Structuring Logic with Sequent Calculus

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Alternative Rules for ∧ and ∨ Γ, A, B ⊢ ∆ Γ, A ∧ B ⊢ ∆ L∧′ Γ 1 ⊢ A, ∆ 1 Γ 2 ⊢ B, ∆ 2 Γ 1 , Γ 2 ⊢ A ∧ B, ∆ 1 , ∆ 2 R∧ ′ Γ 1 , A ⊢ ∆ 1 Γ 2 , B ⊢ ∆ 2 Γ 1 , Γ 2 , A ∨ B ⊢ ∆ 1 , ∆ 2 L∨ ′ Γ ⊢ A, B, ∆ Γ ⊢ A ∨ B, ∆ R∨′ These new rules are called multiplicative rules while the original rules of LK are called additive rules. Both sets of rules are equivalent thanks to the structural rules. Alexis Saurin – ENS Paris & École Polytechnique & CMI Structuring Logic with Sequent Calculus
Gentzen’s Cut Elimination Result Gentzen’s Hauptsatz The Cut rule is admissible in LK. In fact, Gentzen’s result was more than simply a proof of admissibility of the cut since he gave an explicit procedure to eliminate the cuts from a proof : starting with a proof with cuts, we can step by step transform it into a cut-free proof, and this procedure is algorithmic. Alexis Saurin – ENS Paris & École Polytechnique & CMI Structuring Logic with Sequent Calculus
Page 1 and 2: Structuring Logic with Sequent Calc
Page 3 and 4: Natural Deduction 1: Rules Γ, A
Page 5 and 6: Defects of Natural Deduction The no
Page 7 and 8: LK Rules (2) Logical Rules (¬, ∧
Page 9: Formal Theorems A ∨ B ⊢ B ∨ A
Page 13 and 14: Symmetry of LK (1) Sequents are now
Page 15 and 16: Non-Constructivism of LK Propositio
Page 17 and 18: Intuitionism All began with Brouwer
Page 19 and 20: LJ Sequent Calculus Identity Rules
Page 21 and 22: Disjunction and Existence Propertie
Page 23 and 24: Correspondence between classical an
Page 29 and 30: Motivations Does Classical Logic al
Page 31 and 32: LL Sequent Calculus Identity Rules:
Page 33 and 34: Important Characteristics of Linear

Structuring Logic with Sequent Calculus

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