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arXiv:2410.06588 · Analysis

Distribution can be dropped: Reply to Rumfitt

Iulian D. Toader

Published Oct 2, 2024• 1 citation
arXivHTMLPDF

Abstract

Abstract Most believe that there are no empirical grounds that make the adoption of quantum logic necessary. Ian Rumfitt has further argued that this adoption is not possible either, for the proof that distribution fails in quantum mechanics is rule-circular or unsound. I respond to Rumfitt, by showing that neither is the case: rule-circularity disappears when an appropriate semantics is considered, and soundness is restored by slightly modifying standard quantum mechanics. Thus, although this is indeed not necessary, it is however possible to rationally adjudicate against classical logic.

DOI
10.1093/analys/anae093

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Abstract Most believe that there are no empirical grounds that make the adoption of quantum logic necessary. Ian Rumfitt has further argued that this adoption is not possible either, for the proof that distribution fails in quantum mechanics is rule-circular or unsound. I respond to Rumfitt, by showing that neither is the case: rule-circularity disappears when an appropriate semantics is considered, and soundness is restored by slightly modifying standard quantum mechanics. Thus, although this is indeed not necessary, it is however possible to rationally adjudicate against classical logic.
Putnam and Finkelstein have proposed the abandonment of distributivity in the logic of quantum theory. This change results from defining the connectives, not truth-functionally, but in terms of a certain empirical ordering of propositions. Putnam has argued that the use of this ordering (“implication”) to govern proofs resolves certain paradoxes. But his resolutions are faulty; and in any case, the paradoxes may be resolved with no changes in logic. There is therefore no reason to regard the partially ordered set of propositions as a logic—i.e. as embodying a criterion for soundness of proofs. Its role in quantum theory ought to be understood in an entirely different way.
In a recent paper, Rovelli responds to our critical assessment of Relational Quantum Mechanics (RQM). His main argument is that our assessment lacks merit, because we fail to understand, or cope with, the premises of his theory; instead, he argues, we judge his proposal, blinded by the preconceptions inherent to ``our camp''. Here, we explicitly show that our assessment judges RQM on its own terms, together with the basic requirements of precision, clarity, logical soundness and empirical suitability. Under those circumstances, we prove false Rovelli's claim that RQM provides a satisfactory, realistic, non-solipsistic description of the world. Moreover, his reply serves us to further exhibit the serious problems of the RQM proposal, as well as the failures of its author to understanding the basic conceptual difficulties of quantum theory.
A recent proof [H. P. Stapp, Am. J. Phys. 65, 300 (1997)], formulated in the symbolic language of modal logic, claims to show that contemporary quantum theory, viewed as a set of rules that allow us to calculate statistical predictions among certain kinds of observations, cannot be imbedded in any rational framework that conforms to the principles that (1) the experimenters' choices of which experiments they will perform can be considered to be free choices, (2) outcomes of measurements are unique, and (3) the free choices just mentioned have no backward-in-time effects of any kind. This claim is similar to Bell's theorem, but much stronger, because no reality assumption alien to quantum philosophy is used. The paper being commented on [W. Unruh, Phys. Rev. A 59, 126 (1999)] argues that some such reality assumption has been ``smuggled'' in. That argument is examined here and shown, I believe, to be defective.
We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others (Among whom we may count or not E. W. Beth, depending on how we interpret some of his statements), our answer is a clear “no”. Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth’s proposal of applying Tarski’s semantical methods to the analysis of physical theories, with an empirical–experimental approach to Logic, as advocated by both Beth and Putnam, but understood by us in the view of the operational- realistic tradition of Jauch and Piron, i.e. as an investigation of “the logic of yes–no experiments” (or “questions”). Technically, we use the recently-developed setting of Quantum Dynamic Logic (Baltag and Smets 2005, 2008) to make explicit the operational meaning of quantum-mechanical concepts in our formal semantics. Based on our recent results (Baltag and Smets 2005), we show that the correct interpretation of quantum-logical connectives is dynamical, rather than purely propositional. We conclude that there is no contradiction between classical logic and (our dynamic reinterpretation of) quantum logic. Moreover, we argue that the Dynamic-Logical perspective leads to a better and deeper understanding of the “non-classicality” of quantum behavior than any perspective based on static Propositional Logic.
Although quantum mechanics is generally considered to be fundamentally incompatible with classical logic, it is argued here that the gap is not as great as it seems. Any classical, discrete, time reversible system can be naturally described using a quantum Hubert space, operators, and a Schrödinger equation. The quantum states generated this way resemble the ones in the real world so much that one wonders why this could not be used to interpret all of quantum mechanics this way. Indeed, such an interpretation leads to the most natural explanation as to why a wave function appears to "collapse" when a measurement is made, and why probabilities obey the Born rule. Because it is real quantum mechanics that we generate, Bell's inequalities should not be an obstacle.