14.31 Open Cooper™s World. Suppose we have expanded fol by adding the following expressions:

∀b1 , meaning all but one,

Few, interpreted as meaning at most 10%, and

Most, interpreted as meaning more than half.

Translate the following sentences into this extended language. Then say which are true in

Cooper™s World. (You will have to use paper to write out your translations, since Tarski™s World

does not understand these quanti¬ers. If the sentence is ambiguous”for example, sentence 5”

give both translations and say whether each is true.)

1. Few cubes are small.

2. Few cubes are large.

3. All but one cube is not large.

4. Few blocks are in the same column as b.

5. Most things are adjacent to some cube.

6. A cube is adjacent to most tetrahedra.

7. Nothing is adjacent to most things.

8. Something is adjacent to something, but only to a few things.

9. All but one tetrahedron is adjacent to a cube.

Chapter 14

The logic of generalized quantification / 389

14.32 Once again open Cooper™s World. This time translate the following sentences into English and

say which are true in Cooper™s World. Make sure your English translations are clear and

unambiguous.

1. Most y (Tet(y), Small(y))

2. Most z (Cube(z), LeftOf(z, b))

3. Most y Cube(y)

4. Most x (Tet(x), ∃y Adjoins(x, y))

5. ∃y Most x (Tet(x), Adjoins(x, y))

6. Most x (Cube(x), ∃y Adjoins(x, y))

7. ∃y Most x (Cube(x), Adjoins(x, y))

8. Most y (y = b)

9. ∀x (Most y (y = x))

10. Most x (Cube(x), Most y (Tet(y), FrontOf(x, y))

Section 14.5

The logic of generalized quanti¬cation

In this section we look brie¬‚y at some of the logical properties of determin-

ers. Since di¬erent determiners typically have di¬erent meanings, we expect

them to have di¬erent logical properties. In particular, we expect the logical

truths and valid arguments involving determiners to be highly sensitive to the

particular determiners involved. Some of the logical properties of determiners

fall into nice clusters, though, and this allows us to classify determiners in

logically signi¬cant ways.

We will assume that Q is some determiner of English and that we have

introduced a formal counterpart Q into fol in the manner described at the

end of the last section.

Conservativity

As it happens, there is one logical property that holds of virtually all single-

word determiners in every natural language. Namely, for any predicates A and

B, the following are logically equivalent:

Q x (A(x), B(x)) ” Q x (A(x), (A(x) § B(x)))

This is called the conservativity property of determiners. Here are two instances conservativity property

of the ⇐ half of conservativity, followed by two instances of the ’ half:

If no doctor is a doctor and a lawyer, then no doctor is a lawyer.

If exactly three cubes are small cubes, then exactly three cubes

Section 14.5

390 / More about Quantification

are small.

If few actors are rich, then few actors are rich and actors.

If all good actors are rich, then all good actors are rich and good

actors.

It is interesting to speculate why this principle holds of single word deter-

miners in human languages. There is no logical reason why there could not be

determiners that did not satisfy it. (See, for example, Exercise 14.52.) It might

have something to do with the di¬culty of understanding quanti¬cation that

does not satisfy the condition, but if so, exactly why remains a puzzle.

There is one word which has the super¬cial appearance of a determiner

that is not conservative, namely the word only. For example, it is true that

only actors are rich actors but it does not follow that only actors are rich, as

it would if only were conservative. There are independent linguistic grounds

for thinking that only is not really a determiner. One piece of evidence is the

fact that determiners can™t be attached to complete noun phrases. You can™t

say Many some books are on the table or Few Claire eats pizza. But you can

say Only some books are on the table and Only Claire eats pizza, suggesting

that it is not functioning as a determiner. In addition, only is much more

versatile than determiners, as is shown by the sentences Claire only eats pizza

and Claire eats only pizza. You can™t replace only in these sentences with a

determiner and get a grammatical sentence. If only is not a determiner, it is

not a counterexample to the conservativity principle.

Monotonicity

The monotonicity of a determiner has to do with what happens when we

increase or decrease the set B of things satisfying the verb phrase in a sen-

tence of the form Q A B. The determiner Q is said to be monotone increasing

monotone increasing

provided for all A, B, and B , the following argument is valid:

Q x (A(x), B(x))

∀x (B(x) ’ B (x))

Q x (A(x), B (x))

In words, if Q(A, B) and you increase B to a larger set B , then Q(A, B ).

