How to Use the Counterexample Method to Identify Invalid Arguments

A counterexample is an exception to the rule. It is a specific instance of the falsity of a universal quantification. In philosophy, counterexamples are used to argue that a certain philosophical position is wrong by showing that it does not apply in certain cases.

Instructions

1. Identify the most logically sensitive form of the argument. Use capital letters, numbers, or some other variable to represent statements or terms.

2. 1. No famous celebrities are poor people.
   2. Some computer geniuses are not poor people.
   So, 3. Some computer geniuses are not famous celebrities.

3. Let A stand for famous celebrities.
   B can stand for poor people.
   C will represent computer geniuses.

4. Find statements or terms to substitute that produce a well-known falsehood when substituted for the variables in the argument.

5. If we use dogs in the place of (C) computer geniuses, and animals are substituted for (A), famous celebrities in the conclusion, our counterexample will look like this:

   1. No A are B.
   2. Some C are not B.
   So, Some cats are not animals.

6. Substitute these terms or statements for the associated variables uniformly throughout the argument form.

7. 1. No animals are B.
   2. Some cats are not B.
   So, 3. Some cats are not animals.

8. Find terms or statements that, if substituted for the associated remaining variables in the argument, produce premises that are well-known truths.

9. 1. No animals are volcanoes.
   2. Some cats are not volcanoes.
   So, 3. Some cats are not animals.

10. Check your work. If you have formed a good counterexample, you have shown the argument to be invalid. The premises will be true and the conclusion is obviously false!

Tips & Warnings

• Arguments with animal classes and numerical examples are easy to prove.

Resources

• The Power of Logic 4e
• Online Logic Lecture
• Wikipedia - Counter Example