What is Logic?

Logic is the study of correct and incorrect reasoning. Logicians want to understand what makes good reasoning good and what makes bad reasoning bad. Understanding this helps us to avoid making mistakes in our own reasoning, and it allows us to evaluate the reasoning of others. It makes us better thinkers.

The first logician was Aristotle (384-322 BCE). But the logic we use and study these days—sometimes called “Modern Logic” or “Contemporary Logic”—was developed in the 19th and early 20th centuries. Important early works include George Boole’s The Laws of Thought (1854), and Gottlob Frege’s Begriffsschrift (“Concept Script”) (1879) and Grundgesetze der Arithmetik ("Basic Laws of Arithmetic) (1892).

Logic lies at the foundation of mathematics, where it allows us to provide a clear and rigorous account of mathematical proof. It also plays a central role in philosophy, where we use it to help reason as clearly and rigorously as possible about hard questions about ourselves, about knowledge, reality, truth, and beauty, and about right and wrong, good and bad. It also lies at the foundation of computer science: a computer is a logic machine. And a mind is, at least in part, a logic machine too, so logic lies at the foundation of cognitive science and philosophy of mind. It also lies at the foundation of linguistics, providing the tools we use for thinking about linguistic structure (syntax) and linguistic meaning (semantics).

Logic is one of the traditional sub-disciplines of Philosophy and one of the seven traditional “liberal arts”, alongside arithmetic, geometry, astronomy, music, grammar, and rhetoric. In a medieval university, students would begin by studying grammar, logic, and rhetoric, before going on to study the four other liberal arts. Advanced students might then study philosophy and theology. (At the time, “philosophy” included disciplines we would now categorize as “sciences”, like physics, biology, and psychology.)

So there are many good reasons to study logic. It will make you more medieval. It will give you insight into linguistics, the foundations of mathematics, and computer science. It will make you a better philosopher. And it will make you a better thinker.

Also, it can be a lot of fun.

Representing Reasoning

Logic aims to develop a theory of good and bad reasoning. But before we can evaluate a given piece of reasoning, we need a way to represent it.

The Structure of Reasoning

When we reason, we reason from some given information to some new information. We might represent this using a visual diagram:

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We will call a piece of reasoning an argument. We will call the information you reason from the premises, and the information you reason to the conclusion:

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So, for example, given the information that Andrew is vegetarian, you might reason to the conclusion that Andrew does not eat chicken:

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But this diagram leaves a lot of important information out. When you reasoned from the premise to the conclusion, you relied your background knowledge about vegetarians and chickens. This gave you additional information, and so additional premises. Here is a more explicit representation of your reasoning:

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Notice that the conclusion does not follow from any one of the premises taken by itself. It only follows from all the premises taken together. So we cannot represent this as three separate arguments for the same conclusion.

We can also reason from one or more premises to one or more conclusions:

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Notice that here each conclusion follows separately from the premises. So we can represent this as two separate arguments, from the same premises, but to different conclusions:

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When we reason from some premises to more than one conclusion, it is always possible to represent the reasoning in this way, with separate arguments. So, for the sake of simplicity, we will do this, and assume that an argument always has exactly one conclusion.

These diagrams provide a nice visual way of representing arguments. We began with them to help emphasize the basic structure of an argument: an argument is a piece of reasoning, from some given information, to a conclusion.

Premise-Conclusion Form

In practice, it is much easier to represent an argument as a simple list: first the premises, followed by the conclusion. So, for example, we can represent the argument above as,

1. Alice, Bob, and Christina are teammates.
2. Alice is the tallest player on her team.
3. ∴ Alice is taller than Christina.

Here the premises are (1) and (2), and the conclusion is (3). We know that (3) is the conclusion because it is the last sentence in the list. We also know that it is the conclusion because it is marked by a special symbol, ‘∴’ which stands for “therefore”. We will always mark conclusions using ‘∴’.

When we represent arguments like this—as a list of premises, followed by the conclusion—we say that we have put the argument in premise-conclusion form. Putting arguments in premise-conclusion form is a common philosophical exercise. It forces you to clearly separate out the premises and the conclusion.

Arguments are everywhere. You can find them in blogs, magazine articles, textbooks, and newspaper editorials. They often pop up in conversations. But they are not usually found in premise-conclusion form, and important premises are often left unstated. Sometimes even the conclusion is left unstated, as in this example, from Confucius:

If there be righteousness in the heart,
there will be beauty in the character.
If there be beauty in the character,
there will be harmony in the home.
If there be harmony in the home,
there will be order in the nation.
If there be order in the nation,
there will be peace in the world.¹

Can you put this argument in premise-conclusion form? What is the conclusion? When you think you know the answer, click on the gray box below.

Here is an argument:

Some pigs have wings. Everything with wings can fly. So, some pigs can fly.

What are the premises of this argument? What is its conclusion? Can you put it in premise-conclusion form? When you think you know the answer, click on the gray box below:

Here is another argument:

We need to raise the capital gains tax. We need to do this because a low capital gains tax provides a disproportionate benefit to the wealthiest citizens. It serves only to increase the gap between the super-wealthy and the rest of us, and we need to decrease that gap.

What are the premises of this argument? What is the conclusion?

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:

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There are two ways things can go wrong here.

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

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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:

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 bad reasoning!

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!

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 bad reasoning!

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

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 good reasoning!

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.²

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 Loose Ends

Weve introduced five key terms: argument, premise, conclusion, valid, sound. The first three are defined in terms of each other:

Finally, we should briefly wrap up two loose ends.

First Loose End: Sentences?

First, the sentences that play the role of premises or conclusions must be sentences that express information. So they have to be declarative sentences—the kinds of sentences that are true or false. They cannot be questions or commands, since questions and commands are not used to express information.

So this is not an argument:

1. It is important to eat fruits and vegetables.
2. ∴ Eat your fruits and vegetables!

Why not?

Second Loose End: Good Reasoning that is not Valid

Second, we said above that good reasoning is valid reasoning, and we gave an official definition of validity. But not all good reasoning is valid in our sense.

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:

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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.

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1. Borrowed from Pospesel, p. 76.↩

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