Projection postulate and superposition principle in non-lattice quantum logics

The main object of the study of the present paper is the pair (L, P) consisting of two nonempty sets: the logic L of experimentally verifiable propositions (also called questions, events, yes-no measurements) and the set P of pure states, and satisfying several physically plausible postulates formulated recently (here, the postulates (A1), (A3) and (PP)). (L, P), endowed with the axioms mentioned above, was recently shown to give a proper non-lattice frame replacing the well-known quantum logic axiomatic scheme. In the present paper we consider in detail two important concepts, the projection postulate and the superposition principle, both within the non-lattice framework of the pair (L, P)