LCA

Welcome to the interactive logic textbook demo!

Logic is a tool that helps us solve problems. Scientists use logic to figure out what theory their data supports. Mathematicians use logic to prove things like the Pythagorean theorem. Children use logic to solve Sudoku puzzles and win chess games.

You can’t learn logic without using it. That’s one reason why this textbook is interactive. You will learn about logic by applying it.

All of the problems in this book are for practice. They aren’t worth points, and if you get them wrong, you get to try again. (Problem sets for your homework are not done in the textbook; your instructor will tell you where to complete those.)

Look for key concepts in the call-out boxes.

Why retry problems? Because the harder you work, the more you will learn and the better you will do.

Let’s give it a try.

Some chapters are built around a theme. For example, in Chapters 1-3 you are an investigative reporter trying to solve a criminal case. The theme gives context to the material you are learning. Plus it makes learning more fun. Now it’s time to learn some logic!

The world is full of problems. Let’s go solve some.

Chapter 1. The Adventure Begins

Learning Objectives

In this chapter you will learn…

• Different meanings of “logic”
• What it means to call logic formal or symbolic
• Why formal logic is a powerful tool
• What logical entailment is
• What makes an argument valid
• The difference between deductive and inductive logic

1.1 Welcome to the Newsroom

A file folder lands on your desk with a thud. You’re the rookie journalist, so the Chief Editor has put you on the story no one else wants. A bank robbery. Inside the file you see pictures of three suspects: Pia, Quinn, and Raquel.

“The police have all three of them in custody,” the Chief says. “We need to break this story first. You need to figure out who did it by 8 a.m.”

Now you know why no one else wanted this story.

It takes you all night but you finally get through the file. By 7 a.m., here’s what you’ve figured out:

1. One or more of these people is guilty: Pia, Quinn and Raquel.
2. No one else was involved.
3. Pia never works without Quinn. So if Pia is guilty, so is Quinn.
4. Raquel never works alone.

Logic is the rules and principles of reasoning that we use everyday. Like when we figure out someone is guilty.

Logic is the rules and principles of reasoning that we use everyday.

Even if you don’t realize it, you are constantly figuring out what to do, what to think, and how to navigate the world. That means you are constantly using logic.

We aren’t normally aware of what principles we are relying on, though, or why they are good. In this class we will learn tools to understand and assess our reasoning.

Logic can also mean the study of our reasoning.

The word “logic” can refer to more than just our reasoning. It can also refer to the study of our reasoning. In that sense, logic is an academic discipline or area of knowledge just like physics, biology, or mathematics.

Physicists build models to study particles and forces. Similarly, in this course we will build models to study reasoning.

The things we will build are called formal logical systems, or just logical systems for short. A logical system is a model that allows us to study how reasoning works.

A logical system is a model we create to study our reasoning.

There is not one logical system, just like there is not one model in physics. Physicists create many different models for many different purposes. We will also create different formal logical systems to study different kinds of reasoning.

The comparison extends further: just like physicists try to discover the basic physical principles that govern the universe, we will seek basic logical principles that have the power to explain our reasoning.

Since every area of knowledge relies on reasoning and problem solving, logic is relevant to just about anything, including reasoning in physics!

1.2 Active Learning

This textbook is interactive for a reason: humans learn best by doing. Research supports that approach for just about every type of learning, from playing the piano to speaking Cantonese.

It is especially true for logic, which is why you must continually answer questions as we go along.

We know this method can be frustrating. For example, you had to figure out who was guilty before we even explained how to approach the problem. Even though it’s frustrating, it is a very effective way to learn.

With your new knowledge of learning, see if you can figure this out:

Even though active learning can be stressful, there are several things you can do to make it more fun and effective. The first is to not make it personal. How you do isn’t a judgment on you as a person.

Constructivism: Students must actively build their understanding of the material.

Solving crimes in a fictional story is one way to get you into the right mindset. It’s not your fault you got stuck in this office with a jerk for a editor. Just remember: we are like your coaches, helping you solve crimes and survive your terrible boss.

The more you practice at active learning the less stressful it becomes, so let’s help decrease the stress by trying another problem.

