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The opposition between objectivists and subjectivists actually stems from funda-

mental differences in the types of problems they try to solve and in their models.

Objectivist frequentists search for frequencies in a set, which implies the possibility

of infinite repetitions in similar conditions, but also provides an operational means

of calculation. The probability is specific to the set and does not exist without it, but

the data can be hypothetical (it is not always necessary to conduct all of the repe-

titions). This leads objectivists to refuse problems that are devoid of meaning, such

as events that only occur once. Statements are considered objective if they can be

refuted (with counter-examples), even if they cannot be rigorously proven [MAT 78].

On the contrary, subjectivists consider probabilities as measures of confidence, of

reasonable expectation [COX 46], of the numerical coding of a state of knowledge

[DEM 93], of an appropriate mental subtlety and can therefore deal with problems

for which there is no set, particularly unique phenomena. For such phenomena, there

is no probability
per se
, but only probabilistic models [MAT 78]. The hypotheses are

evaluated according to the observed data and the prior probability, even if the knowl-

edge is incomplete. Subjectivists do not try to achieve the best asymptotic behavior, as

statisticians do, but try instead to make the best possible inference given the available

data [DEM 93, KEM 42]. In other words, frequentists deal with random probabili-

ties and subjectivists deal with epistemological probabilities [SHA 78, SHA 86]. The

former are specific to the event itself and are not modified when knowledge changes

[KEM 42]. The latter, on the other hand, are always conditional and change accord-

ing to knowledge. They enable possible conclusions to be drawn, between certainty

and impossibility, and therefore, constitute an extended logic [KEY 29]. Subjectivists

reject the principle according to which the same causes produce the same effects, not

because they consider it to be false, but because it has no meaning, since the causes

are never identical. Oddly enough, objectivity was introduced in order to eliminate the

arbitrary and subjective nature of Bayes and Laplace, but it required the use of statis-

tical criteria that are not universal and which have to be chosen somewhat arbitrarily

[DEM 93].

Finally, the last difference between the two methods, both mathematical and in

meaning, is fundamental because it involves additivity. Random probabilities are nec-

essarily additive, since they are related to the frequentist aspect, while epistemological

probabilities do not have to be, although there is still controversy over this. We will

discuss this further in the following sections.

To sum up, there are three types of people concerned with probabilities: mathe-

maticians who suggest models without worrying about whether they fit reality or how

they will be used, physicists who infer laws from observations and experiments, and

philosophers who wonder about the meaning of all this.

A.3. Fundamental postulates for an inductive logic

Rather than accepting the “axioms” of probabilities such as they are presented, for

example, in Kolmogorov's traditional approach, more subjectivist methods start off

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