Represent the following sentences in first order logic.

Question

Q) Represent the following sentences in first order logic is this the right?

1) John is intelligent than all other student in his class.

FOL: ∀X∶Student(X)→Intelligent(john,X)

2) Neiher Alex nor John is unhappy.

FOL: ¬(happy(alex) ∨ happy(john))

3) Some dogs are domestic pets. (which one is correct)

FOL: i) ∃X∶Dog(X)→DomesticPets(X) (or)

ii) ∃X∶(Dogs(X) ∧ DomesticPets(X))

4) Some turtles are faster than rabbits.

FOL: ∃X∶Turtle(X) ∧ Rabbit(Y)→faster(X,Y)

Answer 1

• Some dogs are domestic pets. Should be the second one $$ ∃X∶\text{Dogs(X)} ∧ \text{DomesticPets(X)}$$ To see why it's not the first one, apply distributive law of $\exists$ over $\to$ we have \begin{align} &∃X∶\text{Dog(X)}→\text{DomesticPets(X)}\\ \equiv&\forall X:\text{Dog(X)}→\exists X:\text{DomesticPets(X)} \end{align} It basicly means if everything are dog, then something is a DomesticPet, clearly isn't what we want to say.
• Some turtles are faster than rabbits. Same mistake: \begin{align} &∃X∶\text{Turtle(X)} ∧ \text{Rabbit(Y)}→\text{faster(X,Y)}\\ \equiv&\forall X∶\text{Turtle(X)} ∧ \text{Rabbit(Y)}→\exists X:\text{faster(X,Y)} \end{align} That is if everything are Tutle and $Y$ is Rabbit, then $\dots$ just make no sense. Instead we write $$\exists X:\text{Turtle(X)} ∧ \forall Y:\text{Rabbit(Y)}\to\text{faster(X,Y)}$$ This is some Turtle faster than all the rabbits.