I have a new paper that strengthens the case for strong Bayesianism, a.k.a. One Magisterium Bayes. The paper is entitled "From propositional logic to plausible reasoning: a uniqueness theorem." (The preceding link will be good for a few weeks, after which only the preprint version will be available for free. I couldn't come up with the $2500 that Elsevier makes you pay to make your paper open-access.)
Some background: E. T. Jaynes took the position that (Bayesian) probability theory is an extension of propositional logic to handle degrees of certainty -- and appealed to Cox's Theorem to argue that probability theory is the only viable such extension, "the unique consistent rules for conducting inference (i.e. plausible reasoning) of any kind." This position is sometimes called strong Bayesianism. In a nutshell, frequentist statistics is fine for reasoning about frequencies of repeated events, but that's a very narrow class of questions; most of the time when researchers appeal to statistics, they want to know what they can conclude with what degree of certainty, and that is an epistemic question for which Bayesian statistics is the right tool, according to Cox's Theorem.
You can find a "guided tour" of Cox's Theorem here (see "Constructing a logic of plausible inference"). Here's a very brief summary. We write A | X for "the reasonable credibility" (plausibility) of proposition A when X is known to be true. Here X represents whatever information we have available. We are not at this point assuming that A | X is any sort of probability. A system of plausible reasoning is a set of rules for evaluating A | X. Cox proposed a handful of intuitively-appealing, qualitative requirements for any system of plausible reasoning, and showed that these requirements imply that any such system is just probability theory in disguise. That is, there necessarily exists an order-preserving isomorphism between plausibilities and probabilities such that A | X, after mapping from plausibilities to probabilities, respects the laws of probability.
Here is one (simplified and not 100% accurate) version of the assumptions required to obtain Cox's result:
- A | X is a real number.
- (A | X) = (B | X) whenever A and B are logically equivalent; furthermore, (A | X) ≤ (B | X) if B is a tautology (an expression that is logically true, such as (a or not a)).
- We can obtain (not A | X) from A | X via some non-increasing function S. That is, (not A | X) = S(A | X).
- We can obtain (A and B | X) from (B | X) and (A | B and X) via some continuous function F that is strictly increasing in both arguments: (A and B | X) = F((A | B and X), B | X).
- The set of triples (x,y,z) such that x = A|X, y = (B | A and X), and z = (C | A and B and X) for some proposition A, proposition B, proposition C, and state of information X, is dense. Loosely speaking, this means that if you give me any (x',y',z') in the appropriate range, I can find an (x,y,z) of the above form that is arbitrarily close to (x',y',z').
- If X and Y are logically equivalent, and A and B are logically equivalent assuming X, then (A | X) = (B | Y).
- We may define a new propositional symbol s without affecting the plausibility of any proposition that does not mention that symbol. Specifically, if s is a propositional symbol not appearing in A, X, or E, then (A | X) = (A | (s ↔ E) and X).
- Adding irrelevant background information does not alter plausibilities. Specifically, if Y is a satisfiable propositional formula that uses no propositional symbol occurring in A or X, then (A | X) = (A | Y and X).
- The implication ordering is preserved: if A → B is a logical consequence of X, but B → A is not, then then A | X < B | X; that is, A is strictly less plausible than B, assuming X.
- a false, b false: (a or b) is false, a is false.
- a false, b true : (a or b) is true, a is false.
- a true, b false: (a or b) is true, a is true.
- a true, b true : (a or b) is true, a is true.
The probability of an event is the ratio of the number of cases favorable to it, to the number of possible cases, when there is nothing to make us believe that one case should occur rather than any other, so that these cases are, for us, equally possible.