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in Kolmogorov theory. However, the works of Cox show that these axioms can be
inferred from a certain number of basic postulates prompted by intuition (see section
In section 6.10, we gave a few examples showing the limits of additive probabili-
ties, which are often the results of the strict constraints that they impose. The modifi-
cation of basic postulates in order to overcome these limits leads to different numerical
theories, which no longer satisfy the same properties, and thus leads us again to meth-
ods such as fuzzy sets or Dempster-Shafer belief theory. The latter has often been
criticized on the grounds that Dempster's orthogonal combination rule had no theoret-
ical justifications. Several authors have responded to this criticism, and in Appendix
B we will present Smets's arguments, which allow this rule to be inferred from more
easily justifiable axioms. We will then establish the relation between these axioms and
those given by Cox, in order to explain the origins of the differences between the two
A.1. Probabilities through history
This study was inspired by Shafer's works and particularly by his remarkable
review articles on the history of science [SHA 78, SHA 86]. The following histori-
cal presentation is largely based on these works. They constitute the basic points, and
additional information was added from articles or topics [COX 46, DEM 93, DUB 88,
GOO 59, HOR 86, JAY 57, JEF 61, KEM 42, NEA 92, SOM 89, TRI 72].
A.1.1. Before 1660
The conflict between knowledge and opinion appeared as early as the Classical
era, in particular with Plato, in roughly 400 BC, and terms such as necessary, possible,
probable, began to be defined. We find for example with Aristotle (roughly 350 BC)
assertions of the type “if an event is necessary, then its opposite is impossible” (until
possibility theory was developed, no consistent theory was capable of modeling this
sentence) or also “what is probable is what usually occurs” (in reference to phenomena
repeating themselves, which are the basis of frequentist theory). For the Ancients,
there were three epistemological categories. In the first, certain knowledge is possible.
This is equivalent to Plato's concept of knowledge or science. The second category is
comprised of events for which knowledge is probable or possible. This corresponds
to Plato's concept of opinion and, in this sense, probability is seen as an attribute
of opinion. The third category, which is absent in Plato's philosophy, corresponds to
events for which no knowledge is possible, hence to the realm of chance. This term
is used to mean the lack of statistical regularity. Transposed in more modern terms,
these concepts correspond, in our opinion, to deductive reasoning for the first category
and to inductive reasoning for the second. The third corresponds to phenomena that
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