Alan Hajek, Australian National University, School of Philosophy
'Regularity' conditions provide bridges between possibility and probability. They have the form:
If X is possible, then the probability of X is positive (or equivalents). Especially interesting are the conditions we get when we understand 'possible' doxastically, and 'probability' subjectively. I characterize these senses of 'regularity' in terms of a certain internal harmony of an agent's probability space (omega, 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 omega; by being an element of F; and by receiving positive probability from P. An agent's space is regular if these three grades collapse into one.
I review several arguments for regularity as a rationality norm. An agent could violate this norm in two ways: by assigning probability zero to some doxastic possibility, and by failing to assign probability altogether to some doxastic possibility. I argue for the rationality of each kind of violation.
Both kinds of violations of regularity have serious consequences for traditional Bayesian epistemology. I consider their ramifications for:
- conditional probability
- probabilistic independence
- decision theory