  In this work we discuss the failure of the principle of truth functionality
in the quantum formalism. By exploiting this failure, we import the formalism
of N-matrix theory and non-deterministic semantics to the foundations of
quantum mechanics. This is done by describing quantum states as particular
valuations associated to infinite non-deterministic truth tables. This allows
us to introduce a natural interpretation of quantum states in terms of a
non-deterministic semantics. We also provide a similar construction for
arbitrary probabilistic theories based in orthomodular lattices, allowing to
study post-quantum models using logical techniques.
