∃x ∃y (Cube(x) § Cube(y) § x = y § ∀z (Cube(z) ’ (z = x ∨ z = y)))

14.13 ∃x ∃y (Cube(x) § Cube(y) § x = y § ∀z (Cube(z) ’ (z = x ∨ z = y)))

‚

∃x ∃y (Cube(x) § Cube(y) § x = y) § ∀x ∀y ∀z ((Cube(x) § Cube(y) § Cube(z))

’ (x = y ∨ x = z ∨ y = z))

The next two exercises contain arguments with similar premises and the same conclusion. If the argument

is valid, turn in an informal proof. If it is not, submit a world in which the premises are true but the

conclusion is false.

14.14 14.15

There are exactly four cubes. There are exactly four cubes.

‚| ‚|

Any column that contains a cube con- Any column that contains a cube con-

tains a tetrahedron, and vice versa. tains a tetrahedron, and vice versa.

No tetrahedron is in back of any other No column contains two objects of the

tetrahedron. same shape.

There are exactly four tetrahedra. There are exactly four tetrahedra.

Section 14.2

378 / More about Quantification

The following exercises state some logical truths or valid arguments involving numerical quanti¬ers. Give

informal proofs of each. Contemplate what it would be like to give a formal proof (for speci¬c values of

n and m) and be thankful we didn™t ask you to give one!

14.16

∃¤0 x S(x) ” ∀x ¬S(x)

[The only hard part about this is ¬guring out what ∃¤0 x S(x) abbreviates.]

14.17 14.18 ∃¤n x A(x)

∃¤m x B(x)

¬∃≥n+1x S(x) ” ∃¤n x ¬S(x)

∃¤n+m x (A(x) ∨ B(x))

14.19 14.20

∃≥n x A(x) ∀x [A(x) ’ ∃!y R(x, y)]

∃≥m x B(x) ∃¤n y ∃x [A(x) § R(x, y)]

¬∃x (A(x) § B(x))

∃¤n x A(x)

∃≥n+m x (A(x) ∨ B(x))

14.21 We have seen that ∃x ∃y R(x, y) is logically equivalent to ∃y ∃x R(x, y), and similarly for ∀.

What happens if we replace both of these quanti¬ers by some numerical quanti¬er? In partic-

ular, is the following argument valid?

∃!x ∃!y R(x, y)

∃!y ∃!x R(x, y)

If so, give an informal proof. If not, describe a counterexample.

The following exercises contain true statements about the domain of natural numbers 0, 1, . . . . Give

informal proofs of these statements.

14.22 ∃!x [x2 ’ 2x + 1 = 0] 14.23 ∃!2 y [y + y = y — y]

14.24 ∃!2 x [x2 ’ 4x + 3 = 0] 14.25 ∃!x [(x2 ’ 5x + 6 = 0) § (x > 2)]

Chapter 14

The, both, and neither / 379

Section 14.3

The, both, and neither

The English determiners the, both, and neither are extremely common. Indeed,

the is one of the most frequently used words in the English language. (We

used it twice in that one sentence.) In spite of their familiarity, their logical

properties are subtle and, for that matter, still a matter of some dispute.

To see why, suppose I say “The elephant in my closet is not wrinkling my

clothes.” What would you make of this, given that, as you probably guessed,

there is no elephant in my closet? Is it simply false? Or is there something else

wrong with it? If it is false, then it seems like its negation should be true. But

the negation seems to be the claim that the elephant in my closet is wrinkling

my clothes. Similar puzzles arise with both and neither:

Both elephants in my closet are wrinkling my clothes.

Neither elephant in my closet is wrinkling my clothes.

What are you to make of these if there are no elephants in my closet, or if

there are three?

Early in the twentieth century, the logician Bertrand Russell proposed an

analysis of such sentences. He proposed that a sentence like The cube is small the

should be analyzed as asserting that there is exactly one cube, and that it is

small. According to his analysis, the sentence will be false if there is no cube,

or if there is more than one, or if there is exactly one, but it™s not small. If

Russell™s analysis is correct, then such sentences can easily be expressed in

¬rst-order logic as follows:

∃x [Cube(x) § ∀y (Cube(y) ’ y = x) § Small(x)]

More generally, a sentence of the form The A is a B, on the Russellian analysis,

would be translated as:

∃x [A(x) § ∀y (A(y) ’ x = y) § B(x)]

Noun phrases of the form the A are called de¬nite descriptions and the above de¬nite descriptions

analysis is called the Russellian analysis of de¬nite descriptions.

While Russell did not explicitly consider both or neither, the spirit of his both, neither

analysis extends naturally to these determiners. We could analyze Both cubes

are small as saying that there are exactly two cubes and each of them is small:

∃!2 x Cube(x) § ∀x [Cube(x) ’ Small(x)]

Section 14.3

380 / More about Quantification

Similarly, Neither cube is small would be construed as saying that there are

exactly two cubes and each of them is not small:

∃!2 x Cube(x) § ∀x [Cube(x) ’ ¬Small(x)]

More generally, Both A™s are B™s would be translated as:

∃!2 x A(x) § ∀x [A(x) ’ B(x)]

and Neither A is a B would be translated as:

∃!2 x A(x) § ∀x [A(x) ’ ¬B(x)]

Notice that on Russell™s analysis of de¬nite descriptions, the sentence The

cube is not small would be translated as:

∃x [Cube(x) § ∀y (Cube(y) ’ y = x) § ¬Small(x)]

This is not, logically speaking, the negation of The cube is small. Indeed

de¬nite descriptions

and negation both sentences could be false if there are no cubes or if there are too many.

