  The present paper demonstrates the failure of the principle of excluded
middle (PEM) in the lattice of all closed linear subspaces of a Hilbert space
(usually defined as quantum logic). Namely, it is shown that for a qubit, a
proposition and its negation can be both false. Since PEM is the assumed
theorem of quantum logic, this raises the question: If PEM holds in the
orthocomplemented lattice of all propositions of the quantum system, then how
the failure of PEM in quantum logic can be explained? Alternatively, if the
propositions relating to the quantum system do not obey PEM, then what is the
semantics of those propositions? Possible answers to these questions are
analyzed in the present paper.
