In the axiomatic development of the logic of nonrelativistic quantum mechanics it is not difficult to set down certain plausible axioms which ensure that the quantum logic of propositions has the structure of an orthomodular poset. This can be done in a number of ways, for example, as in Gunson [2], Mackey [4], Piron [5], Varadarajan [7] and Zierler [8], and we summarise one of these ways in Section II below. It is customary to impose further axioms which ensure that this logic is a complete atomic orthomodular lattice so that easy access is obtained to a representation of the logic as the lattice of closed subspaces of Hilbert space, for example, Jauch [3], Piron [5] and Zierler [8]. Not much is known about the structure of orthomodular posets and in Section I we show how they arise naturally in the study of certain sets of Boolean logics in which one can define common operations of implication and negatioa In Section III we show that every completely orthomodular poset arises in this way.