To analyze an argument with a truth table:
- Represent each of the premises symbolically
- Create a conditional statement, joining all the premises with “and” to form the antecedent, and using the conclusion as the consequent.
- Create a truth table for that statement. If it is always true, then the argument is valid.
Example
Consider the argument:
Premise: |
If you bought bread, then you went to the store |
Premise: |
You bought bread |
Conclusion: |
You went to the store |
Use a truth table to determine if it is a valid argument.
Answer:
While this example is hopefully fairly obviously a valid argument, we can analyze it using a truth table by representing each of the premises symbolically. We can then look at the implication 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. In other words, is it true that:
[(B→S)∧B]→S?
B |
S |
B→S |
(B→S)∧B |
S |
[(B→S)∧B]→S |
T |
T |
T |
T |
T |
T |
T |
F |
F |
F |
F |
T |
F |
T |
T |
F |
T |
T |
F |
F |
T |
F |
F |
T |
Since the truth table for
[(B→S)∧B]→S is always true, this is a valid argument. [Note that we inserted an extra copy of the column for
S to make it easier to evaluate the final column.]
Example
Consider the following 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. |
Use a truth table to determine if it is a valid argument.
Answer:
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)
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 truth table, we can see this is a valid argument.
The previous problem is an example of a syllogism.
Syllogism
A
syllogism is an implication derived from two others, where the consequence of one is the antecedent to the other. The general form of a syllogism is:
Premise: |
p→q |
Premise: |
q→r |
Conclusion: |
p→r |
This is sometimes called the
transitive property for implication.
Example
Premise: |
If I work hard, I’ll get a raise. |
Premise: |
If I get a raise, I’ll buy a boat. |
Conclusion: |
If I don’t buy a boat, I must not have worked hard. |
Answer:
If we let W = working hard, R = getting a raise, and B = buying a boat, then we can represent our argument symbolically:
Premise: |
W→R |
Premise: |
R→B |
Conclusion: |
∼B→∼W |
We could construct a truth table for this argument, but instead, we recognize that the conclusion,
∼B→∼W, is just the contrapositive of the implication
W→B. Since an implication and its contrapositive are logically equivalent, we can see that this conclusion is indeed a logical syllogism derived from the premises.
Example
Solve the following famous Lewis Carroll puzzle. In other words, find a logical conclusion from these premises.
All babies are illogical.
Nobody who can manage a crocodile is despised.
Illogical persons are despised.
Answer:
Let B = is a baby, D = is despised, I = is illogical, and M = can manage a crocodile.
Then we can write the premises as:
B→IM→∼DI→D
- From the first and third premises, we can conclude (by Syllogism) that B→D; that babies are despised.
- Using the contrapositive of the second premised, D→∼M, we can conclude (by Syllogism) that B→∼M; that is, babies cannot manage crocodiles.
- While silly, this is a logical conclusion from the given premises.