  In quantum mechanics, the selfadjoint Hilbert space operators play a triple
role as observables, generators of the dynamical groups and statistical
operators defining the mixed states. One might expect that this is typical of
Hilbert space quantum mechanics, but it is not. The same triple role occurs for
the elements of a certain ordered Banach space in a much more general theory
based upon quantum logics and a conditional probability calculus (which is a
quantum logical model of the Lueders-von Neumann measurement process). It is
shown how positive groups, automorphism groups, Lie algebras and statistical
operators emerge from one major postulate - the non-existence of third-order
interference (third-order interference and its impossibility in quantum
mechanics were discovered by R. Sorkin in 1994). This again underlines the
power of the combination of the conditional probability calculus with the
postulate that there is no third-order interference. In two earlier papers, its
impact on contextuality and nonlocality had already been revealed.