Another way to prepare yourself for active learning is to realize that logic isn’t a fixed ability you have. Logic is an area that everyone improves at by training.

Logic is an area that everyone improves at by training.

What matters is whether you learn the material, not whether you get a problem right the first time. There’s nothing bad about getting a problem in this textbook incorrect. You can even redo the homework problems and improve your score, up until the deadline set by your instructor.

You might be thinking: then I can game the system; I’ll just take notes and ace the section on the second try!

Go for it. Research suggests that taking notes aids learning regardless of whether you ever look at them again. Plus, strategic planning like that is a great use of logic.

That’s not gaming the system; that is the system.

Lastly, before we go answer this question.

1.3 Logic and Form

There are many ways to study logic. The way we take in this book is called formal or symbolic logic because we will study the form of reasoning and inferences.

Form: a structure or pattern.

The form of something is its structure, a repeatable pattern. For example, consider a form used to make something out of cement or concrete: you can reuse the form to make many objects that have the same shape.

Sentences, arguments, and rules of reasoning all have form.

This is easiest to see with an example. Here’s a form we can reuse to make sentences that share a structure:

If ___ didn’t bring an umbrella, then ___ is going to be wet.

We insert a name in the gaps, sort of like pouring concrete into a form, in order to build a sentence:

If Pia didn’t bring an umbrella, then Pia is going to be wet.

Then we can reuse that form to make another sentence with the same structure.

If Quinn didn’t bring an umbrella, then Quinn is going to be wet.

In order to make it clear that the same name has to go into both gaps, we put a symbol there. For example,

If X didn’t bring an umbrella, then X is going to be wet.

Replacing words with symbols is how we reveal the form of a sentence. Let’s practice.

Replacing words with symbols is how we reveal form. That is why this material is sometimes called symbolic logic.

Here’s why form is a powerful tool.

Imagine you know Quinn. He’s an old friend from high school, but you rarely see him these days. Even though he’s always been a bit wild and sometimes rubs people the wrong way, you’ve always thought he’s a decent guy.

The Chief knows this. He also thinks you are new and inexperienced, and can’t suppress your personal feelings when you need to.

So instead of giving you the original file, he gives you a copy with all the names replaced with letters.

After a long night of work, this time what you’ve figured out is this:

1. One or more of these people is guilty: P, Q, and R.
2. No one else was involved.
3. If P is guilty, then so is Q.
4. R never works alone.

When we replaced the names with letters or symbols, we ignored parts of those sentences.

But we preserved something too: we kept the structural connections between the sentences. That is what we mean by form. The letter Q appeared in the first two sentences, creating a relation between them.

Form is the structural connections in the sentence or argument.

The amazing thing is that to reason the way you did you only needed those structural relations. It’s not just that you could still figure out who is guilty, by reasoning in a more difficult or complicated way. You could figure it out in the exact same way.

The fact that reasoning depends on form or structure in this way was a huge intellectual discovery, sort of like the discovery that the earth revolves around the sun. It is what makes the whole body of knowledge that you learn in this course possible.

Reasoning depends on form: the key idea that makes this course possible.

Formal or symbolic logic is not the only way to study logic, but it has proven to be particularly powerful. In this course we will learn how the same formal tools that allow us to study how a Sudoku puzzle works, or to prove that some politician’s argument is bad, are also what make the microprocessor in your phone or computer work.

At the end of some sections we will have additional practice problems, when mastering some concept essential for progressing.

The concept of form is critical to the rest of this book–after all, books like this are often given names like “Formal Logic”–so you have to figure out this puzzle before you finish the section:

1.4 Entailment = Validity

Later that day you see the Chief talking at the water cooler with Murphy, another editor.

“I think all criminals are felons,” the Chief says.

“Hey, I had a feeling Quinn was guilty,” Murphy replies, “but why do you think that?”

“Because all criminals have broken the law,” says the Chief, “and all felons have broken the law. So all criminals are felons.”

Your job is to evaluate the Chief’s argument:

1. All criminals have broken the law.
2. All felons have broken the law.
So:
3. All criminals are felons.

If you understand why that argument is bad, then you understand the fundamental concept of logic: in logic we want to know when some information guarantees that something is true.

