Section 3.5 Write arguments in symbolic form and valid arguments

Section 3.5
Write arguments in symbolic form and valid arguments
Writing an argument in symbolic form
I have a college degree (p)
I am lazy (q)
If I have a college degree, then I am not lazy
I don’t have a college degree
Therefore, I am lazy
Symbolic form:
If I have a college degree, then I am not lazy ( p →~ q)
I don’t have a college degree (~ p)
Therefore, I am lazy q
Hypothesis: (( p →~ q)∧ ~ p)
Conclusion: q
Argument in symbolic form: (( p →~ q )∧ ~ p ) → q
To test to see if the argument is valid, we take the argument in symbolic form and
construct a truth table. If the last column in the truth table results in all true’s, then the
argument is valid
p
T
T
F
F
q
T
F
T
F
~ p
~q
F
F
T
T
F
T
F
T
( p →~ q )
F
T
T
T
(( p →~ q)∧ ~ p )
F
F
T
T
(( p →~ q)∧ ~ p) → q
T
T
T
F
Therefore, this argument is invalid because the last column has a false item.
Example 1
Symbolize the argument, construct a truth table, and determine if the argument is valid.
If I pass the exam, then I will graduate.
I graduated
Therefore, I passed the exam
p = pass the exam
g = I will graduate
If I pass the exam, then I will graduate. ( p → g )
I graduated (g)
Therefore, I passed the exam (p)
Argument: (( p → g ) ∧ g ) → p
P
T
T
F
F
g
T
F
T
F
(p→g)
T
F
T
T
(( p → g ) ∧ g )
T
F
T
F
(( p → g ) ∧ g ) → p
T
T
F
T
This argument is invalid
Example 2
Symbolize the argument, construct a truth table, and determine if the argument is valid.
Jen and Bill will be at the party
Bill was at the party.
Therefore, Jen was at the party
J = Jen will be at the party
B = Bill will be at the party
Jen and Bill will be at the party J ∧ B
Bill was at the party. B
Therefore, Jen was at the party J
Argument in symbolic form: (( J ∧ B) ∧ B) → J
J ∧B
J
B
( J ∧ B) ∧ B (( J ∧ B) ∧ B) → J
T
T
T
T
T
T
F
F
F
T
F
T
F
F
T
F
F
F
F
T
Since the last column is all true, the argument is valid
Example 3
Symbolize the argument, construct a truth table, and determine if the argument is valid.
It will be sunny or cloudy today
It isn’t sunny
Therefore, it will be cloudy
S = It will be sunny
C = It will be cloudy
It will be sunny or cloudy today S ∨ C
It isn’t sunny
~S
Therefore, it will be cloudy C
Hypothesis: ( S ∨ C )∧ ~ S
Conclusion: C
S
T
T
F
F
C
T
F
T
F
~S
F
F
T
T
S ∨C
T
T
T
F
This is a valid argument
( S ∨ C )∧ ~ S
F
F
T
F
(( S ∨ C )∧ ~ S ) → C
T
T
T
T
Example 4
Write in symbolic form
p: The senator supports new taxes.
q: The senator is reelected
The senator is not reelected if she supports new taxes
The senator does not support new taxes
Therefore, the senator is reelected
Symbolic form:
The senator is not reelected if she supports new taxes p →~ q
The senator does not support new taxes ~ p
Therefore, the senator is reelected q
Hypothesis: ( p →~ q)∧ ~ p
Conclusion: q
Argument: (( p →~ q)∧ ~ p) → q
Determine if the argument in problem 6 above is valid
p
T
T
F
F
q
T
F
T
F
~q
F
T
F
T
~p
F
F
T
T
p →~ q
F
T
T
T
( p →~ q ) ∧ ~ p
F
F
T
T
(( p →~ q )∧ ~ p ) → q
T
T
T
F
Since the last row results in false, the argument is invalid
Example 5
If you complete your computer training, you can be eligible to work in the computer lab.
You did not finish your computer training. Therefore, you can not work in the computer
lab.
p : You finished your computer training
q : You can work in the computer lab
Argument in symbolic:
(( p → q )∧ ~ p ) →~ q
p
q
~ p
~q
p→q
(( p → q )∧ ~ p )
(( p → q )∧ ~ p ) →~ q
T
T
F
F
T
F
T
F
F
F
T
T
F
T
F
T
T
F
T
T
F
F
T
T
T
T
F
T
Invalid Arguments