First Order Logic to CNF for Knowledge Base

Question

I am doing some Homework for an Artificial Intelligence Course, we are covering some First Order Logic and Conjuctive Normal Form.

Here are the questions that I have to answer that I am having trouble with

Q10. [20] Suppose that the sentence A in Q9 is changed to:

A1. Some great chefs are French.

1) [6] Write it in the FOL sentence.

a. Existential(x): GC(x) and F(x)

2) [6] Convert 1) to the the definite clause in CNF, suitable for Knowledge_Base through Skolemization, etc. if necessary.

a. Existential(x): GC(x) and F(x)

b. ¬ (Existential(x): GC(x) and F(x))

c. Universal(x): ¬GC(x) or ¬F(x)

d. Universal(x): GC(x) therefore ¬F(x)

3) [8] Prove how the same query can be answered (or not). Justify your answer step by step.

So my question here is, I feel like I am doing the conversion from FOL here to CNF wrong unless you can actually have a negative predicate as a conclusion for CNF for the knowledge base. How would I go about changing this to make it work?

And then I have no idea how to approach #3 to answer the question.

Answer 1

Your symbolization of 'some great chefs are French' is wrong. That should simply be:

$\exists x (GC(x) \land F(x))$

After negation:

$\neg \exists x (GC(x) \land F(x))$

You get:

$\forall x \neg (GC(x) \land F(x))$

And thus:

$\forall x (\neg GC(x) \lor \neg F(x))$

Answer 2

Q10. [20] Suppose that the sentence A in Q9 is changed to:

A1. Some great chefs are French.

1) [6] Write it in the FOL sentence.

a. Existential(x) Universal(y): GC(y) and F(x)

That says: "Everything is a great chef, and something is French."

You want to claim that there is a thing that is a great chef, and this same thing is also French.   So you must use the same token for it.

Existential(x): GC(x) and F(x)

$$\exists x:(\operatorname{GC}(x)\land \operatorname{F}(x))$$