It is important to have an adequate model of uncertainty, since decisions must be made before the uncertainty can be resolved. For instance, flood defenses must be designed before we know the future distribution of flood events. It is standardly assumed that probability theory offers the best model of uncertain information. I think there are reasons to be sceptical of this claim. I criticise some arguments for the claim that probability theory is the only adequate model of uncertainty. In particular (...) I critique Dutch book arguments, representation theorems, and accuracy based arguments. Then I put forward my preferred model: imprecise probabilities. These are sets of probability measures. I offer several motivations for this model of uncertain belief, and suggest a number of interpretations of the framework. I also defend the model against some criticisms, including the so-called problem of dilation. I apply this framework to decision problems in the abstract. I discuss some decision rules from the literature including Levi’s E-admissibility and the more permissive rule favoured by Walley, among others. I then point towards some applications to climate decisions. My conclusions are largely negative: decision making under such severe uncertainty is inevitably difficult. I finish with a case study of scientific uncertainty. Climate modellers attempt to offer probabilistic forecasts of future climate change. There is reason to be sceptical that the model probabilities offered really do reflect the chances of future climate change, at least at regional scales and long lead times. Indeed, scientific uncertainty is multi-dimensional, and difficult to quantify. I argue that probability theory is not an adequate representation of the kinds of severe uncertainty that arise in some areas in science. I claim that this requires that we look for a better framework for modelling uncertainty. (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)

The conspicuous similarities between interpretive strategies in classical statistical mechanics and in quantum mechanics may be grounded on their employment of common implementations of probability. The objective probabilities which represent the underlying stochasticity of these theories can be naturally associated with three of their common formal features: initial conditions, dynamics, and observables. Various well-known interpretations of the two theories line up with particular choices among these three ways of implementing probability. This perspective has significant application to debates on primitive ontology (...) and to the quantum measurement problem. (shrink)

Despite its age, quantum theory still suffers from serious conceptual difficulties. To create clarity, mathematical physicists have been attempting to formulate quantum theory geometrically and to find a rigorous method of quantization, but this has not resolved the problem. In this article we argue that a quantum theory recursing to quantization algorithms is necessarily incomplete. To provide an alternative approach, we show that the Schrödinger equation is a consequence of three partial differential equations governing the time evolution of a given (...) probability density. These equations, discovered by Madelung, naturally ground the Schrödinger theory in Newtonian mechanics and Kolmogorovian probability theory. A variety of far-reaching consequences for the projection postulate, the correspondence principle, the measurement problem, the uncertainty principle, and the modeling of particle creation and annihilation are immediate. We also give a speculative interpretation of the equations following Bohm, Vigier and Tsekov, by claiming that quantum mechanical behavior is possibly caused by gravitational background noise. (shrink)

We prove that under some technical assumptions on a general, non-classical probability space, the probability space is extendible into a larger probability space that is common cause closed in the sense of containing a common cause of every correlation between elements in the space. It is argued that the philosophical significance of this common cause completability result is that it allows the defence of the Common Cause Principle against certain attempts of falsification. Some open problems concerning possible strengthening of the (...) common cause completability result are formulated. (shrink)

We present a generalization of the problem of pattern recognition to arbitrary probabilistic models. This version deals with the problem of recognizing an individual pattern among a family of different species or classes of objects which obey probabilistic laws which do not comply with Kolmogorov’s axioms. We show that such a scenario accommodates many important examples, and in particular, we provide a rigorous definition of the classical and the quantum pattern recognition problems, respectively. Our framework allows for the introduction of (...) non-trivial correlations between the different species involved, opening the door to a new way of harnessing these physical resources for solving pattern recognition problems. Finally, we present some examples and discuss the computational complexity of the quantum pattern recognition problem, showing that the most important quantum computation algorithms can be described as non-Kolmogorovian pattern recognition problems. (shrink)

In this article we present a possible way to make usual quantum mechanics fully compatible with physical realism, defined as the statement that the goal of physics is to study entities of the natural world, existing independently from any particular observer’s perception, and obeying universal and intelligible rules. Rather than elaborating on the quantum formalism itself, we propose a new quantum ontology, where physical properties are attributed jointly to the system, and to the context in which it is embedded. In (...) combination with a quantization principle, this non-classical definition of physical reality sheds new light on counter-intuitive features of quantum mechanics such as the origin of probabilities, non-locality, and the quantum-classical boundary. (shrink)

"Chance" crops up all over philosophy, and in many other areas. It is often assumed -- without argument -- that chances are probabilities. I explore the extent to which this assumption is really sanctioned by what we understand by the concept of chance.

