Staying Regular?

Prof Alan Hájek (Australian National University)

‘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 ‘regularity’—one for each agent—in terms of a certain internal harmony of the agent’s 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 agent’s 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 not assigning probability at all to some doxastic possibility (a probability gap). 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:

-        conditional probability

-        conditionalization

-        probabilistic independence

-        decision theory