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5. Some large cube is to the left of b.
6. A large cube is to the left of b.
7. b has a large cube to its left.
8. b is to the right of a large cube. [Hint: This translation should be almost the same as
the last, but it should contain the predicate symbol RightOf.]
9. Something to the left of b is in back of c.




Section 9.6
248 / Introduction to Quantification


10. A large cube to the left of b is in back of c.
11. Some large cube is to the left of b and in back of c.
12. Some dodecahedron is not large.
13. Something is not a large dodecahedron.
14. It™s not the case that something is a large dodecahedron.
15. b is not to the left of a cube. [Warning: This sentence is ambiguous. Can you think of
two importantly di¬erent translations? One starts with ∃, the other starts with ¬. Use
the second of these for your translation, since this is the most natural reading of the
English sentence.]

Now let™s check the translations against a world. Open Montague™s World.

—¦ Notice that all the English sentences above are true in this world. Check that all your
translations are also true. If not, you have made a mistake. Can you ¬gure out what is
wrong with your translation?

—¦ Move the large cube to the back right corner of the grid. Observe that English sentences
5, 6, 7, 8, 10, 11, and 15 are now false, while the rest remain true. Check that the same
holds of your translations. If not, you have made a mistake. Figure out what is wrong
with your translation and ¬x it.

—¦ Now make the large cube small. The English sentences 1, 3, 4, 5, 6, 7, 8, 10, 11, and
15 are false in the modi¬ed world, the rest are true. Again, check that your translations
have the same truth values. If not, ¬gure out what is wrong.

—¦ Finally, move c straight back to the back row, and make the dodecahedron large. All the
English sentences other than 1, 2, and 13 are false. Check that the same holds for your
translations. If not, ¬gure out where you have gone wrong and ¬x them.

When you are satis¬ed that your translations are correct, submit your sentence ¬le.

9.17 (Translating universal noun phrases) Start a new sentence ¬le, and enter translations of the
‚ following sentences. This time each translation will contain exactly one ∀ and no ∃.
1. All cubes are small.
2. Each small cube is to the right of a.
3. a is to the left of every dodecahedron.
4. Every medium tetrahedron is in front of b.




Chapter 9
Translating complex noun phrases / 249



5. Each cube is either in front of b or in back of a.
6. Every cube is to the right of a and to the left of b.
7. Everything between a and b is a cube.
8. Everything smaller than a is a cube.
9. All dodecahedra are not small. [Note: Most people ¬nd this sentence ambiguous. Can
you ¬nd both readings? One starts with ∀, the other with ¬. Use the former, the one
that means all the dodecahedra are either medium or large.]
10. No dodecahedron is small.
11. a does not adjoin everything. [Note: This sentence is ambiguous. We want you to
interpret it as a denial of the claim that a adjoins everything.]
12. a does not adjoin anything. [Note: These last two sentences mean di¬erent things,
though they can both be translated using ∀, ¬, and Adjoins.]
13. a is not to the right of any cube.
14. ( ) If something is a cube, then it is not in the same column as either a or b. [Warning:
While this sentence contains the noun phrase “something,” it is actually making a
universal claim, and so should be translated with ∀. You might ¬rst try to paraphrase
it using the English phrase “every cube.”]
15. ( ) Something is a cube if and only if it is not in the same column as either a or b.

Now let™s check the translations in some worlds.

—¦ Open Claire™s World. Check to see that all the English sentences are true in this world,
then make sure the same holds of your translations. If you have made any mistakes, ¬x
them.

—¦ Adjust Claire™s World by moving a directly in front of c. With this change, the English
sentences 2, 6, and 12“15 are false, while the rest are true. Make sure that the same holds
of your translations. If not, try to ¬gure out what is wrong and ¬x it.

—¦ Next, open Wittgenstein™s World. Observe that the English sentences 2, 3, 7, 8, 11, 12,
and 13 are true, but the rest are false. Check that the same holds for your translations.
If not, try to ¬x them.

—¦ Finally, open Venn™s World. English sentences 2, 4, 7, and 11“14 are true; does the same
hold for your translations?

When you are satis¬ed that your translations are correct, submit your sentence ¬le.




Section 9.6
250 / Introduction to Quantification


9.18 (Translation) Open Leibniz™s World. This time, we will translate some sentences while looking
‚ at the world they are meant to describe.
—¦ Start a new sentence ¬le, and enter translations of the following sentences. Each of the
English sentences is true in this world. As you go, check to make sure that your translation
is indeed a true sentence.

1. There are no medium-sized cubes.
2. Nothing is in front of b.
3. Every cube is either in front of or in back of e.
4. No cube is between a and c.
5. Everything is in the same column as a, b, or c.

—¦ Now let™s change the world so that none of the English sentences is true. We can do this
as follows. First change b into a medium cube. Next, delete the leftmost tetrahedron and
move b to exactly the position just vacated by the late tetrahedron. Finally, add a small
cube to the world, locating it exactly where b used to sit. If your answers to 1“5 are
correct, all of the translations should now be false. Verify that they are.

