should amount to all statements that are true every conceivable situation. So logic cannot depend on physical laws and cannot be revised by empirical science.

A famous example of a philosopher how thought about quantum logic was Hilary Putnam. Hey was known for rethinking, and revising his own positions. He famously supported the idea that empirical conformation of quantum theory might lead to a refutation of classical logic, but later changed his opinion.

See https://courses.washington.edu/philo482/%2316a_Putnam%28again%29.pdf for reference.

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