Quantum mechanics is a fundamentally probabilistic theory (at least so far as the empirical predictions are concerned). It follows that, if one wants to properly understand quantum mechanics, it is essential to clearly understand the meaning of probability statements. The interpretation of probability has excited nearly as much philosophical controversy as the interpretation of quantum mechanics. 20th century physicists have mostly adopted a frequentist conception. In this paper it is argued that we ought, instead, to adopt a logical or Bayesian (...) conception. The paper includes a comparison of the orthodox and Bayesian theories of statistical inference. It concludes with a few remarks concerning the implications for the concept of physical reality. (shrink)

We investigate how Dutch Book considerations can be conducted in the context of two classes of nonclassical probability spaces used in philosophy of physics. In particular we show that a recent proposal by B. Feintzeig to find so called “generalized probability spaces” which would not be susceptible to a Dutch Book and would not possess a classical extension is doomed to fail. Noting that the particular notion of a nonclassical probability space used by Feintzeig is not the most common employed (...) in philosophy of physics, and that his usage of the “classical” Dutch Book concept is not appropriate in “nonclassical” contexts, we then argue that if we switch to the more frequently used formalism and use the correct notion of a Dutch Book, then all probability spaces are not susceptible to a Dutch Book. We also settle a hypothesis regarding the existence of classical extensions of a class of generalized probability spaces. (shrink)

I defend an analog of probabilism that characterizes rationally coherent estimates for chances. Specifically, I demonstrate the following accuracy-dominance result for stochastic theories in the C*-algebraic framework: supposing an assignment of chance values is possible if and only if it is given by a pure state on a given algebra, your estimates for chances avoid accuracy-dominance if and only if they are given by a state on that algebra. When your estimates avoid accuracy-dominance (roughly: when you cannot guarantee that other (...) estimates would be more accurate), I say that they are sufficiently coherent. In formal epistemology and quantum foundations, the notion of rational coherence that gets more attention requires that you never allow for a sure loss (or “Dutch book”) in a given sort of betting game; I call this notion full coherence. I characterize when these two notions of rational coherence align, and I show that there is a quantum state giving estimates that are sufficiently coherent, but not fully coherent. (shrink)

Quantum logic understood as a reconstruction program had real successes and genuine limitations. This paper offers a synopsis of both and suggests a way of seeing quantum logic in a larger, still thriving context.

Kochen and Specker’s theorem can be seen as a consequence of Gleason’s theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason’s theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated. r 2003 Elsevier Ltd. All rights reserved.

Kochen and Specker's theorem can be seen as a consequence of Gleason's theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason's theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated.

We discuss the meaning of probabilities in the many worlds interpretation of quantum mechanics. We start by presenting very briefly the many worlds theory, how the problem of probability arises, and some unsuccessful attempts to solve it in the past. Then we criticize a recent attempt by Deutsch to derive the quantum mechanical probabilities from the nonprobabilistic parts of quantum mechanics and classical decision theory. We further argue that the Born probability does not make sense even as an additional probability (...) rule in the many worlds theory. Our conclusion is that the many worlds theory fails to account for the probabilistic statements of standard (collapse) quantum mechanics. (shrink)

The paper discusses objections against non-hidden variable versions of the epistemic conception of quantum states—the view that quantum states do not describe the properties of quantum systems but reflect, in some way to be specified, the epistemic conditions of agents assigning them. In the first half of the paper, the main motivation for the epistemic conception of quantum states is sketched, and a version of it is outlined, which combines ideas from an earlier study of it with elements of Richard (...) Healey's recent pragmatist interpretation of quantum theory. In the second half, various objections against epistemic accounts of quantum states are discussed in detail, which are based on criticisms found in the literature. Possible answers by the version outlined here are compared with answers from the quantum Bayesian point of view, which is at present the most discussed version of the epistemic conception of quantum states. (shrink)

The paper investigates the epistemic conception of quantum states---the view that quantum states are not descriptions of quantum systems but rather reflect the assigning agents' epistemic relations to the systems. This idea, which can be found already in the works of Copenhagen adherents Heisenberg and Peierls, has received increasing attention in recent years because it promises an understanding of quantum theory in which neither the measurement problem nor a conflict between quantum non-locality and relativity theory arises. Here it is argued (...) that the main challenge for proponents of this idea is to make sense of the notion of a state assignment being performed correctly without thereby acknowledging the notion of a true state of a quantum system---a state it is in. An account based on the epistemic conception of states is proposed that fulfills this requirement by interpreting the rules governing state assignment as constitutive rules in the sense of John Searle. (shrink)

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. (shrink)

In this article, Savage’s theory of decision-making under uncertainty is extended from a classical environment into a non-classical one. The Boolean lattice of events is replaced by an arbitrary ortho-complemented poset. We formulate the corresponding axioms and provide representation theorems for qualitative measures and expected utility. Then, we discuss the issue of beliefs updating and investigate a transition probability model. An application to a simple game context is proposed.

