TruthTables

1
Truth-Tables
Three ways of using truth-tables to answer if KB
Q a) by the definition of entailment KB
Q iff for every interpretation I, if I
satisfies KB then I satisfies Q. b) by
transforming into a unsatisfiability problem KB
Q iff KB U ? Q is unsatisfiable c) by
transforming into a validity problem w1, , wn
Q iff ((w1 ? ? wn) ? Q)
2
Deductive System

• Language
• Inference Rules (R)
• Logical axioms (AX)

3
Inference Rules

• Inference rules allow us to deduce new wffs from
• known ones
• Notation
• ltgiven wffs that match these patternsgt
• --------------------------------------------------
  -
• ltwe can deduce thisgt

4
Modus Ponens

• If we believe a rule, and we believe that its
  antecedent is true, we can believe that its
  conclusion is true.
• Let A, B be wffs.
• A , A ? B
• --------------------------------------------------
  -
• B

5
Unit resolution

• If at least one of two wffs is true (A or B) we
  know one is false, then the other must be true
• Let A, B be wffs.
• ?A , A ? B
• --------------------------------------------------
  -
• B
• Really, just a variant of modus ponens

6
Resolution

• Case analysis on the possible values of B
• Because B cannot be both true and false, one of
  the other disjuncts must be true in one of the
  premisses
• ?B ? C , A ? B
• --------------------------------------------------
  -
• C ? A
• Alternatively (implication is transitive)
• Given ?A ? B, and also B ? C
• ?A ? C

7
Logical Axioms

• Valid wffs
• Examples
• A ? A
• (A ? B) ? B
• (A ? B) ? A
• A ? (B ? A)

8
Proof Systems
KB - Q iff there is a sequence of wffs D1,
..., Dn such that Dn is Q and for each Di in
the sequence a) either Di is in KB or b) Di can
be inferred from a wff (or wffs) earlier in the
sequence by using one of the rules of inference
in R, or c) Di is an instance of a logical axiom
in AX The sequence (if exists) D1, ..., Dn is
called a proof or a deduction of Q from KB. Q is
said to be a theorem of KB. KB - Q a) by the
definition of entailment