  In this paper we consider the phenomenon of superpositions in quantum
mechanics and suggest a way to deal with the idea in a logical setting from a
syntactical point of view, that is, as subsumed in the language of the
formalism, and not semantically. We restrict the discussion to the
propositional level only. Then, after presenting the motivations and a possible
world semantics, the formalism is outlined and we also consider within this
schema the claim that superpositions may involve contradictions, as in the case
of the Schr\"odinger's cat, which (it is usually said) is both alive and dead.
We argue that this claim is a misreading of the quantum case. Finally, we
sketch a new form of quantum logic that involves three kinds of negations and
present the relationships among them. The paper is a first approach to the
subject, introducing some main guidelines to be developed by a `syntactical'
logical approach to quantum superpositions.
