Sweet Reason

A Field Guide to Modern Logic

James M. Henle, Jay L. Garfield, Thomas Tymoczko, Emily Altreuter

• Publisher: Wiley-Blackwell
• Year: 2011
• ISBN:
  9781118078686
• Edition: 2
• Topic: Filosofia
• Subtopic: Logica in filosofia

Sweet Reason: A Field Guide to Modern Logic, 2nd Edition offers an innovative, friendly, and effective introduction to logic. It integrates formal first order, modal, and non-classical logic with natural language reasoning, analytical writing, critical thinking, set theory, and the philosophy of logic and mathematics.

• An innovative introduction to the field of logic designed to entertain as it informs
• Integrates formal first order, modal, and non-classical logic with natural language reasoning, analytical writing, critical thinking, set theory, and the philosophy of logic and mathematics
• Addresses contemporary applications of logic in fields such as computer science and linguistics
• A web-site ( www.wiley.com/go/henle ) linked to the text features numerous supplemental exercises and examples, enlightening puzzles and cartoons, and insightful essays

Chapter One

First, a word about this chapter. Let’s say you’re going to learn to swim. You’re 5 years old and a little afraid of the water. Your swimming teacher tells you not to be afraid, and picks you up and throws you into the pool!

You immediately start thrashing about with your arms and legs. You’re really scared, but after a few seconds, you notice that you’re not drowning, you’re keeping your head above water. In a few more seconds, you’ve made your way to the side of the pool and you’re hanging on to the edge trying to figure out what happened.

You didn’t drown because everyone is born with swimming reflexes and instincts. When your teacher threw you in, those reflexes took command and saved you. Now that it’s over, you’re not as frightened of the water. You’ve been in the middle of the pool and survived.

This chapter is a little like that first swimming lesson. You may never have studied logic, but you do, in fact, know quite a bit. If you didn’t, you could hardly speak, let alone make your way in the world.

We’re going to throw everything at you. You’ll be surprised at how easy it is to understand the symbols. It’s easy because the logical ideas represented by the symbols are basic ideas that you’ve worked with all your life.

Logic can seem scary at first. If you don’t know what they mean, strange symbols

can appear frightening …

But don’t panic. The “∀” symbol just means “everything.” You’ll see how it works in a moment. It’s not as mean as it looks.

1.1 Introducing Formal Logic

There was only one catch and that was Catch 22, which specified that a concern for one’s own safety in the face of dangers that were real and immediate was the process of a rational mind. Orr was crazy and could be grounded. All he had to do was ask; and as soon as he did, he would no longer be crazy and would have to fly more missions. Orr would be crazy to fly more missions and sane if he didn’t, but if he was sane he had to fly them. If he flew them he was crazy and didn’t have to but if he didn’t want to he was sane and had to. (Joseph Heller, Catch-22)

We begin with connectives, the logical operations that link sentences to each other. We don’t have many connectives; they’re all familiar to you. You know them as “and”, “or”, “not”, “if … then”, and “if and only if”. Connectives allow us to create complex statements from simple statements. Suppose A and B are statements. Then we’ll use

to say that both A and B are true. We’ll use

to mean that at least one of A, B is true (A is true or B is true or both are true). We’ll use

to mean that A is not true. We’ll use

to mean that if A is true then so is B. And finally we’ll use

to mean that A is true if and only if B is true, that is, A and B have the same truth value.

Let’s say we have these statements:

P: George is late to the meeting.

Q: The meeting is in Detroit.

R: George brings a casserole.

Example

How do we say that either George will be late or he’ll bring a casserole?

Answer:

Example

What does Q ⇒ P mean?

Answer: If the meeting is in Detroit then George will be late.

Example

Represent the following with symbols: The meeting is in Detroit and either George doesn’t bring a casserole or George is late.

Answer: Q ∧ (¬R ∨ P) Note the use of parentheses here. We’ll say more about this later.

Exercises Introducing Formal Logic

Odd-numbered solutions begin on page 350

Translate the following sentences using P, Q, and R from above.

1. George is late and the meeting is in Detroit.

2. If the meeting is in Detroit, then George brings a casserole.

3. Either George is late or he does not bring a casserole.

4. George brings a casserole if and only if the meeting is in Detroit.

5. If George does not bring a casserole, he is not late.

6. If the meeting is in Detroit then George brings a casserole, and if George brings a casserole then he is late.

7. The meeting is in Detroit if and only if both George is late and he doesn’t bring a casserole.

8. The meeting is in Detroit, and either George is late or he brings a casserole.

Determine the meaning of each of the following sentences.

9. P ∨ R

10. R ∧ ¬Q

11. Q ⇒ P

12. R ⇔ ¬Q

13. ¬P ∨ (¬Q ∧ R)

14. P ∧ (Q ∨ R)

15. R ∧ (Q ⇒ P)

16. Q ∨ (¬P ⇔ R)

The Greek philosopher Epimenides is credited with formulating a paradox that has stimulated some of the most important advances in logic from the classical period right up to yesterday afternoon (we guarantee this, no matter when you are reading these words). He, a Cretan, put it this way:

All Cretans are Liars.

Since Epimenides was a Cretan, he was asserting that he is a liar, meaning that what he says is...