I need help translating these sentences because I am having trouble understandin

Question

I need help translating these sentences because I am having trouble understanding the proper format and would like to compare with current work especially with sentences #10-20.

Translate each of the following English sentences into rst-order logic, using only the following constants, functions, and predicates ...
Constants:
• IronMan — a hero named Iron Man
• TonyStark — a person named Tony Stark
Functions:
• Height(x) — the height of x
Predicates:
• Book(x) — x is a book
• Day(x) — x is a day
• Dog(x) — x is a dog
• Human(x) — x is a human
• Student(x) — x is a student
• Useful(x) — x is useful
• Boss(x,y) — x is the boss of y
• Coworker(x,y) — x is a coworker of y
• Has(x,y) — x has y
• Likes(x,y) — x likes y
• Equal(x,y) — x and y are the same object
• GT(x,y) — the height x is greater than the height y

Exercises:
1. “All students are human.”
2. “Some books are useful.”
3. “Not all books are useful.”
4. “No dogs are students.”
5. “Every dog has his day.”
6. “Some humans like all dogs.”
7. “Coworkers are humans who have the same (human) boss.”
8. “Tony Stark is Iron Man.”
9. “There is a human who likes all the dogs he has, but no others.”
10. “There is exactly one useful book.”
11. “There is only one dog that likes Tony Stark.”
12. “The tallest dog is taller than the shortest human.”
13. “All dogs are useful.”
14. “Some humans are students.”
15. “Not all humans are students.”
16. “No books are useful.”
17. “Every student has a book.”
18. “There is at least one human who has all books.”
19. “There is a human who has all of the books that he likes.”
20. “Some students have only one dog.”

Explanation / Answer

English sentences are converted to first order logic based on the given function, constant and predicate.

Every FOL supplies 3 premitives-

(i) variable symbol- Eg. x,y

(ii) connectives- not, and, or, implies.

(iii) quantifiers- Universal and existential

1)

x ( Student(x) Human(x) )

2)

x (Books(x) Useful(x))

3)

¬ x (Books(x) Useful(x))

4)

¬ x (Dogs(x) Student(x))

5)

x (Dog(x) y ( Day(y) Has(x,y) ))

6)

x ( Human(x) y ( Dog(y) ¬ (x = y) Like(x,y) ))

13)

x ( Dog(x) Useful(x) )

14)

x (Human(x) Student(x))

15)

¬ x (Human(x) Student(x))

16)

¬ x (Book(x) Useful(x))

17)

x (Student(x) y ( Book(y) Has(x,y) ))

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