0.1 Sequent Calculus
0.1 Sequent Calculus Differences between ND and Sequent Calculus 1. Rules applies to sequents 2. connectives are always introduced 3. four kinds of rules: axioms, cut, structural rules, logical rules 0.1.1 Sequent Calculi LK and LJ Definition 0.1.1 (Sequent). A1,...,An ⊢ B. S S S Definition 0.1.2 (Inference Rule). 1 or 1 2 where S ,S ,S are sequents. S S 1 2 Definition 0.1.3 (Rules of the LJ sequent calculus). • Axioms: Γ ⊢ A (derivability assertion) if A ∈ Γ Γ ⊢ A Σ, A ⊢ B • Cut: (CUT) Γ, Σ ⊢ B • Structural Rules: Γ,B,A, ∆ ⊢ C (exchange,l) Γ,A,B, ∆ ⊢ C Γ ⊢ B Γ ⊢ (weakening,l) (weakening,r) empty RHS is a contradiction (⊥) Γ, A ⊢ B Γ ⊢ B Γ, A, A, ∆ ⊢ C (contraction,l) Γ, A, ∆ ⊢ C • Logical Rules: Γ, A ⊢ C Γ, B ⊢ C Γ ⊢ A1 Γ ⊢ A2 (∨,l) (∨,r)1 (∨,r)2 Γ, A ∨ B ⊢ C Γ ⊢ A1 ∨ A2 Γ ⊢ A1 ∨ A2 Γ, A1 ⊢ C Γ, A2 ⊢ C Γ ⊢ A Γ ⊢ B (∧,l)1 (∧,l)2 (∧,r) Γ, A1 ∧ A2 ⊢ C Γ, A1 ∧ A2 ⊢ C Γ ⊢ A ∧ B Γ, B ⊢ C Γ ⊢ A Γ, A ⊢ B (→,l) (→,r) Γ, A → B ⊢ C Γ ⊢ A → B Γ ⊢ A Γ, A ⊢ (¬,l) (¬,r) Γ, ¬A ⊢ Γ ⊢ ¬A 1 The calculus above is sound and complete for IL. Note that, except for (CUT), the following holds Corollary 0.1.1 (Subformula property). everything that appears in the rules’s premises appears in the conclusion To obtain a calculus for classical logic we need to introduce a rule corresponding to (RAA), such as Γ, ¬A ⊢ (RAA) Γ ⊢ A Example 0.1.1.
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