2026-10-20

Chapter 2: Propositional Logic

• Tautology
• Contradiction
• Entailment-unsatisfiability Duality
• Clause finite set of literals
• Clause Set non-empty set of clauses
• Entailment (Propositional Logic)
• Resolvent
  • • defined for ,
  • Corollary:
  • Example:
  • , interpretation, defined, then:
• Resolution
  • Definition: Clause set closed under resolution if
  • Theorem:
    • Call closure (Abschluss)
  • Construction (1):
    3. Das sind alle.
  • Construction (2):
  • Corollary:
    • Proof by Induction
  • Corollary: is closed
  • Corollary:
  • Theorems:
    4. unsatisfiable unsatisfiable
    5. unsatisfiable
  • Theorem:
    • Partial interpretation :
      • for a clause
    • Empty interpretation: (never assigns anything)
    • -Interpretation
      • forbids :
      • Corollary:
    • If then every is allowed by
      1. Case: If every :
         • allow
         • Let
      2. Case: not (1) and for all
         • allow
      3. Base:
         • not (1):
         • not (2):
      • Recursion step: If is not forbidden by any , then
        • we find
        • such that all allow
          1. Case: If all allow
          2. Case: not (1), all allow =:
          3. not (1) and not (2)

Appendix

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