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Once this is done, check to see if you have a well-formed sentence. Does it look
like a proper translation of the original English? It should.]
2. Every small cube is in back of a large cube.
3. Some cube is in front of every tetrahedron.
4. A large cube is in front of a small cube.
5. Nothing is larger than everything.
6. Every cube in front of every tetrahedron is large.
7. Everything to the right of a large cube is small.
8. Nothing in back of a cube and in front of a cube is large.
9. Anything with nothing in back of it is a cube.
10. Every dodecahedron is smaller than some tetrahedron.

Save your sentences as Sentences 11.16.

—¦ Open Peirce™s World. Notice that all the English sentences are true in this world. Check
to see that all of your translations are true as well. If they are not, see if you can ¬gure
out where you went wrong.

—¦ Open Leibniz™s World. Note that the English sentences 5, 6, 8, and 10 are true in this
world, while the rest are false. Verify that your translations have the same truth values.
If they don™t, ¬x them.

—¦ Open Ron™s World. Here, the true sentences are 2, 3, 4, 5, and 8. Check that your trans-
lations have the right values, and correct them if they don™t.




Section 11.3
300 / Multiple Quantifiers


11.17 (More multiple quanti¬er sentences) Now, we will try translating some multiple quanti¬er
‚ sentences completely from scratch. You should try to use the step-by-step procedure.
—¦ Start a new sentence ¬le and translate the following English sentences.

1. Every tetrahedron is in front of every dodecahedron.
2. No dodecahedron has anything in back of it.
3. No tetrahedron is the same size as any cube.
4. Every dodecahedron is the same size as some cube.
5. Anything between two dodecahedra is a cube. [Note: This use of two really can be
paraphrased using between a dodecahedron and a dodecahedron.]
6. Every cube falls between two objects.
7. Every cube with something in back of it is small.
8. Every dodecahedron with nothing to its right is small.
9. ( ) Every dodecahedron with nothing to its right has something to its left.
10. Any dodecahedron to the left of a cube is large.

—¦ Open Bolzano™s World. All of the above English sentences are true in this world. Verify
that all your translations are true as well.

—¦ Now open Ron™s World. The English sentences 4, 5, 8, 9, and 10 are true, but the rest are
false. Verify that the same holds of your translations.

—¦ Open Claire™s World. Here you will ¬nd that the English sentences 1, 3, 5, 7, 9, and 10
are true, the rest false. Again, check to see that your translations have the appropriate
truth value.

—¦ Finally, open Peano™s World. Notice that only sentences 8 and 9 are true. Check to see
that your translations have the same truth values.




Section 11.4
Paraphrasing English

Some English sentences do not easily lend themselves to direct translation
using the step-by-step procedure. With such sentences, however, it is often
quite easy to come up with an English paraphrase that is amenable to the
procedure. Consider, for example, If a freshman takes a logic class, then he
or she must be smart. The step-by-step procedure does not work here. If we
try to apply the procedure we would get something like




Chapter 11
Paraphrasing English / 301



∃x (Freshman(x) § ∃y (LogicClass(y) § Takes(x, y))) ’ Smart(x)

The problem is that this “translation” is not a sentence, since the last occur-
rence of x is free. However, we can paraphrase the sentences as Every freshman
who takes a logic class must be smart. This is easily treated by the procedure,
with the result being

∀x [(Freshman(x) § ∃y (LogicClass(y) § Takes(x, y))) ’ Smart(x)]

There is one particularly notorious kind of sentence that needs paraphras-
ing to get an adequate ¬rst-order translation. They are known as donkey donkey sentences
sentences, because the ¬rst and most discussed example of this kind is the
sentence

Every farmer who owns a donkey beats it.

What makes such a sentence a bit tricky is the existential noun phrase “a
donkey” in the noun phrase “every farmer who owns a donkey.” The existential
noun phrase serves as the antecedent of the pronoun “it” in the verb phrase;
its the donkey that gets beaten. Applying the step-by-step method might lead
you to translate this as follows:

∀x (Farmer(x) § ∃y (Donkey(y) § Owns(x, y)) ’ Beats(x, y))

This translation, however, cannot be correct since it™s not even a sentence; the
occurrence of y in Beats(x, y) is free, not bound. If we move the parenthesis
to capture this free variable, we obtain the following, which means something
quite di¬erent from our English sentence.

∀x (Farmer(x) § ∃y (Donkey(y) § Owns(x, y) § Beats(x, y)))

This means that everything in the domain of discourse is a farmer who owns
and beats a donkey, something which neither implies nor is implied by the
original sentence.
To get a correct ¬rst-order translation of the original donkey sentence, it
can be paraphrased as

Every donkey owned by any farmer is beaten by them.

This sentence clearly needs two universal quanti¬ers in its translation:

∀x (Donkey(x) ’ ∀y ((Farmer(y) § Owns(y, x)) ’ Beats(y, x)))




Section 11.4
302 / Multiple Quantifiers



Remember

In translating from English to fol, the goal is to get a sentence that has
the same meaning as the original. This sometimes requires changes in the
surface form of the sentence.


Exercises


11.18 (Sentences that need paraphrasing before translation) Translate the following sentences by ¬rst
‚ giving a suitable English paraphrase. Some of them are donkey sentences, so be careful.
1. Only large objects have nothing in front of them.
2. If a cube has something in front of it, then it™s small.
3. Every cube in back of a dodecahedron is also smaller than it.
4. If e is between two objects, then they are both small.
5. If a tetrahedron is between two objects, then they are both small.

