<<

. 56
( 107 .)



>>

depends on context.
The ambiguities become especially vexing with quanti¬ed noun phrases.
Consider, for example, the following joke, taken from Saturday Night Live:



Chapter 11
Ambiguity and context sensitivity / 305



Every minute a man is mugged in New York City. We are going to
interview him tonight.

What makes this joke possible is the ambiguity in the ¬rst sentence. The most
natural reading would be translated by

∀x (Minute(x) ’ ∃y (Man(y) § MuggedDuring(y, x)))

But the second sentence forces us to go back and reinterpret the ¬rst in a
rather unlikely way, one that would be translated by

∃y (Man(y) § ∀x (Minute(x) ’ MuggedDuring(y, x)))

This is often called the strong reading, the ¬rst the weak reading, since
this one entails the ¬rst but not vice versa.
Notice that the reason the strong translation is less likely is not determined
by the form of the original sentence. You can ¬nd examples of the same form context sensitivity
where the strong reading is more natural. For example, suppose you have been
out all day and, upon returning to your room, your roommate says, “Every
ten minutes some guy from the registrar™s o¬ce has called trying to reach
you.” Here it is the strong reading where the existential “some guy” is given
wide scope that is most likely the one intended.
There is another important way in which context often helps us disam-
biguate an ambiguous utterance or claim. We often speak about situations
that we can see, and say something about it in a way that makes perfectly
clear, given that what we see. Someone looking at the same scene typically
¬nds it clear and unambiguous, while someone to whom the scene is not visible
may ¬nd our utterance quite unclear. Let™s look at an example.


You try it
................................................................
1. It is a hard to get too many blocks adjacent to a single block in Tarski™s
World, because many of the blocks over¬‚ow their squares and so do not
leave room for similar sized blocks on adjacent squares. How many medium
dodecahedra do you think it is possible to have adjacent to a single medium
cube?

2. Open Anderson™s First World. Notice that this world has four medium do-
decahedra surrounding a single medium cube.

3. Imagine that Max makes the following claim about this situation:



Section 11.5
306 / Multiple Quantifiers


At least four medium dodecahedra are adjacent to a medium cube.

The most natural understanding of Max™s claim in this context is as the
claim that there is a single cube to which at least four dodecahedra are
adjacent.

4. There is, however, another reading of Max™s sentence. Imagine that a
tyrant tetrahedron is determined to assassinate any medium dodecahe-
dron with the e¬rontery to be adjacent to a medium cube. Open Ander-
son™s Second World and assume that Max makes a claim about this world
with the above sentence. Here a weaker reading of his claim would be the
more reasonable, one where Max is asserting that at least four medium
dodecahedra are each adjacent to some medium cube or other.

5. We would ask you to translate these two readings of the one sentence
into fol, but unfortunately you have not yet learned how translate “at
least four” into fol yet; this will come in Chapter 14 (see Exercise 14.5 in
particular). Instead consider the following sentence:

Every medium dodecahedron is adjacent to a medium cube.

Write the stronger and weaker translations in a ¬le, in that order. Check
that the stronger reading is only true in the ¬rst of Anderson™s worlds,
while the weaker reading is true in both. Save your ¬le as Sentences Max 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Congratulations
The problems of translation are much more di¬cult when we look at ex-
tended discourse, where more than one sentence comes in. To get a feeling for
extended discourse
the di¬culty, we start of with a couple of problems about extended discourse.
Remember

A important source of ambiguity in English stems from the order in which
quanti¬ers are interpreted. To translate such a sentence into fol, you
must know which order the speaker of the sentence had in mind. This
can often be determined by looking at the context in which the sentence
was used.




Chapter 11
Ambiguity and context sensitivity / 307



Exercises


11.24 If you skipped the You try it section, go back and do it now. Save your sentence ¬le as
‚ Sentences Max 1.

11.25 (Translating extended discourse)
‚ —¦ Open Reichenbach™s World 1 and examine it. Check to see that all of the sentences in the
following discourse are true in this world.

There are (at least) two cubes. There is something between them. It is a medium
dodecahedron. It is in front of a large dodecahedron. These two are left of a small
dodecahedron. There are two tetrahedra.

Translate this discourse into a single ¬rst-order sentence. Check to see that your trans-
lation is true. Now check to see that your translation is false in Reichenbach™s World
2.

—¦ Open Reichenbach™s World 2. Check to see that all of the sentences in the following
discourse are true in this world.

There are two tetrahedra. There is something between them. It is a medium
dodecahedron. It is in front of a large dodecahedron. There are two cubes. These
two are left of a small dodecahedron.

Translate this into a single ¬rst-order sentence. Check to see that your translation is
true. Now check to see that your translation is false in Reichenbach™s World 1. However,
note that the English sentences in the two discourses are in fact exactly the same; they
have just been rearranged! The moral of this exercise is that the correct translation of a
sentence into ¬rst-order logic (or any other language) can be very dependent on context.
Submit your sentence ¬le.

