3.8: Analyzing Arguments with Truth Tables
Students will be able to:
- Determine the validity of an argument using truth tables
The validity of arguments also be determined using truth tables.
To analyze an argument with a truth table:
- Represent each of the premises symbolically.
- Create a conditional statement, joining all the premises to form the antecedent and using the conclusion as the consequent.
- Create a truth table for the statement. If it is always true, then the argument is valid.
Consider the argument
Premise: If you bought bread, then you went to the store.
Premise: You bought bread.
Conclusion: You went to the store.
Solution
While this example is fairly obviously a valid argument, we can analyze it using a truth table by representing each of the premises symbolically. We can then form a conditional statement showing that the premises together imply the conclusion. If the truth table is a tautology (always true), then the argument is valid.
We’ll let b represent “you bought bread” and s represent “you went to the store”. Then the argument becomes:
Premise: b → s
Premise: b
Conclusion: s
To test the validity, we look at whether the combination of both premises implies the conclusion; is it true that [(b → s) ⋀ b] → s ?
|
b |
s |
b → s |
|---|---|---|
|
T |
T |
T |
|
T |
F |
F |
|
F |
T |
T |
|
F |
F |
T |
|
b |
s |
b → s |
(b → s) ⋀ b |
|---|---|---|---|
|
T |
T |
T |
T |
|
T |
F |
F |
F |
|
F |
T |
T |
F |
|
F |
F |
T |
F |
|
b |
s |
b → s |
(b → s) ⋀ b |
[(b → s) ⋀ b] → s |
|---|---|---|---|---|
|
T |
T |
T |
T |
T |
|
T |
F |
F |
F |
T |
|
F |
T |
T |
F |
T |
|
F |
F |
T |
F |
T |
Since the truth table for [(b → s) ⋀ b] → s is always true, this is a valid argument.
Premise: If I go to the mall, then I’ll buy new jeans.
Premise: If I buy new jeans, I’ll buy a shirt to go with it.
Conclusion: If I go to the mall, I’ll buy a shirt.
Solution
Let m = I go to the mall, j = I buy jeans, and s = I buy a shirt.
The premises and conclusion can be stated as:
Premise: m → j
Premise: j → s
Conclusion: m → s
We can construct a truth table for [(m → j) ⋀ (j → s)] → (m → s). Try to recreate each step and see how the truth table was constructed.
|
m |
j |
s |
m→j |
j→s |
(m→j) ⋀ (j→s) |
m→s |
[(m→j) ⋀ (j→s)] → (m→s) |
|---|---|---|---|---|---|---|---|
|
T |
T |
T |
T |
T |
T |
T |
T |
|
T |
T |
F |
T |
F |
F |
F |
T |
|
T |
F |
T |
F |
T |
F |
T |
T |
|
T |
F |
F |
F |
T |
F |
F |
T |
|
F |
T |
T |
T |
T |
T |
T |
T |
|
F |
T |
F |
T |
F |
F |
T |
T |
|
F |
F |
T |
T |
T |
T |
T |
T |
|
F |
F |
F |
T |
T |
T |
T |
T |
From the final column of the truth table, we can see this is a valid argument.
Contributors and Attributions
-
Saburo Matsumoto
CC-BY-4.0