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11slides(2Gentzen)

# 11slides(2Gentzen) - Chapter 11(Part 2 Gentzen Sequent...

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Chapter 11 (Part 2) Gentzen Sequent Calculus GL The proof system GL for the classical propo- sitional logic is a version of the original Gentzen (1934) systems LK . A constructive proof of the completeness the- orem for the system GL is very similar to the proof of the completeness theorem for the system RS . Expressions of the system like in the original Gentzen system LK are Gentzen sequents . Hence we use also a name Gentzen sequent calculus. 1

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Language of GL: L = L {∪ , , , ¬ , } . We add a new symbol to the alphabet: -→ . It is called a Gentzen arrow. The sequents are built out of finite sequences (empty included) of formulas, i.e. elements of F * , and the additional sign -→ . We denote, as in the RS system, the finite sequences of formulas by Greek capital let- ters Γ , Δ , Σ, with indices if necessary. Sequent definition: a sequent is the expres- sion Γ -→ Δ , where Γ , Δ ∈ F * .
Meaning of sequents Intuitively, we interpret a sequent A 1 , ..., A n -→ B 1 , ..., B m , where n, m 1 as a formula ( A 1 ... A n ) ( B 1 ... B m ) . The sequent: A 1 , ..., A n -→ (where n 1) means that A 1 ... A n yields a contra- diction. The sequent -→ B 1 , ..., B m (where m 1) means | = ( B 1 ... B m ). The empty sequent: -→ means a contra- diction. 2

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Given non empty sequences: Γ, Δ, we de- note by σ Γ any conjunction of all formulas of Γ, and by δ Δ any disjunction of all formulas of Δ.
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