The super¬cial form of the English sentences makes them look like negations

of one another, but according to Russell, the negation of The cube is small

is something like Either there is not exactly one cube or it is not small. Or

perhaps more clearly, If there is exactly one cube then it is not small. Similarly,

the negation of Both cubes are small would not be Both cubes are not small

but If there are exactly two cubes then they are not both small.

Russell™s analysis is not without its detractors. The philosopher P. F.

Strawson, for example, argued that Russell™s analysis misses an important

feature of our use of the determiner the. Return to our example of the ele-

phant. Consider these three sentences:

The elephant in my closet is wrinkling my clothes.

The elephant in my closet is not wrinkling my clothes.

It is not the case that the elephant in my closet is wrinkling my

clothes.

It seems as if none of these sentences is appropriate if there is no elephant in

my closet. That is to say, they all seem to presuppose that there is a unique

elephant in my closet. According to Strawson, they all do presuppose this,

but they do not claim it.

Strawson™s general picture is this. Some sentences carry certain presuppo-

presuppositions

sitions. They can only be used to make a claim when those presuppositions

are ful¬lled. Just as you can™t drive a car unless there is a car present, you

Chapter 14

The, both, and neither / 381

cannot make a successful claim unless the presuppositions of your claim are

satis¬ed. With our elephant example, the sentence can only be used to make

a claim in case there is one, and only one, elephant in the speaker™s closet.

Otherwise the sentence simply mis¬res, and so does not have a truth value at

all. It is much like using an fol sentence containing a name b to describe a

world where no object is named b. Similarly, on Strawson™s approach, if we

use both elephants in my closet or neither elephant in my closet, our statement

simply mis¬res unless there are exactly two elephants in my closet.

If Strawson™s objection is right, then there will be no general way of trans-

lating the, both, or neither into fol, since fol sentences (at least those without

names in them) always have truth values. There is nothing to stop us from

enriching fol to have expressions that work this way. Indeed, this has been

proposed and studied, but that is a di¬erent, richer language than fol.

On the other hand, there have been rejoinders to Strawson™s objection. For

example, it has been suggested that when we say The elephant in my closet is conversational

implicature

not wrinkling my clothes, the suggestion that there is an elephant in my closet

is simply a conversational implicature. To see if this is plausible, we try the

cancellability test. Does the following seem coherent or not? “The elephant

in my closet is not wrinkling my clothes. In fact, there is no elephant in my

closet.” Some people think that, read with the right intonation, this makes

perfectly good sense. Others disagree.

As we said at the start of this section, these are subtle matters and there is

still no universally accepted theory of how these determiners work in English.

What we can say is that the Russellian analysis is as close as we can come

in fol, that it is important, and that it captures at least some uses of these

determiners. It is the one we will treat in the exercises that follow.

Remember

1. The Russellian analysis of The A is a B is the fol translation of There

is exactly one A and it is a B.

2. The Russellian analysis of Both A™s are B™s is the fol translation of

There are exactly two A™s and each of them is a B.

3. The Russellian analysis of Neither A is a B is the fol translation of

There are exactly two A™s and each of them is not a B.

4. The competing Strawsonian analysis of these determiners treats them

as having presuppositions, and so as only making claims when these

presuppositions are met. On Strawson™s analysis, these determiners

cannot be adequately translated in fol.

Section 14.3

382 / More about Quantification

Exercises

14.26 (The Russellian analysis of de¬nite descriptions)

‚ 1. Open Russell™s Sentences. Sentence 1 is the second of the two ways we saw in the You

try it section on page 368 for saying that there is a single cube. Compare sentence

1 with sentence 2. Sentence 2 is the Russellian analysis of our sentence The cube is

small. Construct a world in which sentence 2 is true.

2. Construct a world in which sentences 2-7 are all true. (Sentence 7 contains the Rus-

sellian analysis of The small dodecahedron is to the left of the medium dodecahedron.)

Submit your world.

14.27 (The Strawsonian analysis of de¬nite descriptions) Using Tarski™s World, open a sentence ¬le

‚ and write the Russellian analysis of the following two sentences:

1. b is left of the cube.

2. b is not left of the cube.

Build a world containing a dodec named b and one other block in which neither of your

translations is true. To do so, you will need to violate what Strawson would call the common

presupposition of these two sentences. Submit both the sentence and world ¬les.

14.28 (The Russellian analysis of both and neither) Open Russell™s World. Notice that the following

‚ sentences are all true:

1. Both cubes are medium.

2. Neither dodec is small.

3. Both cubes are in front of the tetrahedron.

4. Both cubes are left of both dodecahedra.

5. Neither cube is in back of either dodecahedron.

Start a new sentence ¬le and write the Russellian analysis of these ¬ve sentences. Since Tarski™s

World doesn™t let you use the notation ∃!2 , you may ¬nd it easier to write the sentences on

paper ¬rst, using this abbreviation, and then translate them into proper fol. Check that your

translations are true in Russell™s World. Then make some changes to the sizes and positions of

the blocks and again check that your translations have the same truth values as the English

sentences.

14.29 Discuss the meaning of the determiner Max™s. Notice that you can say Max™s pet is happy, but

also Max™s pets are happy. Give a Russellian and a Strawsonian analysis of this determiner.

Which do you think is better?

Chapter 14

Adding other determiners to fol / 383