Propositions in propositional logic provable only by indirect proofs.

*(English)*Zbl 0906.03056D. 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) |

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##### References:

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