Truth tables can be used to determine the validity of propositional arguments. In a valid argument, if the premises are true, then the conclusion must be true. The truth table for a valid argument will not have any rows in which the premises are true and the conclusion is false. For example, here is a truth table of a modus tollens argument, with the final columns, showing it to be valid:
The fourth row down is the only row with true premises, and in that row it also has a true conclusion. So this argument is valid.
An argument is invalid when there is at least one row with true premises and a false conclusion, such as in this affirming the consequent truth table:
The third row down has true premises and a false conclusion, so this argument form is invalid. One question I often get from students at this point is, “But Mr. Nance, look at the first row. It has true premises and a true conclusion. Doesn’t that show the argument to be valid?”
The answer is to remember what was learned in categorical logic about counterexamples. To show an argument form such as AAA-2 to be invalid, we used a counterexample, in which terms were inserted to make the premises true and the conclusion false, e.g.
All dogs are mammals.
All whales are mammals.
Therefore, all whales are dogs.
But remember this important fact: If you can make a counterexample, it doesn’t matter that it is possible for an argument of that form to have true premises and a true conclusion. We could use terms to make the AAA-2 have true premises and a true conclusion as well, such as this:
All dogs are mammals.
All poodles are mammals.
Therefore, all poodles are dogs.
This does not show the argument to be valid (after all, it has an undistributed middle term), any more than the first row of the truth table for affirming the consequent shows that argument to be valid.
Here is the key point: In a truth table, as in a categorical syllogism, only one counterexample is needed to show an argument to be invalid. Therefore it does not matter that there are any rows in the truth table that makes it appear valid (true premises and true conclusion), as long as there is at least one row (true premises and a false conclusion) that shows it to be invalid.