Solutions to Homework 3, problems 14-17
Problem 14.
1. GaveScruffy(Max,Claire)
GaveScruffy(Claire,Max)
GaveScruffy(Max,Max)
GaveScruffy(Claire,Claire)
GaveCarl(Max,Claire)
GaveCarl(Claire,Max)
GaveCarl(Max,Max)
GaveCarl(Claire,Claire)
2. 64 sentences can be expressed in language 2. Basically we can figure this out by
listing them and counting them, such as
Gave(Max,Max,Max)
Gave(Max,Max,Claire)
Gave(Max,Max,Scruffy)
Gave(Max,Max,Carl)
…
Gave(Max,Claire,Max)
And so on. But of course this is tedious. However, since there are 4 names and the
predicate has an arity of 3, we can conclude that the number of statements expressible
in language 2 is 43.
3. You would need exactly the same number of names to express all of the sentences of
language 2 in our modified version of language 1. You would also need 4 predicates:
GaveMax, GaveClaire, GaveScruffy, and GaveCarl.
Problem 15.
1.
2.
3.
4.
5.
6.
Owned(Claire, Folly, 2:00
Gave(Claire, Silly, Max, 2:00)
Student(Max)
Erased(Claire, Folly, 2:00)
Owned(Max, Folly, 3:05)
2:00 < 2:05
Problem 16.
1. Max owned Silly at 2:00 pm.
2. Max erased Silly at 2:30 pm.
3. Max gave Silly to Claire at 2:00 pm.
4. 2:00 pm is earlier than 2:00 pm.
Problem 17.
There are any number of ways of translating these sentences, depending upon what
sort of language you construct. The thing to keep in mind is the necessity of
translating accurately the meaning of the English phrase.
1. AIDS is less contagious than influeza.
Names:
English
AIDS
Influenza
FOL
AIDS
Flu
Comment
Predicates:
English
x is less contagious than y
FOL
LessContagious(x, y)
Comment
LessContagious(AIDS, Flu)
2. Spain is between France and Portugal in size
Names:
English
Spain
France
Portugal
FOL
Spain
France
Portugal
Comment
Predicates:
English
x is between y and z in size
FOL
BetweenInSize(x, y, z)
Comment
BetweenInSize(Spain, France, Portugal)
3. Misery loves company.
Note: In order to translate this phrase correctly, we must first specify its meaning in
English, which is something like "Miserable people prefer that others are miserable as
well." It is this problem that makes problem 17 tricky, hence the "*" in the book.
Names:
English
A person
People other than "Person"
FOL
Person
Others
Comment
Predicates:
English
x is Miserable
x prefers a situation such that y
FOL
Miserable(x)
Prefers(x, y)
Comment
Prefers(Miserable(Person), Miserable(Others))
Another way of doing this would be to avoid specifying any names but to use variables
instead with the above predicates and say that
Prefers(Miserable(x), Miserable(y)
xy
4. Max shook Claire's father's hand.
Names:
English
Max
Claire
FOL
Max
Claire
Comment
Predicates
English
x shook y
the hand of x
FOL
Shook(x, y)
Hand(x)
Comment
Function Symbols
English
x's father
FOL
Father(x, y)
Comment
Shook(Max, Hand(Father(Claire)))
5. John and Nancy's eldest child is younger than John and Mary Ellen's.
Note that there a lot of other solutions to this problem. This is one of the simpler ones:
Names:
English
John
Nancy
Mary Ellen
Jon
Predicates
English
x is younger than y
FOL
John
Nancy
Mary Ellen
Jon
Comment
FOL
Younger(x, y)
Comment
Function Symbols
English
the eldest child of y and z
FOL
Eldest(y, z)
Younger(Eldest(John, Nancy), Eldest(Jon Mary Ellen))
Comment