Section Progress:

9.3 Counterexamples

So far we've just looked at proofs that show an argument is valid. But a proof can also show an argument is invalid.

All proofs of invalidity use the same technique: the counterexample.

Counterexample: a specific case showing the premises can be true and the conclusion false.

A counterexample is a specific case showing that it is possible for the premises to be true and the conclusion false.

The definition of validity, recall, means that it is impossible for the premises to be true and the conclusion false. So if we can show with a counterexample that it is possible, then we've shown that the argument is invalid.

The counterexample technique works because you only need one single example of true premises and false conclusion to refute validity. Validity, on the flip side, requires that for every case where the premises are true, the conclusion is also true.

You've already learned how to use truth tables to find a counterexample. For example, consider this argument:

1. It's not the case that Pia and Quinn are guilty.
Thus,
2. Pia is innocent.

In BOOL:

1. ~(P&Q)
Thus,
2. ~P

This argument is invalid. Here's your first question:

We know that the premise can be true and the conclusion false because of row 2.

How do we use that fact to prove that the argument invalid? Remember, a proof is a step-by-step explanation, so all we have to do is explain why that row means that the argument is invalid.

Here's how it looks:

Proof by counterexample: If Pia is guilty and Quinn is innocent, then the premise is true and the conclusion is false. Thus the argument is invalid. Done.

That is a model proof of invalidity. We said that all proofs should start with the opening "proof". Now we can add: sometimes you need to be even more specific, and explain exactly what proof method you are using.

All proofs of invalidity should start: "Proof by counterexample:"

Whenever you are doing a proof of invalidity, is should start out by saying "proof by counterexample", since that is the method that all proofs of invalidity use.

Now consider this question.

Since that proof is written in English, it is informal.

But what if the proof uses some sentences from BOOL? Consider this proof:

Proof by counterexample: On row 2 of the truth table, P is T and Q is F. On that row the premise, ~(P&Q), is T, and the conclusion, ~P, is F. Thus the argument is invalid. Done.

This is also a good proof: it too shows in a step-by-step way that the argument is invalid. Even though this proof makes some use of BOOL, it is still informal.

As you will see in the next chapter, a formal proof is written entirely in BOOL using specific rules.

This proof, by contrast, still makes some use of English, so it is informal. In fact, every proof of invalidity, or proof by counterexample, will be informal. There are no formal proofs of invalidity, only formal proofs of validity.

That point will be easier to understand, though, once you've learned about formal proofs.

It's your turn to practice a proof of invalidity. Here's an argument:

Not Darwin's Argument
1. Every species of orchid has a pollinator.
2. No known moth could pollinate AS.
3. AS is a type of orchid.
Thus,
4. There exists an unknown moth that pollinates AS.

 

9.3 Counterexamples