Counterexamples

Chapter 2

Soundness

Sometimes a person might argue badly from good premises. Other times that person might argue well from rotten premises. In logic we are generally concerned with the quality of the inference – the connection between premises and conclusion – and not with the quality (truth or falsity) of the premises or conclusion.

This perspective may take some time to get accustomed to. In most of your other coursework, there is careful attention to whether claims are true or false. For example in a history course it will make a difference whether Thomas Jefferson really did own slaves. In a psychology course it will make a difference whether the unconscious mind really does express its desires in dreams. But in logic none of these things matter. What matters in logic is what claims can be inferred soundly from what other claims.

We can put this differently. In an argument there are premises and conclusions. You might care about the truth or falsity of the premises, or about the truth or falsity of the conclusions. In logic (overstating slightly) we couldn't care less about either of those. We care about how strong the connection is between premises and conclusions.

We hope to identify kinds of arguments that will guarantee that when the premises are true, the conclusion must also be true. Such arguments will be called sound. Soundness is the single most important concept in logic. A sound argument is guaranteed to be truth preserving. We can let all the other fields figure out which premises are true. Then logic will guarantee that when true premises are plugged into a sound argument, the conclusion must also be true. From this point of view there are only two things worth studying: logic (which tells you which arguments preserve truth) and everything else (which tells you which premises are worth arguing from).

Fortunately, it is very easy to define soundness. Nevertheless, we will give two different formulations. They both say the same thing, but in different ways. Sometimes one formulation will be more useful than the other.

Soundness A: A sound argument is one in which the truth of the premises (if they were true) would guarantee the truth of the conclusion.

Soundness B: A sound argument is one for which there is no possible world in which the premises are true and the conclusion false.

It is easy to see that these definitions are equivalent. For if the truth of the premises would guarantee the truth of the conclusion then in any possible world in which the premises were true, the conclusion would also have to be true. Conversely, if there is no possible world in which the premises are true and the conclusion false, then any situation that made the premises true would also make the conclusion true, so the truth of the premises would guarantee the truth of the conclusion.

Sidebar: In logic, our world is just one of many possible worlds. A possible world is one just like this one except that some of the facts that hold in this world may not hold in another. For example there are possible worlds in which the most recent presidential election in the U.S. went the other way. There are possible worlds in which your parents didn't meet and in which you therefore don't exist. Whatever could happen in this world does happen in some possible world, and vice versa. So talking about what happens in possible worlds is just a convenient way of talking about what is possible in this world.

Sometimes philosophers will talk about what is, for example, physically possible, or morally possible. In doing so, they are not talking about all the (logically) possible worlds, but only the ones in which our physical laws, or moral laws, are maintained. The study of the logic of possible worlds is called modal logic.

Exploring soundness

From definition A it is easy to see that it doesn't matter for soundness whether the premises are true, so long as the premises couldn't be true and the conclusion false. Neither definition says anything about what happens when the premises are false; it doesn't matter for soundness whether the premises are false. Soundness has to do with the connection between the premises and conclusion, not with either the premises or conclusion separately. (In speaking of the truth and falsity of the premises of an argument, since arguments usually have several premises, we will say the premises of an argument are true when all of its premises are true. So in the case of an argument with only one premise, its premises are true just in case its premise is true.)

It will help to give a number of examples. First some examples of sound arguments.

2.1: Sound, with true premise and a true conclusion

Some people are kind. Therefore, not everybody is unkind.

Note that if it were true that some people are kind, then it would also have to be true that not everybody is unkind. For the premise says that there are at least a few kind people, and those people aren't unkind. The truth of the premises, if they were true, would guarantee the truth of the conclusion. It is not possible (there is no possible world) for the premises to be true and the conclusion false. Now in this particular argument it just so happens that the premise ("Some people are kind") is true. And since the argument is sound, the conclusion is true too.

Arguing soundly from true premises is our ultimate goal. But from the point of view of logic, we don't care a whit whether the premises are true, so long as the argument is sound. Thus the following argument is just as sound as argument 2.1 was.

2.2: Sound, with false premise and true conclusion:

Everybody likes baseball. Therefore, President George W. Bush likes baseball.

Now obviously not everybody likes baseball. But that doesn't matter for the soundness of the argument, because if everybody did like baseball, then it would have to be true that George W. Bush likes baseball. If the premise ("Everybody likes baseball") were true, it would guarantee the truth of the conclusion. Or put differently, you cannot imagine a possible world in which everybody likes baseball but George W. Bush doesn't. If he doesn't, then obviously it would have been false that everybody does. So this argument is sound, even though the premise is false.

Now it just so happens that George W. Bush does like baseball. But that doesn't matter for the soundness of the argument. The argument would be just as sound if he didn't like baseball. We can also argue soundly from false premises to false conclusions. Here is an example.

2.3: Sound, with false premises and false conclusion.

No poor people care about their health. But all nurses care about their health. Therefore, no nurses are poor.

Well, this is a little bit harder, but not too hard. I hope you can see that this argument is sound. Here's one way to see it. First you have to suppose that the premises are true. Now I know that they aren't true. In fact, both of the premises are false. But that doesn't matter. Just suppose they are true. Now in addition suppose that there were a poor nurse, say his name is Chris. Then by the first premise ("No poor people care about their health") it would follow that Chris doesn't care about his health, since he is poor. But by the second premise ("All nurses care about their health.") it would follow that Chris does care about his health, since he is a nurse. But now Chris both does and does not care about his health, and that can't happen.

