### 2.3 Valid and Invalid Arguments

```Discrete Structures
Chapter 2: The Logic of Compound Statements
2.3 Valid and Invalid Arguments
“Contrawise,” continued Tweedledee, “if it was so, it might be; and
if it were so, it would be; but as it isn’t, it ain’t. That’s logic.”
– Lewis Carroll, 1832 – 1898
Through the Looking Glass, 1865
2.3 Valid and Invalid Arguments
1
Definitions
• Argument
– An argument is a sequence of statements.
• Argument Form
– An argument form is a sequence of statement forms.
• Premises
– All statements in an argument and all statement forms
in an argument form are called premises except for the
last one.
2.3 Valid and Invalid Arguments
2
Definitions
• Conclusion
– The final statement or statement form is called the
conclusion.
• Valid
– If an argument form is valid that means no matter
what particular statements are substituted for the
statement variable in its premises, if the resulting
premises are all true, then the conclusion is true.
2.3 Valid and Invalid Arguments
3
Testing an Argument for Validity
1. Identify the premises and conclusion of the
argument form.
2. Construct a truth table showing the truth values
of all the premises and the conclusion.
3. A row of the truth table in which all the premises
are true is called a critical row.
a. If there is a critical row in which the conclusion is
false, the argument form is invalid.
b. If the conclusion in every row is true, then the
argument form is valid.
2.3 Valid and Invalid Arguments
4
Example – pg. 62 # 12b
• Use truth tables to show that the following forms of arguments are invalid.
pq
p
Premises
Conclusion
 q
p
q
1.
T
T
2.
T
F
3.
F
T
4.
F
F
pq
p
2.3 Valid and Invalid Arguments
q
5
Valid Argument Forms
The table below summarizes the rules of inference. You are
Name
Example
Name
Example
Modus Ponens
(mode that
affirms)
pq
p
q
Elimination
a.
Modus Tollens
(mode that
denies)
pq
q
p
Transitivity
pq
qr
pr
Generalization
a.
Proof by
Division into
Cases
pq
pr
qr
r
p
pq
b. q
p
q
Specialization
Conjunction
a.
q
pq
p
p
q
p  q
b.
p
pq
q
p
b.
pq
q
q
q
n Rule
2.3 Valid and Invalid Arguments
p  c
p
6
Examples
• For the next three examples, use truth tables to
show that the argument forms referred to are
valid. Indicate which column represents the
premises and which represent the conclusion,
and include a sentence explaining how the
explanation should show that you understand
what it means for a form of an argument to be
valid.
2.3 Valid and Invalid Arguments
7
Example – pg. 62 # 17
pq
q
1.
2.
3.
4.
p
q
T
T
T
F
F
F
T
F
Premises
Conclusion
pq
q
2.3 Valid and Invalid Arguments
8
Example – pg. 62 # 19
pq
p
q
Premises
1.
2.
3.
4.
p
q
T
T
F
F
T
F
T
F
pq
Conclusion
p
2.3 Valid and Invalid Arguments
q
9
Example – pg. 62 # 21
pq
pr
qr
r
2.3 Valid and Invalid Arguments
10
Example Continued – pg. 62 # 21
Premises
p
q
r
1.
T
T
T
2.
T
T
T
T
F
F
F
T
F
7.
F
F
F
T
T
F
T
F
T
8.
F
F
F
3.
4.
5.
6.
pq
pr
2.3 Valid and Invalid Arguments
Conclusion
qr
r
11
Fallacies – Converse Error
This claim is most simply put as
•
pq
q
p
• It's a fallacy because at no point is it shown
that p is the only possible cause of q; therefore,
even if q is true, p can still be false.
2.3 Valid and Invalid Arguments
12
Fallacies – Converse Error
Example:
– If my car was Ferrari, it would be able to travel at
over a hundred miles per hour.
– I clocked my car at 101 miles per hour.
–  my car is a Ferrari.
Here the fallacy is fairly obvious; given the evidence, the car
might be a Ferrari, but it might also be a Bugatti,
Lamborghini, or any other model of performance car, since
the ability to travel that fast is not unique to Ferraris.
2.3 Valid and Invalid Arguments
13
Fallacies – Inverse Error
This claim is most simply put as
•
pq
p
q
• It's a fallacy because replacement is not
allowed because a conditional statement is not
logically equivalent to its inverse.
2.3 Valid and Invalid Arguments
14
Fallacies – Inverse Error
Example
– If I hit my professor with a cream pie, he will flunk me.
– I will not hit my professor with a cream pie.
–  he will not flunk me.
• Again, it is intuitively obvious that this reasoning does not work. While
many professors may consider being nailed with a cream pie a sufficient
reason to assign a grade of "F" to a student, there are an overwhelming
number of other reasons for which you might flunk (cheating, not studying,
not showing up for tests, etc.).
2.3 Valid and Invalid Arguments
15
Example – pg. 62 # 30
• The argument might be valid or it might exhibit the converse or inverse
error. Use symbols to write the logical form of each argument. If the
argument is valid, identify the rule of inference that guarantees its validity.
Otherwise, state whether the converse or inverse error is made.
– If this computer program is correct, then it produces the correct output
when run with the test data my teacher gave me.
– This computer program produces the correct output when run with the
test data my teacher gave me.
–  This computer program is correct.
2.3 Valid and Invalid Arguments
16
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