2.6: Method of Conterexample

We can use the method of counterexample to show that an argument is invalid. The method involves telling a consistent story in which all the premises of the argument are true, but the conclusion is false.

The idea behind the method is this. It is impossible for a deductively valid argument to have all true premises while having a false conclusion. A counterexample is a possible scenario in which the premises are true and the conclusion is false. So, it shows that it is possible for the argument to have all true premises and a false conclusion. And this proves that it is deductively invalid.

Consider the argument:

1. 1. If Jones drove the getaway car, he’s guilty of the robbery.
2. 2. Jones did not drive the getaway car.

So, Jones is not guilty of the robbery.

Counterexample: Suppose that Jones did not drive the getaway car, but he was one of the other robbers. In this case, both premises would be true and the conclusion would be false. This means the argument is invalid.

Now here is one for you to try:

1. 1. If Musa overslept, he would have been late to work.
2. 2. Musa was late to work.

So, Musa did oversleep.

Can you construct a counterexample to show that this argument deductively invalid?

Exercises on Validity

1. What conclusions are obvious consequences of the following sets of premises? (The first one is worked for you, as an example)

Example:

1. If Ivan is from Texas, then Darius is from Alabama.
2. Ivan is from Texas.

So, Darius is from Alabama.

Exercise 1.1:

1. If Ivan is from Texas, then Darius is from Florida.
2. Darius is not from Florida.

So…

Exercise 1.2:

1. Either Sara is from Texas or she is from Florida.
2. Sara is not from Florida.

So…

2. Which of the following arguments are valid? In many cases, you won’t know whether the premises are true or not, but in those cases where you do, say whether the argument is sound.

Exercise 2.1:

1. All Republicans hate the poor.
2. Marco Rubio is a Republican.

So, Marco Rubio hates the poor.

Exercise 2.2:

1. All Democrats cheat on their spouses.
2. All men are Democrats.

Therefore, all men cheat on their spouses.

Exercise 2.3:

1. If my alarm breaks, I’m late to work.
2. I was late to work.

Therefore, my alarm broke.

Exercise 2.4:

1. If my alarm breaks, I’m late to work.
2. I was not late to work.

Therefore, my alarm did not break.

Exercise 2.5:

1. Many Fords run for years without any problems.
2. My car is a Ford.

Therefore, my car will run for years without any problems.

Exercise 2.6:

1. Batman and Captain America can’t both be the best superhero.
2. Batman is the best superhero.

Therefore, Captain American isn’t the best superhero.

Exercise 2.7:

1. OU and OSU can’t both win the Big 12 outright.
2. OU will not win the Big 12 outright.

Therefore, OSU will win the Big 12 outright.

Exercise 2.8:

1. If we don’t have free will, we can’t be blamed for our actions.
2. If we can’t be blamed for our actions, we shouldn’t be punished.

Therefore, if we don’t have free will, we shouldn’t be punished.

Exercise 2.9:

1. If Sam committed first degree murder, then he intended to kill Ivan.
2. And he did intend to kill him (he admitted it in his testimony).

So, Sam is guilty of murder in the first degree.

3. Use the method of counterexample to show that the following arguments could each have all true premises while having a false conclusion. If you succeed, this will prove that they are invalid.

Exercise 3.1:

1. Some politicians are honest.
2. Will is a politician.

So, Will is honest.

Exercise 3.2:

1. If Jennifer Lawrence is the U.S. President, then she is famous.
2. Jennifer Lawrence is famous.

So, Jennifer Lawrence is President.

Exercise 3.3:

1. Some ethics teachers are not honest people.

So, some honest people are not ethics teachers.

Exercise 3.4:

1. Whenever Bill is home his car is in the garage.
2. Bill’s car is in the garage.

So, Bill must be home.

4. Construct a valid argument with at least one false premise.

5. Construct a valid argument with a false conclusion.

6. Is it possible to construct a valid argument with all true premises and a false conclusion? If not, why not?

7. Describe the method of counterexample.

8. When it’s properly employed, what does the method of counterexample show?

Answer

1.1: Ivan is not from Texas.

1.2: Sara is from Texas.

2.1: Valid; but it is unsound because the first premise is false.

2.2: Valid; but it is unsound because both premises are false.

2.3: Invalid; it would be possible for both premises to be true and the conclusion false. This would happen if the two premises were both true, but I was late for some other reason, e.g., my car didn’t start or it broke down on the way to work.

2.4: Valid; this one is harder. Drawing a picture will help. But don’t worry a lot about it yet; we’ll study arguments like this in Chapter 3.

2.5: Invalid; the premises could be true but the conclusion false. This would be the case if my car were one of the exceptions, one of the few Fords that were lemons.

2.6: Invalid; the premises could both be true and the conclusion could still be false. This would be the case if any of the ten teams not from Oklahoma won. It would be the case, for example, if Nebraska won outright.

2.7: Valid; tracing the if-then statements shows that each leads to the other in a way that cannot be false. We’ll learn more about these types of arguments in the next chapter.

2.8: Invalid; the premises could both be true yet the conclusion false.

3.1: Here is one counterexample (there are many others). Imagine that Will is a politician who accepts bribes from his constituents. In this case, the premises are true, but the conclusion is false. This provides a counterexample that shows this argument is invalid.

4: Both the arguments about Republicans and the argument about Democrats are valid.

5: Each of them has at least one false premise, so they both also have a false conclusion.

6: This is impossible. The definition of validity does not allow this possibility.