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Discussion #9 Tautologies and Contradictions

Discussion #9 Tautologies and Contradictions. Topics. Definitions Sound reasoning with tautologies and contradictions. Definitions. Tautology – a logical expression that is true for all variable assignments. Contradiction – a logical expression that is false for all variable assignments.

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Discussion #9 Tautologies and Contradictions

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  1. Discussion #9Tautologies and Contradictions

  2. Topics • Definitions • Sound reasoning with tautologies and contradictions

  3. 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. P P P  P (P  P) T F T F F T T F P Q P  Q (P  Q) (P  Q)  Q T T T F T T F F T T F T F T T F F F T T Tautologies Since P  P is true for all variable assignments, it is a tautology.

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

  6. Sound Reasoning • A logical argument has the form: A1 A2  …  An  B and is sound if when Ai = T for all i, B = T. (i.e. If the premises are all true, then the conclusion is also true.) • This happens when A1 A2  …  An  B is a tautology.

  7. Intuitive Basis for Sound Reasoning If (A1 A2  …  An  B) is a tautology, and Ai = T for all i then B must necessarily be true! B = T is the only possibility for the conclusion!

  8. A B (A  B) (A  B) A (A  B) A  B T T T T T T F F F T F T T F T F F T F T A  B A B Modus Ponens Hence, modus ponens is sound.

  9. A B A  B A (A  B) A (A  B) A  B T T T F F T T F T F F T F T T T T T F F F T F T A  B A B Disjunctive Syllogism Hence, disjunctive syllogism is sound.

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