  In this paper we analyze and discuss the historical and philosophical
development of the notion of logical possibility focusing on its specific
meaning in classical and quantum mechanics. Taking into account the logical
structure of quantum theory we continue our discussion regarding the
Aristotelian Square of Opposition in orthomodular structures enriched with a
monadic quantifier. Finally, we provide an interpretation of the Orthomodular
Square of Opposition exposing the fact that classical possibility and quantum
possibility behave formally in radically different manners.
