Wed Mar 21, 2012
Dennes Room, 234 Moses Hall, 6–8 PM
|Working Group in the History and Philosophy of Logic, Mathematics, and Science
Alan Hájek (School of Philosophy, ANU)
Regularity conditions provide nice bridges between the various box/diamond modalities and various notions of probability. Schematically, they have the form:
If X is possible, then the probability of X is positive
(or equivalents). Of special interest are the conditions we get when possible is understood doxastically (i.e. in terms of binary belief), and probability is understood subjectively (i.e. in terms of degrees of belief). I characterize these senses of regularityone for each agentin terms of a certain internal harmony of the agents probability space <W, F, P>. I distinguish three grades of probabilistic involvement. A set of possibilities may be recognized by such a probability space by being a subset of W; by being an element of F; and by receiving positive probability from P. These are non-decreasingly committal ways in which the agent may countenance a proposition. An agents space is regular if these three grades collapse into one.
I briefly review several of the main arguments for regularity as a rationality norm, due especially to Lewis and Skyrms. There are two ways an agent could violate this norm: by assigning probability zero to some doxastic possibility, and by failing to assign probability altogether to some doxastic possibility. Authors such as Williamson have argued for the rationality of the former kind of violation, and I give an argument of my own. So I think that the second and third grades of probabilistic involvement may come apart for a rational agent. I then argue for the latter kind of violation: the first and second grades may also come apart for such an agent.
Both kinds of violations of regularity have serious consequences for traditional Bayesian epistemology. I consider especially their ramifications for: