形式逻辑与推理基础

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"该文本是为初次接触形式逻辑的学员设计的全面且严谨的入门教程。通过丰富的习题集和清晰的解释，帮助学生掌握基本概念，并从命题逻辑到一阶逻辑逐步深入。涵盖的内容包括有效论证形式、推理规则、等价替换规则以及间接证明方法等，涉及人工智能和哲学领域的基础逻辑知识。" 在逻辑学中，特别是命题逻辑和一阶逻辑，理解和掌握有效的论证形式至关重要。这些论证形式是逻辑推理的基础，它们在人工智能和哲学领域广泛应用。以下是标题和描述中提到的一些关键知识点： 1. **有效论证（Valid Argument）**：一个论证如果其前提为真，则结论必然也为真，那么这个论证就是有效的。有效论证的形式是逻辑推理的核心。 2. **推理形式**： - **模态 Ponens (MP)**：如果已知 "如果 p，则 q"（p → q）并且已知 "p"，那么可以推导出 "q"。 - **模态 Tollens (MT)**：如果已知 "如果 p，则 q"（p → q）并且已知 "非 q"（¬q），则可以推导出 "非 p"（¬p）。 - **析取三段论 (DS)**：如果已知 "p 或 q"（p ∨ q），并且已知 "非 p"（¬p），则可以推导出 "q"；反之亦然。 - **简化 (Simp)**：如果已知 "p 且 q"（p ∧ q），则可以推导出 "p" 或 "q"。 - **合取 (Conj)**：如果已知 "p" 和 "q"，则可以推导出 "p 且 q"。 - **假设性三段论 (HS)**：如果已知 "如果 p，则 q" 且 "如果 q，则 r"，那么可以推导出 "如果 p，则 r"。 - **加法 (Add)**：如果已知 "p"，则可以推导出 "p 或 q"。 - **构造性二难困境 (CD)**：如果已知 "p 或 q" 且 "如果 p，则 r" 以及 "如果 q，则 s"，那么可以推导出 "r 或 s"。 3. **有效等价替换形式（Rule of Replacement）**： - **双重否定 (DN)**：任何命题 "p" 都可以等价于 "非非 p"（¬¬p）。 - **德摩根定律 (DeM)**："p 且 q" 的否定等价于 "非 p 或非 q"，而 "p 或 q" 的否定等价于 "非 p 且非 q"。 - **交换律 (Comm)**："p 或 q" 可以等价地写作 "q 或 p"，"p 且 q" 可以等价地写作 "q 且 p"。 - **结合律 (Assoc)**："p 或 (q 或 r)" 可以等价地写作 "((p 或 q) 或 r)"，"p 且 (q 且 r)" 可以等价地写作 "((p 且 q) 且 r)"。 - **分配律 (Dist)**："p 且 (q 或 r)" 可以等价地写作 "(p 且 q) 或 (p 且 r)"，"p 或 (q 且 r)" 可以等价地写作 "(p 或 q) 且 (p 或 r)"。 - **反证法 (Contra)**："如果 p，则 q" 等价于 "如果非 q，则非 p"。 - **蕴含 (Impl)**："如果 p，则 q" 等价于 "非 p 或 q"。 - **出口律 (Exp)**："如果 (p 且 q) 则 r" 等价于 "p 且 (q 则 r)"。 - **恒等式 (Taut)**："p" 等价于 "p 且 p"，也等价于 "p 或 p"。 - **等价 (Equiv)**："如果 p，则 q" 等价于 "(如果 p，则 q) 且 (如果 q，则 p)"，也等价于 "(p 且 q) 或 (非 p 且非 q)"。 4. **条件证明 (Conditional Proof)**：如果已知 "如果 p，则 q"（→p），并使用假设 p 进行一系列推理得到 q，那么可以得出 q（CP）。 5. **间接证明 (Indirect Proof)**：如果要证明 "非 p"，可以通过假设 p 是真的，然后推理出矛盾来完成证明（→~p，AP，/∴p）。这些逻辑规则和论证形式是理解命题逻辑和一阶逻辑的基础，对于学习者来说，掌握它们能够更好地进行逻辑推理和批判性思考。在人工智能领域，这些规则用于构建和分析算法的逻辑结构；在哲学中，它们则用于严密论证和批判性分析。

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Chapter One

Introduction

The Elements of an Argument

Consider the following simple example of reasoning:

Identical twins often have different IQ test scores. Yet such twins inherit the same

genes. So environment must play some part in determining IQ.

Logicians call this kind of reasoning an argument. (But they don’t have in mind shouting

or fighting. Rather, their concern is arguing for or presenting reasons for a conclusion.) In

this case, the argument consists of three statements:

1. Identical twins often have different IQ test scores.

2. Identical twins inherit the same genes.

3. So environment must play some part in determining IQ.

The first two statements in this argument give reasons for accepting the third. In logic

terms, they are said to be premises of the argument, and the third statement is called the

argument’s conclusion. An argument can be defined as a series of statements, one of

which is the conclusion (the thing argued for) and the others are the premises (reasons for

accepting the conclusion).

