More Notes on Shorter Truth Tables

Shorter truth tables take some time to learn. Do not rush through them. Students need lots of examples to see how they work.

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 invariably to a contradiction, then the argument cannot be invalid, so it must be valid. But if you can assume the argument is invalid and 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: 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 would result in the following:

~[(p • q) ⊃ r]
T   T T T  F  F

Feel free to comment if you have any questions.

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