The only relevant thing is uncertainty —the extent of our knowledge and ignorance. The actual fact of whether or not the events considered are in some sense determined, or known by other people, and so on, is of no consequence.
The numerous, different, opposed attempts to put forward particular points of view which, in the opinion of their supporters, would endow Probability Theory with a ‘nobler’ status, or a ‘more scientific’ character, or ‘firmer’ philosophical or logical foundations, have only served to generate confusion and obscurity, and to provoke well-known polemics and disagreements —even between supporters of essentially the same framework.
The main points of view that have been put forward are as follows:
The classical view, based on physical considerations of symmetry, in which one should be obliged to give the same probability to such ‘symmetric’ cases. But which symmetry? And, in any case, why? The original sentence becomes meaningful if reversed: the symmetry is probabilistically significant, in someone’s opinion, if it leads him to assign the same probabilities to such events.
The logical view is similar, but much more superficial and irresponsible inasmuch as it is based on similarities or symmetries which no longer derive from the facts and their actual properties, but merely from the sentences which describe them, and from their formal structure or language.
The frequentist (or statistical) view presupposes that one accepts the classical view, in that it considers and event as a class of individual events, the latter being ‘trials’ of the former. The individual events not only have to be ‘equally probable’, but also ‘stochastically independent’ (these notions when applied to individual events are virtually impossible to define or explain in terms of the frequentist interpretation). In this case, also, it is straightforward, by means of the subjective approach, to obtain, under the appropriate conditions, in a perfectly valid manner, the result aimed at (but unattainable) in the statistical formulation. It suffices to make use of the notion of exchangeability. The result, which acts as a bridge connecting the new approach with the old, has often been referred to by the objectivists as “de Finetti’s representation theorem”.
It follows that all the three proposed definitions of ‘objective’ probability, although useless per se, turn out to be useful and good as valid auxiliary devices when included as such in the subjectivist theory.
—de Finetti (1970)