Solutions to Homework 3, Problems 14-17

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. Owned(Claire, Folly, 2:00 2. Gave(Claire, Silly, Max, 2:00) 3. Student(Max) 4. Erased(Claire, Folly, 2:00) 5. Owned(Max, Folly, 3:05) 6. 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 FOL Comment AIDS AIDS Influenza Flu

Predicates: English FOL Comment x is less contagious than y LessContagious(x, y)

LessContagious(AIDS, Flu)

2. Spain is between France and Portugal in size

Names: English FOL Comment Spain Spain France France Portugal Portugal

Predicates: English FOL Comment x is between y and z in size BetweenInSize(x, y, z)

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 FOL Comment A person Person People other than "Person" Others Predicates: English FOL Comment x is Miserable Miserable(x) x prefers a situation such that y Prefers(x, y)

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 FOL Comment Max Max Claire Claire

Predicates English FOL Comment x shook y Shook(x, y) the hand of x Hand(x)

Function Symbols English FOL Comment x's father Father(x, y)

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 FOL Comment John John Nancy Nancy Mary Ellen Mary Ellen Jon Jon

Predicates English FOL Comment x is younger than y Younger(x, y) Function Symbols English FOL Comment the eldest child of y and z Eldest(y, z)

Younger(Eldest(John, Nancy), Eldest(Jon Mary Ellen))