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:

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For example, we might reason from the information that Susan is a vegetarian to the information that there is nothing on the menu that Susan can eat.

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”>

In logic, we call a chunk of reasoning like this an argument, and we call this sort of diagram an argument map.

The information you reason from is a reason that supports your conclusion. A reason is an answer to a certain sort of “why?” question. If someone asks you, “Why do you think there isn’t anything on the menu that Susan can eat?”, you might answer: “Because she is a vegetarian.”

Several reasons can support the same conclusion. Each reason gives you a separate argument, but we can represent each of those arguments with a single map. For example, here are three reasons students commonly give when asked why they think it is morally okay to eat meat:¹

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Each of these claims offers an independent reason in support of the same conclusion. For example, suppose scientists discover that it is not true that eating meat is necessary for our health. Then that middle reason isn’t any good, and doesn’t support the conclusion. Still, the other two reasons might be good reasons in support of the conclusion.

Let’s review what we’ve said so far:

• Logic is about good and bad reasoning.
• We can represent a piece of reasoning as an argument, and we can represent arguments using argument maps.
• An argument map involves a conclusion (at the top), which is then supported by one or more independent reasons that are meant to support that conclusion. We indicate this structure of support by drawing the reasons beneath the conclusion, and drawing arrows from the reasons to the conclusion.
• Each independent reason is a separate argument. So a single argument map can represent several arguments.

Articulating Reasons into Premises

Let’s go back to that argument about Susan:

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To make this a bit less abstract, let’s imagine a conversation where this argument comes up. Suppose Susan is your vegetarian friend, and she is visiting from out of town. You mention to your friend Ted that you are planning to take her out to dinner at the Rock, and Ted says, “Don’t do that. There is nothing on the menu at the Rock she can eat, because she is a vegetarian.”

Ted’s wrong about something. What is it? (Click on the grey box when you think you know. Hint: if you are not familiar with the Rock, here is their menu.)

So Ted was wrong about something, and it was something that his argument depended on: he was wrong about what was on the menu. But the thing Ted was wrong about wasn’t something he said in conversation: it was an implicit unstated assumption he was making And it doesn’t show up in our argument map. This means that our argument map is not complete: it only presents part of Ted’s reason for thinking that Susan won’t be able to find anything to eat at the Rock—the part Ted made explicit. But if we want a complete understanding of his line of reasoning, we also need to represent his implicit unstated assumption.

Here is a better argument map:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”>

This is a better argument map because it gives us a more complete representation of Ted’s reason. In particular, it allows us to point to where Ted went wrong: part of his reason was false.

So reasons have parts. Each part of a reason is a claim. The parts work together to support the conclusion. So if any of the parts are bad, then the reason is a bad reason. Logicians call the parts of a reason—the claims that work together to support a conclusion—the premises of an argument. And they call the parts that are important but go unstated—like Ted’s assumption about the Rock’s menu—hidden premises.

Above, we saw that the same conclusion can be supported by more than one independent reason:

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Notice how these three reasons are represented as three separate reasons, while Ted’s reason is represented as a single reason that has two parts working together.

Each of these reasons for eating meat is similarly incomplete. Consider the first: Someone who says that it is morally okay to eat meat because people have always eaten meat has made a mistake. Here is how Dan Lowe explains their mistake:

The problem with this argument is that just because people have done something throughout history doesn’t mean it’s morally acceptable. Consider a short list of things which have gone on throughout human history which aren’t morally acceptable: despotism, racism, sexism, and homophobia. In the case of homophobia, consider the way gays and lesbians have been treated throughout human history. Ancient Hebraic law commands gays to be put to death. In the middle ages, gays were punished for homosexuality by castration and death by being burned alive. In the late 19th century, the poet and playwright Oscar Wilde was sent to prison for homosexual acts. Gays were among the groups targeted for extermination during the holocaust. Up until the 1970’s, the American Psychiatric Association classified homosexuality as a mental disorder. And as I write this in 2015, homosexuality is illegal in more than 70 countries worldwide. (Dan Lowe, “Common Arguments for the Moral Acceptability of Eating Meat: A Discussion for Students.” Between the Species 19:1 2016, p. 178)

Dan is pointing out that it is not generally true that it is morally okay to do things that people have always done. So he is suggesting that the argument contains a hidden premise. The complete reason being given in support of eating meat looks something like this:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”>

This is a better argument map because it is more complete. Using this map, we can explain why the reason does not support the conclusion: one part of the reason—the part that went unstated—is false.

Here is an important fact about us and how we reason. Usually, when we give reasons, the statements we make—the premises we speak out loud—are fairly plausible. It’s our unstated (“hidden”) premises that get us in trouble. It’s like we want to hide our lazy reasoning from ourselves. It is often a short step from exposing someone’s hidden premises and showing why they are wrong.

Let’s remind ourselves of what we’ve learned:

• Several independent reasons can support the same conclusion.
• A single reason can have several parts that work together to support the conclusion: we call these parts premises.
• When we give reasons, we often leave some of those parts unexpressed: we call these hidden premises.
• A reason is only good if all of its parts are good.

Let me end with a diagram showing 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”>

Exercises

1. Create complete argument maps for each of the other two commonly given reasons in support of eating meat.
2. What is something controversial that you believe? What are your reasons? Can you construct a map that represents each your independent reasons, articulating each reason into its premises?

A Plug for Some Cool Free Online Software

If you like making argument maps, check out MindMup’s argument visualization tool. It is a lot of fun to play around with.

Done!

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1. It take this example from Dan Lowe, “Common Arguments for the Moral Acceptability of Eating Meat: A Discussion for Students.” Between the Species 19:1 2016. If you are interested in reading more about these arguments, I highly recommend Dan’s paper. He↩