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