Citations of:
Bertrand’s Paradox and the Principle of Indifference
Philosophy of Science 74 (2):150175 (2007)
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If the laws of nature are as the Humean believes, it is an unexplained cosmic coincidence that the actual Humean mosaic is as extremely regular as it is. This is a strong and wellknown objection to the Humean account of laws. Yet, as reasonable as this objection may seem, it is nowadays sometimes dismissed. The reason: its unjustified implicit assignment of equiprobability to each possible Humean mosaic; that is, its assumption of the principle of indifference, which has been attacked on (...) 

In this paper we present a concept of similarity in games, on which to ground alternative solution concepts, some of which differ from the classical notions in the field. In order to do this we impose a constraint on players’ beliefs that amounts to a variant of the wellknown symmetry principle in classical bargaining theory. We show how this similarity relation helps to identify different Nash equilibria in games, and how these “similar Nash equilibria” can be extended to nonsymmetric games. (...) 

This paper refutes two important and influential views in one fell stroke. The first is G.E. Moore’s view that assertions of the form ‘Q but I don’t believe that Q’ are inherently “absurd.” The second is Gareth Evans’s view that justification to assert Q entails justification to assert that you believe Q. Both views run aground the possibility of being justified in accepting eliminativism about belief. A corollary is that a principle recently defended by John Williams is also false, namely, (...) 

An important suggestion of objective Bayesians is that the maximum entropy principle can replace a principle which is known to get into paradoxical difficulties: the principle of indifference. No one has previously determined whether the maximum entropy principle is better able to solve Bertrand’s chord paradox than the principle of indifference. In this paper I show that it is not. Additionally, the course of the analysis brings to light a new paradox, a revenge paradox of the chords, that is unique (...) 

The paper starts by describing and clarifying what Williamson calls the consequence fallacy. I show two ways in which one might commit the fallacy. The first, which is rather trivial, involves overlooking background information; the second way, which is the more philosophically interesting, involves overlooking prior probabilities. In the following section, I describe a powerful form of sceptical argument, which is the main topic of the paper, elaborating on previous work by Huemer. The argument attempts to show the impossibility of (...) 

Ravit Dotan argues that a No Free Lunch theorem from machine learning shows epistemic values are insufficient for deciding the truth of scientific hypotheses. She argues that NFL shows that the best case accuracy of scientific hypotheses is no more than chance. Since accuracy underpins every epistemic value, nonepistemic values are needed to assess the truth of scientific hypotheses. However, NFL cannot be coherently applied to the problem of theory choice. The NFL theorem Dotan’s argument relies upon is a member (...) 

In his Bayesian Nets and Causality, Jon Williamson presents an ‘Objective Bayesian’ interpretation of probability, which he endeavours to distance from the logical interpretation yet associate with the subjective interpretation. In doing so, he suggests that the logical interpretation suffers from severe epistemological problems that do not affect his alternative. In this paper, I present a challenge to his analysis. First, I closely examine the relationship between the logical and ‘Objective Bayesian’ views, and show how, and why, they are highly (...) 

Darrell P. Rowbottom reviews the book "In defense of objective Bayesianism" by Jon Williamson. 



This paper shows that Bertrand's proposed ‘solutions’ to his own question, which generates his chord paradox, are inapplicable. It uses a simple analogy with cake cutting. The problem is that none of Bertrand's solutions considers all possible cuts. This is no solace for the defenders of the principle of indifference, however, because it emerges that the paradox is harder to solve than previously anticipated. 

Knowledgefirst epistemology includes a knowledge norm of action: roughly, act only on what you know. This norm has been criticized, especially from the perspective of socalled standard decision theory. Mueller and Ross provide example decision problems which seem to show that acting properly cannot require knowledge. I argue that this conclusion depends on applying a particular decision theory which is illmotivated in this context. Agents’ knowledge is often most plausibly formalized as an ambiguous epistemic state, and the theory of decision (...) 

We often use symmetries to infer outcomes’ probabilities, as when we infer that each side of a fair coin is equally likely to come up on a given toss. Why are these inferences successful? I argue against answering this with an a priori indifference principle. Reasons to reject that principle are familiar, yet instructive. They point to a new, empirical explanation for the success of our probabilistic predictions. This has implications for indifference reasoning in general. I argue that a priori (...) 

I discuss the formal implementation, interpretation, and justification of likelihood attributions in cosmology. I show that likelihood arguments in cosmology suffer from significant conceptual and formal problems that undermine their applicability in this context. 



The classical interpretation of probability together with the principle of indifference is formulated in terms of probability measure spaces in which the probability is given by the Haar measure. A notion called labelling invariance is defined in the category of Haar probability spaces; it is shown that labelling invariance is violated, and Bertrand’s paradox is interpreted as the proof of violation of labelling invariance. It is shown that Bangu’s attempt to block the emergence of Bertrand’s paradox by requiring the relabelling (...) 

The classical interpretation of probability together with the principle of indifference is formulated in terms of probability measure spaces in which the probability is given by the Haar measure. A notion called labelling invariance is defined in the category of Haar probability spaces; it is shown that labelling invariance is violated, and Bertrand’s paradox is interpreted as the proof of violation of labelling invariance. It is shown that Bangu’s attempt to block the emergence of Bertrand’s paradox by requiring the relabelling (...) 

Decisions, whether moral or prudential, should be guided at least in part by considerations of the consequences that would result from the various available actions. For any given action, however, the majority of its consequences are unpredictable at the time of decision. Many have worried that this leaves us, in some important sense, clueless. In this paper, I distinguish between ‘simple’ and ‘complex’ possible sources of cluelessness. In terms of this taxonomy, the majority of the existing literature on cluelessness focusses (...) 

Bertand’s paradox is a fundamental problem in probability that casts doubt on the applicability of the indifference principle by showing that it may yield contradictory results, depending on the meaning assigned to “randomness”. Jaynes claimed that symmetry requirements solve the paradox by selecting a unique solution to the problem. I show that this is not the case and that every variant obtained from the principle of indifference can also be obtained from Jaynes’ principle of transformation groups. This is because the (...) 



The Principle of Indifference is a central element of the ‘classical’ conception of probability, but, for all its strong intuitive appeal, it is widely believed that it faces a devastating objection: the socalled (by Poincare´) ‘Bertrand paradoxes’ (in essence, cases in which the same probability question receives different answers). The puzzle has fascinated many since its discovery, and a series of clever solutions (followed promptly by equally clever rebuttals) have been proposed. However, despite the longstanding interest in this problem, an (...) 

In this paper I show that the validity of the Indifference Principle in light of its related paradoxes, is still an open question. I do so by offering an analysis of IP and its related paradoxes in the way they are manifested within the framework of Kolmogorov's probability theory. I describe the conditions that any mathematical formalization of IP must satisfy. Consequently, I show that IP's mathematical formalization has to be a set of constrains on probability spaces which mathematically describe (...) 

In robustness analysis, hypotheses are supported to the extent that a result proves robust, and a result is robust to the extent that we detect it in diverse ways. But what precise sense of diversity is at work here? In this paper, I show that the formal explications of evidential diversity most often appealed to in work on robustness – which all draw in one way or another on probabilistic independence – fail to shed light on the notion of diversity (...) 