We tend to forget that mathematical constructs operate only within their assumed system, and that the probability calculus
assumes all events will take a value of either 1 or 0.
(4) The standard Bayesian defence of the requirement of probabilistic coherence is the 'Dutch-book argument': if your degrees of belief violate the probability calculus
, then there exists a set of bets which you will judge to be fair, but is guaranteed to lose you money.
My claim that the standard theory of probability may have heuristic value in historical investigation does not presuppose that the standard probability calculus
is the only valid paradigm of inferential argument.
How are we to construe the probability calculus
? According to radical Bayesians ("subjectivists"), it constitutes the entire story about rationality.
Secondly, the interpretation of conditional probabilities as what your degrees of belief would be were you to learn the truth of the conditioning proposition is not one that satisfies the probability calculus
in conjunction with some unexceptionable factual statements.
So it's just a cheat to assume that we are talking about the same sort of thing - "probability" - in both The fact is that the only relation that so-called conditional probability", as defined by (1), has to any kind of probability - where the latter is understood as a notion explicated by the axioms of the standard probability calculus
(Kolmogorov's axioms) - is that the former is defined as a ratio of probabilities.
Furthermore, to the extent that her credences cohere with the axioms of the probability calculus
the credence that she attaches to such hypotheses have already exerted their influence on the other hypotheses to which they are probabilistically relevant given her particular personal probability function.
Earman devotes the greatest attention to the criticisms of Thomas Bayes and Richard Price who argued, on the basis of the probability calculus
, that Hume's argument is flawed.
theories in which one or more axioms of the probability calculus
(usually the additivity axiom) is infringed.
The rules of consistency are those of the probability calculus
(including countable additivity).
This looseness will feature when we come to assess how knowable probabilities are, in section 6, but we shall see that the more determinate aspects of the probability calculus
are themselves already sufficient to tell us a great deal about knowledge and belief, in Section 5.