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394 / More about Quantification


You want to be rich, right? Well, according to this report, few actors
have incomes above the federal poverty level. Hence, few actors are
rich.
Your father™s argument depends on the fact that few is monotone decreasing.
The set of rich people is a subset of those with incomes above the poverty
level, so if few actors are in the second set, few are in the ¬rst. Notice that
we immediately recognize the validity of this inference without even thinking
twice about it.
Suppose you were to continue the discussion by pointing out that the actor
Brad Pitt is extraordinarily rich. Your father might go on this way:
Several organic farmers I know are richer than Brad Pitt. So even
some farmers are extraordinarily rich.
This may seem like an implausible premise, but you know fathers. In any
case, the argument is valid, though perhaps unsound. Its validity rests on the
fact that Several is both persistent and monotone increasing. By persistence,
we can conclude that several farmers are richer than Brad Pitt (since the
organic farmers are a subset of the farmers), and by monotonicity that several
farmers are extraordinarily rich (since everyone richer than Brad Pitt is).
Finally, from the fact that several farmers are extraordinarily rich it obviously
follows that some farmers are (see Exercise 14.51).
There are many other interesting topics related to the study of determiners,
but this introduction should give you a feel for the kinds of things we can
discover about determiners, and the large role they play in everyday reasoning.


Remember

1. There are three properties of determiners that are critical to their
logical behavior: conservativity, monotonicity, and persistence.

2. All English determiners are conservative (with the exception of only,
which is not usually considered a determiner).

3. Monotonicity has to do with the behavior of the second argument of
the determiner. All basic determiners in English are monotone increas-
ing or decreasing, with most being monotone increasing.

4. Persistence has to do with the behavior of the ¬rst argument of the
determiner. It is less common than monotonicity.




Chapter 14
The logic of generalized quantification / 395



Exercises

For each of the following arguments, decide whether it is valid. If it is, explain why. This explanation
could consist in referring to one of the determiner properties mentioned in this section or it could consist
in an informal proof. If the argument is not valid, carefully describe a counterexample.

14.33 14.34
Few cubes are large. Few cubes are large.
 
Few cubes are large cubes. Few large things are cubes.

14.35 14.36
Many cubes are large. Few cubes are large.
 
Many cubes are not small. Few cubes are not small.

14.37 14.38
Few cubes are not small. Most cubes are left of b.
 
Few cubes are large. Most small cubes are left of b.

14.39 14.40
At most three cubes are left of b. Most cubes are not small.
 
At most three small cubes are left of b. Most cubes are large.

14.41 14.42
∃x [Dodec(x) § Most y (Dodec(y), y = x))] At least three small cubes are left of
  b.
∃!x Dodec(x)
At least three cubes are left of b.

14.43 14.44
Most small cubes are left of b. Most tetrahedra are left of b.
  a is a tetrahedron in the same column
Most cubes are left of b.
as b.
a is not right of anything in the same
row as b.
Most tetrahedra are not in the same
row as b.

14.45 14.46
Only cubes are large. Only tetrahedra are large tetrahedra.
 
Only cubes are large cubes. Only tetrahedra are large.




Section 14.5
396 / More about Quantification


14.47 14.48
Most of the students brought a snack Most of the students brought a snack
 
to class. to class.
Most of the students were late to Most of the students were late to
class. class.
Most of the students were late to class At least one student was late to class
and brought a snack. and brought a snack.

14.49 14.50
Most former British colonies are Many are called.
 
democracies. Few are chosen.
All English speaking countries were
Most are rejected.
formerly British colonies.
Most English speaking countries are
democracies.

14.51 In one of our example arguments, we noted that Several A B implies Some A B. In general, a
 determiner Q is said to have existential import if Q A B logically implies Some A B. Classify
each of the determiners listed in Table 14.2 as to whether it has existential import. For those
that don™t, give informal counterexamples. Discuss any cases that seem problematic.

14.52 Consider a hypothetical English determiner “allbut.” For example, we might say Allbut cubes
 are small to mean that all the blocks except the cubes are small. Give an example to show
that “allbut” is not conservative. Is it monotone increasing or decreasing? Persistent or anti-
persistent? Illustrate with arguments expressed in English augmented with “allbut.”

14.53 (Only) Whether or not only is a determiner, it could still be added to fol, allowing expressions
 of the form Only x (A, B), which would be true if and only if only A™s are B™s.
1. While Only is not conservative, it does satisfy a very similar property. What is it?
2. Discuss monotonicity and persistence for Only. Illustrate your discussion with argu-
ments expressed in English.

