Effects and Propositions

Presented to the 2007 conference, New Directions in the Foundations of Physics. To appear in Foundations of physics with the proceedings of the conference. Version of July 17, 2008 EFFECTS AND PROPOSITIONS William Demopoulos* Abstract The quantum logical and quantum information-theoretic traditions have exerted an especially powerful influence on Bub’s thinking about the conceptual foundations of quantum mechanics. This paper discusses both the quantum logical and information- theoretic traditions from the point of view of their representational frameworks. I argue that it is at this level—at the level of its framework—that the quantum logical tradition has retained its centrality to Bub’s thought. It is further argued that there is implicit in the quantum information-theoretic tradition a set of ideas that mark a genuinely new alternative to the framework of quantum logic. These ideas are of considerable interest for the philosophy of quantum mechanics, a claim which I defend with an extended discussion of their application to our understanding of the philosophical significance of the no hidden variable theorem of Kochen and Specker. 0. Introduction I should begin with two remarks about the nature and scope of this study. The first remark concerns my use of the notion of an effect and its juxtaposition with the concept of a proposition in my title. The connection between propositions and effects is a subject about which I will have a great deal to say in the course of the development of my positive proposals. But to forestall possible misunderstandings, and as a precaution against raising false expectations, I want to say at the outset that this paper does not address the similarities and differences between effect algebras and orthomodular posets of propositions; nor does it concern positive operator valued measures (POVMs) and their relation to projection valued measures (PVMs). Indeed, I make no use of the notion of an effect algebra, or that of a POVM. My concerns are philosophical and conceptual rather than mathematical and foundational. The paper neither contains nor promises

• I have been influenced in more ways than I have been able to record in the text by my friends and colleagues, Jeffrey Bub, Itamar Pitowsky, Robert Di Salle, and most recently, Christopher A. Fuchs, My thanks to them for many hours of conversation and many pages of correspondence. My research was supported by the Social Sciences and Humanities Research Council of Canada. 2 results of the sort one might expect of a study of effect algebras and their relation to the axiomatic tradition of quantum logic. It should go without saying that by ignoring effect algebras and POVMs, I do not imply a judgement about their utility, interest or importance for the foundations or philosophy of the subject; it is merely that my interests lie elsewhere. Unfortunately the term ‘effect,’ although already well-established with a technical meaning, happens to suit my purposes almost perfectly, and I have been unable to find an alternative that works as well. The source of my interest in the concept of an effect, in the sense in which it occurs here, is the result of reflection on conceptual aspects of the work of Fuchs and his collaborators; as I came to realize only later, it also depends on an idea implicit in the work of Pitowsky (especially his 2003 and 2006). As will become clear in the sequel, the central mathematical concept on which this paper rests is familiar from the tradition of axiomatic quantum logic. Effects and propositions are investigated for their suitability as the objects which, from the point of view of the conceptual foundations of the theory, most naturally realize this mathematical structure. My second remark concerns the scope of the discussion to follow. There are at least three groups of conceptual issues associated with the interpretation of quantum mechanics. There is first the cluster of “puzzling phenomena” permitted by the theory. The most famous among these are the 2-slit experiment, the EPR thought experiment, and the latter’s striking realization in work carried out in response to Bell’s (1964) discoveries. Secondly, there is the problem, variously formulated, of reconciling the apparent conflict between the discontinuous state transition, which occurs in the context of a measurement, and the continuous evolution to which a quantum mechanical system conforms in the absence of measurement. Thirdly, there is the quantum-theoretical possibility of a finite family of properties associated with a physical system that are so interrelated that there is no 0-1 generalized probability measure definable on them. Following tradition, I call the puzzling phenomena which comprise the first group, quantum paradoxes, and the second, the measurement problem. I depart from tradition by calling the problem of hidden variables the essential content of one of the two principal theorem

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