Validity and Soundness

How does one set about evaluating an argument? I suggest, first, numbering the premises. But where does one go from there?

Two ways all arguments can fail

A good argument must satisfy two conditions:

I. The Content Condition: The premises are true or reasonable to believe.

II. The Structure Condition: If the premises were true, they would support the conclusion.

Here is an example of an argument that satisfies I but not II:

1. If John is a mammal, then John is warm-blooded.

2. Jupiter is a gas giant.

3. So, Kramer is the neighbor of Seinfeld.

This argument contains only true claims, but it is bad because it fails condition II.

Here is an argument that satisfies II but fails I:

1. If Beyoncé is a kangaroo, then Beyoncé can lay eggs.

2. Beyoncé is a kangaroo.

3. So, Beyoncé can lay eggs.

This argument contains all false claims, but it satisfies II. If the premises were true—they aren’t, but if they were—then they would support the conclusion.

These are zany examples, but imagine this playing out as follows. Someone might give this argument against the existence of God:

1. Evil exists.

2. If God exists, evil would not exist.

3. So, God does not exist.

This is a simple version of the argument from evil. Does it satisfy the Structure Condition? Surely it does. If these premises were true, they would provide genuine support for the conclusion. The only question, then, is whether the premises really are true. If you want to resist the conclusion, you have to find a premise to reject. Which would you reject? Surely, it is plausible that evil exists. There are people who murder for fun. Most likely, then, we should reject premise 2. Knowing the argument has good structure shows us we must evaluative its content.

Validity

A valid argument is one that satisfies the Structure Condition in a special way. Here are three equivalent definitions of validity:

Definition 1: A valid argument is one such that if the premises were true, the conclusion would have to follow.

Definition 2: A valid argument is one such that the truth of the premises would guarantee the truth of the conclusion.

Definition 3: A valid argument is one such that there is no possible case in which the premises are true but the conclusion is false.

However different these definitions may seem, they are in fact equivalent. (Still, if you need to just pick one to memorize, I suggest the first.)

Notice that valid is here a technical word. It does not mean ‘plausible’ or ‘defensible’ or ‘understandable’. In this technical sense, viewpoints and statements cannot be valid or invalid; only arguments can, and arguments are not statements. Rather, an argument is some statement(s) meant to support some other statement.

Conversely, arguments cannot be true or false. Only statements can be true or false, and arguments are not statements. Arguments are sets of statements. The premises of an argument can be true or false, and so can the conclusion; but not the argument itself.

Validity is a special way of satisfying the Structure Condition, because here, the premises, if true, provide very powerful support for the conclusion. Indeed, if true, the premises guarantee the conclusion, where other arguments might simply make their conclusions likely.

Deduction and Induction

An argument that aims at making its conclusion certain is a deductive argument. An arguments that aims at making its conclusion likely is an inductive argument. Deductive arguments seek to be valid. But inductive arguments can be perfectly good even if they are not valid. For example:

1. Every day so far the sun has risen.

2. So, tomorrow it will rise again.

That’s a perfectly good inductive argument, but it doesn’t make its conclusion certain. After all, there will come a day, alas, when the premise is true but the conclusion is false. So, the premise does not guarantee the conclusion, and the argument, taken as deductive, would not be valid.

For another sort of inductive argument, check out inference to the best explanation.

Why validity matters

Validity is important in evaluating arguments. Here’s an example. People often reason like this:

Other cultures have differing moral beliefs. In Eskimo tribes, they leave unwanted babies to die in the cold. In some South American tribes, it is considered acceptable to kill a stranger in the forest and take his head as a trophy. Who are we to say our morality is any better? There is no objective moral truth.

This bit of reasoning—which, make no mistake, I reject—can be formulated like this:

1. Moral beliefs vary from culture to culture.

2. So, there is no objective moral truth.

This argument, however, is not valid. Even if 1 is true, it does not guarantee 2. We have to add a premise to make it valid. Can you think what premise we should add?

It’s this one (I’ll call it 1.5):

1. Moral beliefs vary from culture to culture.

1.5. If moral beliefs vary from culture to culture, then there is no objective moral truth.

2. So, there is no objective moral truth.

Now the argument is valid. Note that we can always fix the validity of an argument by adding a “bridge” premise such as 1.5, an ‘if-then’ statement that takes you from another premise to the conclusion.

Of course, while adding a premise might fix the validity, there will then be a question as to whether the added premise is true. That is exactly what we should ask in this case. Is 1.5 true? Why should we accept it? For now, I’ll leave it to you to consider.

Soundness

A sound argument is one that satisfies two conditions:

A. It is valid.

B. All of its premises are true.

Deductive arguments aim at soundness. Just consider: Can a sound argument have a false conclusion?

If you think about it, the answer is No. After all, a sound argument is valid, which means if the premises are true, the conclusion has to be true; and in a sound argument, the premises are true. So, it’s clear why soundness would be a desirable feature of an argument.

However, soundness isn’t the ultimate aim. The ultimate aim is persuasiveness or convincingness. And not all sound arguments are convincing. Consider:

1. Some fire trucks are red.

2. So, some fire trucks are red.

This argument is valid and its premise is true. So, the argument is sound. But obviously, it is not convincing. If someone had reasonable doubts about the conclusion, he could not resolve them by considering the premise. This premise does not offer a way of acquiring justification for the conclusion. In sum, then, all convincing deductions are sound, but not all sound deductions are convincing.

Self-test

1. What is the definition of validity? I gave three. Can you recall just one?

2. What is the definition of soundness?

3. Write an example of a valid argument with absurdly false premises.

4. Can a sound argument have a false conclusion? Explain your answer.