14.54 (Adverbs of temporal quanti¬cation) It is interesting to extend the above discussion of quan-
 ti¬cation from determiners to so-called adverbs of temporal quanti¬cation, like always, often,
usually, seldom, sometimes, and never. To get a hint how this might go, let™s explore the
ambiguities in the English sentence Max usually feeds Carl at 2:00 p.m.
Earlier, we treated expressions like 2:00 as names of times on a particular day. To
interpret this sentence in a reasonable way, however, we need to treat such expres-
sions as predicates of times. So we need to add to our language a predicate 2pm(t)
that holds of those times t (in the domain of discourse) that occur at 2 p.m., no mat-
ter on what day they occur. Let us suppose that Usually means most times. Thus,




Chapter 14
Other expressive limitations of first-order logic / 397



Usually t (A(t), B(t))

means that most times satisfying A(t) also satisfy B(t).

1. One interpretation of Max usually feeds Carl at 2:00 p.m. is expressed by

Usually t (2pm(t), Feeds(max, carl, t))

Express this claim using an unambiguous English sentence.
2. A di¬erent interpretation of the sentence is expressed by

Usually t (Feeds(max, carl, t), 2pm(t))

Express this claim using an unambiguous English sentence. Then elucidate the di¬er-
ence between this claim and the ¬rst by describing situations in which each is true
while the other isn™t.
3. Are the same ambiguities present in the sentence Claire seldom feeds Folly at 2:00
p.m.? How about with the other adverbs listed above?
4. Can you think of yet a third interpretation of Max usually feeds Carl at 2:00 p.m.,
one that is not captured by either of these translations? If so, try to express it in our
language or some expansion of it.



Section 14.6
Other expressive limitations of ¬rst-order logic

The study of generalized quanti¬cation is a response to one expressive limi-
tation of fol, and so to its inability to illuminate the full logic inherent in
natural languages like English. The determiners studied in the preceding sec-
tions are actually just some of the ways of expressing quanti¬cation that we
¬nd in natural languages. Consider, for example, the sentences

More cubes than tetrahedra are on the same row as e.
Twice as many cubes as tetrahedra are in the same column as f.
Not as many tetrahedra as dodecahedra are large.

The expressions in bold take two common noun expressions and a verb ex-
pression to make a sentence. The techniques used to study generalized quan-
ti¬cation in earlier sections can be extended to study these determiners, but
we have to think of them as expressing three place relations on sets, not just three place
quanti¬cation
two place relations. Thus, if we added these determiners to the language, they
would have the general form Q x (A(x), B(x), C(x)).




Section 14.6
398 / More about Quantification


A related di¬erence in expressive power between fol and English comes in
the ability of English to use both singular and plural noun phrases. There is a
plurals
di¬erence between saying The boys argued with the teacher and saying Every
boy argued with the teacher. The ¬rst describes a single argument between
a teacher and a group of boys, while the second may describe a sequence of
distinct arguments. Fol does not allow us to capture this di¬erence.
Quanti¬cation is just the tip of an iceberg, however. There are many ex-
pressions of natural languages that go beyond ¬rst-order logic in various ways.
Some of these we have already discussed at various points, both with exam-
ples and exercises. As one example, we saw that there are many uses of the
natural language conditional if. . . then. . . that are not truth functional, and
so not captured by the truth-functional connective ’.
Another dimension in which fol is limited, in contrast to English, comes in
the latter™s ¬‚exible use of tense. Fol assumes a timeless domain of unchanging
tense
relationships, whereas in English, we can exploit our location in time and
space to say things about the present, the past, and locations around us. For
example, in fol we cannot easily say that it is hot here today but it was cool
yesterday. To say something similar in fol, we need to allow quanti¬ers over
times and locations, and add corresponding argument positions to our atomic
predicates.
Similarly, languages like English have a rich modal structure, allowing us
modality
not only to say how things are, but how they must be, how they might be,
how they can™t be, how they should be, how they would be if we had our
way, and so forth. So, for example, we can say All the king™s horses couldn™t
put Humpty Dumpty together again. Or Humpty shouldn™t have climbed on the
wall. Or Humpty might be dead. Such statements lie outside the realm of fol.
All of these expressions have their own logic, and we can explore and try to
understand just which claims involving these expressions follow logically from
others. Building on the great success of fol, logicians have studied (and are
continuing to study) extensions of fol in which these and similar expressive
de¬ciencies are addressed. But as of now there is no single such extension of
fol that has gained anything like its currency.

Exercises


14.55 Try to translate the nursery rhyme about Humpty Dumpty into fol. Point out the various
 linguistic mechanisms that go beyond fol. Discuss this in class.




Chapter 14
Other expressive limitations of first-order logic / 399



14.56 Consider the following two claims. Does either follow logically from the other? Are they logically
 equivalent? Explain your answers.
1. I can eat every apple in the bowl.
2. I can eat any apple in the bowl.

14.57 Recall the ¬rst-order language introduced in Table 1.2, page 30. Some of the following can be
 given ¬rst-order translations using that language, some cannot. Translate those that can be.
For the others, explain why they cannot be faithfully translated, and discuss whether they
could be translated with additional names, predicates, function symbols, and quanti¬ers, or if
the shortcoming in the language is more serious.

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