Stating the strength of ``pragmatic beliefs'' by odds

From proposition (1) we finally understand very well Butterworth's beliefs, in spite of the contradiction with (0). In fact, since ancient times betting has been recognized to be the best way to check how much one really believes something, as well stated by Kant when he talks about pragmatic beliefs:[24]

``The usual touchstone, whether that which someone asserts is merely his persuasion - or at least his subjective conviction, that is, his firm belief - is betting. It often happens that someone propounds his views with such positive and uncompromising assurance that he seems to have entirely set aside all thought of possible error. A bet disconcerts him. Sometimes it turns out that he has a conviction which can be estimated at a value of one ducat, but not of ten. For he is very willing to venture one ducat, but when it is a question of ten he becomes aware, as he had not previously been, that it may very well be that he is in error.''

And, in fact, in the mathematical theory of probability of Laplace all probabilistic statements can be mapped into betting statements, like his famous one concerning his evaluation of the uncertainty on the value of the mass of Saturn:[17]

``To give some applications of this method I have just availed myself of the opus magnus that Mr. Bouvard has just finished on the motions of Jupiter and Saturn, of which he has given very precise tables. ... His calculations give him the mass of Saturn as 3,512th part of that of the sun. Applying my probabilistic formulae to these observations, I find that the odds are 11,000 to 1 that the error in this result is not a hundredth of its value.''

That is

$$\begin{eqnarray*} P(3477 \le M_{Sun}/M_{Sat} \le 3547\,\vert\,I(\mbox{Laplace})) &=& 99.99\%\,, \end{eqnarray*}$$

where $I(\mbox{Laplace})$ stands for all information available to Laplace (probabilistic statements are always conditioned by a state of information). The Laplace's result is a very clear statement and there is a perfect match between beliefs, odds and probabilistic statement. Instead, I ensure you, a ``95% C.L. lower limit'' result cannot be turned into a 19:1 bet that the quantity in object is above that limit (see footnote 9), neither a p-value of e.g $10^{-4}$ can be turned into a 10000:1 bet in favor of a discovery (see also the last minute reference [30].)