In everyday life, few of us bother to explicitly label premises or conclusions. We usu-

ally don’t even bother to distinguish one argument from another. But good writing pro-

vides clues that signal the presence of an argument. Such words as “because,” “since,” and

“for” usually indicate that what follows is a premise. And words such as “therefore,”

“hence,” “consequently,” and “so” usually signal a conclusion. Similarly, such expressions

as “It has been observed that . . . ,” “In support of this . . . ,” and “The relevant data . . .”

generally introduce premises, whereas expressions such as “It follows that . . . ,” “The

result is . . . ,” “The point of all this is . . . ,” and “The implication is . . .” usually signal

conclusions. Here is a simple example:

Since it’s wrong to kill a human being, it follows that abortion is wrong, because

abortion takes the life of (kills) a human being.

In this example, the words “since” and “because” signal premises offered in support of the

conclusion signaled by the phrase “it follows that.” Put into textbook form, the argument

reads,

1. It’s wrong to kill a human being.

2. Abortion takes the life of (kills) a human being.

∴ 3. Abortion is wrong.

The symbol “∴” represents the word “therefore” and indicates that what follows is a

conclusion. This particular argument has two premises, but an argument may have any

number of premises (even only one!) and may be surrounded by or embedded in other

arguments.

Not just any group of sentences makes an argument. The sentences in an argument

must express statements—that is, say something that is either true or false. Many sen-

tences are used for other purposes: to ask questions, to issue commands, or to give vent to

emotions. In ordinary contexts none of the following express statements:

Open the door. (command)

Who’s the boss here? (question)

Thank goodness! (expression of emotion)

Of course, sometimes nondeclarative sentences are indeed used to make statements.

“Who’s the boss here?” can be used to make a statement, particularly if the boss is talk-

ing. In this case the boss is not really asking a question at all, but rather is saying, “I am

the boss here,” thus declaring a fact under the guise of asking a question.

But even if every sentence in a group of sentences expresses a statement, the result is

not necessarily an argument. The statements must be related to one another in the appro-

priate way. Something must be argued for (the conclusion), and there must be reasons for

accepting the conclusion. Thus, mere bald assertions are not arguments, anecdotes gener-

ally are not arguments, nor are most other forms of exposition or explanation. It’s impor-

tant to understand the difference between rhetoric that is primarily expository or explana-

tory and rhetoric that is basically argumentative. A passage that contains only exposition

gives us no reason to accept the “facts” in it other than the authority of the writer or

speaker, whereas passages that contain arguments give reasons for some of their claims

(conclusions) and call for a different sort of evaluation than merely an evaluation of the

authority of the writer.

Examples

Two of the following groups of statements constitute arguments, and two do not.

These examples also illustrate that although words such as “therefore” and “because”

usually signal the presence of an argument, this is not always the case.

1. I believe in God because that is how I was raised. (This is biography, not an argu-

ment. “Because” is used here to indicate the cause of the speaker’s belief—to give

an explanation—not to signal a premise.)

2 Introduction

2. I believe in God because life has meaning. If there is no God, life would be mean-

ingless. (This is an argument. The speaker is advancing a reason to believe that

God exists.) Here is the argument put into textbook form:

1. Life has meaning.

2. If there were no God, life would be meaningless.

∴ 3. God exists.

(Notice that in this case the word “because” does signal that a premise is to follow.)

3. Biff was obviously afraid of making a commitment to a long-term relationship.

Therefore, Susie was not surprised when they eventually broke up. (This is not an

argument. This is an explanation of why Susie was not surprised.)

4. We’llget a tax break if we marry before the end of the year. Therefore, I think we

should move our wedding date up and not wait until January. (This is an argument.)

1. We’ll get a tax break if we marry before the end of the year.

∴ 2. We should move our wedding date up and not wait until January.

Exercise 1-1

Here are fifteen passages (the first six are from student papers and exams, modestly

edited). Determine which contain arguments and which do not. Label the premises and

conclusions of those that do, and explain your answers. Paraphrase if that makes things

clearer. (Even-numbered items in most exercise sets are answered in a section at the back

of the book.)

1. I don’t like big-time college football. I don’t like pro football on TV either. In fact,

I don’t like sports, period.

2. My summer vacation was spent working in Las Vegas. I worked as a waitress at the

Desert Inn and made tons of money. But I guess I got addicted to the slots and

didn’t save too much. Next summer my friend Hal and I are going to work in Reno

if we can find jobs there.

3. Well, I have a special reason for believing in big-time college football. After all, I

wouldn’t have come here if Ohio State’s football team hadn’t gone to the Rose Bowl,

because that’s how I heard about this place to begin with.

4. At the present rate of consumption, the oil will be used up in 20 to 25 years. And

we’re sure not going to reduce consumption in the near future. So we’d better start

developing solar power, windmills, and other “alternative energy sources” pretty

soon.

5. The abortion issue is blown all out of proportion. How come we don’t hear nearly as

much about the evils of birth control pills? After all, a lot more potential people are

“killed” by the pill than by abortion.

