Some probabilities are known to us indirectly—for example, as a result of applying the theorems of the probability calculus; but first, of course, some probabilities must be known directly. Where a probability is known to us directly, it is known to us in the way that the validity of a syllogistic argument is known, whatever that way is. The probability relation is not an empirical one. If it is true that
p/q >
r/s, or that
p/q > 1/3, or that
r/s = 1/2, then it is true a priori, and not in virtue of any matter of fact. In particular, the truth of such statements is independent of the factual truth of
p, q, r, and
s. Finally,
p/q = 0 if
p is inconsistent with
q, and
p/q = 1 if
q entails
p.
Keynes's fundamental thesis, of which the above statements are developments, is that there are inferences in which the premises do not entail the conclusion but are nevertheless, just by themselves, objectively more or less good reason for believing it. This thesis seems to require the existence of different degrees of implication. Such degrees are Keynes's probabilities. Thus, for Keynes the study of probability coincides exactly with the study of inference, demonstrative and nondemonstrative.
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