We discuss the following problems, plaguing the present search for the “final theory”: (1) How to find a mathematical structure rich enough to be suitably approximated by the mathematical structures of general relativity and quantum mechanics? (2) How to reconcile nonlocal phenomena of quantum mechanics with time honored causality and reality postulates? (3) Does the collapse of the wave function contain some hints concerning the future quantum gravity theory? (4) It seems that the final theory cannot avoid the problem of (...) dynamics, and consequently the problem of time. What kind of time, if this theory is supposed to be background free? (5) Will the dynamics of the “final theory” be probabilistic? Quantum probability exhibits some essential differences as compared with classical probability; are they but variations of some more general probabilistic measure theory? (6) Do we need a radically new interpretation of quantum mechanics, or rather an entirely new theory of which the present quantum mechanics is an approximation? (7) If the final theory is to be background free, it should provide a mechanism of space-time generation. Should we try to explain not only the generation of space-time, but also the generation of its material content? (8) As far as the existence of the initial singularity is concerned, one usually expects either “yes” or “not” answers from the final theory. However, if the mathematical structure of the future theory is supposed to be truly more general that the mathematical structures of the present general relativity and quantum mechanics, is a “third answer“ possible? Could this third answer be related to the probabilistic character of the final theory? We discuss these questions in the framework of a working model unifying gravity and quanta. The analysis reveals unexpected aspects of these rather wildly discussed issues. (shrink)

This article investigates the nature of entangled correlations in algebraic quantum field theory (AQFT). We define a notion of local disentanglement, expressing the possibility of destroying entanglement by means of local operations. Contrary to the case of ordinary quantum mechanics, local disentanglement cannot be achieved in general in relativistic quantum field theory. However, we show that if the split property holds, there exists a local operation that can destroy entanglement between spacelike-separated quantum field systems.

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)

A widely shared view in the cognitive sciences is that discovering and assessing explanations of cognitive phenomena whose production involves uncertainty should be done in a Bayesian framework. One assumption supporting this modelling choice is that Bayes provides the best approach for representing uncertainty. However, it is unclear that Bayes possesses special epistemic virtues over alternative modelling frameworks, since a systematic comparison has yet to be attempted. Currently, it is then premature to assert that cognitive phenomena involving uncertainty are best (...) explained within the Bayesian framework. As a forewarning, progress in cognitive science may be hindered if too many scientists continue to focus their efforts on Bayesian modelling, which risks to monopolize scientific resources that may be better allocated to alternative approaches. (shrink)

This paper explores the issues of holism and nonseparability in relativistic quantum field theory by focusing on an analog of the typical model featuring in many discussions of holism and nonseparability in nonrelativistic quantum mechanics. It is argued that the quantum field theoretic model does exhibit holism in a metaphysical sense and that there are plausible grounds to view QFT holistic in an epistemological sense. However, the complexities arising from the fact that quantum fields have infinite degrees of freedom prohibit (...) the exploitation of the elaborated analogy toward demonstrating that the QFT model exhibits the kind of state nonseparability familiar from ordinary quantum mechanics. Still, it is argued that the QFT model does satisfy a rather weak epistemological criterion for state nonseparability. (shrink)

We interpret the philosophy of Niels Bohr as related to the so called ''linguistic turn'' and consider paraconsistency in the light of the Bohrian notion of complementarity. Following [16], Jean-Yves Beziau has discussed the seemingly contradictory perspectives found in the quantum mechanical double slit experiment in terms of paraconsistent view-points [7, 8]. This interpretation goes in line with the well known Bohrian Neo-Kantian epistemological account of quantum mechanics. In the present paper, we put forward the idea that one can also (...) consider, within quantum mechanics and departing from the philosophy of the danish physicist, a more radical paraconsistency found within one of the main formal elements of the theory, namely, quantum superpositions. We will argue that, rather than epistemological, the contradictions found within quantum superpositions could be interpreted as ontological contradictions. (shrink)

