once you get the hang of it. The method involves two steps:
(1) Determine the logical pattern, or form, of the argument that you’re testing for invalidity, using letters (A,B,C, etc.) to represent the various terms in the
The method works because with a valid argument you can never have true premises and a false conclusion. Thus, any time you find an argument that has true premises and a false conclusion, you know immediately that it is invalid. And because validity is determined by the logical form of an argument rather than by the actual truth or falsity of the premises and conclusion, you also know immediately that any argument that has that form must also be invalid. Thus, if you can find just one counterexample (i.e., one argument of that form that has all true premises and a false conclusion), then you can prove that all arguments that have that form are invalid.
Let’s now apply the counterexample method to a few simple examples. Suppose we want to determine whether the following argument is valid or invalid:
Example 1: Some Republicans are conservatives, and some
Republicans are pro-choice. So, some conservatives are pro-choice.
The first step in the counterexample method is to determine the logical form of the argument. To make sure that we are representing the form of the argument correctly, it is helpful if we begin by numbering the steps in the argument, with the conclusion stated last. We thus get:
1. Some Republicans are conservatives. 2. Some Republicans are pro-choice.
3. Therefore, some conservatives are pro-choice.
(Note that in logic “some” always means “at least one,” i.e. “some and perhaps all.” “Some” never means “some but not all.” Thus, in logic, when we say “Some dogs
are animals” we mean “At least one dog is an animal” (which is true), not “Some, but not all, dogs are animals” (which is false).)
Next, we assign letters to represent the various terms in the argument. Using “As” to represent “Republicans,” “Bs” to represent “conservatives,” and “Cs” to represent “people who are pro-choice,” what we get is:
1. Some As are Bs. 2. Some As are Cs.
3. Therefore, some Bs are Cs.
This is the logical form of the argument in Example 1. Having determined this form, we have completed the first step of the counterexample method.
The second step in the counterexample method involves attempting to construct a
second argument that has exactly the same form as our argument being tested, but which
(unlike the first) has obviously true premises and an obviously false conclusion. With most invalid argument forms, it is possible to construct such a counterexample using a few stock terms, such as “dogs,” “cats,” “animals,” “men,” “women,” “people,” “apples,” “pears,” and “fruit.” Using the terms “animals” as a substitute for “As,” “dogs” as a substitute for “Bs,” and “cats” as a substitute for “Cs,” we can construct an argument that has the same form as the argument in Example 1, but which has clearly true premises and a clearly false conclusion:
1. Some animals are dogs. (true) 2. Some animals are cats. (true)
3. Therefore, some dogs are cats. (false)
We have thus constructed a counterexample to the argument in Example 1. Other counterexamples that would work equally well include:
1. Some fruits are apples. (true) 2. Some fruits are pears. (true)
Let’s now try another example:
Example 2: If God exists, then life has meaning. Hence, God does exist, since life has meaning.
First, we begin by numbering the premises and the conclusion.
1. If God exists, then life has meaning. 2. Life has meaning.
3. Therefore, God exists.
We next identify the form of the argument. Using “A” to represent the statement “God exists,” and “B” to represent the statement “Life has meaning” we get:
1. If A then B. 2. B
3. Therefore, A.
Finally, attempt to find an argument that has exactly the same form, but one that has obviously true premises and an obviously false conclusion.
For starters, we might try:
1. If Lassie is a dog, then Lassie is an animal. (true) 2. Lassie is an animal. (true)
3. Therefore, Lassie is a dog. (true)
But this won’t work, because we need a false conclusion in order to get a good counterexample.
Next, we might try:
1. If JFK was killed by a lone assassin, then JFK is dead. (true) 2. JFK is dead. (true)
3. Therefore, JFK was killed by a lone assassin. (true??false??)
But this won’t work either, because for an effective counterexample we need a conclusion that is obviously false (i.e., one that is known to be false by practically everyone), and it is far from obviously false that JFK was killed by a lone assassin. So, back to the drawing board one more time:
1. If Kansas City is in California, then Kansas City is in the United States. (true) 2. Kansas City is in the United States. (true)
3. Therefore, Kansas City is in California. (false)
Bingo! This gives us the counterexample we’re looking for. Now we’ve proven that Example 2 does not have a valid argument form, and thus is invalid. Another counterexample might be.
1. If George Washington died in a car crash, then George Washington is dead. (true)
2. George Washington is dead. (true)
3. Therefore, George Washington died in a car crash. (false)
1. Some senators are Republicans. 2. All senators are politicians.
3. Therefore, some Republicans are politicians.
Next, we identify the form of the argument:
1. Some As are Bs. 2. All As are Cs.
3. Therefore, some Bs are Cs.
Finally, we try to create a counterexample. First, we might try:
1. Some dogs are animals. (true) 2. All dogs are mammals. (true)
3. Therefore, some animals are mammals. (true)
But this won’t work, since the conclusion is true. Next, we might try:
1. Some apples are red. (true) 2. All apples are fruits. (true)
3. Therefore, some red things are fruits. (true)
But this won’t work either, since again the conclusion is true. Finally, we might try:
1. Some students love jazz. (true) 2. All students love hot wings. (false)
3. Therefore, some people who love jazz are people who like hot wings. (true)
But this fails, too, because not only is the conclusion true, but also one of the premises is false. At this point we might begin to suspect that the reason we can’t find a
counterexample is that there is no counterexample to be found, because the blessed thing is valid. And this is in fact the case. One could work until Doomsday trying to come up with a counterexample to the argument in Example 3 and never succeed, since the form of the argument is such that it guarantees a true conclusion if you plug in true premises. And this raises the urgent question: At what point should you throw in the towel? At what point is it safe to conclude that the reason you can’t find a good counterexample to a particular argument is not that you have been insufficiently imaginative or persevering in your attempts, but rather that the argument is simply valid?
Alas, there is no such perfect security with the counterexample method. As a rule, however, it’s generally safe to assume after three or four unsuccessful tries that no
counterexample can be found. On exams, however, the rule is always: Try as many attempted counterexamples at time permits.
EXERCISE
Use the counterexample method to determine whether the following argument forms are valid or invalid.
1. All Albanians are beer-drinkers. Therefore, since all Catalonians are beer-drinkers, all Albanians are Catalonians.
2. If Josef is an anarchist, then he’s a Bolshevik. Hence, Josef is not a Bolshevik, since he’s not an anarchist.
3. No ales are brandies. Some ales are not champagnes. Therefore, some champagnes are not brandies.
6. If Alice is a Presbyterian, then she believes in God. Hence, Alice is a Presbyterian, since she believes in God.
7. No Argentinians are Bolivians. Therefore, some Cubans are not Bolivians, since some Argentinians are not Cubans.
8. Some animals are brown. Some animals are cows. So, some animals are brown cows.
9. No Alsatians are Burgundians. Some Channel-watchers are not Alsatians. So, some Burgundians are not Channel-watchers.
10. All aphids are bugs. Some aphids are cranky. So, some cranky things are not bugs.
11. No apple-pickers are broncobusters. Some broncobusters are cellists. So, some cellists are apple-pickers.