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Propositions in propositional logic provable only by indirect proofs. (English) Zbl 0906.03056
D. Prawitz noticed [Natural deduction. A proof-theoretical study (1965; Zbl 0173.00205), p. 95] that normalization fails for “naive” set theory with the standard introduction and elimination rules for $$\in$$: $\in I:\quad \Gamma \Rightarrow A[x/t]/ \Gamma \Rightarrow t\in \{x| A\}; \qquad \in E:\quad \Gamma \Rightarrow t\in \{x| A\}/ \Gamma \Rightarrow A[x/t]$ L. Hallnäs developed this observation in his dissertation [“On normalization of proofs in set theory”, Diss. Math. 261 (1988; Zbl 0667.03041)]. The present paper brings this observation down to the propositional level (a similar observation was made by N. Tennant). By translating the “naive” comprehension axiom $$t\in t\leftrightarrow t\notin t$$ (where $$t\equiv \{x| x\notin x\})$$ by $$P\rightarrow \neg P$$ it is shown that the intuitionistically deducible formula $$\neg(P \leftrightarrow \neg P)$$ does not have a normal deduction $$d$$ which is also “extensively normal”: every subdeduction $$d'$$ of $$d$$ deriving a sequent $$\Gamma' \Rightarrow A$$ contains no proper subdeduction $$d''$$ of a sequent $$\Gamma'' \Rightarrow A$$ with $$\Gamma'' \subseteq \Gamma'$$. In particular normalization according to standard reductions plus reduction of $$d'$$ to $$d''$$ does not terminate. This observation is extended to classical logic and to a wider class of formulas. One of the corollaries: a provable propositional sequent having no extensively normal deduction has infinitely many different deductions.
Reviewer: G.Mints (Stanford)

##### MSC:
 03F05 Cut-elimination and normal-form theorems 03B20 Subsystems of classical logic (including intuitionistic logic)
##### Keywords:
natural deduction; normalization; comprehension
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##### References:
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