Hempel's qualitative criteria of converse consequence and special consequence for confirmation are examined, and the resulting paradoxes traced to the general intransitivity of confirmation. Adopting a probabilistic measure of confirmation, a limiting form of transitivity of confirmation from evidence to predictions is derived, and it is shown to what extent its application depends on prior probability judgments. In arguments involving this kind of transitivity therefore there is no necessary "convergence of opinion" in the sense claimed by some personalists. The conditions (...) of application of the limiting transitivity theorem are most perspicuously described in terms of De Finetti's notion of exchangeability, which leads to a suggested revaluation of the function of theories in relation to confirmation and explanation. (shrink)