# Counter-examples

How do you show that an argument is invalid or weak?

Remember that an argument is valid if it’s impossible for the premises to be true and the conclusion false, and it is strong if it’s very unlikely that the premises are true and the conclusion false.

So to show that an argument is invalid, you only need to find a case in which the premises are true and the conclusion false, whereas to show that it is weak, you have to show that it is quite possible for the premises to be true and the conclusion false. To do so, you will construct counter-examples.

*Definition*: A counter-example to an argument is a situation which shows that the argument can have true premises and a false conclusion.

If the argument being evaluated is deductive, then we can show it to be invalid and, therefore, bad if we can describe a counter-example.

Recall this lovely example:

Everybody loves a winner, so nobody loves me.

In standard form, the argument looks like this – with the suppressed premise:

\(\begin{array}{ll} \text{P1} & \text{Everybody loves a winner.}\\ \text{P2} & \text{[I am not a winner.]}\\ &\text{Therefore,}\\ \text{C} & \text{Nobody loves me.} \end{array}\)

Here’s a counter-example to this argument:

Suppose that everybody loves all winners and that I am not a winner (so both premises are true.) Still, the conclusion can be false if one of the people out there who love all the winners also loves the occasional non-winner, including me. We can imagine such a person saying: “I love all winners, but I love you too, even though you’re not a winner.”

## Counter-examples and validity

If there are no counter-examples to a particular argument, then it is valid, as it is then impossible to find a situation in which the premises of the argument are true and the conclusion is false. That means that in every situation in which the premises are true, then the conclusion is also true, and this is what we need to know to conclude that an argument is valid.

This is an important link between the concepts of validity and counter-example:

- An argument is valid if and only if there are no counter-examples to the argument.

## Counter-examples for Non-deductive Arguments

Can a counter-example be used to show that a non-deductive argument is weak? It can, but only if the counter-example itself represents a plausible way things might have been.

Since a non-deductive argument acknowledges that the conclusion might be false when the premises are true, but only in exceptional circumstances, you need to find a counter-example that is not so exceptional.

That is, you need to find a reasonable situation in which the premises are all true and the conclusion false. If you want to attack a non-deductive argument with a counter-example and do real damage, your counter-example must describe a situation which not only makes the premises true and the conclusion false, but is also quite likely to come about.

If all you can come up is too far fetched, then you’re probably describing one of the exceptional cases that were excluded to begin with, and so you would fail to provide a counter-example, and you could conclude that the argument is strong.

Here’s an example of an argument, one that you may very well have heard before:

Smoking marijuana is no more dangerous to your health or to society than drinking alcohol is. And drinking alcohol is legal. Therefore, smoking marijuana should probably become legal.

In standard form, the argument looks like this:

\(\begin{array}{ll} \text{P1} & \text{Smoking marijuana is no more dangerous to your health or to}\\ &\text{society than drinking alcohol is. }\\ \text{P2} & \text{Drinking alcohol is legal.}\\ \text{P3} & \text{[If the smoking of marijuana and the drinking of alcohol have}\\ &\text{similar impact on health and society, they should have the}\\ &\text{same legal status.]}\\ &\text{Therefore, probably,}\\ \text{C} & \text{Smoking marijuana should become legal.} \end{array}\)

Notice that we have introduced what we considered to be an essential suppressed premise in the standard form. Now, can we find a plausible counter-example?

Yes we can, here’s one:

The negation of the conclusion is consistent with the premises. Indeed, we might as well say that alcohol should become illegal, precisely because the impact of drinking on the society is similar to the impact of smoking marijuana – maybe even worse!

This constitutes a counter-example, because it is possible for the premises to be true and the conclusion false, and given the premises provided, neither situation seems more plausible than the other.

Implausible counter-examples are counter-examples all the same, but they can only show that a deductive argument is invalid. However, only a plausible counter-example can show that a non-deductive argument is weak.

## Deductive or Non-deductive?

You might also think of the search for a counter-example as a method that helps you determine whether an argument is deductive or non-deductive. If finding a counter-example to an argument makes us want to say that argument is no good, then the argument must be deductive, because in deductive arguments the premises are intended to give conclusive support for the conclusion.

On the other hand, if generating a counter-example does not incline us to give up the argument, then it is a non-deductive argument because non-deductive arguments have conclusions which are only meant to be strongly suggested by their premises, and leave it open that the conclusion may be false.

Here’s another way of making the point. If the only counter-examples you can find are far-fetched stories, then you may have an indication to think that the argument is non-deductive.

Here’s an example:

Wolfgang robbed the safe. Wolfgang’s fingerprints were found on the burgled safe. Lots of money, which was in the safe, was found hidden in Wolfgang’s house. Wolfgang was seen by several witnesses near the scene of the burglary when it was committed.

Now, we can come up with counter-examples to the argument:

Wolfgang didn’t rob the safe. Sigmund did, but he was perfectly disguised as Wolfgang, had a copy of Wolfgang’s fingerprints that he put on the safe, and hid some of the money in Wolfgang’s house.

Space aliens robbed the safe and engraved Wolfgang’s fingerprints on the safe with lasers coming out of their eyes.

The safe was never robbed. The whole thing was invented by the bank to get back at Wolfgang for switching banks. They really don’t like it when they lose customers.

So it’s easy to come up with counter-examples, but much harder to come up with plausible counter-examples. This indicates that the argument is meant to be non-deductive.

© Patrick Girard, University of Auckland