The standard logico-algebraic approach to quantum mechanics singled out a class of structures, called state-event-probability structures and underlying orthodox Hilbert space approach. These structures constitued the so called logic of quantum mechanics.With the development of the unsharp formulation the corresponding structures underlying generalized quantum mechanics, and called state-effect-probability structures, have been introduced. Two possible axiomatic formulations for the logic of unsharp (i.e., generalized) quantum mechanics are presented; the mutual relationships are investigated and some steps are taken in the direction of proving their equivalence.