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 Lüders-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 Sorkin (Mod Phys Lett A 9:3119–3127, 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. (shrink)