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.
Yet its truth-value obtains necessarily, and to satisfy the probability calculus
its probability must be 0 or 1.
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.
One should also note that Earman's own treatment of Hume is relatively brief (86 out of 217 pages) and some readers may find tedious the extensive use of symbolic logic and mathematical equations in the discussion of the probability calculus
After all, they point out, many ways of assigning numbers to events satisfy the probability calculus
, not all of which therefore represent the genuine probabilities of those events.
theories in which one or more axioms of the probability calculus
(usually the additivity axiom) is infringed.