  The failure of distributivity in quantum logic is motivated by the principle
of quantum superposition. However, this principle can be encoded differently,
i.e., in different logico-algebraic objects. As a result, the logic of
experimental quantum propositions might have various semantics. E.g., it might
have either a total semantics, or a partial semantics (in which the valuation
relation -- i.e., a mapping from the set of atomic propositions to the set of
two objects, 1 and 0 -- is not total), or a many-valued semantics (in which the
gap between 1 and 0 is completed with truth degrees). Consequently, closed
linear subspaces of the Hilbert space representing experimental quantum
propositions may be organized differently. For instance, they could be
organized in the structure of a Hilbert lattice (or its generalizations)
identified with the bivalent semantics of quantum logic or in a structure
identified with a non-bivalent semantics. On the other hand, one can only
verify -- at the same time -- propositions represented by the closed linear
subspaces corresponding to mutually commuting projection operators. This
implies that to decide which semantics is proper -- bivalent or non-bivalent --
is not possible experimentally. Nevertheless, the latter allows simplification
of certain no-go theorems in the foundation of quantum mechanics. In the
present paper, the Kochen-Specker theorem asserting the impossibility to
interpret, within the orthodox quantum formalism, projection operators as
definite {0,1}-valued (pre-existent) properties, is taken as an example. The
paper demonstrates that within the algebraic structure identified with
supervaluationism (the form of a partial, non-bivalent semantics), the
statement of this theorem gets deduced trivially.
