  In his book `Physics and Philosophy', Heisenberg suggested that the quantum
world is one of ``potentialities or possibilities'' and that the classical
realm is one of ``things or facts''. After ascertaining that his categories
most naturally have the structure of ontological equivalence classes, we show
that they cannot be seriously introduced into the quantum formalism without
rendering it incoherent, as under some circumstances the formalism permits an
equality between members of distinct equivalence classes. This is labeled the
`equality of inequivalents problem'. Three possible reactions are discussed:
First, one could deny the problem by challenging some of the assumptions that
underlie its formulation; second, one could accept the problem and take it to
be grounds for dismissing Heisenberg's or similar distinctions altogether;
third, one could accept the problem but take its source to be not Heisenberg's
distinction but the quantum formalism. A plausibility argument in support of
the third reaction is given by analogy: it is shown that axiomatic probability
suffers from a similar problem if one wishes to seriously distinguish between
things which are possibilities and things which are not. An enriched
axiomatization of probability is proposed which captures the concept of
probability as a measure over possibilities and thereby overcomes the equality
of inequivalents problem in axiomatic probability.
