How to Prove an Argument Invalid by a Counterexample

An argument is invalid if the conclusion doesn't follow necessarily from the premises. Whether or not the premises are actually true is irrelevant. So is whether or not the conclusion is true. The only question that matters is this: Is it possible for the premises to be true and the conclusion false? If this is possible, then the argument is invalid.

Proving Invalidity

The "counterexample method" is a powerful way of exposing what is wrong with an argument that is invalid. If we want to proceed methodically, there are two steps: 1) Isolate the argument form; 2) Construct an argument with the same form that is obviously invalid. This is the counterexample.

Let's take an example of a bad argument.

1. Some New Yorkers are rude.
2. Some New Yorkers are artists.
3. Therefore Some artists are rude.

Step 1: Isolate the Argument Form

This simply means replacing the key terms with letters, making sure that we do this in a consistent way. If we do this we get:

1. Some N are R
2. Some N are A
3. Therefore some A are R

Step 2: Create the counterexample

For instance:

1. Some animals are fish.
2. Some animals are birds.
3. Therefore some fish are birds

This is what is called a "substitution instance" of the argument form laid out in Step 1. There is an infinite number of these that one could dream up. Every one of them will be invalid since the argument form is invalid. But for a counterexample to be effective, the invalidity must shine forth. That is, the truth of the premises and the falsity of the conclusion must be beyond question.

Consider this substitution instance:

1. Some men are politicians
2. Some men are Olympic champions
3. Therefore some politicians are Olympic champions.

The weakness of this attempted counterexample is that the conclusion isn't obviously false. It may be false right now, but one can easily imagine an Olympic champion going into politics.

Isolating the argument form is like boiling an argument down to its bare bones--its logical form. When we did this above, we replaced specific terms like "New Yorker" with letters. Sometimes, though, the argument is revealed by using letters to replace whole sentences or sentence-like phrases. Consider this argument, for instance:

1. If it rains on election day the Democrats will win.
2. It won't rain on election day.
3. Therefore the Democrats won't win.

This is a perfect example of a fallacy known as "affirming the antecedent." Reducing the argument to its argument form, we get:

1. If R then D
2. Not R
3. Therefore not D

Here, the letters don't stand for descriptive words like "rude" or "artist". Instead, they stand for an expression like, "the Democrats will win" and "it will rain on election day." These expressions can themselves be either true or false. But the basic method is the same. We show the argument s invalid by coming up with a substitution instance where the premises are obviously true and the conclusion is obviously false. For instance:

1. If Obama is older than 90, then he's older than 9.
2. Obama is not older than 90.
3. Therefore Obama is not older than 9.

The counterexample method is effective at exposing the invalidity of deductive arguments. It doesn't really work on inductive arguments since, strictly speaking, these are always invalid.