Abstract The main quantum phenomena-indeterminism and interference-are explained on a purely logical basis by extending classical mathematical logic. In this extension, statements are represented by Hermitian matrices that, it is postulated, can undergo unitary transformations of truth values (the principle of relativity of truth values). The formula for quantum probability is deduced. A new method of quantization is then introduced by the additional postulate that invariants of translations in some space can be described by statements represented by matrices that are invariants of the same translations in the same space. The formula p̂ = − ih∂/∂ q is deduced (up to an unknown multiplicative constant). The connection between the existence of the continuum and the existence of quantum mechanics is demonstrated.