D is a proper proof-theoretic solution to genuine paradoxes. In the next section, we move on to the second question on why RND is a convincing requirement and claim that a clear answer is not yet given. 4.2 Why Should We Accept Only a Normalizable Derivation? Our proof-theoretic analysis of the paradoxes uses a natural deduction system developed by Prawitz (1965, 1971) but first introduced by Gentzen (1935, 2008). When Gentzen (1936) attempted to show the consistency of arithmetic, he believed that if there was a fault in the paradoxes, it must be sought in the logical reasoning employed. One of the main purposes of the proof-theoretic analysis of the paradoxes is to find any errors in the reasoning. It appears to be convincing that a suitable proof-theoretic solution to the paradoxes can be the reasoning-rejection solution. RND may be the reasoning-rejection solution in a broad sense. However, as our second question asks, it should be explained why a nonnormalizable derivation is unacceptable. The Second Question (ii): Why should we accept only a normalizable derivation? A promising answer is that a non-normalizable derivation would not be proof-theoretically valid. In this section, we will briefly investigate Prawitz's idea of proof-theoretic validity and the relation between proof-theoretic validity and normalizability. If proof-theoretic validity implies (full) normalizability, then paradoxical derivations, i.e. T-paradoxes which are not (fully) normalizable, are not proof-theoretically valid. Then, since there is no nonnormalizable derivation which is proof-theoretically valid, it can be explained why any non-normalizable derivations are unacceptable. The name 'proof theory' was originally coined by David Hilbert. The aim of his proof theory is to obtain a reduction of mathematics to some more elementary part of it, such 144 as finitistic or constructive mathematics, by analyzing the proofs of mathematical theories. Prawitz (1971, 1973) thinks that Hilbert's proof theory is only a tool to obtain the reductio