1. Making Inferences with Propositions (Rules of Inference)
Section 1.10 and 1.11
Making Inferences with Propositions
(Rules of Inference)

2. What can you conclude if you know each of the following two “facts”
Suppose that you know this and …
this
What can you conclude? 1
p  q p 2
p ∨ q 3
p  q 4
¬ p 5 6 7 q 8
¬ q

3. Rules of Inference Rule of inference Name p p → q ∴ q Modus ponens
¬q  p → q
∴ ¬p
Modus tollens P
∴ p ∨ q
Addition
p ∧ q
∴ p
Simplification
p  q
∴ p ∧ q
Conjunction

4. Rules of Inference Rule of inference Name p → q q → r ∴ p → r
Hypothetical syllogism
p ∨ q  ¬ p
∴ q
Disjunctive syllogism
p ∨ q  ¬p ∨ r
∴ q ∨ r
Resolution

5. Activity #2
For each exercise, decide what conclusion, if any, can be reached from the given hypotheses. Defend WHY.
If the car was involved in the hit-and-run, then the paint would be chipped. But the paint is not chipped.
Either the weather will turn bad or we will leave on time. If the weather turns bad, then the flight will be cancelled.
If the bill was sent today, then you will be paid tomorrow. You will be paid tomorrow.
The grass needs mowing and the trees need trimming. If the grass needs mowing then we need to rake the leaves.

6. Using a truth table to show induction
Show that Hypothetical Syllogism is valid: p q r
p → q
q → r
p → r T F

7. The Deduction Method Suppose you are trying to prove:
P1  P2  P3 → ( Q → R)
You can instead prove:
P1  P2  P3  Q → R

8. Activity #4 Justify each step in the following proof sequence.
Prove A  ( B → C) → ( B → ( A  C) ) A
B → C B C
A  C

9. Activity #4 Justify each step in the following proof sequence.
Prove B  [( B C) → ¬ A ]  ( B → C) → ¬ A B
( B C) → ¬ A
B → C C
B C
¬ A

10. Activity #4 Justify each step in the following proof sequence.
Prove ¬ A  B  [B → (A  C)] → C
¬ A B
B → (A  C)
A  C
¬ (¬ A)  C
(¬ A) → C C