This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly pick (...) out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum. (shrink)

Supererogatory acts are those that lie “beyond the call of duty.” There are two standard ways to define this idea more precisely. Although the definitions are often seen as equivalent, I argue that they can diverge when options are infinite, or when there are cycles of better options; moreover, each definition is acceptable in only one case. I consider two ways out of this dilemma.

David Builes presents a paradox concerning how confident you should be that any given member of an infinite collection of fair coins landed heads, conditional on the information that they were all flipped and only finitely many of them landed heads. We argue that if you should have any conditional credence at all, it should be 1/2.

Branden Fitelson and Alan Hájek have suggested that it is finally time for a “revolution” in which we jettison Kolmogorov’s axiomatization of probability, and move to an alternative like Popper’s. According to these authors, not only did Kolmogorov fail to give an adequate analysis of conditional probability, he also failed to give an adequate account of another central notion in probability theory: probabilistic independence. This paper defends Kolmogorov, with a focus on this independence charge. I show that Kolmogorov’s sophisticated theory (...) of conditional distributions, advocated by authors such as Kenny Easwaran but often overlooked by philosophers, has the resources to define the notion of independence Fitelson and Hájek seek. I then argue that Kolmogorov’s original definition of independence, once suitably reinterpreted as capturing a slight logical weakening of this notion, is not so problematic after all. Indeed, rather than discarding Kolmogorov’s definition, we should make room for two distinct notions of probabilistic independence, one stronger and one weaker, which play different theoretical roles. (shrink)