  The word \textit{proposition} is used in physics with different meanings,
which must be distinguished to avoid interpretational problems. We construct
two languages $\mathcal{L}^{\ast}(x)$ and $\mathcal{L}(x)$ with classical
set-theoretical semantics which allow us to illustrate those meanings and to
show that the non-Boolean lattice of propositions of quantum logic (QL) can be
obtained by selecting a subset of \textit{p-testable} propositions within the
Boolean lattice of all propositions associated with sentences of
$\mathcal{L}(x)$. Yet, the aforesaid semantics is incompatible with the
standard interpretation of quantum mechanics (QM) because of known no-go
theorems. But if one accepts our criticism of these theorems and the ensuing SR
(semantic realism) interpretation of QM, the incompatibility disappears, and
the classical and quantum notions of truth can coexist, since they refer to
different metalinguistic concepts (\textit{truth} and \textit{verifiability
according to QM}, respectively). Moreover one can construct a quantum language
$\mathcal{L}_{TQ}(x)$ whose Lindenbaum-Tarski algebra is isomorphic to QL, the
sentences of which state (testable) properties of individual samples of
physical systems, while standard QL does not bear this interpretation.
