By ed van der Geer at al Birkhaeuser

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1) The wff is a theorem; (2) the wff is an element of Γ ; (3) the wff is obtained from preceding elements of the sequence by an application of the rule (MP). The wff A is a logical consequence of the set of wff’s Γ , which is denoted by Γ A , when there is a derivation of A from Γ . Notice that although both proofs and derivations are finite sequences of wff’s, the premise set Γ itself does not need to be finite. That is, the consequence relation can hold between an infinite set of wff’s and a wff.

A ∧ B) > δ (A) , that is, the degree of the cut formula is reduced. (d) ( ⊃) , (⊃ ) . The original and the transformed proof look like the following. Chapter 2. . ⊃ cut Γ .. Classical first-order logic 39 .. A, Γ Δ, B Θ1 Λ1 , A B , Θ2 Λ2 ⊃ Δ, A ⊃ B A ⊃ B , Θ1 , Θ2 Λ1 , Λ2 Γ , Θ1 , Θ2 Δ, Λ1 , Λ2 .. . .. Θ1 Λ1 , A A, Γ Δ, B . cut Θ1 , Γ Λ1 , Δ, B B , Θ2 Λ2 cut Θ1 , Γ , Θ2 Λ1 , Δ, Λ2 C ’s, C’s Γ , Θ1 , Θ2 Δ, Λ1 , Λ2 It might seem that the transformation is a step backward, because we have two applications of the cut rule instead of one.

27. (Cut theorem) The cut rule is admissible in LK . Proof: We prove that if a sequent is provable with applications of the cut rule, then it is provable without using the cut rule. The cut rule is not a derived rule, that is, there is no way, in general, to obtain the lower sequent from the premises by applications of the rules of LK . 10. Prove the claim in the last sentence. ] The proof we give is constructive, that is, it is effective, and it is prooftheoretical, that is, it does not appeal to the interpretation of the sequents.