We prove new results on common cause closedness of quantum probability spaces, where by a quantum probability space is meant the projection lattice of a non-commutative von Neumann algebra together with a countably additive probability measure on the lattice. Common cause closedness is the feature that for every correlation between a pair of commuting projections there exists in the lattice a third projection commuting with both of the correlated projections and which is a Reichenbachian common cause of the correlation. The (...) main result we prove is that a quantum probability space is common cause closed if and only if it has at most one measure theoretic atom. This result improves earlier ones published in [1]. The result is discussed from the perspective of status of the Common Cause Principle. Open problems on common cause closedness of general probability spaces (L, ϕ) are formulated, where L is an orthomodular bounded lattice and ϕ is a probability measure on L. (shrink)

The role of measure theoretic atomicity in common cause closedness of general probability theories with non-distributive event structures is raised and investigated. It is shown that if a general probability space is non-atomic then it is common cause closed. Conditions are found that entail that a general probability space containing two atoms is not common cause closed but it is common cause closed if it contains only one atom. The results are discussed from the perspective of the Common Cause Principle.

In this work we present an epistemic analysis of time phenomenon using the mathematical machinery of information theory and modular theory. By adopting limited commitment to the ontology of time evolution, and instead by mainly relying on the information that is in principle accessible to the observer, we find that the most primary aspect of the temporal experience, the perceived distinctiveness across the states of the world, emerges as a purely epistemic function. By analyzing the mathematical properties of this epistemic (...) function, we interpret it to be in principle insensitive to any ontic state of the world, which leads to the conclusion that the observer is subject to temporal experience irrespective of whether the underlying state of the world is dynamical or invariant. On the ground of the presented analysis, we also provide a solution to the conceptual challenge of non-equilibrium phenomena that faces the thermal time hypothesis. (shrink)

Some naturalistic philosophers of mind subscribing to the predictive processing theory of mind have adopted a realist attitude towards the results of Bayesian cognitive science. In this paper, we argue that this realist attitude is unwarranted. The Bayesian research program in cognitive science does not possess special epistemic virtues over alternative approaches for explaining mental phenomena involving uncertainty. In particular, the Bayesian approach is not simpler, more unifying, or more rational than alternatives. It is also contentious that the Bayesian approach (...) is overall better supported by the empirical evidence. So, to develop philosophical theories of mind on the basis of a realist interpretation of results from Bayesian cognitive science is unwarranted. Naturalistic philosophers of mind should instead adopt an anti-realist attitude towards these results and remain agnostic as to whether Bayesian models are true. For continuing on with an exclusive focus and praise of Bayes within debates about the predictive processing theory will impede progress in philosophical understanding of scientific practice in computational cognitive science as well as of the architecture of the mind. (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)

Classical (Bayesian) probability (CP) theory has led to an influential research tradition for modeling cognitive processes. Cognitive scientists have been trained to work with CP principles for so long that it is hard even to imagine alternative ways to formalize probabilities. However, in physics, quantum probability (QP) theory has been the dominant probabilistic approach for nearly 100 years. Could QP theory provide us with any advantages in cognitive modeling as well? Note first that both CP and QP theory share the (...) fundamental assumption that it is possible to model cognition on the basis of formal, probabilistic principles. But why consider a QP approach? The answers are that (1) there are many well-established empirical findings (e.g., from the influential Tversky, Kahneman research tradition) that are hard to reconcile with CP principles; and (2) these same findings have natural and straightforward explanations with quantum principles. In QP theory, probabilistic assessment is often strongly context- and order-dependent, individual states can be superposition states (that are impossible to associate with specific values), and composite systems can be entangled (they cannot be decomposed into their subsystems). All these characteristics appear perplexing from a classical perspective. However, our thesis is that they provide a more accurate and powerful account of certain cognitive processes. We first introduce QP theory and illustrate its application with psychological examples. We then review empirical findings that motivate the use of quantum theory in cognitive theory, but also discuss ways in which QP and CP theories converge. Finally, we consider the implications of a QP theory approach to cognition for human rationality. (shrink)

A novel conceptual framework for theoretical psychology is presented and illustrated for the example of bistable perception. A basic formal feature of this framework is the non-commutativity of operations acting on mental states. A corresponding model for the bistable perception of ambiguous stimuli, the Necker–Zeno model, is sketched and some empirical evidence for it so far is described. It is discussed how a temporal non-locality of mental states, predicted by the model, can be understood and tested.

One among many misleading quotations about the alleged mysteries of quantum theory is from Feynman (1965): Today we know that quantum theory describes many aspects of our world in a fully intelligible fashion. Pothos & Busemeyer (P&B) propose ways in which this may include psychology and cognitive science.