Structuring Logic with Sequent Calculus

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Subformula Property Subformula Property A provable Sequent can be proved using only subformulas of the formulas appearing in the sequent. (A cut-free proof only makes use of subformulas of the root sequent) It reduces the search space a lot! Alexis Saurin – ENS Paris & École Polytechnique & CMI Structuring Logic with Sequent Calculus
Symmetry of LK (1) Sequents are now of the form: ⊢ ′ Γ. Implication is a defined connective: A ⇒ B ≡ ¬A ∨ B Negation only appears on atomic formulas, thanks to de Morgan’s laws: ¬(A ∨ B) ≡ (¬A ∧ ¬B) ¬∀xA ≡ ∃x¬A ¬(A ∧ B) ≡ (¬A ∨ ¬B) ¬∃xA ≡ ∀x¬A More precisely, when writing ¬A, we will always mean the negation normal form of this formula for the obviously terminating and confluent rewriting system: ¬(A ∨ B) → (¬A ∧ ¬B) ¬(A ∧ B) → (¬A ∨ ¬B) ¬¬A → A ¬∀xA → ∃x¬A ¬∃xA → ∀x¬A 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: Gentzen’s Cut Elimination Result
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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