RND with the assumption that proof-theoretic validity implies normalizability. The restriction of ¬E−rule via Definition 4.3.1 is executed by the requirement of a P−valid inference that only P−valid inferences are to be used, but not by RND. The requirement of a P−valid inference can be requested independently of RND. Therefore, if a reasoningrejection solution is a constraint on a particular inference rule, RND is not a reasoningrejection solution. 155 4.4 Should We Consider Only ⊥ as an Unacceptable Conclusion? In this section, by examining Petrolo and Pistone's case of a (full) normal derivation of a contradiction, we shall deal with the last question of this chapter, 'should we consider only ⊥ as an unacceptable conclusion?' The Third Question (iii): Should we consider only ⊥ as an unacceptable conclusion? As we have examined in the last section 4.3, a derivation of ⊥ formalizing Russell's paradox consists of two normal derivations Σ10 and Σ11. Σ10 ¬a ∈ a Σ11 a ∈ a ¬E⊥ Instead of applying ¬E−rule, we can apply ∧I−rule and have Σ10 ¬a ∈ a Σ11 a ∈ a ∧I¬a ∈ a∧a ∈ a Since Σ10 and Σ11 are normal derivations, either the above derivation is in normal form. One of interesting points is that we often call the formula ¬a ∈ a∧ a ∈ a a contradiction. Let us distinguish between a contradiction and an absurdity. For any formula having the form ¬φ ∧φ a contradiction. A contradiction is often regarded as an unacceptable conclusion. Then, it seems convincing that the above derivation is paradoxical because it derives an unacceptable conclusion through acceptable reasoning with acceptable premises. RND imposes a constraint on the application of non-normalizable derivations. If there is a paradoxical derivation which is a (closed) full normal derivation of an unacceptable conclusion, such as ¬φ ∧φ , then RND cannot be the proof-theoretic solution to the paradoxes. Petrolo and Pistone (2018) have considered the very case. 156 Under the distinction between a c