  The structure of a complete lattice formed by closed linear subspaces of a
Hilbert space (i.e., a Hilbert lattice) entails some unreasonable consequences
from the physical point of view. Specifically, this structure seems to
contradict to the localized variant of the Kochen-Specker theorem according to
which the bivaluation of a proposition represented by a closed linear subspace
that does not belong to a Boolean algebra shared by the state, in which a
quantum-mechanical system is prepared, must be value indefinite. For this
reason, the Hilbert lattice structure seems to be too strong and needs to be
weakened. The question is, how should it be weakened so that to support the
quantum uncertainty principle and the Kochen-Specker theorem? Which logic will
a weakened structure identify? The present paper tries to answer these
questions.
