How Would We Translate this Sentence Into Logic?

Question

I am trying to do a proof by resolution but I don't think I have the correct translation into logic for one of the sentences given.

Sentence: There is a SUV that is bigger than every car.

My current answer is this: ∃ x ∀ y [SUV(x) ∧ Car(y)] --> Bigger(x, y)

This is my answer in CNF: ¬SUV(f) ∨ ¬Car(g) ∨ Bigger(f, g)

Is this correct? Or am I doing something wrong?

Answer 1

Your association is a bit off.

"There is a SUV that is bigger than every car."

"There is something (call it $x$) that is a SUV and, for anything (call it $y$), $x$ is bigger than $y$ if $y$ is a car."

$$\exists x~\Big(\operatorname{SUV}(x)\land \forall y~\big(\operatorname{Car}(y)\to\operatorname{Bigger}(x,y)\big)\Big)$$

Which is equivalent to the PRENEX form:

$$\exists x~\forall y~\Big(\operatorname{SUV}(x)\land \big(\operatorname{Car}(y)\to\operatorname{Bigger}(x,y)\big)\Big)$$

So try your resolution encoding again from that.

Answer 2

Try to avoid pulling the all the quantifers outside as this can change the meaning in some cases. If the SUV is a car, then that is something you need to include. As a result of saying the SUV is a car, $x\neq y$ must be mentioned as something cannot be bigger than itself.

You can paraphrase the sentence to say:

There is an SUV and for everything that is a car, then that SUV is bigger than that car.

Or assuming that the SUV is a car

There is a car that is an SUV and for everything that is a car, that is not this SUV, then that SUV is bigger than that car.