Gott (Nature 363:315–319, 1993) considers the problem of obtaining a probabilistic prediction for the duration of a process, given the observation that the process is currently underway and began a time t ago. He uses a temporal Copernican principle according to which the observation time can be treated as a random variable with uniform probability density. A simple rule follows: with a 95% probability.

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 (...) similar. Second, I argue that the logical interpretation is not manifestly inferior, at least for the reasons that Williamson offers. I suggest that the key difference between the logical and ‘Objective Bayesian’ views is in the domain of the philosophy of logic; and that the genuine disagreement appears to be over Platonism versus nominalism. (shrink)

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 re-labelling (...) of random events to preserve randomness cannot succeed non-trivially. A non-trivial strategy to preserve labelling invariance is identified, and it is argued that, under the interpretation of Bertrand’s paradox suggested in the paper, the paradox does not undermine either the principle of indifference or the classical interpretation and is in complete harmony with how mathematical probability theory is used in the sciences to model phenomena. It is shown in particular that violation of labelling invariance does not entail that labelling of random events affects the probabilities of random events. It also is argued, however, that the content of the principle of indifference cannot be specified in such a way that it can establish the classical interpretation of probability as descriptively accurate or predictively successful. 1 The Main Claims2 The Elementary Classical Interpretation of Probability3 The General Classical Interpretation of Probability in Terms of Haar Measures4 Labelling Invariance and Labelling Irrelevance5 General Bertrand’s Paradox6 Attempts to Save Labelling Invariance7 Comments on the Classical Interpretation of Probability. (shrink)

About a decade ago, Adam Elga introduced philosophers to an intriguing puzzle. In it, Sleeping Beauty, a perfectly rational agent, undergoes an experiment in which she becomes ignorant of what time it is. This situation is puzzling for two reasons: First, because there are two equally plausible views about how she will change her degree of belief given her situation and, second, because the traditional rules for updating degrees of belief don't seem to apply to this case. In this dissertation, (...) my goals are to settle the debate concerning this puzzle and to offer a new rule for updating some types of degrees of belief. Regarding the puzzle, I will defend a view called "the Lesser view," a view largely favorable to the Thirders' position in the traditional debate on the puzzle. Regarding the general rule for updating, I will present and defend a rule called "Shifted Jeffrey Conditionalization." My discussions of the above view and rule will complement each other: On the one hand, I defend the Lesser view by making use of Shifted Jeffrey Conditionalization. On the other hand, I test Shifted Jeffrey Conditionalization by applying it to various credal transitions in the Sleeping Beauty problem and revise that rule in accordance with the results of the test application. In the end, I will present and defend an updating rule called "General Shifted Jeffrey Conditionalization," which I suspect is the general rule for updating one's degrees of belief in so-called tensed propositions. (shrink)

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 so-called (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 long-standing interest in this problem, an (...) important assumption, necessary for its generation, has been overlooked. My aim in this paper is to identify this assumption. Since what it claims turns out to be prima facie problematic, I will urge that the burden of proof now shifts to the objectors to PI; they have to provide reasons why this assumption holds. (shrink)

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, (...) that justification to believe Q entails justification to believe that you believe Q. (shrink)

Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the (...) philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. (shrink)

O argumento zumbi negativo parte das premissas de que p ∧ ¬q é idealmente negativamente concebível, de que o que é idealmente negativamente concebível é possível e de que o fisicalismo é incompatível com a possibilidade de p ∧ ¬q para concluir que o fisicalismo é falso. No argumento, p é a conjunção das verdades e leis físicas fundamentais e q é uma verdade fenomenal qualquer. Uma sentença φ é idealmente negativamente concebível sse um raciocinador ideal não acredita que ¬φ (...) em reflexão priori. Uma versão da tese da escrutabilidade pressuposta pelo argumento afirma que, para todo φ que sobrevém em p, o raciocinador ideal acredita que p → φ em reflexão a priori. Nesse artigo, argumento que, dado algumas interpretações relevantes da noção de probabilidade (pr(.)), o raciocinador ideal acredita verdadeiramente, para todo φ, que p → pr(φ) = x em reflexão a priori. Mas então, dependendo do valor de pr(q) e das correlações entre q e outras sentenças, o raciocinador ideal também acredita (provavelmente, verdadeiramente) que p → q em reflexão a priori. Então, para alguns qs relevantes, p ∧ ¬q não é idealmente negativamente concebível e o argumento zumbi tem uma premissa falsa. A escolha de um q adequado dependeria de informação empírica, o que faria o argumento zumbi não ser nem conclusivo, nem a priori. ENGLISH: The negative zombie argument has as premises that p∧¬q is ideally negatively conceivable, that what is ideally negatively conceivable is possible, and that Physicalism is incompatible with p∧¬q being possible and as conclusion that Physicalism is false. In the argument, p is the conjunction of the fundamental physical truths and laws and q is an arbitrary phenomenal truth. A sentence φ is ideally negatively conceivable if and only if an ideal reasoner does not believe that ¬φ on a priori reflection. The argument presupposes a version of the scrutability thesis stating that, for all φ that supervene on p, the ideal reasoner believes that p → φ on a priori reflection. In this paper, I argue that, given relevant interpretation of probabilities, the ideal reasoner believes truly, for all φ, that p → pr(φ) = x on a priori reflection. But then, depending on the value of pr(q) and the correlations between q and other sentences, the ideal reasoner also believes (probably, truly) that p → q on a priori reflection. For some relevant qs, p∧¬q is not ideally negatively conceivable and the zombie argument has a false premise. The choice of an adequate q depends on empirical information, what makes the zombie argument neither conclusive nor a priori. (shrink)