0.1 Equivalence between Natural Deduction and Axiomatic Systems
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1 0.1 Equivalence between Natural Deduction and Axiomatic Systems Theorem Γ ND P iff Γ AS P ( ) it is enough to prove that all axioms are theorems in ND, as MP corresponds to ( e). ( ) by induction on the depth of Γ ND P. P depth base case: depth=1, i.e. P Γ = Γ AS P. assume that the claim holds for all trees of depth n. We distinguish cases according to the last rule applied. A B ( e 1 ) Γ AS A B and ((A B) A) + MP = Γ AS A A [A] [B] A B C C ( e) C by induction hypothesis: Γ AS A B; Γ,A AS C; Γ,B AS C by deduction theorem: Γ A C; Γ B C by ( e): (A C) ((B C) ((A B) C)) and three applications of MP follows Γ AS C 0.2 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 Sequent Calculi LK and LJ Definition (Sequent). A 1,...,A n B. Definition (Inference Rule). S 1 S or S 1 S 2 S where S 1,S 2,S are sequents. 1
2 Definition (Rules of the LJ sequent calculus). Axioms: Γ A (derivability assertion) if A Γ Cut: Γ A Σ,A B (CUT) Γ,Σ B Structural Rules: Γ,B,A, C (exchange,l) Γ,A,B, C Γ B (weakening,l) Γ,A B Γ Γ B (weakening,r) empty RHS is a contradiction ( ) Γ,A,A, C (contraction,l) Γ,A, C Logical Rules: Γ,A C Γ,B C (, l) Γ,A B C Γ A 1 1 Γ A 1 A 2 Γ A 2 2 Γ A 1 A 2 Γ,A 1 C (, l)1 Γ,A 1 A 2 C Γ,A 2 C (, l)2 Γ,A 1 A 2 C Γ A Γ B (, r) Γ A B Γ,B C Γ A (, l) Γ,A B C Γ,A B Γ A B Γ A (, l) Γ, A Γ,A Γ A (, r) The calculus above is sound and complete for IL. Note that, except for (CUT), the following holds Corollary (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 Γ A (RAA) Example The above rule is derivable in LJ if we assume that A A. Indeed Γ, A (w,r) A A (w,l) Γ, A A Γ,A A (, l) Γ,A A A A A (CUT) Γ A Note that (RAA) does not satisfy the subformula property. 2
3 0.2.2 Sequent Calculus LK To find a calculus for classical logic in which 1. all the rules (but (CUT)) satisfy the subformula property and 2. (CUT) is eliminable from derivations, we have to change the notion of sequent as follows: Definition (Sequents for LK). A 1,...,A n B 1,...,B m. Definition (Inference rules for LK). Are as those of LJ in which a context (and ) is added everywhere on the right hand side of sequents (RHS). E.g. Γ,B Γ A, ( l) Γ,A B Γ,B,A,Σ (exchange,l) Γ,A,B,Σ Γ B,A, (exchange,r) Γ A,B, Γ (weakening,l) Γ,A Γ Γ,A (weakening,r) Γ,A,A (contraction,l) Γ,A Γ A,A, (contraction,r) Γ A, Γ A, Γ,A (CUT) Γ,Γ, Note that the exchange rules are not needed, if in sequents Γ, Γ and are finite multisets of formulas. Definition Γ, are called side formulas or context. In the conclusion of each rule, the formula not occurring in the context is called principal formula. The formulas in the premises from which the principal formula derives are called active formulas. Definition Proofs (in LJ or LK) are labeled finite trees with a single root, with axioms at the top nodes and each node-label is connected with the labels of the (immediate) successor nodes (if any) according to one of the rules. Example Example Example A,B A A B A A (B A) A A (, r) A, A (, l) A A A A 3
4 Example Example A A B C,A A,C A C,B C,A A,C A C,B,C C A C,B B (, l) A C,B C,B C (, l) A C,B C,A B C A C,B C (A B) C A C (B C) ((A B) C) (A C) ((B C) ((A B) C)) A A (, r) A, A A,A A A A,A A (c,r) A A A,B A,B A B,B A A B,(A B) (B A) 2 (A B) (B A),(A B) (B A) (c,r) (A B) (B A) Note that the rules (,r) i and (,l) i are problematic for proofs search. We therefore consider the following calculus Definition LK is the sequent calculus in which (,r) i are replaced by (,r) and (,l) i are replaced by (,l) below: Γ A 1,A 2, Γ A 1 A 2, Γ,A 1,A 2 (, l) Γ,A 1 A 2 Theorem LK and LK are equivalent Γ A,B, 1 Γ A B,B, 2 Γ A B,A B, (c,r) Γ A B, (,r) 1,2 and (c,r) simulate (,r) 4
5 Γ A, (w,r) Γ A,B, Γ A B, (w,r) and (,r) simulate (,r) 1,2 LK + (w) = LK LK + (c) = LK Soundness and Completeness of LK Let be derivable in LK Γ = A Γ A Theorem (Soundness). If Γ then Γ =. Axiom A A = A = A (,l) rule: by induction: Γ, A = Γ,A Γ,B (, l) Γ,A B every interpretation is also an interpretation for A B Γ,A B = Γ,B = similarly for the other rules Definition A rule S or S 1 S 2 S S 1 and S 2 (applied bottom up). S 1 is invertible, if whenever S is valid so are Lemma The logical rules of LK are invertible. Theorem (Completeness). If Γ = then Γ. Proof of Completeness. Assume that Γ = and apply the logical rules of LK backwards to the sequent Γ. 5
6 S 1... S n Γ Γ 1 = 1,...,Γ n = n, by Lemma 0.2.1, where S i = Γ i = i. A Γ n and A n and weakening A A (weakening) S n. Hence model theoretic approach proof theoretic approach Corollary Contraction rules and (CUT) are redundant in LK Constructing CNF via LK Theorem For each proposition P a proposition P in CNF can be found such that P P. see p. 73 (Chapter 3) in Gallier s book. 0.3 Equivalence between Axiomatic Systems and LK Axiomatic Systems Natural Deduction Sequent Calculus sound complete Classical Logic SC Ax. S. ND Theorem Γ H A = Γ LK ( ) A axioms: proof in LK all the axioms MP in LK: A A B B A A,B B,A B (, l) A B,A B (CUT) A B (CUT) Theorem Γ H A = Γ LK ( ) A proceeds as the proof Axiomatic Systems by rule). Natural Deduction proof (rule 6
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