Deductive Reasoning: Propositional Logic

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Presentation transcript:

Deductive Reasoning: Propositional Logic Chapter 6 Deductive Reasoning: Propositional Logic

4 Logical Connectives & Conjunction (and)—as in p & q (Alice rode her bike, and John walked.) V Disjunction (or)—as in p v q (Either Alice rode her bike, or John walked.) ~ Negation (not)—as in ~p (Alice did not ride her bike. Or: It is not the case that Alice rode her bike.) â Conditional (if-then)—as in p â q (If Alice rode her bike, then John walked.)

Truth table for a conjunction: p q p & q T T T T F F F T F F F F

Truth table for a disjunction: p q p v q T T T T F T F T T F F F

Truth table for a negation: p ~ p T F F T

Truth table for a conditional: p q p â q T T T T F F F T T F F T

Some Rules of Inference 1. Modus Ponens (MP) 3. Hypothetical Syllogism (HS) p → q p → q p q → r  q  p → r 2. Modus Tollens (MT) 4. Simplification (SIMP) p → q p & q ~ q  p  ~ p

Some Rules of Replacement 10. Tautology (TAUT) 15. De Morgan’s Law (DM) p ≡ (p & p) ~ (p & q) ≡ (~ p v ~ q) p ≡ (p v p) ~ (p v q) ≡ (~ p & ~ q) 11. Double Negation (DN) 16. Contraposition (CONTRA) ~ ~ p ≡ p (p → q) ≡ (~ q → ~ p) p ≡ ~ ~ p