6. I’ve often wondered how they make lead pencils. Of course, they don’t use lead, they

use graphite. But I mean, How do they get the graphite into the wood? That’s my

Introduction 3

problem. The only thing I can think of is maybe they cut the lead into long round

strips and then cut holes in the wood and slip the lead in.

7. Punishment, when speedy and specific, may suppress undesirable behavior, but it

cannot teach or encourage desirable alternatives. Therefore, it is crucial to use posi-

tive techniques to model and reinforce appropriate behavior that the person can use

in place of the unacceptable response that has to be suppressed.

—Walter and Harriet Mischel, Essentials of Psychology

8. There was no European language that Ruth could not speak at least a little bit. She

passed the time in the concentration camp, waiting for death, by getting other pris-

oners to teach her languages she did not know. Thus did she become fluent in

Romany, the tongue of the gypsies. —Kurt Vonnegut, Jailbird

9. The death of my brother was another instance in which I realized the inadequacy of

the superstition of progress in regard to life. A good, intelligent, serious man, he was

still young when he fell ill. He suffered for over a year and died an agonizing death

without ever understanding why he lived and understanding even less why he was

dying. No theories could provide any answers to these questions, either for him or

for me, during his slow and painful death. —Leo Tolstoy, Confession

10. To be sustained under the Eighth Amendment, the death penalty must “comport with

the basic concept of human dignity at the core of the Amendment”; the objective in

imposing it must be “consistent with our respect for the dignity of other men.” Under

these standards, the taking of life “because the wrongdoer deserves it” surely must

fail, for such a punishment has as its very basis the total denial of the wrongdoer’s

dignity and worth.

—Justice Thurgood Marshall, Dissenting Opinion in Gregg v. Georgia

11. The electoral college should be abolished. Everyone’s vote should count the same.

With the electoral college the votes of those who live in small states count for more.

That’s how Bush won the election, even though Gore got more votes. So I think we

should do away with it.

12. Every event must have a cause. Since an infinite series of causes is impossible, there

must be a first uncaused cause of everything: God.

13. If God were all good, he would want his creatures to always be happy. If God were

all powerful, he would be able to accomplish anything he wants. Therefore, God

must be lacking in either power or goodness or both.

14. It is now some years since I detected how many were the false beliefs that I had from

my earliest youth admitted as true, and how doubtful was everything I had since con-

structed on this basis: and from that time I was convinced that I must once for all

seriously undertake to rid myself of all the opinions which I had formerly accepted,

and commence to build anew from the foundation, if I wanted to establish any firm

and permanent structure in the sciences.

—René Descartes, Meditations

15. If you can discover a better way of life than office holding for your future rulers, a well-

governed city becomes a possibility. For only in such a state will those rule who are

truly rich, not in gold, but in the wealth that makes happiness—a good and wise life.

—Plato, The Republic

4 Introduction

Deduction and Induction

Deduction and induction are commonly thought to be the cornerstones of good reasoning.

The fundamental logical property of a deductively valid argument is this: If all its prem-

ises are true, then its conclusion must be true. In other words, an argument is valid just in

case it is impossible for all its premises to be true and yet its conclusion be false. The truth

of the premises of a valid argument guarantees the truth of its conclusion.

To determine whether or not an argument is valid, one must ask whether there are any

possible circumstances under which the premises could all be true and yet the conclusion

be false. If not, the argument is valid. If it is possible for the premises to be true and the

conclusion false, the argument is invalid. An invalid argument is simply an argument that

is not valid.

The question naturally arises as to why it is impossible for the conclusion of a valid

argument to be false if all its premises are true. Why do its premises, if true, “guarantee”

the truth of its conclusion? Unfortunately, there is no simple answer to this question. A

clear answer for some types of deductive arguments is given in this book. It is revealing

to notice that in a typical case the information contained in the conclusion of a deductively

valid argument is already “contained” in its premises. We tend not to notice this fact

because it is usually contained in the premises implicitly (along with other information not

contained in the conclusion). Indeed, cases in which the conclusion is explicitly men-

tioned in a premise tend to be rather trivial.

Examples

Here is an example of a deductively valid argument whose conclusion is implicitly

contained in its premises:

1. All wars are started by miscalculation.

2. The Iraq conflict was a war.

∴ 3. The Iraq conflict was started by miscalculation.

Having said in the first premise that all wars are started by miscalculation and in the

second that the Iraq conflict was a war, we have implicitly said that the Iraq conflict

was started by miscalculation. And this is what is asserted by the argument’s

conclusion.

Here is another example:

1. If Bonny has had her appendix taken out, then she doesn’t have to worry

about getting appendicitis.

2. She has had her appendix taken out.

∴ 3. She doesn’t have to worry about getting appendicitis.

The first premise states that if she has had her appendix out, then she doesn’t have to

worry about appendicitis, and the second, that she has in facthad her appendix out, which

implicitly asserts the conclusion that she doesn’t have to worry about appendicitis.

Introduction 5

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