Shorter Truth Tables for Validity

Mr. Nance,

As I am teaching shorter truth tables for validity, I noticed that sometimes (on a valid argument) I get the contradiction in a different place than the answer key does. Is that okay, or am I making a mistake?

You are probably not making a mistake.

The shorter truth table method used on a valid argument will always result in a contradiction, but where that contradiction appears depends on the order of the propositions you work with, which can certainly vary.

For example, on Exercise 8, problem #1, the answer key shows the contradiction in two places, which happens if you find all of the truth values in the conclusion first, before going back to the premises.

STT1

But you might start by getting the truth values for S and W from the antecedent of the conclusion first, and then going directly to the premises. That would bring you to this point:STT2

Now, for the premises to be true, the consequents of each (P and F) must be true as well. That gives the contradiction in the conclusion, instead of in the premises as before:STT1

This is a perfectly legitimate answer. In the answer key, I tried to place the truth values in the positions I thought most likely for other who did the problem correctly. Typically, after making the premises true and the conclusion false, I try to start on the right side (the conclusion) and work my way left.

Here are a few more thoughts.

Shorter truth tables take some time to learn. Do not rush through them. Students need lots of examples to see how they work. Also, make sure you and they understand the concept behind them. You are assuming that the argument is invalid (by making the premises true and the conclusion false). If this assumption leads to an unavoidable contradiction, then the argument cannot be invalid, so it must be valid. But if you assume the argument is invalid and can fill out all the truth values without any contradiction, you have shown that the premises can be true and the conclusion false, i.e. you have shown it to be invalid.

Keep this in mind also: For each proposition (premise or conclusion), you must place the truth values under the main logical operator. The main logical operator is the operator in the column that would be the last to be filled out in the larger truth table. For example, consider this compound proposition:

~(p • q) ⊃ r

If this were a premise of an argument, the T would be placed under the conditional. But for the proposition

~[(p • q) ⊃ r]

the T would be placed under the negation. Working the truth values all the way out for this proposition would result in the truth values shown here:

~[(p • q) ⊃ r]
T   T T T  F  F

Feel free to comment if you have any questions.

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