THE ROLE OF PROBABILITY IN EPISTEMOLOGY

Kyburg takes high probability to be logically necessary and a logically sufficient condition for justified acceptance. This leads to a very simple and elegant theory, but I will argue that he is wrong on both counts. Supposing that high probability is a necessary condition for justified acceptance has the effect of dramatically truncating the role of deductive reasoning. The difficulty is that a rule like adjunction, according to which a conjunction can be inferred from its conjuncts, can lead to conclusions that are less probable than the premises to which it is applied. The only logical inference rules that can be employed by a Kyburgian epistemic agent are those that are probabilistically valid, in the sense that the conclusion of the inference is guaranteed to be as probable as the least probable premise to which the rule appeals. This is a severe constraint, because no rule that makes essential use of multiple premises can be probabilistically valid. Thus the epistemic agent is precluded from using not just adjunction, but also modus ponens, modus tollens, and so forth. Surprisingly, the same constraint results from the supposition that high probability is a sufficient condition for justified acceptance, provided we accept a principle on which Kyburg and I agree, namely, that an epistemic agent cannot be justified in accepting an explicit contradiction. I will refer to this as the consisten~yprinciple. As the lottery paradox illustrates, if high probability is sufficient to justify acceptance, then an epistemic agent will be led to accept sets of beliefs that are jointly inconsistent. Given the consistency principle, it follows once more that deductive reasoning does not lead automatically to the justified acceptance of beliefs. Otherwise, by adjunction, the episternic agent could conjoin contradictory beliefs and accept an explicit contradiction. Accordingly, Kyburg eschews adjunction. But the rejection of deductive reasoning does not stop here. Although Kyburg himself is well aware of this, others have often overlooked that it is not just adjunction that must be abandoned. Truths of logic have probability 1. so by the sufficiency of high probability for acceptance, truths of logic can always be accepted (a principle I would endorse anyway). Because ( P 3 (Q 3 ( P & Q ) ) ) is a truth of logic, to avoid accepting explicit contradictions while accepting the individual contradictory propositions, Kyburg must also reject modus ponens. Otherwise, any application of adjunction could be replaced by two applications of modus ponens. For similar reasons he must reject modus tollens, and all familiar logical inference rules that proceed from more than one premise to a conclusion. It seems to be the case, once again, that the only logical inference rules that can be employed by a Kyburgian epistemic