Lecture 12 Logic II

#notes#cs471

Logical Equivalence §

• Two sentances logically equivalent iff they are true in the same set of models:

  • $a\equiv b$ iff $\alpha \models \beta$ and $\beta \models \alpha$

• Validity: True in all models

• Satisfiable: True in some models

• Unsatisfiable: True in no models

Proof By Inference Rules §

• Rule 1 (Modus Ponens):
  • $\alpha \Rightarrow \beta ,\alpha$
  • Then, $\beta$
• Rule 2 (And Elimination):
  • $\alpha \wedge \beta$
  • Then, $\alpha$
• Rule 3: All logical equivalences can be used as inference rules
• Monotonicity: Set of entailed sentences can only increase as information is added to the Knowledge Base (KB).

Resolution §

Every sentence is logically equivalent to a conjunction of clauses Literals: Can be any symbol or it’s negation Clauses: $Literal\vee Literal\vee Literal\vee Literal$