Syntax & Semantics
symbols, signatures structuresterms, formulae, sentences their interpretations over structures
formal proof — consequence
syntactic derivation semantic implication derivability — validity
consistency — satisfiability
⊢
|=towards G¨odel’s Completeness Theorem
Intr.Math.Log. Winter 12/13 M Otto 17/25
Kurt G¨ odel, 1906–1978
Intr.Math.Log. Winter 12/13 M Otto 18/25
formal proof system: a sequent calculus
sequent: finite string of formulae Γ ϕ
antecendent Γ ⊆ FO finite sequence (unordered, possibly empty) succedent ϕ ∈ FO
semantics of sequent (validity): Γ ϕ valid if Γ |= ϕ
sequent calculus: rule-based calculus for the syntactic generation of the derivable sequents
soundness (correctness): only valid sequents are derivable completeness (weak form): all valid sequents are derivable
Intr.Math.Log. Winter 12/13 M Otto 19/25
sequent calculus
sequent calculus rules:
premise sequents conclusion sequent
idea: sequents as proof snapshots;
sequent rules as legitimate proof steps
examples:
Γ (ϕ
∧ψ) Γ ϕ
Γϕ
1ϕ Γϕ
2ϕ Γ(ϕ
1 ∨ϕ
2) ϕ
here: a sequent calculus S for FO(σ) with =,¬,∨,∃ (without ∧,→,↔,∀)
Intr.Math.Log. Winter 12/13 M Otto 20/25
sequent calculus rules
types of rules:
• rules for assumption/antecedent (weakening)
• propositional rules for ¬,∨
• quantifier rules for ∃
• equality rules for =
assumption/antecedent rules
(Ass)
Γ ϕ for ϕ
∈Γ (Ant) Γ ϕ
Γ
′ϕ for Γ
⊆Γ
′Intr.Math.Log. Winter 12/13 M Otto 21/25
sequent calculus
propositional rules:(∨A) Γϕ1 ϕ Γϕ2 ϕ
Γ(ϕ1 ∨ϕ2) ϕ (∨S) Γ ϕi
Γ (ϕ1 ∨ϕ2) for i = 1,2
(CD) Γψ ϕ Γ¬ψ ϕ
Γ ϕ (Ctr) Γ¬ϕ ψ Γ¬ϕ ¬ψ
Γ ϕ
Intr.Math.Log. Winter 12/13 M Otto 22/25
sequent calculus
quantifier rules:(∃A) Γϕ(y /x ) ψ
Γ∃x ϕ ψ (∃S) Γ ϕ(t /x )
Γ
∃xϕ
y 6∈ free(Γ,
∃xϕ, ψ )
side condition in (∃A) crucial for correctness!
Intr.Math.Log. Winter 12/13 M Otto 23/25
sequent calculus
equality rules(=) Γ
t=
t(Subst) Γ ϕ(
t/
x) Γ
t=
t′ϕ(t
′/x )
Intr.Math.Log. Winter 12/13 M Otto 24/25
