Using Counterexamples

2006 Edward G. Rozycki

edited 9/5/19

Counterexamples are statements or examples very useful in two important ways:

A. as forensic stratagems. That is, they are devices used in debating or argumentation to offer rebuttal to proposed generalizations or definitions.

B. to help you test and clarify your own generalizations in any variety of disciplines and topics.

Consider the following generalization:

Students with low grade-point averages in high school are invariably failures in life.

This generalization has a Subject-Predicate structure, AB, where

1. the subject, A = Students with low grade-point averages in high school, and

2. the predicate, B = are invariably failures in life

Counter-examples are statements that are both

a. generally recognized to be true, and

b. incompatible with (or, logically contradictory to) the generalization being proposed.

*** [CAUTION: not just any contradiction to a generalization counts as a counter-example!]***

Counter examples are constructed on the model, A (and) not-B.

For example,

Students with low grade-point averages in high school, like Albert Einstein, sometimes become famous scientists.

If the generalization is restrictive, e.g.

Only students with low grade-point averages in high school are invariably failures in life.

The counterexample constructed on the model, not-A and B, works also. For example,

People who always rush into things without planning ahead are invariably failures in life.

It is sometimes best to devise both types of counterexample to use.


Construct counter-examples to rebut each of the statements below:

1. Only penguins are ferocious polar animals.

2. Poor people are, for the most part, dishonest.

3. Only the best educations require involvement with rough sports.

4. No pastry made in France can be delicious.

5. A good teacher is someone with the gift of gab.

6. Crocodiles are the only dangerous North American reptile.

7. Housing prices are the only good indicators of the quality of the neighborhood schools.

8. Unless you get your high school diploma, you'll never get ahead in life.

9. People go into doctoral programs to escape from having to make a living.

10. Only students whose native language is English appreciate Shakespeare.


See, also, IX. Counterexamples