Open Ron™s World. Recall that there are lots of hidden things in this world. Each of the above
English sentences is true in this world, so the same should hold of your translations. Check to
see that it does. Now open Bolzano™s World. In this world, only sentence 3 is true. Check that
the same holds of your translations. Next open Wittgenstein™s World. In this world, only the
English sentence 5 is true. Verify that your translations have the same truth values. Submit
your sentence ¬le.

11.19 (More sentences that need paraphrasing before translation) Translate the following sentences
‚ by ¬rst giving a suitable English paraphrase.
1. Every dodecahedron is as large as every cube. [Hint: Since we do not have anything
corresponding to as large as (by which we mean at least as large as) in our language,
you will ¬rst need to paraphrase this predicate using larger than or same size as.]
2. If a cube is to the right of a dodecahedron but not in back of it, then it is as large as
the dodecahedron.
3. No cube with nothing to its left is between two cubes.
4. The only large cubes are b and c.
5. At most b and c are large cubes. [Note: There is a signi¬cant di¬erence between this
sentence and the previous one. This one does not imply that b and c are large cubes,
while the previous sentence does.]

Open Ron™s World. Each of the above English sentences is true in this world, so
the same should hold of your translations. Check to see that it does. Now open
Bolzano™s World. In this world, only sentences 3 and 5 are true. Check that the




Chapter 11
Paraphrasing English / 303



same holds of your translations. Next open Wittgenstein™s World. In this world, only the English
sentences 2 and 3 are true. Verify that your translations have the same truth values. Submit
your sentence ¬le.

11.20 (More translations) The following English sentences are true in G¨del™s World. Translate them,
o
‚ and make sure your translations are also true. Then modify the world in various ways, and
check that your translations track the truth value of the English sentence.
1. Nothing to the left of a is larger than everything to the left of b.
2. Nothing to the left of a is smaller than anything to the left of b.
3. The same things are left of a as are left of b.
4. Anything to the left of a is smaller than something that is in back of every cube to the
right of b.
5. Every cube is smaller than some dodecahedron but no cube is smaller than every do-
decahedron.
6. If a is larger than some cube then it is smaller than every tetrahedron.
7. Only dodecahedra are larger than everything else.
8. All objects with nothing in front of them are tetrahedra.
9. Nothing is between two objects which are the same shape.
10. Nothing but a cube is between two other objects.
11. b has something behind it which has at least two objects behind it.
12. More than one thing is smaller than something larger than b.

Submit your sentence ¬le.

11.21 Using the symbols introduced in Table 1.2, page 30, translate the following into fol. Do
‚ not introduce any additional names or predicates. Comment on any shortcomings in your
translations. When you are done, submit your sentence ¬le and turn in your comments to your
instructor.
1. Every student gave a pet to some other student sometime or other.
2. Claire is not a student unless she owned a pet (at some time or other).
3. No one ever owned both Folly and Scru¬y at the same time.
4. No student fed every pet.
5. No one who owned a pet at 2:00 was angry.
6. No one gave Claire a pet this morning. (Assume that “this morning” simply means
before 12:00.)
7. If Max ever gave Claire a pet, she owned it then and he didn™t.
8. You can™t give someone something you don™t own.
9. Max fed all of his pets before Claire fed any of her pets.
10. Max gave Claire a pet between 2:00 and 3:00. It was hungry.




Section 11.4
304 / Multiple Quantifiers


11.22 Using the symbols introduced in Table 1.2, page 30, translate the following into colloquial
 English. Assume that each of the sentences is asserted at 2 p.m. on January 2, 2001, and
use this fact to make your translations more natural. For example, you could translate
Owned(max, folly, 2:00) as Max owns Folly.
1. ∀x [Student(x) ’ ∃z (Pet(z) § Owned(x, z, 2:00))]
2. ∃x [Student(x) § ∀z (Pet(z) ’ Owned(x, z, 2:00))]
3. ∀x ∀t [Gave(max, x, claire, t) ’ ∃y ∃t Gave(claire, x, y, t )]
4. ∃x [Owned(claire, x, 2:00) § ∃t (t < 2:00 § Gave(max, x, claire, t))]
5. ∃x ∃t (1:55 < t § t < 2:00 § Gave(max, x, claire, t))
6. ∀y [Person(y) ’ ∃x ∃t (1:55 < t § t < 2:00 § Gave(max, x, y, t))]
7. ∃z {Student(z) § ∀y [Person(y) ’ ∃x ∃t (1:55 < t § t < 2:00 §
Gave(z, x, y, t))]}

11.23 Translate the following into fol. As usual, explain the meanings of the names, predicates, and
 function symbols you use, and comment on any shortcomings in your translations.
1. There™s a sucker born every minute.
2. Whither thou goest, I will go.
3. Soothsayers make a better living in the world than truthsayers.
4. To whom nothing is given, nothing can be required.
5. If you always do right, you will gratify some people and astonish the rest.


Section 11.5
Ambiguity and context sensitivity

There are a couple of things that make the task of translating between English
and ¬rst-order logic di¬cult. One is the sparseness of primitive concepts in
fol. While this sparseness makes the language easy to learn, it also means
that there are frequently no very natural ways of saying what you want to
say. You have to try to ¬nd circumlocutions available with the resources at
hand. While this is often possible in mathematical discourse, it is frequently
impossible for ordinary English. (We will return to this matter later.)
The other thing that makes it di¬cult is that English is rife with ambi-
ambiguity
guities, whereas the expressions of ¬rst-order logic are unambiguous (at least
if the predicates used are unambiguous). Thus, confronted with a sentence
of English, we often have to choose one among many possible interpretations
in deciding on an appropriate translation. Just which is appropriate usually

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