9.4: Refuting General Statements by Finding Counterexamples

Here is a universal generalization about cows: "All cows are brown." You can refute it by pointing out a cow that isn’t brown.

When you say, "Most of the cows are brown," you are also making a generalization, but not a universal generalization, and you cannot refute it by pointing out a cow that isn’t brown.

When you say, "My cow is all brown," you are not generalizing but are making a more specific statement. A generalization about a group is a statement about the group that says some, all, or a percentage of them have some property. The property we have been talking about is brown color.

Here is a list of different kinds of generalizations:

All A are B

No A are B

Most A are B

Many A are B

44% of A are B

Two-thirds of A are B

Some A are B

The letters "A" and "B" stand for groups or things, not whole statements. Only the first item on the list is a universal generalization, but when most people use the term "generalization," they mean a universal generalization, and they don’t realize that other items on the list are also generalizations. You'll have to be alert to this.

Generalizations saying that some percent or fraction of A are B are called " statistical generalizations. "

One version of the straw man fallacy is called quibbling about a generalization. The mother tells her little girl, "Drinking poison will kill you; put that down!" Our quibbler responds, "Wait! It might not kill her. I once heard that someone drank a half teaspoon of some poison and lived to tell about it because the emergency crew arrived in time, and the person was fed intravenously for a week." OK, the quibbler is correct that there is a counterexample to the mother's generalization IF you take her generalization too literally. However, the mother really meant to tell her little girl that drinking poison will very probably kill her. The quibbler didn't pay attention to the spirit of the remark and took it too literally. Critical thinkers are charitable and don’t quibble. When quibblers confront us and our generalizations, we recognize their mistake and we point out that we didn't mean for them to take us so literally.

Your opponent's statement has been refuted when you have made a totally convincing case that the statement is false. A refutation is a successful disproof. If you make a statement that is inconsistent with your opponent’s statement, you don't have to be correct, but if you refute them, you do. Presenting a counterexample is one way to refute a universal generalization. The counterexample will be an exception to the claim.

Exercise \(\PageIndex{1}\)

Fill in the blank. Suppose Chandra Morrison says every U.S. president has been a man, and Stephanie says the third president was female. Stephanie has ________Chandra.

a. refuted
b. given a counterexample to
c. done both a and b to
d. done none of the above to

Answer

Answer (d). Stephanie has said something inconsistent with what Chandra said, but not refuted her, so (a) is incorrect. If you refute someone, you must be correct, but Stephanie was not correct, was she? Now about (b). Stephanie has not given a counterexample to Chandra's statement, because a counterexample must be correct, yet what Stephanie says is incorrect. So (b) is also the wrong answer.

If someone makes the general claim that all As are Bs, one good way to test the claim is to sample some of the As and check to see whether they are also Bs. If you find even one exception, the generalization is refuted.

Refutation is the engine driving science forward. Science progresses by trying to refute statements that are precise enough to be tested. Scientists attempt to refute predictions, conjectures, claims, hypotheses, laws, and theories, provided they are formulated precisely enough that a scientist can figure out how to run a test or experiment which, if failed, would refute them. Statements that fail the tests because they are inconsistent with the observations are declared to have been refuted. The scientific community holds on only to that which it has not yet refuted. The truth is what can stand up to this procedure of attempted refutation.

For an everyday example of this procedure, suppose you flip the switch to turn on the light in your bedroom and nothing happens. Then you try to figure out why nothing happened. Can you think of any explanations? How about “The laws of electricity were just repealed”? No, that is not a likely explanation. Here are four better ones:

• The bulb is burned out.
• A fuse is blown.
• The switch is broken.
• A wire in the circuit is broken.

Which one of these hypotheses is correct? Any of the four could be correct. Well, suppose you screw in a new light bulb and it lights up. That settles it. Your scientific experiment has supported the first hypothesis and refuted the other three.

Exercise \(\PageIndex{1}\)

The following passage describes a scientific test designed to confirm or refute some hypothesis, (i) State the hypothesis to be tested. Hint: It had to do with both Uranus and something beyond Uranus. (ii) Describe the test—that is, state how the hypothesis was tested. (iii) What possible test result would have been consistent with the hypothesis? (iv) What possible test result would have been inconsistent with the hypothesis? (v) Did the test results refute the hypothesis? That’s a lot of questions. OK, here’s the passage.

The success of the English astronomer Edmund Halley in using Newton's laws of mechanics and gravitation to predict the orbits of recurring comets and the success of other astronomers in predicting the positions of the planets convinced almost all astronomers in the Western world that the heavenly bodies are not supernatural beings but are in fact physical objects obeying Newton's laws. In the early 1800s, the outermost planet known to exist in our solar system was Uranus. Unfortunately, the positions of Uranus that were predicted from using Newton's laws did not quite agree with the observed positions, and the deviation was too much to attribute to errors made with the astronomical instruments. Astronomers at the time offered two suggestions for the fact that the predicted positions did not agree with the observations. One hypothesis was that Newton had made some mistake with his laws of mechanics and that the laws should be revised. The other conjecture was that Uranus wasn't the outermost planet after all—that some other unknown planet was attracting Uranus. To check this latter conjecture, the English astronomer J. C. Adams, in 1843, and the French astronomer Leverrier, in 1845, calculated that the positions of Uranus could be explained by Newton's laws if there were another planet nearby of a specific size and orbit. They suggested that astronomers begin looking in a certain place in the night sky for this planet, a place where the planet must be in order to account for Uranus's orbit. The planet was in fact observed there in 1846 by astronomers from several different observatories. That planet is now called Neptune.

Answer

(i) Another planet beyond Uranus was attracting it sufficiently to account for the actually observed positions of Uranus in the sky according to an accounting using Newton's laws. (The hypothesis is not simply that there is another planet.) (ii) It was tested by using Newton's laws to predict where the new planet should be located. (iii) Test results that would consistent with the hypothesis: finding a new planet in the predicted location after a careful search. (iv) Test results that would be inconsistent with the hypothesis: not finding a new planet in the predicted location after a careful search. (v) No, the actual test results were consistent with the hypothesis.