Section 14.4

Adding other determiners to fol

We have seen that many English determiners can be captured in fol, though

by somewhat convoluted circumlocutions. But there are also many determin-

ers that simply aren™t expressible in fol. A simple example is the determiner

Most, as in Most cubes are large. There are two di¬culties. One is that the most, more than half

meaning of most is a bit indeterminate. Most cubes are large clearly implies

More than half the cubes are large, but does the latter imply the former? In-

tuitions di¬er. But even if we take it to mean the same as More than half,

it cannot be expressed in fol, since the determiner More than half is not

expressible in fol.

It is possible to give a mathematical proof of this fact. For example, con-

sider the sentence:

More than half the dodecahedra are small.

To see the problem, notice that the English sentence makes a claim about the

relative sizes of the set A of small dodecahedra and the set B of dodecahe-

dra that are not small. It says that the set A is larger than the set B and

it does so without claiming anything about how many objects there are in

these sets or in the domain of discourse. To express the desired sentence, we

might try something like the following (where we use A(x) as shorthand for

Dodec(x) § Small(x), and B(x) as shorthand for Dodec(x) § ¬Small(x)):

[∃x A(x) § ∀x ¬B(x)] ∨ [∃≥2 x A(x) § ∃¤1 x B(x)] ∨ [∃≥3 x A(x) § ∃¤2 x B(x)] ∨ . . .

The trouble is, there is no place to stop this disjunction! Without some

¬xed ¬nite upper bound on the total number of objects in the domain, we

need all of the disjuncts, and so the translation of the English sentence would

be an in¬nitely long sentence, which fol does not allow. If we knew there

were a maximum of twelve objects in the world, as in Tarski™s World, then we

could write a sentence that said what we needed; but without this constraint,

the sentence would have to be in¬nite.

This is not in itself a proof that the English sentence cannot be expressed

in fol. But it does pinpoint the problem and, using this idea, one can actually unexpressible in fol

give such a proof. In particular, it is possible to show that for any ¬rst-order

sentence S of the blocks language, if S is true in every world where more than

half the dodecahedra are small, then it is also true in some world where less

Section 14.4

384 / More about Quantification

than half the dodecahedra are small. Unfortunately, the proof of this would

take us beyond the scope of this book.

The fact that we cannot express more than half in fol doesn™t mean there

is anything suspect about this determiner. It just means that it does not fall

within the expressive resources of the invented language fol. Nothing stops

us from enriching fol by adding a new quanti¬er symbol, say Most. Let™s

explore this idea for a moment, since it will shed light on some topics from

earlier in the book.

How not to add a determiner

We™ll begin by telling you how not to add the determiner Most to the language.

Following the lead from ∀ and ∃, we might start by adding the following clause

to our grammatical rules on page 231:

If S is a w¬ and ν is a variable, then Most ν S is a w¬, and any

occurrence of ν in Most ν S is said to be bound.

We might then say that the sentence Most x S(x) is true in a world just in

case more objects in the domain satisfy S(x) than don™t.1 Thus the sentence

Most x Cube(x) says that most things are cubes.

How can we use our new language to express our sentence Most dodeca-

hedra are small? The answer is, we can™t. If we look back at ∀, ∃, and the

numerical determiners, we note something interesting. It so happens that we

can paraphrase every cube is small and some cube is small using everything

and something; namely, Everything is such that if it is a cube then it is small

and Something is a cube and it is small. At the end of the section on numerical

quanti¬cation, we made a similar observation. There is, however, simply no

way to paraphrase Most dodecahedra are small using Most things and expres-

sions that can be translated into fol. After all, it may be that most cubes

are small, even when there are only three or four cubes and millions of dodec-

ahedra and tetrahedra in our domain. Talking about most things is not going

to let us say much of interest about the lonely cubes.

These observations point to something interesting about quanti¬cation

and the way it is represented in fol. For any determiner Q, let us mean by

its general form any use of the form Q A B as described at the beginning of

this chapter. In contrast, by its special form we™ll mean a use of the form Q

thing(s) B. The following table of examples makes this clearer.

1 For

the set-theoretically sophisticated, we note that this de¬nition make sense even if

the domain of discourse is in¬nite.

Chapter 14

Adding other determiners to fol / 385

Determiner Special form General form

every everything every cube, every student of logic, . . .

some something some cube, some student of logic, . . .

no nothing no cube, no student of logic, . . .

exactly two exactly two things exactly two cubes, exactly two stu-

dents of logic, . . .

most most things most cubes, most students of logic, . . .

Many determiners have the property that the general form can be reduced

to the special form by the suitable use of truth-functional connectives. Let™s

call such a determiner reducible. We have seen that every, some, no, and the reducibility

various numerical determiners are reducible in this sense. Here are a couple

of the reductions:

Every A B ” Everything is such that if it is an A then it is a B

Exactly two A B ” Exactly two things satisfy A and B

But some determiners, including includes most, many, few, and the, are

not reducible. For non-reducible determiners Q, we cannot add Q to fol by

simply adding the special form in the way we attempted here. We will see how

we can add such determiners in a moment.

There was some good fortune involved when logicians added ∀ and ∃ as

they did. Since every and some are reducible, the de¬nition of fol can get away

with just the special forms, which makes the language particularly simple. On

the other hand, the fact that fol takes the special form as basic also results

in many of the di¬culties in translating from English to fol that we have

noted. In particular, the fact that the reduction of Every A uses ’, while that

of Some A uses §, causes a lot of confusion among beginning students.

