ISAAC LEVI COHERENCE, REGULARITY AND CONDITIONAL
PROBABILITY*
- An admirable feature of Bruno De Finetti's approach to rational probability judgement is his insistence that we should be slow to invoke principles of inductive logic mandatory on all rational agents at all times to impose restrictions on probability judgements. He does, of course, favor some such restrictions; but those he advances are, so he holds, mandated by cogent assumptions concerning how judgements of probability should function in rational decision making. 1 These assumptions mandate relatively little.
Let K be a 13otential body of knowledge or evidence (representable by a deductively closed set of sentences). Let Q(h;e) be defined for every h On the algebra of sentences or propositions under consideration) and for every e in that algebra consistent with K where ifK~-e~e' and K, e~-h +~h', Q(h;e) = Q(h ';e'). Q( h;e ) is a non-negative, finitely additive, normalizable probability measure obeying the multiplication theorem relative to K if and only if it satisfies the following conditions:
(1) Q(h;e) is a non-negative real number (non-negativity). ( )
IlK, e
F--(h&g), O(hvg,'e) = O(h;e) + a(g;e) (finite additivity). (3) IlK, e I-h, Q(h;e) > O. (normalizability). (4)
If f is consistent with K and e, Q(f;f&e)Q(h&f,'e) = Q(h;f&e) Q (f,'e). (multiplication theorem). If Q( h;e ) satisfies ( )-( ), let Q' ( h;e) = a( h;e) /Q(e;e ). The Q'-function must also satisfy ( )-( ). In addition, it satisfies the following two conditions:
If f is consistent with K and e, O(h&f;e) = Q(h;f&e)O(f;e). Q' is a non-negative, finitely additive and normalized probability measure satisfying the multiplication theorem relative to K. In the context of theories of rational choice of the sort favored by De Finetti and other so-called 'Bayesians', it will make no difference to the evaluation ofoptionsin deliberation whether one uses a given normalizable functionor its normalization. It is entirely a matter of convenience. It is convenient or traditional to normalize. Since, however, Bayesian theories of rational choice do not mandate normalization except as a matter of convenience, I shall focus attention on conditions (1)-( ).