  Most quantum logics do not allow for a reasonable calculus of conditional
probability. However, those ones which do so provide a very general and rich
mathematical structure, including classical probabilities, quantum mechanics as
well as Jordan algebras. This structure exhibits some similarities with Alfsen
and Shultz's non-commutative spectral theory, but these two mathematical
approaches are not identical. Barnum, Emerson and Ududec adapted the concept of
higher-order interference, introduced by Sorkin in 1994, into a general
probabilistic framework. Their adaption is used here to reveal a close link
between the existence of the Jordan product and the non-existence of
interference of third or higher order in those quantum logics which entail a
reasonable calculus of conditional probability. The complete characterization
of the Jordan algebraic structure requires the following three further
postulates: a Hahn-Jordan decomposition property for the states, a polynomial
functional calculus for the observables, and the positivity of the square of an
observable. While classical probabilities are characterized by the absence of
any kind of interference, the absence of interference of third (and higher)
order thus characterizes a probability calculus which comes close to quantum
mechanics, but still includes the exceptional Jordan algebras.
