Evaluating Arguments

Good and Bad Arguments

Not all reasoning is good reasoning, and so not all arguments are good arguments. What makes a good argument good and a bad argument bad?

Recall the basic structure of a piece of reasoning:

There are two ways things can go wrong here.

First, we can plug bad information into to the premise box:

You are probably familiar with the saying “Garbage in, garbage out”. When you reason from false premises, it doesn’t matter how well you reason. You’ll have no reason to suppose that your conclusion is true.

But suppose the information you plug into the premise box is true. Still things can go wrong: you might reason badly from those premises to your conclusion:

When you reason badly, it doesn’t matter that you began with true premises. You’ll have no reason to suppose that your conclusion is true.

And, of course, it is possible to reason badly from bad information!

So what we want in a good argument is good reasoning from good information:

When we reason correctly from true premises, we can be confident that our conclusion is true.

Now for some examples. Consider the argument,

1. No mammals lay eggs.
2. All platypus are mammals.
3. ∴ All platypus don’t lay eggs.

Here the reasoning is good. The conclusion follows from the premises: if they were both true, it would have to be true too. But the first premise is false. So this is a case of reasoning well from bad information. As you can see, it led us to a false conclusion.

Or consider,

1. All ducks are mammals.
2. All mammals are birds.
3. ∴ All ducks are birds.

Again, the reasoning is good. The conclusion follows from the premises: if they were both true, it would have to be true too. But again, we are reasoning from false premises. So this is a case of reasoning well from bad information. As you can see, we happened to be led to a true conclusion, but only by accident.

Here is an argument with true premises:

1. Some ducks are birds.
2. No birds are mammals.
3. ∴ No ducks are mammals.

Here the premises are true, and the conclusion is true. But does the conclusion follow from the premises, or is this a case of bad reasoning?

One last example. Consider the argument,

1. Obama is President.
2. If Obama is President, then a Democrat is President.
3. ∴ A Democrat is President.

The premises are both true. And the conclusion follows from the premises: there is no way that they could be true but it false. So the conclusion is true, and it doesn’t just happen to be true. It’s truth was necessitated by the truth of the premises. So, finally, we have an example of good reasoning from good information!

Soundness and Validity

As we have seen, arguments are kind of like friends. A friend can be good in one way but bad in another—a good listener but bad at keeping secrets, for example. And an argument can be good in one way but bad in another—good reasoning but bad premises, or good premises but bad reasoning.

It is helpful to introduce some technical terms. We will call an argument that has good reasoning a valid argument. And we will call an argument that has good reasoning from good premises a sound argument.

So this argument is sound:

1. Obama is President.
2. If Obama is President, then a Democrat is President.
3. ∴ A Democrat is President.

And these arguments are valid but not sound:

1. No mammals lay eggs.
2. All platypus are mammals.
3. ∴ All platypus don’t lay eggs.

1. All ducks are mammals.
2. All mammals are birds.
3. ∴ All ducks are birds.

While this argument is neither valid nor sound:

1. Some ducks are birds.
2. No birds are mammals.
3. ∴ No ducks are mammals.

It is clear enough what it is for an argument has good or bad premises: a good premise is a true premise, and a bad premise is a false premise. But what does it mean to say that an argument has good reasoning?

Reasoning is good when the conclusion “follows”. But even that is just a metaphor. When we put an argument in premise-conclusion form, the conclusion always comes after the premises, so in a sense it always “follows” the premises. But that isn’t what we mean when we say that the conclusion follows from the premises. So what do we mean?

Roughly, what we mean is that, if the premises were true, then the conclusion would have to be true too. That is, we mean that there is no possible situation in which all the premises are true, but the conclusion false. So let this be our official definition of “validity”:

Here is an example of an invalid argument:

1. If Obama is President, then a Democrat is President.
2. A Democrat is President.
3. ∴ Obama is President.

As I write this, (1), (2) and (3) are all true. But the reasoning is bad; the conclusion does not follow from the premises; the argument is not valid. This is because there is a possible situation in which the premises are all true, but the conclusion false. Can you describe such a situation?