That concept is called deductive logical entailment, or just entailment for short.

Here’s the definition we’ll use a lot throughout this textbook:

A set of premises entails a conclusion just in case this condition is met:

• Whenever the premises are true, then the conclusion is also true.

In other words: If those premises are true, the conclusion must be true.

Entailment: Whenever the premises are true, the conclusion is also true.

Previously we said that we will develop a model or tool, called a logical system, in order to study reasoning. Now we can see one of the key aspects of reasoning we want our tool to model: logical entailment. We want our logical system to show when some information does or doesn’t entail a conclusion.

You can test whether you really understand a concept by seeing if you can put it in slightly different words.

Select all the sentences below that express entailment.

The concept of entailment is so common and important that there are many ways of expressing it in English. Make these sentences all say the same thing as: the premises entail the conclusion.

A valid argument means: the premises entail the conclusion.

Saying an argument is valid might sound different, since it refers to the argument rather than the premises, but it’s still another way of saying the same thing: a valid argument is one whose premises entail the conclusion.

It’s worth noting that the word “valid” has other meanings in English. For example, you might say someone has a “valid point”, which means they are expressing a legitimate concern or issue. In this book, though, we won’t use the word in any of those other ways. We’ll just use it in the specific sense as another way of talking about entailment.

Now let’s put your knowledge into action. Drag the sentences into an argument so that it’s valid.

1.5 Deductive vs. Inductive Logic

In the last section we learned about logical entailment, which happens when the premises guarantee that the conclusion is true. That is the fundamental concept of this book. But it’s not the only sort of logic you can study with formal tools.

Your next task is to assess this argument:

1. There are one million tickets in a fair lottery.
2. Chief bought one ticket.
Thus:
3. Chief won’t win.

If you realized this isn’t a fair question, you’re right. The argument is good in one sense but bad in another. We just have to be clear about what we mean.

If you answered “bad”, you probably realized that the argument isn’t valid. You are right: those premises do not entail that he won’t win. Since he has a ticket, at least he has a chance.

If you answered “good”, you probably realized it is still highly likely that the conclusion is true.

Inductive logic: probability and likelihood.

The logic of likelihood and probability is called inductive logic.

The logic of entailment and validity, by contrast, is called deductive logic.

Deductive logic: guarantee and certainty.

So what we can say is that the argument is deductively bad but inductively good.

Let’s see if you’ve got the idea.

Unfortunately, the person making an argument doesn’t always tell you so clearly whether they are doing induction or deduction. Sometimes we have to use contextual clues and the principle of charity to figure it out.

Principle of Charity: give people the benefit of the doubt and interpret their arguments in a reasonable way, if possible.

For example, imagine that a crime scene investigator has examined the bank robbery, and makes the following argument:

“Gunpowder traces were found on the chair in the middle of the room, and the bullet hole is on the north wall. So the perpetrator fired from the south side of the room.”

Inductive and deductive logic can both be studied with the formal tools we learn in this book. Our focus, however, will be on deduction: the notions of entailment and validity.

Our focus is deduction: entailment and validity.

The main reason why is that you have to learn deductive logic first. Inductive logic is more complicated and presupposes a grasp of deductive logic.

That shouldn’t be surprising: in a sense, deduction is just a special case of induction. A good inductive argument makes the conclusion likely to be true. If the premises guarantee that the conclusion is true, then that is maximally likely.

Now here’s a slightly harder question. The American legal standard for a criminal conviction is “beyond a reasonable doubt”.

Finally, let’s apply what you’ve learned.

Great work–you’re almost done with Chapter 1.

1.6 Extra Practice

Regular, daily practice is much better for learning and long-term memory than “massed practice”, which is a fancy word for cramming.

Massed practice: A fancy name for cramming.

That’s why you should login every day and solve a few problems in this textbook. Re-solving old problems is great for learning, but you might also want to keep trying new problems.

That’s what this section is for. These problems are just here to give you more to practice on.

Congratulations! You have completed Chapter 1 and the LCA demo!