Introduction
What you’ll learn to do: Evaluate arguments using truth tables
In the next section, we will use truth tables to evaluate logical arguments.
Learning Outcomes
- Analyze arguments with truth tables
Analyzing arguments with Truth Tables
Some arguments are easily analyzed 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 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.
Show Solution
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.
Show 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: |
[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. |
Show Solution
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.
Lewis Carroll, author of Alice in Wonderland, was a math and logic teacher, and wrote two books on logic. In them, he would propose premises as a puzzle (like the one in the following example), to be connected using syllogisms.
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.
Show Solution
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.
An Important Note
It is important to be aware of the reasons that we study logic. There are three very significant reasons. First, the truth tables we studied tell us the exact meanings of the words such as “and,” “or,” “not,” and so on. For instance, whenever we use or read the “If…, then” construction in a mathematical context, logic tells us exactly what is meant. Second, the rules of inference provide a system in which we can produce new information (statements) from known information. Finally, logical rules help us correctly change certain statements into (potentially more useful) statements with the same meaning. Thus logic helps us understand the meanings of statements and it also produces new meaningful statements.
Logic is the glue that holds strings of statements together and pins down the exact meaning of certain key phrases such as the “If…, then” or “For all” constructions. Logic is the common language that all mathematicians use, so we must have a firm grip on it in order to write and understand mathematics.
