strating nonconsequence, that is, for showing that some purported conclusion

is not a consequence of the premises provided in the argument.

Recall that logical consequence was de¬ned in terms of the validity of

arguments. An argument is valid if every possible circumstance that makes

the premises of the argument true also makes the conclusion true. Put the

other way around, the argument is invalid if there is some circumstance that

makes the premises true but the conclusion false. Finding such a circumstance

is the key to demonstrating nonconsequence.

To show that a sentence Q is not a consequence of premises P1, . . . , Pn ,

we must show that the argument with premises P1 , . . . , Pn and conclusion Q

is invalid. This requires us to demonstrate that it is possible for P1 , . . . , Pn to

be true while Q is simultaneously false. That is, we must show that there is

a possible situation or circumstance in which the premises are all true while

the conclusion is false. Such a circumstance is said to be a counterexample to counterexamples

the argument.

Informal proofs of nonconsequence can resort to many ingenious ways for

Section 2.5

64 / The Logic of Atomic Sentences

showing the existence of a counterexample. We might simply describe what is

informal proofs of

nonconsequence clearly a possible situation, one that makes the premises true and the conclu-

sion false. This is the technique used by defense attorneys, who hope to create

a reasonable doubt that their client is guilty (the prosecutor™s conclusion) in

spite of the evidence in the case (the prosecution™s premises). We might draw

a picture of such a situation or build a model out of Lego blocks or clay.

We might act out a situation. Anything that clearly shows the existence of a

counterexample is fair game.

Recall the following argument from an earlier exercise.

Al Gore is a politician.

Hardly any politicians are honest.

Al Gore is dishonest.

If the premises of this argument are true, then the conclusion is likely. But

still the argument is not valid: the conclusion is not a logical consequence of

the premises. How can we see this? Well, imagine a situation where there are

10,000 politicians, and that Al Gore is the only honest one of the lot. In such

circumstances both premises would be true but the conclusion would be false.

Such a situation is a counterexample to the argument; it demonstrates that

the argument is invalid.

What we have just given is an informal proof of nonconsequence. Are

there such things as formal proofs of nonconsequence, similar to the formal

proofs of validity constructed in F? In general, no. But we will de¬ne the

notion of a formal proof of nonconsequence for the blocks language used in

Tarski™s World. These formal proofs of nonconsequence are simply stylized

counterparts of informal counterexamples.

For the blocks language, we will say that a formal proof that Q is not a

formal proofs of

nonconsequence consequence of P1 , . . . , Pn consists of a sentence ¬le with P1, . . . , Pn labeled

as premises, Q labeled as conclusion, and a world ¬le that makes each of

P1 , . . . , Pn true and Q false. The world depicted in the world ¬le will be called

the counterexample to the argument in the sentence ¬le.

You try it

................................................................

1. Launch Tarski™s World and open the sentence ¬le Bill™s Argument. This

argument claims that Between(b, a, d) follows from these three premises:

Between(b, c, d), Between(a, b, d), and Left(a, c). Do you think it does?

2. Start a new world and put four blocks, labeled a, b, c, and d on one row

of the grid.

Chapter 2

Demonstrating nonconsequence / 65

3. Arrange the blocks so that the conclusion is false. Check the premises. If

any of them are false, rearrange the blocks until they are all true. Is the

conclusion still false? If not, keep trying.

4. If you have trouble, try putting them in the order d, a, b, c. Now you will

¬nd that all the premises are true but the conclusion is false. This world is

a counterexample to the argument. Thus we have demonstrated that the

conclusion does not follow from the premises.

5. Save your counterexample as World Counterexample 1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Congratulations

Remember

To demonstrate the invalidity of an argument with premises P1 , . . . , Pn

and conclusion Q, ¬nd a counterexample: a possible circumstance that

makes P1 , . . . , Pn all true but Q false. Such a counterexample shows that

Q is not a consequence of P1 , . . . , Pn .

Exercises

2.21 If you have skipped the You try it section, go back and do it now. Submit the world ¬le World

‚ Counterexample 1.

2.22 Is the following argument valid? Sound? If it is valid, give an informal proof of it. If it is not

valid, give an informal counterexample to it.

All computer scientists are rich. Anyone who knows how to program a computer is a

computer scientist. Bill Gates is rich. Therefore, Bill Gates knows how to program a

computer.

2.23 Is the following argument valid? Sound? If it is valid, give an informal proof of it. If it is not

valid, give an informal counterexample to it.

Philosophers have the intelligence needed to be computer scientists. Anyone who be-

comes a computer scientist will eventually become wealthy. Anyone with the intelli-

gence needed to be a computer scientist will become one. Therefore, every philosopher

will become wealthy.

Section 2.5

66 / The Logic of Atomic Sentences

Each of the following problems presents a formal argument in the blocks language. If the argument is

valid, submit a proof of it using Fitch. (You will ¬nd Exercise ¬les for each of these in the usual place.)

Important: if you use Ana Con in your proof, cite at most two sentences in each application. If the

argument is not valid, submit a counterexample world using Tarski™s World.

