  Classical countably additive real-valued probabilities come at a
philosophical cost: in many infinite situations, they assign the same
probability value -- namely, zero -- to cases that are impossible as well as to
cases that are possible. There are three non-classical approaches to
probability that can avoid this drawback: full conditional probabilities,
qualitative probabilities and hyperreal probabilities. These approaches have
been criticized for failing to preserve intuitive symmetries that can easily be
preserved by the classical probability framework, but there has not been a
systematic study of the conditions under which these symmetries can and cannot
be preserved. This paper fills that gap by giving complete characterizations
under which symmetries understood in a certain "strong" way can be preserved by
these non-classical probabilities, as well as by offering some results to make
it plausible that the strong notion of symmetry here may be the right one.
Philosophical implications are briefly discussed, but the main purpose of the
paper is to offer technical results to inform more sophisticated further
philosophical discussion.