In general, you cannot tell whether or not an argument is valid just by considering whether or not its premises and conclusion are actually true or false. Validity is about what is or isn’t possible, not just about what happens to be the case. So you have to consider all the possibilities; you have to use your imagination.

Now that we have an official definition of validity, it is easy to provide an official definition of soundness:

Remember, a sound argument is an argument with good reasoning—that is, a valid argument—from good information—that is, true premises.

In philosophy—and, indeed, in daily life—we want our arguments to be sound. As responsible reasoners, we want to be reasoning validly from true premises, since that ensures that we have reached a true conclusion. We want to put good information into our reasoning machine, and we want our machine to process that information correctly, so we get good information out of the machine.

But there is no general logical theory that allows us to assess whether or not a given premise is true or false. If you want to know whether or not all mammals lay eggs, you need to ask a biologist, not a logician. And there is no way to determine, using logic alone, who happens to be President, whether Andrew is vegetarian, and so on.

What logic can provide is a theory of good reasoning—a theory of validity, a theory of what follows from what.

Let’s work through some more examples.

1. No cars are allowed in the park.
2. All police cruisers are cars.
3. Therefore, all police cruisers are not allowed in the park.¹

Is this argument valid? If it is not valid, can you describe a situation in which the premises are all true, but the conclusion false? If is valid, is it also sound?

1. Some pigs have wings.
2. Everything with wings can fly.
3. ∴ Some pigs can fly.

Is this argument valid? If it is not valid, can you describe a situation in which the premises are all true, but the conclusion false? If is valid, is it also sound?

1. A good university must have a good library.
2. ISU has a good library.
3. ∴ ISU is a good university.

Is this argument valid? If it is not valid, can you describe a situation in which the premises are all true, but the conclusion false? If is valid, is it also sound?

Can a valid argument have a false conclusion? If so, what would an example be?

Can a valid argument with all true premises have a false conclusion? If so, what would an example be?

Can a valid argument have false premises but a true conclusion? If so, what would an example be?

Suppose you know that an argument is sound. What do you know about its conclusion? Stop and think about this—the answer is implicit in the definitions of validity and soundness just given.

It is easy to come up with a valid argument for the existence of God, but hard to come up with an uncontroversially sound argument for the existence of God. Try it! Try to formulate a valid argument whose conclusion is ‘God exists’. Are the premises uncontroversially true?

Again, it is easy to come up with a valid argument for the non-existence of God, but hard to come up with an uncontroversially sound argument for that same conclusion. Try it!

Review and a Loose End

We’ve introduced two new key terms: valid, sound:

But here is the loose end. All valid reasoning is good reasoning, but not all good reasoning is valid.

Consider an eighteenth century European biologist, who has made a wide and careful study of mammals, and never encountered a mammal that lays eggs. Suppose she infers that no mammals lay eggs:

This seems like a reasonable inference for her to have made: it was based on a large set of good data, carefully gathered and considered. But it is not a valid inference: it is quite possible that the premises all be true but the conclusion false, as actually happened in this case.

Or consider a scientist who is interested in explaining the Cretaceous–Paleogene extinction event: the mass extinction that wiped out all the dinosaurs (except the birds, of course). The evidence strongly suggests that the extinction was caused, at least in part, by the impact of a large asteroid. So the scientist infers that the extinction was probably caused by a large asteroid.

Again, this seems like a reasonable inference. But it is not a valid inference in our sense. It is possible that, despite all the evidence, the extinction was caused by something else.

These examples show that not all good reasoning is valid reasoning. Some good reasoning is probabilistic or inductive: the given information supports a certain conclusion, but not absolutely, as there is still the chance that the conclusion is false despite the evidence.

The study of this kind of reasoning is the domain of inductive logic and probability theory. We are not going to be studying inductive logic or probability theory. We are going to be studying deductive logic. Deductive logic attempts to provide a theory of validity—iron-clad reasoning where the premises leave no possibility that the conclusion be false.

Done!

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