2.24 2.25

Larger(b, c) FrontOf(a, b)

‚ ‚

Smaller(b, d) LeftOf(a, c)

SameSize(d, e) SameCol(a, b)

Larger(e, c) FrontOf(c, b)

2.26 2.27

SameRow(b, c) SameRow(b, c)

‚ ‚

SameRow(a, d) SameRow(a, d)

SameRow(d, f) SameRow(d, f)

LeftOf(a, b) FrontOf(a, b)

LeftOf(f, c) FrontOf(f, c)

Section 2.6

Alternative notation

You will often see arguments presented in the following way, rather than

in Fitch format. The symbol .·. (read “therefore”) is used to indicate the

conclusion:

All men are mortal.

Socrates is a man.

.·. Socrates is mortal.

There is a huge variety of formal deductive systems, each with its own

notation. We can™t possibly cover all of these alternatives, though we describe

one, the resolution method, in Chapter 17.

Chapter 2

Chapter 3

The Boolean Connectives

So far, we have discussed only atomic claims. To form complex claims, fol pro-

vides us with connectives and quanti¬ers. In this chapter we take up the three

simplest connectives: conjunction, disjunction, and negation, corresponding

to simple uses of the English and, or, and it is not the case that. Because they Boolean connectives

were ¬rst studied systematically by the English logician George Boole, they

are called the Boolean operators or Boolean connectives.

The Boolean connectives are also known as truth-functional connectives. truth-functional

connectives

There are additional truth-functional connectives which we will talk about

later. These connectives are called “truth functional” because the truth value

of a complex sentence built up using these connectives depends on nothing

more than the truth values of the simpler sentences from which it is built.

Because of this, we can explain the meaning of a truth-functional connective

in a couple of ways. Perhaps the easiest is by constructing a truth table, a truth table

table that shows how the truth value of a sentence formed with the connec-

tive depends on the truth values of the sentence™s immediate parts. We will

give such tables for each of the connectives we introduce. A more interesting Henkin-Hintikka game

way, and one that can be particularly illuminating, is by means of a game,

sometimes called the Henkin-Hintikka game, after the logicians Leon Henkin

and Jaakko Hintikka.

Imagine that two people, say Max and Claire, disagree about the truth

value of a complex sentence. Max claims it is true, Claire claims it is false. The

two repeatedly challenge one another to justify their claims in terms of simpler

claims, until ¬nally their disagreement is reduced to a simple atomic claim,

one involving an atomic sentence. At that point they can simply examine the

world to see whether the atomic claim is true”at least in the case of claims

about the sorts of worlds we ¬nd in Tarski™s World. These successive challenges

can be thought of as a game where one player will win, the other will lose. The

legal moves at any stage depend on the form of the sentence. We will explain

them below. The one who can ultimately justify his or her claims is the winner.

When you play this game in Tarski™s World, the computer takes the side

opposite you, even if it knows you are right. If you are mistaken in your initial

assessment, the computer will be sure to win the game. If you are right,

though, the computer plugs away, hoping you will blunder. If you slip up, the

computer will win the game. We will use the game rules as a second way of

explaining the meanings of the truth-functional connectives.

67

68 / The Boolean Connectives

Section 3.1

Negation symbol: ¬

The symbol ¬ is used to express negation in our language, the notion we

commonly express in English using terms like not, it is not the case that, non-

and un-. In ¬rst-order logic, we always apply this symbol to the front of a

sentence to be negated, while in English there is a much more subtle system

for expressing negative claims. For example, the English sentences John isn™t

home and It is not the case that John is home have the same ¬rst-order

translation:

¬Home(john)

This sentence is true if and only if Home(john) isn™t true, that is, just in case

John isn™t home.

In English, we generally avoid double negatives”negatives inside other

negatives. For example, the sentence It doesn™t make no di¬erence is problem-

atic. If someone says it, they usually mean that it doesn™t make any di¬erence.

In other words, the second negative just functions as an intensi¬er of some

sort. On the other hand, this sentence could be used to mean just what it

says, that it does not make no di¬erence, it makes some di¬erence.

Fol is much more systematic. You can put a negation symbol in front of

any sentence whatsoever, and it always negates it, no matter how many other

negation symbols the sentence already contains. For example, the sentence

¬¬Home(john)

negates the sentence

¬Home(john)

and so is true if and only if John is home.

The negation symbol, then, can apply to complex sentences as well as to

atomic sentences. We will say that a sentence is a literal if it is either atomic

literals

or the negation of an atomic sentence. This notion of a literal will be useful

later on.

We will abbreviate negated identity claims, such as ¬(b = c), using =, as

nonidentity symbol (=)

in b = c. The symbol = is available on the keyboard palettes in both Tarski™s

World and Fitch.

Semantics and the game rule for negation

Given any sentence P of fol (atomic or complex), there is another sentence

¬P. This sentence is true if and only if P is false. This can be expressed in

terms of the following truth table.

Chapter 3

Negation symbol: ¬ / 69

P ¬P

truth table for ¬

true false

false true

The game rule for negation is very simple, since you never have to do game rule for ¬

anything. Once you commit yourself to the truth of ¬P this is the same as

committing yourself to the falsity of P. Similarly, if you commit yourself to

the falsity of ¬P, this is tantamount to committing yourself to the truth of

P. So in either case Tarski™s World simply replaces your commitment about

the more complex sentence by the opposite commitment about the simpler

sentence.