Structuring Logic with Sequent Calculus

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Correspondence between classical and intuitionistic provability (4) Theorem Γ ⊢ LK A iff Γ ⋆ ⊢ LJ A ⋆ Lemma ⊢ LK A ⇔ A ⋆ Definition A is said to be stable when ⊢ LJ ¬¬A ⇒ A. Lemma For all formula A, A ⋆ is stable. Lemma If Γ ⊢ LK ∆ then Γ ⋆ , ¬∆ ⋆ ⊢ LJ . Alexis Saurin – ENS Paris & École Polytechnique & CMI Structuring Logic with Sequent Calculus
Correspondence between classical and intuitionistic provability (5) In what sense can we say that LJ is finer (subtler?) than LK? An intuitionistic logician cannot necessarily prove a formula A when a classical mathematician can, BUT he can find another formula (A ⋆ ) that he is able to prove and that the classical mathematician cannot distinguish from the previous one. In particular, in intuitionistic logic, the use of excluded middle (or contraction on the right) shall be explicitly mentioned in the formula thanks to the use of double negation 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 25: 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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