Another epistemological shock from a paper in the junk archive that confuses symbols with their values and that knowing if the limit of a sequence exists does not imply knowing its value.
In the paper “AN ENDLESS HIERARCHY OF PROBABILITIES” the authors only show that if a quantity υ is known then P(q0), i.e. the probability that the probability of q is υo can be calculated at the limit. This means that the probability exists but it cannot be determined in general unless υ is known . This does not resolve the issue whether P(q0) can be known at all.
In the paper the authors violated the condition set by Rescher that “After all, without a categorically established factual basis of some sort, there is no way of assessing probabilities.” by. in a sense. turning a general statement about mathematical convergence of series into a claim about specific values of probabilities.
The authors further claimed that
“In this paper it is shown that the arguments of Rescher, Savage and Hume, plausible and persuasive as they may seem at first sight, are actually misleading. Not only can an endless hierarchy of probabilities be given a clear interpretation, it can also ascertain what the probability value for the original proposition q is.”
Although they carefully selected the wording, the claim is false because I do not believe that Rescher cares so much about “clear interpretations” as everything is fairly clear from start but whether the probability value for the original proposition q can be known at all since there are infinite choices between 0 and 1 for the value of õ.
Going one step further, I think everyone who are talking about “an endless hierarchy of probabilities” is at least three quarters-wrong. Such endless hierarchies make sense only in one of the four definitions of probability, i.e. when probability is a measure of belief of the occurrence of single events and not of averages of mass phenomena, like in physical sciences. (See Papoulis, A., Probability, Random Variables and Stochastic Processes, p. 14). Only then the concept of probability is a form of inductive reasoning. It is well-known in that case that prior estimates have negligible effects on posterior probabilities.
In general, it doesn’t even make sense to talk about the probability that P(q) = υo
is true or false. It is either true or false. But if one wants to talk like that for some reason, then Nicholas Rescher is correct and the authors haven’t proved even remotely that he is wrong. All they have shown is that the probability has some unknown value at the limit.
Another epistemological shock from a paper disposed in the junk archive.