The theory of inductive inference developed by I. Levi in Gambling with Truth has two important limitations. In the first place, it makes acceptability of hypotheses 'question-dependent': any statement of the form 'It is reasonable to believe a hypothesis h on evidence e' has to be expanded so as to include a reference to some specific question which h is supposed to answer, or, alternatively, to a set of possible answers which is associated with that question. Secondly, Levi's account is insensitive to modal distinctions: no difference is made between what is merely reasonable to believe and what is unreasonable not to believe. The aim of this paper is to construct a theory which utilizes Levi's main proposals and, at the same time, avoids the above-mentioned limitations.
Consider two purely probabilistic definitions of reasonable belief: h.
B.
It is reasonable to believe h on evidence e iffP(h, e) >P(~ h, e).
It is reasonable to believe h on evidence e iff P(h, e) > 1 --e
(where e is some small number).
Both (A) and (B) presuppose the existence of a probability measure on the pairs (h, e). But even if we accept this limitation we can still claim that neither definition is very satisfying. (A) is too generous: if the difference between P(h, e) and P(~ h, e) is rather small, then the mere fact that P(h, e) > P (~ h, e) cannot by itself justify believing h. On the other hand, (B) is, in my opinion, too restrictive: it is sometimes reasonable to believe h even ifh is not highly probable. When we consider different hypotheses we are not only interested in their probability but also in their content. It seems reasonable to believe a very informative hypothesis even if its probability is slightly less than 1 --e.
Examples are not very difficult to find. Suppose that hi, h2,..., h lO are ten equally satisfactory (i.e., equally complete), pairwise incompatible and collectively exhaustive answers to some question which I want to settle. Let