There is a simple test to determine whether a determiner is monotone increas-

ing:

Test for monotone increasing determiners: Q is monotone increasing if

and only if the following argument is valid:

Chapter 14

The logic of generalized quantification / 391

Table 14.1: Monotonically increasing and decreasing determiners.

Monotone increasing Monotone decreasing Neither

every no all but one

some

the

both neither

many few

several

most

at least two at most two exactly two

in¬nitely many ¬nitely many

Max™s

Q cube(s) is (are) small and in the same row as c.

Q cube(s) is (are) small.

The reason this test works is that the second premise in the de¬nition of

monotone increasing, ∀x (B(x) ’ B (x)), is automatically true. If we try out

the test with a few determiners, we see, for example, that some, every, and

most are monotone increasing, but few is not.

On the other hand, Q is said to be monotone decreasing if things work in monotone decreasing

the opposite direction, moving from the larger set B to a smaller set B:

Q x (A(x), B (x))

∀x (B(x) ’ B (x))

Q x (A(x), B(x))

The test for monotone decreasing determiners is just the opposite as for

monotone increasing determiners:

Test for monotone decreasing determiners: Q is monotone decreasing

if and only if the following argument is valid:

Q cube(s) is (are) small.

Q cube(s) is (are) small and in the same row as c.

Many determiners are monotone increasing, several are monotone decreas-

ing, but some are neither. Using our tests, you can easily verify for yourself

Section 14.5

392 / More about Quantification

the classi¬cations shown in Table 14.1. To apply our test to the ¬rst column

of the table, note that the following argument is valid, and remains so even if

most is replaced by any determiner in this column:

Most cubes are small and in the same row as c.

Most cubes are small.

On the other hand, if we replace most by any of the determiners in the other

columns, the resulting argument is clearly invalid.

To apply the test to the list of monotone decreasing determiners we observe

that the following argument is valid, and remains so if no is replaced by any

of the other determiners in the second column:

No cubes are small.

No cubes are small and in the same row as c.

On the other hand, if we replace no by the determiners in the other columns,

the resulting argument is no longer valid.

If you examine Table 14.1, you might notice that there are no simple one-

word determiners in the third column. This is because there aren™t any. It

so happens that all the one-word determiners are either monotone increasing

or monotone decreasing, and only a few fall into the decreasing category.

Again, this may have to do with the relative simplicity of monotonic versus

non-monotonic quanti¬cation.

Persistence

Persistence is a property of determiners very similar to monotonicity, but

persistence has to do which what happens if we increase or decrease the set

of things satisfying the common noun: the A in a sentence of the form Q A B.

The determiner Q is said to be persistent provided for all A, A , and B, the

persistence

following argument is valid:2

Q x (A(x), B(x))

∀x (A(x) ’ A (x))

Q x (A (x), B(x))

In words, if Q A B and you increase A to a larger A , then Q A B. On the other

hand, Q is said to be anti-persistent if things work in the opposite direction:

anti-persistence

2 Some authors refer to persistence as left monotonicity, and what we have been calling

monotonicity as right monotonicity, since they have to do with the left and right arguments,

respectively, when we look at Q A B as a binary relation Q(A, B).

Chapter 14

The logic of generalized quantification / 393

Table 14.2: Persistent and anti-persistent determiners.

Persistent Anti-persistent Neither

some every all but one

several few most

at least two at most two exactly two

in¬nitely many ¬nitely many many

Max™s no the

both

neither

Q x (A (x), B(x))

∀x (A(x) ’ A (x))

Q x (A(x), B(x))

To test a determiner for persistence or anti-persistence, try out the two argu-

ment forms given below and see whether the result is valid:

Test for Persistence: The determiner Q is persistent if and only if the

following argument is valid:

Q small cube(s) is (are) left of b.

Q cube(s) is (are) left of b.

Test for Anti-persistence: The determiner Q is anti-persistent if and only

if the following argument is valid:

Q cube(s) is (are) left of b.

Q small cube(s) is (are) left of b.

Applying these tests gives us the results shown in Table 14.2. Make sure

you try out some or all of the entries to make sure you understand how the

tests work. You will want to refer to this table in doing the exercises.

The properties of monotonicity and persistence play a large role in ordinary

reasoning with determiners. Suppose, by way of example, that your father is

trying to convince you to stay on the family farm rather than become an

actor. He might argue as follows:

Section 14.5