1. Deductive Reasoning: Propositional Logic
Chapter 6
Deductive Reasoning: Propositional Logic

2. 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.)

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

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

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

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

7. 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

8. 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