11.26 (Ambiguity) Use Tarski™s World to create a new sentence ¬le and use it to translate the following
‚ sentences into fol. Each of these sentences is ambiguous, so you should have two di¬erent
translations of each. Put the two translations of sentence 1 in slots 1 and 2, the two translations
of sentence 3 in slots 3 and 4, and so forth.
1. Every cube is between a pair of dodecahedra.
3. Every cube to the right of a dodecahedron is smaller than it is.
5. Cube a is not larger than every dodecahedron.




Section 11.5
308 / Multiple Quantifiers


7. No cube is to the left of some dodecahedron.
9. (At least) two cubes are between (at least) two dodecahedra.

Now open Carroll™s World. Which of your sentences are true in this world? You should ¬nd that
exactly one translation of each sentence is true. If not, you should correct one or both of your
translations. Notice that if you had had the world in front of you when you did the translations,
it would have been harder to see the ambiguity in the English sentences. The world would have
provided a context that made one interpretation the natural one. Submit your sentence ¬le.

(Ambiguity and inference) Whether or not an argument is valid often hinges on how some ambiguous
claim is taken. Here are two arguments, each of whose ¬rst premise is ambiguous. Translate each ar-
gument into fol twice, corresponding to the ambiguity in the ¬rst premise. Under one translation the
conclusion follows. Prove it. Under the other, it does not. Describe a situation in which the premises
are true (with this translation) but the conclusion is false.

11.27 11.28
Everyone admires someone who has All that glitters is not gold.
 
red hair. This ring glitters.
Anyone who admires himself is con-
This ring is not gold.
ceited.
Someone with red hair is conceited.



Section 11.6
Translations using function symbols
Intuitively, functions are a kind of relation. One™s mother is one™s mother
because of a certain relationship you and she bear to one another. Similarly,
relations and functions
2 + 3 = 5 because of a certain relationship between two, three, and ¬ve.
Building on this intuition, it is not hard to see that anything that can be
expressed in fol with function symbols can also be expressed in a version of
fol where the function symbols have been replaced by relation symbols.
The basic idea can be illustrated easily. Let us use mother as a unary
function symbol, but MotherOf as a binary relation symbol. Thus, for example,
mother(max) = nancy and MotherOf(nancy, max) both state that Nancy is the
mother of Max.
The basic claim is that anything we can say with the function symbol we
can say in some other way using the relation symbol. As an example, here is
a simple sentence using the function symbol:

∀x OlderThan(mother(x), x)



Chapter 11
Translations using function symbols / 309



It expresses the claim that a person™s mother is always older than the person.
To express the same thing with the relation symbol, we might write
∀x ∃y [MotherOf(y, x) § OlderThan(y, x)]
Actually, one might wonder whether the second sentence quite manages to
express the claim made by the ¬rst, since all it says is that everyone has at
least one mother who is older than they are. One might prefer something like
∀x ∀y [MotherOf(y, x) ’ OlderThan(y, x)]
This says that every mother of everyone is older than they are. But this too
seems somewhat de¬cient. A still better translation would be to conjoin one of
the above sentences with the following two sentences which, together, assert
that the relation of being the mother of someone is functional. Everyone has
at least one, and everyone has at most one.
∀x ∃y MotherOf(y, x)
and
∀x ∀y ∀z [(MotherOf(y, x) § MotherOf(z, x)) ’ y = z]
We will study this sort of thing much more in Chapter 14, where we will
see that these two sentences can jointly be expressed by one rather opaque
sentence:
∀x ∃y [MotherOf(y, x) § ∀z [MotherOf(z, x) ’ y = z]]
And, if we wanted to, we could then incorporate our earlier sentence and
express the ¬rst claim by means of the horrendous looking:
∀x ∃y [MotherOf(y, x) § OlderThan(y, x) § ∀z [MotherOf(z, x) ’ y = z]]
By now it should be clearer why function symbols are so useful. Look at all
the connectives and additional quanti¬ers that have come into translating our
very simple sentence
∀x OlderThan(mother(x), x)
We present some exercises below that will give you practice translating
sentences from English into fol, sentences that show why it is nice to have
function symbols around.

Remember

Anything you can express using an n-ary function symbol can also be
expressed using an n + 1-ary relation symbol, plus the identity predicate,
but at a cost in terms of the complexity of the sentences used.




Section 11.6
310 / Multiple Quantifiers


Exercises


11.29 Translate the following sentences into fol twice, once using the function symbol mother, once
 using the relation symbol MotherOf.
1. Claire™s mother is older than Max™s mother.
2. Everyone™s mother™s mother is older than Melanie.
3. Someone™s mother™s mother is younger than Mary.

11.30 Translate the following into a version of fol that has function symbols height, mother, and
 father, the predicate >, and names for the people mentioned.
1. Mary Ellen™s father is taller than Mary Ellen but not taller than Claire™s father.
2. Someone is taller than Claire™s father.
3. Someone™s mother is taller than their father.
4. Everyone is taller than someone else.
5. No one is taller than himself.
6. Everyone but J.R. who is taller than Claire is taller than J.R.
7. Everyone who is shorter than Claire is shorter than someone who is shorter than
Melanie™s father.
8. Someone is taller than Jon™s paternal grandmother but shorter than his maternal grand-
father.

Say which sentences are true, referring to the table in Figure 9.1 (p. 254). Take the domain of

<<

. 56
( 107 .)



>>