O1 tells us that when we study a quantum system (and in general any physical system) this notion is at disposal: the measure p can be interpreted as a ?degree of belief?(or ?truth value?) of a certain proposition, namely, if a is a proposition, p(a) tell us how much we are sure that a happens in real word. But one must be careful about O2: the degree of belief of a proposition can be tested and compared with the one of another proposition, only if these propositions are associated to observables that are measurable at the same time. Keeping this fact in mind, we can say that, if we are agree on O1, we have an ordering relation at disposal over the set of all the physical propositions. This partial order, allows us to de?ne the met and join between all the propositions (actually this is an assumption, despite it is reasonable), thus we can conclude that it is a lattice
