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.
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?
