CS486 Spring 1997

Lecture 4 Thursday, Jan. 30, 1997

• Reading: Please read Smullyan Chapter II §2, Chapter III §1, Chapter XI §1 (p.101-108).
• Definitions of consistency and completeness.
• Consistency and completeness for truth-table proofs and BDD's.
• Hilbert systems.
  • Axioms and inference rules.
  • Proofs.
• Consistency for Hilbert systems.
• Consistency theorem for analytic tableaux-Smullyan p.25.
  • The idea: counter-example open (by induction on height of tableau).
  • Not open no counter-example.
    (closed tautology.)
• Completeness theorem for analytic tableaux-Smullyan p.25-28.
  • What could go wrong?
  • The idea: open counter-example (by induction on open path lenght).
    (no counter-example not open.)
    tautology closed.
  • Hintikka sets.
  • Hintikka's lemma.
  • Theorem 1 Any complete open branch of a tableau is satisfiable.
  • Theorem 2 Every tautology is provable by the tableau method.

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