Structuring Logic with Sequent Calculus

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LL Sequent Calculus Logical Rules: ⊢ F , G, Γ ⊢ F G, Γ ⊢ F , Γ ⊢ G, ∆ ⊢ F ⊗ G, Γ, ∆ ⊗ ⊢ F , Γ ⊢ G, Γ ⊢ F G, Γ ⊢ F , Γ ⊢ F ⊕ G, Γ ⊕1 ⊢ G, Γ ⊢ F ⊕ G, Γ ⊕2 ⊢ 1 1 ⊢ Γ ⊢ ⊥, Γ ⊥ ⊢ ⊤, Γ ⊤ ⊢ F , Γ ⊢?F , Γ ? ⊢ F , ?Γ ⊢!F , ?Γ ! ⊢ Γ ⊢?F , Γ ?W ⊢?F , ?F , Γ ⊢?F , Γ ?C Alexis Saurin – ENS Paris & École Polytechnique & CMI Structuring Logic with Sequent Calculus
Important Characteristics of Linear Logic Thanks to the additional structure put in the sequents themselves, we can capture more things at the logical level and not at the term level like in classical logic; The control on structural rules allows a careful study of cut-elimination, which via Curry-Howard corresponds to execution of a functional program; Thanks to the richness of the sequents it is possible to consider the sequents as storing a state of the computation in a process of proof-search (logic programming paradigm). Lots of other directions... 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 and 10: Formal Theorems A ∨ B ⊢ B ∨ A
Page 11 and 12: Gentzen’s Cut Elimination Result
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: LL Sequent Calculus Identity Rules:

Structuring Logic with Sequent Calculus

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