We have been using truth tables to define the logical operators. They can also be used to test argument forms for validity.
Remember that a valid argument form is one in which if the premises are true then the conclusion must be true. This relationship can be seen very clearly in a truth table, because the table will show all the logical possibilities. Thus if one or more of the situations has true premises and a false conclusion, then we know that the argument form is not valid. It does not preserve the truth of the premises.
Say we have an argument involving three propositions, such as this:
If John and Mary go to the beach, then it will rain.
John and Mary do not go to the beach.
Therefore, it does not rain.
Since "John and Mary go the beach" is actually two propositions, we will represent the argument in this way:
(J & M) É R
~(J & M)
~R
The truth table for this argument form will have eight lines (23 or 2 x 2 x 2); it will be set up like this:
| Premise 1 | Premise 2 | Conclusion | J | M | R | (J & M) É R | ~(J & M) | ~R |
|---|---|---|---|---|---|---|
| 1) | T | T | T | |||
| 2) | T | T | F | |||
| 3) | T | F | T | |||
| 4) | T | F | F | |||
| 5) | F | T | T | |||
| 6) | F | T | F | |||
| 7) | F | F | T | |||
| 8) | F | F | F |
Since our premises are compound statements, we need to assign truth-values to the simple compounds and atomic statments. That is we will assign truth-values to the two J & M statements and to the unnegated Rs. We will do so by entering the truth-values in lower-case letters. We are using lower-case letters to indicate that this is a preliminary assignment of truth-values:
| Premise 1 | Premise 2 | Conclusion | J | M | R | (J & M) É R | ~(J & M) | ~R |
|---|---|---|---|---|---|---|
| 1) | T | T | T | t | t | t |
| 2) | T | T | F | t | t | f |
| 3) | T | F | T | f | f | t |
| 4) | T | F | F | f | f | f |
| 5) | F | T | T | f | f | t |
| 6) | F | T | F | f | f | f |
| 7) | F | F | T | f | f | t |
| 8) | F | F | F | f | f | f |
Now we can assign truth-values to the compounds as they occur in the premises and conclusion:
| Premise 1 | Premise 2 | Conclusion | J | M | R | (J & M) É R | ~(J & M) | ~R |
|---|---|---|---|---|---|---|
| 1) | T | T | T | T | F | F |
| 2) | T | T | F | F | F | T |
| 3) | T | F | T | T | T | F |
| 4) | T | F | F | T | T | T |
| 5) | F | T | T | T | T | F |
| 6) | F | T | F | T | T | T |
| 7) | F | F | T | T | T | F |
| 8) | F | F | F | T | T | T |
Note that the truth-value of the conclusion is not determined in the truth table by the premises; it is determined by the truth-values of the R column to the left.
Now let's display the truth table again, but this time we will apply several rules. After constructing a truth table, you are to do the following:
| Premise 1 | Premise 2 | Conclusion | J | M | R | (J & M) É R | ~(J & M) | ~R |
|---|---|---|---|---|---|---|---|
| 1) | T | T | T | T | F | F | |
| 2) | T | T | F | F | F | T | |
| 3) | T | F | T | T | T | F | NO |
| 4) | T | F | F | T | T | T | OK |
| 5) | F | T | T | T | T | F | NO |
| 6) | F | T | F | T | T | T | OK |
| 7) | F | F | T | T | T | F | NO |
| 8) | F | F | F | T | T | T | OK |
Remember: Valid argument forms have only true premises and a true conclusion; invalid argument forms have at least one instance in which there are true premises and a false conclusion.
Exercise 8 provides practice opportunities.
Copyright © 1999, Michael Eldridge