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arXiv:1504.06460

Epistemic nature of quantum reasoning

Alfredo B. Henriques, Amı́lcar Sernadas

Published Apr 24, 2015• 1 citation
arXivHTMLPDF

Abstract

Doubts are raised concerning the usual interpretation of the alleged failure, by quantum mechanics, of the distributive law of classical logic. The difficulty raised by incompatible sets of observables is overcome within an epistemic enrichment of classical logic that provides the means for distinguishing between the value of a variable and its observation while retaining the classical connectives.

DOI
10.48550/arxiv.1504.06460

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Doubts are raised concerning the usual interpretation of the alleged failure, by quantum mechanics, of the distributive law of classical logic. The difficulty raised by incompatible sets of observables is overcome within an epistemic enrichment of classical logic that provides the means for distinguishing between the value of a variable and its observation while retaining the classical connectives.
Within the many versions of the Copenhagen interpretation numerous hints can be found about how to reason about quantum mechanics. However, strict guidelines are often missing. The standard quantum logic is sometimes opted as providing such a guideline but, in itself, it is again puzzling due to the failure of the law of distributivity. In this paper I take an instrumentalist approach inspired by the Copenhagen interpretation towards the interpretation of propositions in a physical theory. Starting with so-called elementary propositions for a physical theory I introduce logical connectives in accordance with this approach and relate the thereby expanding set of propositions to mathematical objects within the theory in a consistent way. This is first done for classical mechanics and then for quantum mechanics. In both cases this approach leads to the development of intuitionistic logics rather than classical ones. It also follows that the obtained intuitionistic quantum logic is a proper generalization of the obtained intuitionist logic for classical mechanics. Finally, some hints are provided in how these logics are to be used in practical situations.
We articulate a collection of desiderata for an account of the dynamical quantities of a physical theory, and we present a theory that meets these desiderata in the case of quantum mechanics. Our theory retains a distinction between the values of dynamical quantities and the truth values of sentences asserting that a system has a particular value of a particular quantity. This allows our theory to incorporate the phenomenon of quantum indeterminacy as a pattern in the properties instantiated by a system in a particular quantum state, while also retaining the semantics of classical logic. We contrast our theory with quantum logic, which flattens the distinction between quantity values and truth values, and we address a series of objections that have been raised to previous attempts to reconcile quantum indeterminacy with classical logic.
We have proposed in several recent papers a critical view of some parts of quantum mechanics (QM) that is methodologically unusual because it rests on analysing the language of QM by using some elementary but fundamental tools of mathematical logic. Our approach proves that some widespread beliefs about QM can be questioned and establishes new links with a classical view, which is significant in the debate on the interpretations of QM. We propose here a brief survey of our results, highlighting their common background. We firstly show how quantum logic (QL) can be embedded into classical logic (CL) if the embedding is required to preserve the logical order and not the algebraic structure, and also how QL can be interpreted as a pragmatic sublanguage within a pragmatic extension of CL. Both these results challenge the thesis that CL and QL formalize the properties of different and incompatible notions of truth. We then show that quantum probability admits an epistemic interpretation if contextuality is taken into account as a basic constituent of the language of QM, which overcomes the interpretation of quantum probability as ontic. Finally, we show that the proofs that QM is a contextual theory stand on a supplementary epistemological assumption that is usually unnoticed and left implicit. Dropping such assumption opens the way, at least in principle, to non-contextual interpretations of QM.
This paper treats some of the issues raised by Putnam's discussion of, and claims for, quantum logic, specifically: that its proposal is a response to experimental difficulties; that it is a reasonable replacement for classical logic because its connectives retain their classical meanings, and because it can be derived as a logic of tests. We argue that the first claim is wrong(1), and that while conjunction and disjunction can be considered to retain their classical meanings, negation cruciallydoes not.The argument is conductedviaa thorough analysis of how the meet, join and complementation operations are defined in the relevant logical structures, respectively Boolean- and ortholattices(3). Since Putnam wishes to reinstate a realist interpretation of quantum mechanics, we ask how quantum logic can be a logic of realism. We show that it certainly cannot be a logic ofbivalencerealism (i.e., of truth and falsity), although itisconsistent with some form ofontologicalrealism(4). Finally, we show that while a reasonable explication of the idealized notion of test yields interesting mathematical structure, it by no means yields the rich ortholattice structure which Putnam (following Finkelstein) seeks.
One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations).