We consider a conception of reality that is the following: An object is
'real' if we know that if we would try to test whether this object is present,
this test would give us the answer 'yes' with certainty. If we consider a
conception of reality where probability plays a fundamental role it can be
shown that standard probability theory is not well suited to substitute
'certainty' by means of 'probability equal to 1'. The analysis of this problem
leads us to propose a new type of probability theory that is a generalization
of standard probability theory. This new type of probability is a function to
the set of all subsets of the interval [0, 1] instead of to the interval [0, 1]
itself, and hence its evaluation happens by means of a subset instead of a
number. This subset corresponds to the different limits of sequences of
relative frequency that can arise when an intrinsic lack of knowledge about the
context and how it influences the state of the physical entity under study in
the process of experimentation is taken into account. The new probability
theory makes it possible to define probability on the whole set of experiments
within the Geneva-Brussels approach to quantum mechanics, which was not
possible with standard probability theory. We introduce the structure of a
'state experiment probability system' and derive the state property system as a
special case of this structure. The category SEP of state experiment
probability systems and their morphisms is linked with the category SP of state
property systems and their morphismsComment: 27 page