How to add a determiner

The observations made above show that if we are going to add a quanti¬er

like Most to our language, we must add the general form, not just the special new grammatical form

form. Thus, the formation rule should take two w¬s and a variable and create

a new w¬:

If A and B are w¬s and ν is a variable, then Most ν (A, B) is a w¬,

and any occurrence of ν in Most ν (A, B) is said to be bound.

The w¬ Most x (A, B) is read “most x satisfying A satisfy B.” Notice that the

syntactic form of this w¬ exhibits the fact that Most x (A, B) expresses a binary

relation between the set A of things satisfying A and the set B of things that

satisfy B. We could use the abbreviation Most x (S) for Most x (x = x, S); this

Section 14.4

386 / More about Quantification

is read “most things x satisfy S.” This, of course, is the special form of the

determiner, whereas the general form takes two w¬s.

We need to make sure that our new symbol Most means what we want it

to. Toward this end, let us agree that the sentence Most x (A, B) is true in a

world just in case most objects that satisfy A(x) satisfy B(x) (where by this

we mean more objects satisfy A(x) and B(x) than satisfy A(x) and ¬B(x)).

With these conventions, we can translate our English sentence faithfully as:

Most x (Dodec(x), Small(x))

The order here is very important. While the above sentences says that most

dodecahedra are small, the sentence

Most x (Small(x), Dodec(x))

says that most small things are dodecahedra. These sentences are true under

very di¬erent conditions. We will look more closely at the logical properties

of Most and some other determiners in the next section.

Once we see the general pattern, we see that any meaningful determiner

Q of English can be added to fol in a similar manner.

If A and B are w¬s and ν is a variable, then Q ν (A, B) is a w¬, and

any occurrence of ν in Q ν (A, B) is said to be bound.

The w¬ Q x (A, B) is read “Q x satisfying A satisfy B,” or more simply, “Q

A™s are B™s.” Thus, for example,

Few x (Cube(x), Small(x))

is read “Few cubes are small.”

As for the special form, again we use the abbreviation

Q x (S)

for Q x (x = x, S); this is read “Q things x satisfy S.” For instance, the w¬

Many x (Cube(x)) is shorthand for Many x (x = x, Cube(x)), and is read “Many

things are cubes.”

What about the truth conditions of such w¬s? Our reading of them suggests

new semantic rules

how we might de¬ne their truth conditions. We say that the sentence Q x (A, B)

is true in a world just in case Q of the objects that satisfy A(x) also satisfy

B(x). Here are some instances of this de¬nition:

1. At least a quarter x (Cube(x), Small(x)) is true in a world i¬ at least a quar-

ter of the cubes in that world are small.

Chapter 14

Adding other determiners to fol / 387

2. At least two x (Cube(x), Small(x)) is true in a world i¬ at least two cubes

in that world are small.

3. Finitely many x (Cube(x), Small(x)) is true in a world i¬ ¬nitely many of

the cubes in that world are small.

4. Many x (Cube(x), Small(x)) is true in a world i¬ many of the cubes in

that world are small.

The ¬rst of these examples illustrates a kind of determiner we have not

even mentioned before. The second shows that we could treat the numerical

determiners of the preceding section in a di¬erent manner, by adding them

as new primitives to an expansion of fol. The third example is of another

determiner that cannot be expressed in fol.

But wait a minute. There is something rather unsettling about the fourth

example. The problem is that the English determiner many, unlike the other

examples, is context dependent; just what counts as many varies from one

context to another. If we are talking about a class of twenty, 18 or 19 would

count as many. If we are talking about atoms in the universe, this would count

as very few, not many.

This context dependence infects our de¬nition of the truth conditions for

Many x (Cube(x), Small(x)). What might count as many cubes for one purpose,

or one speaker, might not count as many for another purpose or another

speaker. Logic is supposed to be the science of reasoning. But if we are trying

to be scienti¬c, the incursion of context dependence into the theory is most

unwelcome.

There are two things to say about this context dependence. The ¬rst is dealing with context

dependence

that even with context dependent determiners, there are certain clear logical

principles that can be uncovered and explained. We will take some of these up

in the next section. The second point is that the context dependence problem

has a solution. It is possible to model the meaning of context dependent deter-

miners in a perfectly precise, mathematically rigorous manner. Unfortunately,

the modeling requires ideas from set theory that we have not yet covered. The

basic idea, which will only make sense after the next chapter, is to model the

meaning of any determiner as a binary relation on subsets of the domain of

discourse. Just what relation is the best model in the case of a determiner

like Many will depend on the intentions of the person using the determiner.

But all such relations will have certain features in common, features that help

explain the logical properties of the determiners just alluded to. For further

details, see Exercises 18.5 and 18.6.

Section 14.4

388 / More about Quantification

Remember

Given any English determiner Q, we can add a corresponding quanti¬er

Q to fol. In this extended language, the sentence Q x (A, B) is true in a

world just in case Q of the objects that satisfy A(x) also satisfy B(x).

Exercises

14.30 Some of the following English determiners are reducible, some are not. If they are reducible,

explain how the general form can be reduced to the special form. If they do not seem to be

reducible, simply say so.

1. At least three

2. Both

3. Finitely many

4. At least a third