Lecture Notes on Cut Elimination

The identity theorem of the sequent calculus exhibits one connection between the judgments A left and A right : If we assume A left we can prove A right . In other words, the left rules of the sequent calculus are strong enough so that we can reconstitute a proof of A from the assumption A. So the identity theorem is a global version of the local completeness property for the elimination rules. The cut theorem of the sequent calculus expresses the opposite: if we have a proof of A right we are licensed to assume A left . This can be interpreted as saying the left rules are not too strong: whatever we can do with the antecedent A left can also be deduced without that, if we know A right . Because A right occurs only as a succedent, and A left only as an antecedent, we must formulate this in a somewhat roundabout manner: If Γ =⇒ A right and Γ, A left =⇒ J then Γ =⇒ J . In the sequent calculus for pure intuitionistic logic, the only conclusion judgment we are considering is C right , so we specialize the above property. Because it is very easy to go back and forth between sequent calculus deductions of A right and verifications of A↑, we can use the cut theorem to show that every true proposition has a verification, which establishes a fundamental, global connection between truth and verifications. While the sequent calculus is a convenient intermediary (and was conceived as such