Transcript Section 1.11 Making Inferences with Propositions (Rules of Inference)

Section 1.11
Making Inferences with Propositions
(Rules of Inference)
Activity #1 (Review)
• For each exercise, decide what conclusion, if any, can
be reached from the given hypotheses.
• 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.
Rules of Inference
Rule of inference
p
p→q
∴q
¬q
p→q
∴ ¬p
P
∴p∨q
p∧q
∴p
p
q
∴p∧q
Name
Modus ponens
Modus tollens
Addition
Simplification
Conjunction
Rules of Inference
Rule of inference
p→q
q→r
∴p→r
p∨q
¬p
∴q
p∨q
¬p ∨ r
∴q∨r
Name
Hypothetical syllogism
Disjunctive syllogism
Resolution
Activity #2 (Review)
• For each of these that were valid arguments, justify
your conclusion by naming the rule(s) used.
• 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.
Using a truth table to show
induction
• Show that Hypothetical Syllogism is valid:
p
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
p→q
q→r
p→r
The Deduction Method
• Suppose you are trying to prove:
P1  P2  P3 → ( Q → R)
• You can instead prove:
P1  P2  P3  Q → R
Activity #4
• Justify each step in the following proof sequence.
• Prove A  ( B → C) → ( B → ( A  C) )





A
B→C
B
C
AC
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
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