Quantum cognition in decision making is a recent and rapidly growing field. In this paper, we develop an expected utility theory in a context of non-classical uncertainty. We replace the classical state space with a Hilbert space which allows introducing the concept of quantum lottery. Within that framework, we formulate axioms on preferences over quantum lotteries to establish a representation theorem. We show that demanding the consistency of choice behavior conditional on new information is equivalent to the von Neumann–Lüders postulate (...) applied to beliefs. A dynamically consistent quantum-like agent may violate dynamic recursive consistency, however. This feature suggests interesting applications in behavioral economics as we illustrate in an example of persuasion. (shrink)

In the Bayesian approach to quantum mechanics, probabilities—and thus quantum states—represent an agent’s degrees of belief, rather than corresponding to objective properties of physical systems. In this paper we investigate the concept of certainty in quantum mechanics. Particularly, we show how the probability-1 predictions derived from pure quantum states highlight a fundamental difference between our Bayesian approach, on the one hand, and Copenhagen and similar interpretations on the other. We first review the main arguments for the general claim that probabilities (...) always represent degrees of belief. We then argue that a quantum state prepared by some physical device always depends on an agent’s prior beliefs, implying that the probability-1 predictions derived from that state also depend on the agent’s prior beliefs. Quantum certainty is therefore always some agent’s certainty. Conversely, if facts about an experimental setup could imply agent-independent certainty for a measurement outcome, as in many Copenhagen-like interpretations, that outcome would effectively correspond to a preexisting system property. The idea that measurement outcomes occurring with certainty correspond to preexisting system properties is, however, in conflict with locality. We emphasize this by giving a version of an argument of Stairs [(1983). Quantum logic, realism, and value-definiteness. Philosophy of Science, 50, 578], which applies the Kochen–Specker theorem to an entangled bipartite system. (shrink)

This volume focuses on various questions concerning the interpretation of probability and probabilistic reasoning in biology and physics. It is inspired by the idea that philosophers of biology and philosophers of physics who work on the foundations of their disciplines encounter similar questions and problems concerning the role and application of probability, and that interaction between the two communities will be both interesting and fruitful. In this introduction we present the background to the main questions that the volume focuses on (...) and summarize the highlights of the individual contributions. (shrink)

We argue that there is a simple, unique, reason for all quantum paradoxes, and that such a reason is not uniquely related to quantum theory. It is rather a mathematical question that arises at the intersection of logic, probability, and computation. We give our ‘weirdness theorem’ that characterises the conditions under which the weirdness will show up. It shows that whenever logic has bounds due to the algorithmic nature of its tasks, then weirdness arises in the special form of negative (...) probabilities or non-classical evaluation functionals. Weirdness is not logical inconsistency, however. It is only the expression of the clash between an unbounded and a bounded view of computation in logic. We discuss the implication of these results for quantum mechanics, arguing in particular that its interpretation should ultimately be computational rather than exclusively physical. We develop in addition a probabilistic theory in the real numbers that exhibits the phenomenon of entanglement, thus concretely showing that the latter is not specific to quantum mechanics. (shrink)

Based on a gambling formulation of quantum mechanics, we derive a Gleason-type theorem that holds for any dimension n of a quantum system, and in particular for \. The theorem states that the only logically consistent probability assignments are exactly the ones that are definable as the trace of the product of a projector and a density matrix operator. In addition, we detail the reason why dispersion-free probabilities are actually not valid, or rational, probabilities for quantum mechanics, and hence should (...) be excluded from consideration. (shrink)

Entropy is ubiquitous in physics, and it plays important roles in numerous other disciplines ranging from logic and statistics to biology and economics. However, a closer look reveals a complicated picture: entropy is defined differently in different contexts, and even within the same domain different notions of entropy are at work. Some of these are defined in terms of probabilities, others are not. The aim of this chapter is to arrive at an understanding of some of the most important notions (...) of entropy and to clarify the relations between them, After setting the stage by introducing the thermodynamic entropy, we discuss notions of entropy in information theory, statistical mechanics, dynamical systems theory and fractal geometry. (shrink)

We argue that the intractable part of the measurement problem -- the 'big' measurement problem -- is a pseudo-problem that depends for its legitimacy on the acceptance of two dogmas. The first dogma is John Bell's assertion that measurement should never be introduced as a primitive process in a fundamental mechanical theory like classical or quantum mechanics, but should always be open to a complete analysis, in principle, of how the individual outcomes come about dynamically. The second dogma is the (...) view that the quantum state has an ontological significance analogous to the significance of the classical state as the 'truthmaker' for propositions about the occurrence and non-occurrence of events, i.e., that the quantum state is a representation of physical reality. We show how both dogmas can be rejected in a realist information-theoretic interpretation of quantum mechanics as an alternative to the Everett interpretation. The Everettian, too, regards the 'big' measurement problem as a pseudo-problem, because the Everettian rejects the assumption that measurements have definite outcomes, in the sense that one particular outcome, as opposed to other possible outcomes, actually occurs in a quantum measurement process. By contrast with the Everettians, we accept that measurements have definite outcomes. By contrast with the Bohmians and the GRW 'collapse' theorists who add structure to the theory and propose dynamical solutions to the 'big' measurement problem, we take the problem to arise from the failure to see the significance of Hilbert space as a new kinematic framework for the physics of an indeterministic universe, in the sense that Hilbert space imposes kinematic objective probabilistic constraints on correlations between events. (shrink)