Philosophy 112

Evaluating Arguments

“A deduction is speech in which, certain things having been supposed, something different from those supposed results of necessity because of their being so.” — Aristotle

Before reading this supplement, you might want to watch a couple of short videos:

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?

The last module ended with a diagram representing the general form of an argument map:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G reason1 Premise 1 Premise 2 … conclusion Conclusion reason1->conclusion reason2 Premise 1 Premise 2 … reason2->conclusion reason3 Premise 1 Premise 2 … reason3->conclusion

Here we have three independent reasons, and so three separate arguments, each consisting of several premises, supporting one conclusion. Because the three arguments are independent, we can evaluate each one separately. That means that, when it comes to evaluation, we only need to consider one argument at a time:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise Premise 1 Premise 2 … conclusion Conclusion premise->conclusion

Argument maps are powerful tools for representing arguments. But when we are just looking at a single argument, it is simpler to represent it in premise-conclusion form, by listing each premise, followed by the conclusion:

  1. Premise 1
  2. Premise 2
  3. ∴ Conclusion

The ‘∴’ symbol is read as “therefore”, and marks what follows it as the conclusion.

In logic, we distinguish two ways that an argument can go wrong. First, as we’ve already seen, an argument can go wrong because one or more of the premises is false:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise Susan is a vegetarian There are no vegetarian options on the menu. conclusion There is nothing on the menu that Susan can eat. premise->conclusion

Or, in premise-conclusion form,

  1. Susan is a vegetarian.
  2. There are no vegetarian options on the menu.
  3. ∴ There is nothing on the menu that Susan can eat.

Premise (2) is false: there are vegetarian options on the menu. That is why this argument is a bad argument: it starts from a bad input.

The second way an argument can go wrong has to do with the reasoning itself:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise Premise 1 Premise 2 … conclusion Conclusion premise->conclusion

bad reasoning!

For example, suppose a Getaway driver knows that if he is caught speeding, he will get in trouble. So he drives his buddies slowly away from their latest bank heist, thinking to himself: “If I am caught speeding, I will get in trouble. I will not be caught speeding. So I will not get in trouble.”

He has made a mistake. His mistake does not lie in his premises—they are both true:

  1. If I am caught speeding, I will get in trouble.
  2. I will not be caught speeding.

His mistake instead lies in how he reasoned from the premises to his conclusion:

  1. ∴ I will not get in trouble.

This simply doesn’t follow. In fact, if he doesn’t drive faster, he and his buddies are going to get caught, and then they will get in trouble.

That is, if we want to locate his error on an argument map, it is not that his reason is bad because part of it is false. It is that the link between his reason and the conclusion has gone wrong:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise If I am caught speeding I will get in trouble I will not be caught speeding. conclusion I will not get in trouble premise->conclusion

bad reasoning!

So there are two ways an argument can go bad: it can suffer from bad input—that is, it can have one or more false premises—or it can suffer from bad processing—that is, the premises, even if they were true, wouldn’t support the conclusion.

Validity: Good Processing

When we reason from premises to a conclusion, we want to preserve truth. That is, if our premises are true, we want a guarantee that our conclusion is also true. In logic, we call this relationship validity:

An argument is valid just in case it is impossible for all of its premises to be true but its conclusion false.

Consider the following argument:

  1. A Republican is President.
  2. Trump is a Republican.
  3. ∴ Trump is President.

The premises are both true, and so is the conclusion. But is the argument valid?

It is not valid. Suppose Pence were President. Then (1) and (2) would be true, but (3) would be false. So it is possible for the premises to all be true but the conclusion false.

As this example illustrates, thinking about validity requires imagination. You have to go beyond what is actually true or false, and consider what could be true and what could be false.

Here is another example:

  1. Trump is President.
  2. If Trump is President, then a Republican is President.
  3. ∴ A Republican is President.

The premises are both true, and so is the conclusion. Is this argument valid?

Yes! This argument is valid. There is no way that the premises could both be true but the conclusion false. Imagine a situation in which it is false that a Republican is President. Perhaps you have imagined a situation in which someone other than Trump is President. If so, (1) is false in that situation. So let’s see if we can imagine (3) being false while (1) is true…. We can! Just imagine that Trump were not Republican but Democrat. Then (1) is true and (3) is false. But, in that situation, (2) is false. So there is no way to imagine (3) being false while both (1) and (2) are true together.

Is this argument valid?

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

Yes! It is valid. (1) and (3) are both false. But if (1) and (2) were both true, (3) would have to be true too. So the reasoning—the information processing—is good, even if one of the premises—the information input—is bad.

Valid or invalid?

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

Valid! Both premises are actually false. But if the premises were both true, the conclusion would have to be true too. Don’t be distracted by the fact that the conclusion is actually true: that doesn’t help show that it is supported by these premises. Plenty of invalid arguments have true conclusions.

Can you give an example of an invalid argument with a true conclusion?

That was an open-ended question, and this is just a dumb webpage, so it can’t really give you intelligent feedback. An easy way to construct an example is to pick a true conclusion, and then pick premises that have no bearing on that conclusion. For example:

  1. People can be weird.
  2. Fish have gills.
  3. ∴ There are a lot of rabbits on the ISU campus.

The conclusion is actually true, but it is easy to imagine possible situations where (1) and (2) are both true, but (3) is false.

