This document discusses rules of inference in logic proofs. It defines premises and conclusions, and introduces several common rules of inference like modus ponens, modus tollens, law of syllogism, and rule of disjunction. Examples are provided to demonstrate how each rule is used to establish the validity of arguments. The learning objectives are to prove the validity of arguments using rules of inference. Additional rules of inference are also outlined.

1. z
FORMAL LOGIC
Discrete Structures I
FOR-IAN V. SANDOVAL

2. z
Lesson 7
RULE OF INFERENCE

3. z
LEARNING OBJECTIVES
❑ Prove the validity of the arguments using the rule on
inference

4. z
RULE OF INFERENCE
❑ Proof – is an argument from hypotheses (assumptions) to a
conclusion.
❑ Logic proofs usually begins with premises statement that
are allowed to be assume, while conclusion is the
statement that need to be proven.
❑ Argument – is a sequence of propositions.

5. z
Premises and Conclusion
❑ Premises
“If you have the current password, then you can log onto
the network. You have a current password.”
❑ Conclusion
“You can log onto the network.”

6. z
Rule of Detachment or Modus Ponens
Rules of Inference Related Logical Implications
p
p → q
∴ q
[p ^ (p → q)] → q
(Tautology)
❑ Modus pones means the method of affirming
❑ Example 1
a. Sanczha wins P1,000,000.00 lotto. p
b. If Sanczha wins P1,000,000.00 lotto, then Jomar will
put up a business.
p→q
c. Therefore, Jomar will put up a business. ∴ q

7. z
Law of Syllogism
Rules of Inference Related Logical Implications
p → q
q → r
∴ p → r
[(p→q) ^ (q → r)] p → r
(Tautology)
❑ Example 2
a. If 18 is divisible by 6, then 6 is divisible by 3. p→q
b. If 6 is divisible by 3, then 18 is divisible by 3. q→r
c. Therefore, if 18 is divisible by 6, then 18 is divisible by 3. ∴ p→r

8. z
Modus Tollens
Rules of Inference Related Logical Implications
p → q
~q
∴ p
[(p → q) ^ ~q] → ~p
(Tautology)
❑ Example 3
a. If Manny Pacquiao elected Philippine President, then
Francis Leo Marcos will pledge as cabinet member.
p→q
b. Francis Leo Marcos did not pledge as cabinet member. ~q
c. Therefore, Manny Pacquiao is not elected Philippine
President.
∴ ~p

9. z
Rule of Conjunction
Rules of Inference Related Logical Implications
p
q
∴ p ^ q
❑ Example 4
a. It is sunny. p
b. It is cloudy. q
c. Therefore, it is sunny and cloudy. ∴ p ^ q

10. z
Rule of Disjunctive Syllogism
Rules of Inference Related Logical Implications
p v q
~p
∴ q
[(p v q) ^ ~p] → q
❑ Example 5
a. Anitia’s color pencils are in his bag or it is on her table. p v q
b. Anitia’s color pencils are not in her bag. ~p
c. Therefore, Anitia’s color pencil is in her table. ∴ q

11. z
❑ Example 5: Establish the validity of the arguments.
[(p→~q) ^ (~q→~r) ^ p] → ~r
Solution:
[(p→~q) ^ (~q→~r) ^ p] → ~r
Steps Reasons
1. p→~q Premise
2. ~q→~r Premise
3. p→~r Law of Syllogism (Step 1 and 2)
4. p Premise
5. ∴ r Rule of Detachment (Step 4 and 3)

12. z
❑ Example 6: Establish the validity of the arguments.
[(p→q) ^ (r→s) ^ (w v ~s) ^ (~w v u) ^ ~u] → ~p
Solution:
[(p→q) ^ (r→s) ^ (w v ~s) ^ (~w v u) ^ ~u] → ~p
Steps Reasons
1. p→q Premise
2. r→s Premise
3. p→s Law of Syllogism
(Steps 1 and 2)
4. w v ~s Premise
5. ~s v w Commutative Law
(Step 4)

13. z
❑ Example 6: Establish the validity of the arguments.
[(p→q) ^ (r→s) ^ (w v ~s) ^ (~w v u) ^ ~u] → ~p
Solution:
[(p→q) ^ (r→s) ^ (w v ~s) ^ (~w v u) ^ ~u] → ~p
Steps Reasons
6. s → w Implication (Step 5)
7. p → w Law of Syllogism
(Steps 3 and 6)
8. ~w v u Premise
9. w → u Implication (Step 8)
10. p → u Law of Syllogism
(Steps 7 and 9)

14. z
❑ Example 6: Establish the validity of the arguments.
[(p→q) ^ (r→s) ^ (w v ~s) ^ (~w v u) ^ ~u] → ~p
Solution:
[(p→q) ^ (r→s) ^ (w v ~s) ^ (~w v u) ^ ~u] → ~p
Steps Reasons
11. ~u Premise
12. ∴ ~p Modus Tollens (Steps
10 and 11)

15. z
Other Rules of Inference
Rules of Inference Related Logical Implications
~p → F
∴ p
(~p → F) → p
❑ Rule of Contradiction
Rules of Inference Related Logical Implications
p ^ q
∴ p
(p ^ q) → p
❑ Rule of Conjunctive Simplification

16. z
Other Rules of Inference
Rules of Inference Related Logical Implications
p
∴ p v q
p → (p v q)
❑ Rule of Disjunctive Amplification
Rules of Inference Related Logical Implications
p ^ q
p → (q → r)
∴ r
{(p ^ q) ^ [p → (q→r)]} → r
❑ Rule of Conditional Proof

17. z
Other Rules of Inference
Rules of Inference Related Logical Implications
p → r
q → r
∴ (p v q) → r
[(p→r) ^ (q→r)] → [(p v q) → r]
❑ Rule for Proofs by Cases
Rules of Inference Related Logical Implications
p→q
r→s
p v r
∴ q v s
[(p→q) ^ (r→s) ^ (p v r)] → (q v s)
❑ Rule of Constructive Dilemma

18. z
Other Rules of Inference
Rules of Inference Related Logical Implications
p→q
r→s
~q v ~s
∴~p v ~r
[(p→q) ^ (r→s) ^ (~q v ~s)] → (~p v ~r)
❑ Rule of Destructive Dilemma

19. z
Group Enrichment Activity
A. Consider each of the following arguments. Identify the rule
of inference that establishes its validity.
1. Zandra loves Discrete Structure.
If Zandra loves Discrete Structure, then she also loves
Computer Science courses.
Therefore, she loves English.
2. Renalyn plays guitar or she plays organ.
Renalyn does not play guitar.
Therefore, she plays piano
3. If Redentor can drive a car then he can drive a
motorcycle.
Redentor cannot drive a motorcycle.
Therefore, Redentor cannot drive a car.

20. z
Group Enrichment Activity
B. Establish the validity of the arguments.
1. [p→(q v r)] ^ ~q ^ ~r → ~p
2. [(p→r) ^ (~p→q) ^ (q→s)] → (~r→s)

21. z
• Levin, O. (2019). Discrete Mathematics: An Open Introduction 3rd Edition. Colorado: School of Mathematics Science
University of Colorado.
• Aslam, A. (2016). Proposition in Discrete Mathematics retrieved from https://www.slideshare.net/AdilAslam4/chapter-1-
propositions-in-discrete-mathematics
• Operator Precedence retrieved from http://intrologic.stanford.edu/glossary/operator_precedence.html
REFERENCES