So if it really were true that no poor people care about their health (of course it isn't, but if it were), and if it really were true that all nurses care about their health (it almost certainly isn't – probably there is a nurse somewhere in a coma, for example – but if it were true) then it would also have to be true that no nurses are poor. The truth of the premises, if they were true (which they aren't), would guarantee the truth of the conclusion. It is not possible for the premises to be true and the conclusion false.

So far we have seen sound arguments with true premises and a true conclusion, with false premises and a true conclusion, and with false premises and a false conclusion. Are there sound arguments with true premises and a true conclusion?

No, there aren't. You can see that there are no such arguments by thinking about the definition of soundness. A sound argument is one in which the truth of the premises if they were true would guarantee the truth of the conclusion. So in the case where the premises actually are true, the conclusion would have to be true too. That is the whole point of a sound argument. Hence there can be no sound argument with true premises and a false conclusion.

We can summarize our results so far in a table for sound arguments.

Sound Arguments

Premises	Conclusion	Are there any?
True	True	Yes
True	False	No
False	True	Yes
False	False	Yes

Exploring unsoundness

In the case of sound arguments, we saw that you can tell very little about whether an argument is sound by looking at the truth or falsity of its premises and conclusion. The only thing we could say is that if the premises are true and the conclusion is false, then the argument is definitely unsound. Are there any constraints that truth and falsity place on what kinds of argument can be unsound? The answer is a resounding "no", as some examples will show.

2.4: Unsound, true premises and true conclusion

Most plants have green leaves. Therefore, many people enjoy ice cream.

Well, that was easy. Of course it is true that most plants have green leaves, and that many people enjoy ice cream. But the truth of the first claim is not connected to the truth of the second claim in the right way. The fact that most plants have green leaves does not guarantee that many people enjoy ice cream. It is obviously possible for most plants to have green leaves even in a world in which very few people enjoy ice cream. Imagine, for example, a world that is like this one except that people experience ice cream the way that we experience chalk, or bitter medicine. Plant leaves would still be green, but not many people would enjoy ice cream. So in that world, the premise is true and the conclusion is false. So by definition B of soundness, the argument is not sound.

2.5: Unsound, true premises and false conclusion

We saw above that an argument with true premises and a false conclusion could not be sound. So any argument with true premises and a false conclusion would serve as an example here. Every argument whose premises are true and whose conclusion is false, is unsound.

2.6: Unsound, with one or more false premises and a true conclusion.

A six inch cube of sand weighs more than a six inch cube of gold. Things that weigh more are always more valuable. Therefore, a six inch cube of gold is more valuable than a six inch cube of sand.

Well, this argument is a real mess, and is consequently very confusing. For starters, it seems to get almost everything wrong. The first premise is obviously wrong. The cube of sand would weigh less, not more, than the cube of gold. What about the second premise? Are heavier things always more valuable? Think of the Gutenberg Bible – the first printed book – and a slightly heavier field stone. Which is more valuable? Obviously the book, even though it weighs less. So the second premise is false too. (Things that weigh more are sometimes more valuable than things that weigh less, but not always.) However, the conclusion is true; the gold is more valuable than the sand.

So the argument does manage to get to a true conclusion. Is it sound? No; it gets that wrong too. To see that it is unsound, you have to imagine what it would be like if the premises were true. Suppose that a six inch cube of sand really was heavier than the same volume of gold. Then if (as the second premise says) heavier things were always more valuable, the sand would have to be more valuable than the gold, not (as the conclusion says) less valuable. So this argument is actually what we could call super-unsound. The truth of the premises would not only permit the falsity of the conclusion (that's enough to make it unsound), they actually require the falsity of the conclusion.

Perhaps you have already noticed that this example is a lot more complicated than it needed to be just to show that it is possible to argue unsoundly from false premises to a true conclusion. Can you think of a simpler unsound argument that has a false premise and a true conclusion?

2.7: Unsound, with one or more false premises and a false conclusion.

Every student loves logic. Therefore, nobody ever eats too much.

Is the premise true or false? How about the conclusion? Unfortunately, they are both false. In a better possible world they would both be true, but not in this one. But is the argument sound? No. Even if every student did love logic, it would not follow that nobody ever eats too much. We can imagine a possible world in which every student really does love logic, but some people still eat too much. The argument is unsound.

It is easy to think of unsound arguments; just think of arguments in which there is no connection between the premises and the conclusion. Soundness is a very high standard. For an argument to be sound, there must be no possible way that the premises could be true without the conclusion also being true. As we will soon see, that goal is all too rarely achieved.

We can summarize our exploration of unsoundness in another table:

Unsound Arguments

Premises	Conclusion	Are there any?
True	True	Yes
True	False	Yes
False	True	Yes
False	False	Yes

Conclusion

Putting together the tables for sound and unsound arguments, we can conclude that you can tell almost nothing about the soundness of arguments by looking at the truth or falsity of their premises and conclusions. There is only one anchor: if the premises are true and the conclusion false, then we know that the argument must be unsound. But that's all we can tell.

Then how do we know whether an argument is sound or unsound? For an argument to be sound the premises and conclusion need to be connected in the right way. They need to be connected in such a way that the truth of the premises forces, or entails, the truth of the conclusion – whether or not the premises happen to be true. The remainder of this text will develop a method, called the method of counterexamples, for demonstrating the unsoundness of unsound arguments. Counterexamples will require us to think not about the actual truth or falsity of premises and conclusion, but about their truth and falsity in a range of possible worlds. Thinking about other possible worlds will give us a sort of back-door access to the connection between premises and conclusion.