Apply the Law of Syllogism in logical reasoning

1. Syllogisms
Objectives
The student is able to (I can):
• Apply the Law of Syllogism in logical reasoning

2. Law of Syllogism
• If p → q and q → r are true statements, then p → r is a
true statement.
Examples: Determine if each conjecture is valid by the Law
of Syllogism.
Given: If a number is divisible by 4, then it is divisible by 2. If
a number is even, then it is divisible by 2.
Conjecture: If a number is divisible by 4, then it is even.
p: A number is divisible by 4
q: A number is divisible by 2
r: A number is even
p → q and r → q; therefore, p → r
not validnot validnot validnot valid

3. Determine if each conjecture is valid by the Law of Syllogism.
Given: If an animal is a mammal, then it has hair. If an
animal is a dog, then it is a mammal.
Conjecture: If an animal is a dog, then it has hair.
p: An animal is a mammal
q: It has hair
r: An animal is a dog
p → q and r → p, therefore r → q
or r → p and, p → q therefore r → q validvalidvalidvalid

4. We can also use syllogisms to set up chains of conditionals.
Example: What can we conclude from the following chain?
If you study hard, then you will earn a good grade. (p → q)
If you earn a good grade, then your family will be happy.
(q → r)
Conclusion: If you study hard, then your family will be happy.
(p → r)

5. Write a concluding statement:
a → b
d → ~c
~c → a
b → f

6. Write a concluding statement:
Put the statements in order:
a → b d → ~c
d → ~c ~c → a
~c → a a → b
b → f b → f
Conc.: d → f

7. In a lot of ways, proofs are just expanded syllogisms. We are
still setting up chains of statements; the main difference is
that we also have to provide justifications.
Consider the following:
If 3x + 2 = x + 14, then 2x + 2 = 14. (subtraction prop.)
If 2x + 2 = 14, then 2x = 12. (subtraction prop.)
If 2x = 12, then x = 6. (division prop.)
Now look at this as a proof:
StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons
1. 3x + 2 = x + 14 1. Given statement
2. 2x + 2 = 14 2. Subtr. prop.
3. 2x = 12 3. Subtr. prop.
4. x = 6 4. Division prop.