—¦ Make various changes to the world, so that some of the English sentences come out true
and some come out false. Then check to see that the truth values of your translations
track the truth values of the English sentences.
9.19 Start a new sentence ¬le and translate the following into fol using the symbols from Table 1.2,
‚ page 30. Note that all of your translations will involve quanti¬ers, though this may not be
obvious from the English sentences. (Some of your translations will also require the identity
predicate.)
1. People are not pets.
2. Pets are not people.
3. Scru¬y was not fed at either 2:00 or 2:05. [Remember, Fed is a ternary predicate.]
4. Claire fed Folly between 2:00 and 3:00.
5. Claire gave a pet to Max at 2:00.
6. Claire had only hungry pets at 2:00.
7. Of all the students, only Claire was angry at 3:00.
8. No one fed Folly at 2:00.
9. If someone fed Pris at 2:00, they were angry.
10. Whoever owned Pris at 2:00 was angry ¬ve minutes later.
9.20 Using Table 1.2, page 30, translate the following into colloquial English.
 1. ∀t ¬Gave(claire, folly, max, t)
2. ∀x (Pet(x) ’ Hungry(x, 2:00))




Chapter 9
Quantifiers and function symbols / 251



3. ∀y (Person(y) ’ ¬Owned(y, pris, 2:00))
4. ¬∃x (Angry(x, 2:00) § Student(x) § Fed(x, carl, 2:00))
5. ∀x ((Pet(x) § Owned(max, x, 2:00)) ’ Gave(max, x, claire, 2:00))

9.21 Translate the following into fol, introducing names, predicates, and function symbols as
 needed. As usual, explain your predicates and function symbols, and any shortcomings in
your translations. If you assume a particular domain of discourse, mention that as well.
1. Only the brave know how to forgive.
2. No man is an island.
3. I care for nobody, not I,
If no one cares for me.
4. Every nation has the government it deserves.
5. There are no certainties, save logic.
6. Misery (that is, a miserable person) loves company.
7. All that glitters is not gold.
8. There was a jolly miller once
Lived on the River Dee.
9. If you praise everybody, you praise nobody.
10. Something is rotten in the state of Denmark.



Section 9.7
Quanti¬ers and function symbols

When we ¬rst introduced function symbols in Chapter 1, we presented them
as a way to form complex names from other names. Thus father(father(max))
refers to Max™s father™s father, and (1 + (1 + 1)) refers to the number 3. Now
that we have variables and quanti¬ers, function symbols become much more
useful than they were before. For example, they allow us to express in a very
compact way things like:

∀x Nicer(father(father(x)), father(x))

This sentence says that everyone™s paternal grandfather is nicer than their
father, a false belief held by many children.
Notice that even if our language had individual constants naming every-
one™s father (and their fathers™ fathers and so on), we could not express the
above claim in a single sentence without using the function symbol father.




Section 9.7
252 / Introduction to Quantification


True, if we added the binary predicate FatherOf, we could get the same point
across, but the sentence would be considerably more complex. It would require
three universal quanti¬ers, something we haven™t talked about yet:

∀x ∀y ∀z ((FatherOf(x, y) § FatherOf(y, z)) ’ Nicer(x, y))

In our informal mathematical examples, we have in fact been using func-
tion symbols along with variables throughout the book. For example in Chap-
ter 8, we proved the conditional:

Even(n2 ) ’ Even(n)

This sentence is only partly in our o¬cial language of ¬rst-order arithmetic.
Had we had quanti¬ers at the time, we could have expressed the intended
claim using a universal quanti¬er and the binary function symbol —:

∀y (Even(y — y) ’ Even(y))

The blocks language does not have function symbols, though we could
have introduced some. Remember the four function symbols, fm, bm, lm and
rm, that we discussed in Chapter 1 (page 33). The idea was that these meant
frontmost, backmost, leftmost, and rightmost, respectively, where, for instance,
the complex term lm(b) referred to the leftmost block in the same row as b.
Thus a formula like
lm(x) = x
is satis¬ed by a block b if and only if b is the leftmost block in its row. If we
append a universal quanti¬er to this atomic w¬, we get the sentence

∀x (lm(x) = x)

which is true in exactly those worlds that have at most one block in each
row. This claim could be expressed in the blocks language without function
symbols, but again it would require a sentence with more than one quanti¬er.
To check if you understand these function symbols, see if you can tell
which of the following two sentences is true in all worlds and which makes a
substantive claim, true in some worlds and false in others:

∀x (lm(lm(x)) = lm(x))
∀x (fm(lm(x)) = lm(x))




Chapter 9
Quantifiers and function symbols / 253



In reading a term like fm(lm(b)), remember that you apply the inner func-
tion ¬rst, then the outer. That is, you ¬rst ¬nd the leftmost block in the
row containing b”call it c”and then ¬nd the frontmost block in the column
containing c.
Function symbols are extremely useful and important in applications of
fol. We close this chapter with some problems that use function symbols.

Exercises


9.22 Assume that we have expanded the blocks language to include the function symbols fm, bm, lm
 and rm described earlier. Then the following formulas would all be sentences of the language:
1. ∃y (fm(y) = e)
2. ∃x (lm(x) = b § x = b)
3. ∀x Small(fm(x))
4. ∀x (Small(x) ” fm(x) = x)
5. ∀x (Cube(x) ’ Dodec(lm(x))
6. ∀x (rm(lm(x)) = x)
7. ∀x (fm(bm(x)) = x)
8. ∀x (fm(x) = x ’ Tet(fm(x)))
9. ∀x (lm(x) = b) ’ SameRow(x, b))
10. ∃y (lm(fm(y)) = fm(lm(y)) § ¬Small(y))

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