The Wandering Blog

In the previous article, we introduced the concept of soundness. We said that a sound argument has two things: premisses that are true in the real world, and valid reasoning. We also defined validity: an argument is valid if and only if there is no possible situation where the premisses are true and the conclusion false. So far so good.

It turns out that there is a simple way to test if an argument is invalid. In this article, we are going to learn about this test and how to apply it.

Invalidity

A valid argument is one where there is no possible situation where the premisses are true and the conclusion is false. So, an invalid argument must be one where there is at least one possible situation where the premisses are true and the conclusion is false. Consider this example:

If we come upon a wild boar in the forest, then we are in trouble. We are in trouble. Therefore, we have come upon a wild boar in the forest.

This is an invalid argument, because we can think up a situation where the premisses are true and the conclusion false. Consider this line of thought. It seems reasonable to accept that if we come upon a wild boar in the forest, then we are in trouble. And further, let’s imagine that we are in trouble. Does it necessarily follow that we have come upon a wild boar in the forest? Take a moment and try to answer that question before reading further.

No, it does not necessarily follow that, if we accept the premisses, the conclusion follows. For one thing, it might be the case that although we are in trouble, we are in trouble for an entirely different reason than encountering a wild boar. Perhaps the reason we are in trouble is that we are on a magic island that has a monster that wants to kill us. It’s still true that if we came upon a wild boar in the forest, we would be in trouble; and it’s also true that we are in trouble. But according to our story, it’s not true that we have come upon a wild boar in the forest.

This gives us the outline of a method to test for the invalidity of an argument.

The Invalidity Test

In order to test an argument for invalidity, we must: (1) imagine that the premisses are true; (2) imagine that the conclusion is false; (3) come up with a coherent (non-contradictory) story that explains how (1) and (2) can be the case.

That’s it. Let’s look at another example. Suppose I’m charging money for people to hear me read this article, and I’m standing at the door, letting in people who pay me. And I say:

If you have paid your money, you can come into the room. Anna has not paid her money. Therefore, Anna can not come into the room.

Let’s use our method to show that this argument is invalid. (1) First, imagine that the premisses are true. You see signs in the hallway and on the door that say “if you have paid your money, you can come into the room.” And you know Anna has not paid. (2) Second, imagine that the conclusion is false. You see Anna come into the room.

(3) Come up with a story that explains how these things can all be the case. That is, can you explain how it could be that there are signs in the hallway saying you must pay, and you know Anna hasn’t paid, but she comes in anyway? There are many possibilities. Perhaps Anna is my sister, and I let family members in for free. That could explain it. Or, perhaps Anna is my co-lecturer, and has an important role to play in reading out examples.

In any case, the condition “if you have paid your money, you can come into the room” means that payers can gain entry; but it does not exclude other ways of getting into the room. If we wanted to make sure there was no other way to get into the room, we could say, “You can come into the room only if you have paid your money.” Now, Anna would have to pay to come in.

A few things to note about this method. First, remember that validity is independent of truth. We are testing the reasoning of the argument, not the truth of the premisses or conclusion. Second, the explanation you come up with can be wide-ranging. If you can find any explanation, no matter how outlandish, the argument is invalid.

Examples

Test yourself with these two examples, one fairly easy, the other one a bit more difficult.

1. We need to raise some money for our club. Having a bake sale would raise money. Therefore, we should have a bake sale.

2. Two shots were fired. Two bullets were found in the victim’s body. Therefore, two shots were fired at the victim.

Take a moment and use the invalidity test to show how these arguments are invalid. Remember the test: imagine (1) the premisses are true and (2) the conclusion is false, and then (3) explain how everything can work together.

1. Imagine that we are in a club that needs to raise money, and having a bake sale would certainly raise money. Also imagine that we should not have a bake sale. How can these claims go together? Under what scenario could these premisses be true, but the conclusion false? Here’s one possible answer. Perhaps there are better ways to raise money, such as having a rummage sale. So just because we need to raise money, and a bake sale would raise money, does not necessarily mean that we should have a bake sale.

2. Imagine that it’s true two shots were fired, and two bullets were found in the victim’s body. Also imagine that two shots were not fired at the victim. How can we explain this? Again, under what scenario could the premisses be true (two shots were fired, and two bullets were found in the victim), but the conclusion (two shots were fired at the victim), be false? Here’s one possible answer. Perhaps there were two shots, but these shots were only fired into the air. However, our victim was actually shot twice many years ago, and the surgeons were unable to remove the bullets. In other words, it’s possible that two shots were fired, and that there were two bullets in the victim, but that two shots were not fired at the victim. So the argument is invalid, because there are scenarios in which the premisses are true and the conclusion false.

This should give you a good idea how the test works. Here are a few examples to think about for next time, when we’ll discuss the implications of invalidity in greater depth.

1. The winning ticket is number 500. Bill holds ticket number 501. Therefore, Bill does not hold the winning ticket.

2. Some people smoke cigars. Some people smoke pipes. Therefore, some people smoke cigars and pipes.

3. Anna does not believe there is a cat in the box. Therefore, Anna believes that there is no cat in the box.