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 other four 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:

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:

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

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:

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:

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:

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 from each other, and from 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?

Review and a Loose End

We’ve introduced three key terms: argument, premise, conclusion, They are defined in terms of each other:

And we’ve introduced a way of representing arguments, by putting them in premise-conclusion form.

We need to add one qualification to our definition of an argument. 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?

Done!

Head back to the table of contents.