chapter12

# chapter12 - CHAPTER 12 Gentzen Proof System for...

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CHAPTER 12 Gentzen Proof System for Intuitionistic Logic In 1935 G.Gentzen formulated a ﬁrst syntactically decidable formalization for classical and intuitionistic logic and proved its equivalence with the Heyting’s original Hilbert style formalization (the famous Gentzen’s Hauptsatz). We present here the original version of his work and discuss his original proof of the Hauptsatz Theorem. We deal here, as it has happened historically, with proof theoretical formalizations of the intuitionistic logic only. It means we present here the intuitionistic logic as a proof system only, as it was done in the original papers and leave the model theoretic investigations for later. 1 LI - The Gentzen Sequent Calculus The proof system LI was published by Gentzen in 1935 as a particular case of his proof system LK for the classical logic. We have already discussed a version of the original Gentzen’s system LK in the previous chapter ?? , so we present here the proof system LI ﬁrst and then we show how it can be extended to the original Gentzen system LK . Language of LI Let SEQ = { Γ -→ Δ : Γ , Δ ∈ F * } be the set of all Gentzen sequents built out of the formulas of the language L = L {∪ , , , ¬} and the additional symbol -→ . In the intuitionistic logic we deal only with sequents of the form Γ -→ Δ, where Δ consists of at most one formula. I.e. we assume that all sequents are elements of a following subset IS of the set SEQ of all sequents. IS = { Γ -→ Δ : Δ consists of at most one formula } . (1) The set IS is called the set of all intuitionistic sequents. Axioms of LI As the axioms of LI we adopt any sequent from the set IS deﬁned by ( 1), which contains a formula that appears on both sides of the sequent arrow -→ , 1

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i.e any sequent of the form Γ 1 ,A, Γ 2 -→ A, (2) for any formula A ∈ F and any sequences Γ 1 , Γ 2 ∈ F * . Inference rules of LI The set inference rules is divided into two groups: the structural rules and the logical rules. They are deﬁned as follows. Structural Rules of LI Weakening ( weak ) Γ -→ Γ -→ A . A is called the weakening formula. Contraction ( contr ) A,A, Γ -→ Δ A, Γ -→ Δ , A is called the contraction formula , Δ contains at most one formula. Exchange ( exchange ) Γ 1 ,A,B, Γ 2 -→ Δ Γ 1 ,B,A, Γ 2 -→ Δ , Δ contains at most one formula. Logical Rules of LI Conjunction rules ( ∩ → ) A,B, Γ -→ Δ ( A B ) , Γ -→ Δ , ( → ∩ ) Γ -→ A ; Γ -→ B Γ -→ ( A B ) , Δ contains at most one formula. Disjunction rules ( → ∪ ) 1 Γ -→ A Γ -→ ( A B ) , ( → ∪ ) 2 Γ -→ B Γ -→ ( A B ) , 2
( ∪ → ) A, Γ -→ Δ ; B, Γ -→ Δ ( A B ) , Γ -→ Δ , Δ contains at most one formula. Implication rules

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chapter12 - CHAPTER 12 Gentzen Proof System for...

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