I have a question with a table of FOLs and Predicates and I am given a set of sentences I need to convert from English to FOL. I have posted the image at the bottom of the question.
I am fairly new to this concept of universal quantifiers and I would really appreciate if someone could just take a look and let me know if I am handling these correctly?
Whenever Thabo drives a lorry in Durban, he is involved in an accident
∀x ∀y (Drives(thabo, bloemfontein) ⇒ Accident(thabo))
Kitty is involved in an accident only if she does not obey the traffic rules.
∃x [(accident(kitty) ⇐⇒ ¬ (Obeys(kitty))]
Unless Betty obeys the traffic rules when she drives a lorry in Bloemfontein, she is involved in an accident.
∀x [(obeys(betty) ∧ ∀y(Drives(betty, bloemfontein)) ⇒ accident(betty)]
Durban or Bloemfontein is a city.
∀x [City(durban) ∨ city(Bloemfontein)]
If we assume that Kitty drives a lorry in Durban, either Thabo or Betty drives a lorry in Bloemfontein.
∀x [Drives(kitty, Durban) ⇒ ∀x ∀y ((Drives(thabo, bloemfontein) ∨ (Drives(betty, bloemfontein))]
Neither Kitty nor Betty obeys the traffic rules.
∀x [¬(obeys(kitty) ∨ obeys(betty))]
Thabo is involved in accident if and only if both he and Betty do not obey the traffic rules.
∀x [Accident(thabo) ⇐⇒ ¬ (obeys(thabo) ∨ obeys(kitty))]
None of Thabo, Kitty or Betty is involved in an accident.
∀x [¬(Accident(thabo) ∧ Accident(kitty) ∧ Accident(betty)]
If Kitty obeys the traffic rules and Betty does not drive a lorry in Bloemfontein, Thabo is not involved in an accident.
∃x [(obeys(kitty) ∧ ¬∀x ∀y (Drives(betty, bloemfontein)) ⇒ Accident(thabo))]
Thabo obeys the traffic rules only if he drives a lorry in Durban.
∃x [(obeys(thabo) ⇐⇒ (Drives(thabo, durban))]
I'll point out one thing:
None of Thabo, Kitty or Betty is involved in an accident.
∀x [¬(Accident(thabo) ∧ Accident(kitty) ∧ Accident(betty)]
Why do you have ∀x when x doesn't appear anywhere in your formula? You are not making a generic statement "every person that", your statement only includes three determined people.
Also, what you wrote translates to:
For every person,
It is not the case that Thabo is in an accident and Kitty is in an accident and Betty is in an accident.because the negation binds the case that all three are simultaneously in an accident.However, by you translation something like
Thabo is in an accident and Kitty is not in an accident and Betty is not in an accident is valid.Makes sense? :)
Update: Second thing I'll point out
Kitty is involved in an accident only if she does not obey the traffic rules.
∃x [(accident(kitty) ⇐⇒ ¬ (Obeys(kitty))]
Are you sure that you can determine that if kitty does not obey the traffic she will be involved in an accident? What the sentence suggests to be is that if she is involved in an accident then she is not obeying the traffic rules. But, it does not guarantee that not obeying the rules will cause an accident.
Again, unnecessary ∃x statement. You are not making a generic statement about somebody, you are referring explicitly to kitty.