Dariusz Łukasiewicz has investigated in depth the “Argument from Chance” which argues that the data revealing chance in the world are incompatible with Divine Providence. Łukasiewicz agrees that these data undermine the traditional model of Providence—a fine-grained model in which every single detail is controlled by God—but maintains that they are not incompatible with a coarse-grained model—in which God leaves to chance many aspects of history (including some horrendous evils). Furthermore, Łukasiewicz provides independent reasons to prefer this coarse-grained model. Even (...) though I agree that a maximally fine-grained model is undermined by the scientific data, I argue that this is no sufficient reason to adopt a model as coarse-grained as Łukasiewicz’s. I propose a model of intermediate level of fine-grainedness which could avoid the drawbacks of both extremes, and seems to me to provide a more traditional approach to the problem of evil. (shrink)

In October 2009 I decided to stop doing philosophy. This meant, in particular, stopping work on the book that I was writing on the nature of probability. At that time, I had no intention of making my unfinished draft available to others. However, I recently noticed how many people are reading the lecture notes and articles on my web site. Since this draft book contains some important improvements on those materials, I decided to make it available to anyone who wants (...) to read it. That is what you have in front of you. The account of Laplace’s theory of probability in Chapter 4 is very different to what I said in my seminar lectures, and also very different to any other account I have seen; it is based on a reading of important texts by Laplace that appear not to have been read by other commentators. The discussion of von Mises’ theory in Chapter 7 is also new, though perhaps less revolutionary. And the final chapter is a new attempt to come to grips with the popular, but amorphous, subjective theory of probability. The material in the other chapters has mostly appeared in previous articles of mine but things are sometimes expressed differently here. I would like to say again that this is an incomplete draft of a book, not the book I would have written if I had decided to finish it. It no doubt contains poor expressions, it may contain some errors or inconsistencies, and it doesn’t cover all the theories that I originally intended to discuss. Apart from this preface, I have done no work on the book since October 2009. (shrink)

Bertrand's random-chord paradox purports to illustrate the inconsistency of the principle of indifference when applied to problems in which the number of possible cases is infinite. This paper shows that Bertrand's original problem is vaguely posed, but demonstrates that clearly stated variations lead to different, but theoretically and empirically self-consistent solutions. The resolution of the paradox lies in appreciating how different geometric entities, represented by uniformly distributed random variables, give rise to respectively different nonuniform distributions of random chords, and hence (...) to different probabilities. The principle of indifference appears consistently applicable to infinite sets provided that problems can be formulated unambiguously. (shrink)

Two old problems in probability theory involving the concept of randomness are considered. Data obtained with one of them--Bertrand's chord problem--demonstrate the equivocality of this term in the absence of a definition or explication of assumptions underlying its use. They also support two propositions about probabilistic thinking: (1) upon obtaining an answer to a question of probability, people tend to see it as the answer, overlooking tacit assumptions on which it may be based, and tend not to consider the possibility (...) of other assumptions from which different answers would follow; (2) perceptual balance--moderate uniformity of density of distribution--is a determinant of the degree to which a two-dimensional pattern is perceived as random. (shrink)

It is not only overtly probabilistic illatives like ‘makes it certain that’ but also apparently non-probabilistic ones like ‘therefore’ that have probabilistic import. Illatives like ‘therefore’ convey the meaning that the premise confers on the conclusion a probability not only greater than 0 but also greater than 1/2. But because they do not say whether that probability is equal to or less than 1, these illatives are appropriately called ‘neutral’.