  This paper is motivated by the questions of how to give the concept of
probability an adequate real-world meaning, and how to explain a certain type
of phenomenon that can be found, for instance, in Ellsberg's paradox. It
attempts to answer these questions by constructing an alternative theory to one
that was proposed in earlier papers on the basis of various important
criticisms that were raised against this earlier theory. The conceptual
principles of the corresponding definition of probability are laid out and
explained in detail. In particular, what is required to fully specify a
probability distribution under this definition is not just the distribution
function of the variable concerned, but also an assessment of the internal
and/or the external strength of this function relative to other distribution
functions of interest. This way of defining probability is applied to various
examples and problems including, perhaps most notably, to a long-running
controversy concerning the distinction between Bayesian and fiducial inference.
The characteristics of this definition of probability are carefully evaluated
in terms of the issues that it sets out to address.