Is this argument valid?

  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?

The conclusion does not follow. Suppose that some ducks were birds and some ducks were mammals. Then the premises would be true, but the conclusion false. So this is an example of bad reasoning from good information. The conclusion happens to be true, but only by accident. So the argument is not valid.

Soundness

As we’ve seen, there are lots of valid arguments that have false premises. In philosophy—and, indeed, in daily life—we want our arguments to be valid AND have all true premises. We call such arguments sound:

An argument is sound just in case it is valid and all of its premises are true.

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. And logic won’t tell you whether or not the menu at the Rock is vegetarian friendly.

What logic can provide is a theory of validity—a theory of good reasoning; a theory of good information processing; a theory of what follows from what. So that’s what we will do in this course.

Augustus De Morgan (1806-1871)
Augustus De Morgan (1806-1871)

Whether the premises be true or false, is not a question of logic, but of morals, philosophy, history or any other knowledge to which their subject-matter belongs. The question of logic is: does the conclusion certainly follow if the premises be true?

Let’s work through some more examples.

No Cars Allowed
No Cars Allowed
  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?

It is valid. It is hard to say whether or not it is sound. Premise (2) is true, but without further information about the park and its rules, we don’t know whether or not premise (1) is true. But if we can trust the sign, then the premises are both true, and the argument is valid, so it is 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?

It is valid, but it is not sound, because both of the premises are false (can you explain why is (2) false?)

  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?

It is not valid. Consider a possible situation in which ISU has a good library, and a good university must have a good library and a good philosophy department, but ISU does not have a good philosophy department. In that situation, both premises are true, but the conclusion is false.

(Since it is not valid, it can’t be sound, so we don’t need to worry about whether or not the premises are actually true.)

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

Yes, a valid argument can have a false conclusion. We’ve already seen an example:

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

To check if the example you came up with was correct, ask yourself two questions:

  1. Is the conclusion false?
  2. If all the premises were true, would the conclusion have to be true?

If the answer to both of these question is yes, then your example was correct.

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

No! If the argument is valid, then there is no possible situation in which all the premises are true, but the conclusion false.

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

Yes! Here is one example:

  1. Fish are mammals.
  2. Whales are fish.
  3. ∴ Whales are mammals.

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.

If an argument is sound, all of its premises are true and it is valid. That is (plugging in the definition of validity), all of its premises are true and it is impossible for all of its premises to be true but its conclusion false. So if an argument is sound, we know that its conclusion is true.

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’. (You might start by constructing an argument map representing each independent reason you can think of in support of the conclusion.) Are the premises of your argument uncontroversially true?

That was an open-ended question, so I hope you didn’t click on this box hoping for an answer! Here is one example of a valid argument for the existence of God:

  1. The moon is made of green cheese.
  2. If the moon is made of green cheese, then God exists.
  3. ∴ God exists.

This argument is valid—-if the premises were true, the conclusion would have to be true too. But the first premise is clearly false, so the argument is not sound.

Here is another example of a valid argument for the existence of God:

  1. The Bible says God exists.
  2. Everything the Bible says is true.
  3. ∴ God exists.

Again, this is valid—if the premises were true, the conclusion would have to be true too. But is it sound? Are the premises true? Some Christians believe that it is—they believe that both premises are true. But many Christians and most non-Christians will reject premise (2), claiming that it is false. Who’s right and who’s wrong? Ask the philosophers and theologians: that’s not a question that logic can answer.

Famously, Paul says, in Titus 1:12-13,

One of themselves, even a prophet of their own, said, the Cretans are always liars, evil beasts, slow bellies. This witness is true.

But what that Cretan said cannot be true, if all Cretans always lie! This is an example of the Liar Paradox (though it is not clear whether Paul meant it to be so). When Paul says “This witness is true”—in other words, that what the Cretan said was true—he says something false.

So perhaps logic can show that not everything said by Paul in the Bible is true. Does that show that (2) is false?

I don’t mean to pick on Theists here. 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! (Again, you might find it helps to begin by constructing an argument map, and then focusing in on the more promising reasons you’ve come up with, and attempting to articulate their premises.)

Here is an example of a valid argument that is obviously unsound:

  1. The moon is made of green cheese.
  2. If the moon is made of green cheese, then God doesn’t exist.
  3. ∴ God doesn’t exist.

Here is an example of a valid argument whose soundness is unclear:

  1. If God existed, there would be no evil in the world.
  2. There is evil in the world.
  3. ∴ God does not exist.

Some philosophers argue that (1) and (2) are both true, and so this argument is sound. Others reject the truth of (1) or (2), claiming that the argument is valid but not sound. Who’s right and who’s wrong? Again, this is not a question that can be answered by logic alone.

Review and a Loose End

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

Validity
An argument is valid just in case it is impossible that all the premises be true but the conclusion false.
Soundness
An argument is sound just in case it is valid and all of its premises are true.

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:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise all mammals that I have observed gave live birth conclusion no mammals lay eggs premise->conclusion

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. In fact, in this case, the conclusion turned out to be actually false, as the discovery of the Platypus showed.

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, who are dinosaurs). 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 in this course. 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. That’s not because inductive logic and probability theory are unimportant!

Done!

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  1. I borrow this example, and the associated photograph, from Pospesel and Marans, Arguments: Deductive Logic Exercises.↩