Art of Reasoning : An Introduction to Logic (9780393421774) by Kelley David; Hutchins Debby

Author:Kelley, David; Hutchins, Debby
Language: eng
Format: epub
Publisher: W W Norton College

We can adapt this procedure to test for the validity of an argument. Let’s work with the argument about Shakespeare:

~W ⊃ D

~D

* * *

W

If we were creating a truth table for the first premise alone, it would look like this:

~ W ⊃ D

F T T T

F T T F

T F T T

T F F F

To create a truth table for the argument, we create additional columns to represent the second premise and the conclusion, with a forward slash before the conclusion in order to separate it from the premises:

Premise 1 Premise 2 / Conclusion

~ W ⊃ D ~ D / W

T T T

T F F

F T T

F F F

As with truth tables for individual statements, each atomic component that appears more than once must have the same truth value on each row. So in the columns under premise 2 and the conclusion, the atomic statements D and W must have the same truth values on each row that each has in premise 1.

We now compute the truth values of each premise and of the conclusion, putting the result under the main connective for each statement. The procedure is exactly the same as with truth tables for single statements, except that now we have three different statements, whose truth values are represented in the three shaded columns.

Premise 1 Premise 2 / Conclusion

~ W ⊃ D ~ D / W

F T T T F T T

F T T F T F T

T F T T F T F

T F F F T F F

The test for validity is whether the conclusion can be false while the premises are both true. So the first step is to look at the column under the conclusion and flag the rows on which it is false—rows 3 and 4:

Premise 1 Premise 2 / Conclusion

~ W ⊃ D ~ D / W

F T T T F T T

F T T F T F T

T F T T F T F

T F F F T F F

Now we check the premises on those rows in order to see whether they are both true.

Premise 1 Premise 2 / Conclusion

~ W ⊃ D ~ D / W

F T T T F T T

F T T F T F T

T F T T F T F

T F F F T F F

In row 3, the premise ~D is false, and in line 4 the premise ~W ⊃ D is false. Since the truth table shows us every combination of possible truth values, we have just established that there’s no way this argument could have true premises and a false conclusion, which is the standard for validity. So this argument is valid.

When we use truth tables to determine whether an argument is valid or invalid, we are concerned only with the truth values of the premises and conclusion as whole statements—the truth values in the shaded columns. For compound statements such as premises 1 or 2 in the previous argument, we disregard the truth values of their components and take account only of the columns under their main connectives.

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