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. The above rule is derivable in LJ if we assume that ⊢ A ∨ ¬A. Indeed

Γ, ¬A ⊢ A A (w,r) ⊢ (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.

0.1.2 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 0.1.4 (Sequents for LK). A1,...,An ⊢ B1,...,Bm. Deﬁnition 0.1.5 (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 ⊢ ∆ Γ ⊢ B,A, ∆ (exchange,l) (exchange,r) Γ,A,B, Σ ⊢ ∆ Γ ⊢ A,B, ∆

Γ ⊢ ∆ (weakening,l) Γ ⊢ ∆ (weakening,r) Γ, A ⊢ ∆ Γ ⊢ ∆, A

Γ, A, A ⊢ ∆ Γ ⊢ A, A, ∆ (contraction,l) (contraction,r) Γ, A ⊢ ∆ 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.

2 Deﬁnition 0.1.6. 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 0.1.7. 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.

A, B ⊢ A (→,r) Example 0.1.2. A B A ⊢ → (→,r) ⊢ A → (B → A) A A ⊢ (¬,r) ⊢ A, ¬A Example 0.1.3. (¬,l) A A ¬¬ ⊢ (→,r) ⊢ ¬¬A → A Example 0.1.4.

A ⊢ A B → C, A ⊢ A, C A → C,B,C ⊢ C A → C, B ⊢ B (→,l) A → C, B → C, A ⊢ A, C A → C, B → C, B ⊢ C (∨,l) A → C, B → C, A ∨ B ⊢ C (→,r) A → C, B → C ⊢ (A ∨ B) → C (→,r) A → C ⊢ (B → C) → ((A ∨ B) → C) (→,r) ⊢ (A → C) → ((B → C) → ((A ∨ B) → C)) A A ⊢ (¬,r) ⊢ A, ¬A (∨,r) Example 0.1.5. ⊢ A, A ∨ ¬A (∨,r) ⊢ A ∨ ¬A, A ∨ ¬A (c,r) ⊢ A ∨ ¬A A, B ⊢ A, B 2 × (→,r) ⊢ A → B, B → A (∨,r) Example 0.1.6. ⊢ A → B, (A → B) ∨ (B → A) (∨,r) ⊢ (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 0.1.8. LK is the sequent calculus in which (∨,r)i are replaced by (∨,r) ′ and (∧,l)i are replaced by (∧,l) below:

Γ ⊢ A1, A2, ∆ ′ (∨,r) Γ ⊢ A1 ∨ A2, ∆

3 Γ, A1, A2 ⊢ ∆ ′ (∧,l) Γ, A1 ∧ A2 ⊢ ∆

Theorem 0.1.1. LK and LK′ are equivalent

Proof.

Γ ⊢ A,B, ∆ (∨,r)1 Γ ⊢ A ∨ B,B, ∆ ′ (∨,r)2 (∨,r)1,2 and (c,r) simulate (∨,r) Γ ⊢ A ∨ B, A ∨ B, ∆ (c,r) Γ ⊢ A ∨ B, ∆

Γ ⊢ A, ∆ (w,r) ′ Γ ⊢ A,B, ∆ (w,r) and (∨,r) simulate (∨,r)1,2 (∨,r)′ Γ ⊢ A ∨ B, ∆

LK′ +(w) =⇒ LK

LK +(c) =⇒ LK′

0.1.3 Soundness and Completeness of LK′ Let ⊢ be ”derivable in LK′” Γ |= A ⇐⇒ Γ ⊢ A

Theorem 0.1.2 (Soundness). If Γ ⊢ ∆ then Γ |= ∆.

Proof.

• Axiom A ⊢ A =⇒ A |= A Γ, A ⊢ ∆ Γ, B ⊢ ∆ • (∨,l) rule: (∨,l) Γ, A ∨ B ⊢ ∆ every interpretation is also by induction: Γ, A |= ∆ Γ, B |= ∆ i) an interpretation for A ∨ B 5u i) i) 5u 5u i) i) 5u 5u i) i) 5u 5u i) i) 5u 5u i) i) 5u 5u Γ, A ∨ B |= ∆

4 • similarly for the other rules

S S S Deﬁnition 0.1.9. A rule 1 or 1 2 is invertible, if whenever S is valid so are S S S1 and S2 (applied bottom up). Lemma 0.1.1. The logical rules of LK′ are invertible.

Theorem 0.1.3 (Completeness). If Γ |=∆ then Γ ⊢ ∆.

Proof of Completeness. Assume that Γ |= ∆ and apply the logical rules of LK′ back- wards to the sequent Γ ⊢ ∆.

S1 ... Sn GG w GG ww GG ww GG ww G ww Γ ⊢ ∆

Γ1 |= ∆1,..., Γn |= ∆n, by Lemma 0.1.1, where Si =Γi |= ∆i. A ⊢ A A ∈ Γn and A ∈ ∆n and weakening (weakening). Sn

Hence model theoretic approacho /proof theoretic approach

Corollary 0.1.2. Contraction rules and (CUT) are redundant in LK′.