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: |
[latex]B{\rightarrow}S[/latex] |
| Premise: |
[latex]B[/latex] |
| Conclusion: |
[latex]S[/latex] |
To test the validity, we look at whether the combination of both premises implies the conclusion. In other words, is it true that:
[latex]\left[\left(B{\rightarrow}S\right){\wedge}B\right]{\rightarrow}S[/latex]?
| [latex]B[/latex] |
[latex]S[/latex] |
[latex]B{\rightarrow}S[/latex] |
[latex]\left(B{\rightarrow}S\right){\wedge}B[/latex] |
[latex]S[/latex] |
[latex]\left[\left(B{\rightarrow}S\right){\wedge}B\right]{\rightarrow}S[/latex] |
| 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 [latex]\left[\left(B{\rightarrow}S\right){\wedge}B\right]{\rightarrow}S[/latex] is always true, this is a valid argument. [Note that we inserted an extra copy of the column for [latex]S[/latex] 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: |
[latex]M{\rightarrow}J[/latex] |
| Premise: |
[latex]J{\rightarrow}S[/latex] |
| Conclusion: |
[latex]M{\rightarrow}S[/latex] |
We can construct a truth table for [latex]\left[\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)\right]{\rightarrow}\left(M{\rightarrow}S\right)[/latex]
| [latex]M[/latex] |
[latex]J[/latex] |
[latex]S[/latex] |
[latex]M{\rightarrow}J[/latex] |
[latex]J{\rightarrow}S[/latex] |
[latex]\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)[/latex] |
[latex]M{\rightarrow}S[/latex] |
[latex]\left[\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)\right]{\rightarrow}\left(M{\rightarrow}S\right)[/latex] |
| 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: |
[latex]p{\rightarrow}q[/latex] |
| Premise: |
[latex]q{\rightarrow}r[/latex] |
| Conclusion: |
[latex]p{\rightarrow}r[/latex] |
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: |
[latex]W{\rightarrow}R[/latex] |
| Premise: |
[latex]R{\rightarrow}B[/latex] |
| Conclusion: |
[latex]\sim{B}{\rightarrow}{\sim}W[/latex] |
We could construct a truth table for this argument, but instead, we recognize that the conclusion, [latex]{\sim}B{\rightarrow}{\sim}W[/latex], is just the contrapositive of the implication [latex]W{\rightarrow}B[/latex]. 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:
[latex]B{\rightarrow}I\\M{\rightarrow}{\sim}D\\I{\rightarrow}D[/latex]
- From the first and third premises, we can conclude (by Syllogism) that [latex]B{\rightarrow}D[/latex]; that babies are despised.
- Using the contrapositive of the second premised, [latex]D{\rightarrow}{\sim}M[/latex], we can conclude (by Syllogism) that [latex]B\rightarrow\sim{M}[/latex]; that is, babies cannot manage crocodiles.
- While silly, this is a logical conclusion from the given premises.
