1. Discussion #9 1/9 Discussion #9 Tautologies and Contradictions

2. Discussion #9 2/9 Topics Definitions Sound reasoning with tautologies and contradictions

3. Discussion #9 3/9 Definitions Tautology – a logical expression that is true for all variable assignments. Contradiction – a logical expression that is false for all variable assignments. Contingent – a logical expression that is neither a tautology nor a contradiction.

4. Discussion #9 4/9 Tautologies F F T T T F F T  (P   P)P   P PP P T T T T T T T F F F F T FF TF FT TT  (P  Q)  Q  (P  Q)P  Q QP Since P   P is true for all variable assignments, it is a tautology.

5. Discussion #9 5/9 Tautological Derivation by Substitution Since A   A is a tautology, so are(P  Q)   (P  Q) and(P  Q  R)   (P  Q  R) T T T T F F F T T T F T FF TF FT TT A  (A  B)  BA  (A  B)A  B BA Since A  (A  B)  B is a tautology, so is (P   Q)  ((P   Q)  R)  R Using schemas that are tautologies, we can get other tautologies by substituting expressions for schema variables.

6. Discussion #9 6/9 Sound Reasoning A logical argument has the form: A 1  A 2  …  A n  B and is sound if when A i = T for all i, B = T. (i.e. If the premises are all true, then the conclusion is also true.) This happens when A 1  A 2  …  A n  B is a tautology.

7. Discussion #9 7/9 Intuitive Basis for Sound Reasoning If(A 1  A 2  …  A n  B) is a tautology, and A i = T for all i then B must necessarily be true! AB A  B T?T B = T is the only possibility for the conclusion!

8. Discussion #9 8/9 Modus Ponens T T T T F F F T T T F T FF TF FT TT (A  B)  A  B(A  B)  A(A  B) BA AB A  B T?T Hence, modus ponens is sound. A  B A B

9. Discussion #9 9/9 Disjunctive Syllogism T T F F T T T T F T F F F T T T AA FF TF FT TT (A  B)   A  B(A  B)   AA  B BA Hence, disjunctive syllogism is sound. A  B  A B