# How to state sentences for KB ∧ ¬ α given existing KB?

How do I state the sentences for KB ∧ ¬ α when I already have KB.

KB:

∀xTourist(x) => Person(x): Every tourist is a person.

∀xTourist(x) ∧ visits(x, Malaysia) => walksCanopy(x): Every tourist who visits Malaysia walks the canopy.

∀xPerson(x) ∧ has(x, Acrophobia) => fallSick(x, walksCanopy): Every person who has acrophobia falls sick when they walk the canopy.

∃xPerson(x) => has(x, Acrophobia): There are some people who have acrophobia.

Friend(Abu, Bill): Abu and Bill are friends.

Person(Abu) => livesIn(Abu, Malaysia): Abu is a person who lives in Malaysia.

Person(Bill) => livesIn(Bill, Canada): Bill is a person who lives in Canada.

∀xFriend(x) ∧ Friend(Bill, x) ∧ visitsCountry(Bill, x): Bill visits the countries of all his friends.

has(Bill, Acrophobia): Bill has acrophobia.

Prove that "Bill will fall sick".

I already looked at my lecture notes and even looked it up online but I can't seem to understand what KB ∧ ¬ α is.

Thanks.

• You need the formula $\alpha$. – Mauro ALLEGRANZA Apr 12 at 7:45
• Then you have to negate it and add to the four formulas of KB. – Mauro ALLEGRANZA Apr 12 at 7:46
• Having said that, maybe you can add detials about the problem you are working on... – Mauro ALLEGRANZA Apr 12 at 7:47
• Maybe you are working with Resolution ? – Mauro ALLEGRANZA Apr 12 at 8:06
• @MauroALLEGRANZA Hi, I've edited my question and changed my KB. If I am understanding it correctly α is "Bill will fall sick"? – Mike Mann Apr 12 at 9:21

Let $$\text {KB}$$ a set of sentences and $$\alpha$$ a sentence.

In order to prove that $$\text {KB} \vDash \alpha$$ we consider $$\text {KB} \cup \{ \lnot \alpha \}$$ and apply the Resolution proof-procedure for predicate logic.

If we succeed deriving the empty clause, we have shown that $$\text {KB} \cup \{ \lnot \alpha \}$$ is unsatisfiable, that is equivalent to $$\text {KB} \vDash \alpha$$.

In your example, you are asked to prove that "Bill will fall sick" is implied by $$\text {KB}$$.

This means that "Bill will fall sick" is "your $$\alpha$$" and thus you have to add "Bill will not fall sick" to $$\text {KB}$$.

• So basically if I have to state my sentences for KB it'd be: ∀xTourist(x) => Person(x) ∧ ¬ fallSick(Bill): Every tourist is a person and Bill will not fall sick. and so on. Is that right? – Mike Mann Apr 12 at 14:11
• @MikeMann - Yes $\lnot \alpha$ is ¬ fallSick(Bill). – Mauro ALLEGRANZA Apr 12 at 14:14
• thanks! I'm gonna work on CNF conversion and resolution now and see if I can figure it out. – Mike Mann Apr 12 at 14:23