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In Elements of Intuitionism (p. 145), Dummett wrote:

Cut-elimination is directly connected with establishing consistency, and was so intended by Gentzen. Given the equivalence of N and L, acceptance of Ex Falso Quodlibet [i.e the intuitionistic absurdity rule $\bot E$] is the form of $\lnot - $ in N or thinning on the right in L makes the consistency of N equivalent to the non-derivability of the empty sequent '$\emptyset : \emptyset$' in L. But simply examination of the rules of L, it is clear that there is no rule by which this sequent could be possibly derived. Recognition of $\lnot -$ [i.e. Ex Falso Quodlibet] as a correct principle thus amounts to recognizing the consistency of N

My question is twofold:

  1. In the explanation of the Cut elimination theorem, it is often said that the derivability of the empty sequent is only possible in an inconsistent system, but seldom is given a simple explanation of the mechanism by which one could terminate on such a result with an inconsistent system; could someone provide a simple explanation of this mechanism?
  2. Does the Cut elimination theorem allows to reject the objection according to which the absurdity rule presupposed to prove consistency: the need of rule $\bot E$ would be therefore a petitio principii ?
Joseph Vidal-Rosset
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    Maybe useful this post – Mauro ALLEGRANZA Sep 22 '21 at 13:10
  • I am not sure I understand what kind of mechanism you're looking for in Question 1. In an inconsistent system, you can derive $A \vdash$ and $\vdash A$ for some $A$, and then the empty sequent $\vdash$ follows by applying the cut rule. Of course, you need thinning to obtain inconsistency from the derivability of the empty sequent. – Z. A. K. Sep 22 '21 at 13:38
  • Thanks Mauro for this first reply. Note that mentionning the rule of thinning one the right, you say that if any atomic formula is derivable from $\mathcal{T}$ ,, then $\mathcal{T}$ is inconsistent, and, by contraposition, $\mathcal{T}$ is consistent, only if there is at least one atomic formula underivable from $\mathcal{T}$. It seems therefore that the intuitionistic absurdity rule is necessary to decide consistency. – Joseph Vidal-Rosset Sep 22 '21 at 16:17

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