Axioms --> Clausal Form

Question

Question :

Consider the following sentences and prove that "Diana will win the game"

1.All Players are clever.

2.Anyone who is clever and dedicated can play the game well.

3.Anyone who is playing the game well will win his/her game.

4.Diana is a dedicated player.

So my first attempt was representing the using axioms as follows(Please note that I'll be using the following notations as follows:

VX - For all X(Universal quantifier)

EX - For some X(Existential quantifier)

Let P(X) be X is a player

C(X) be X is clever

D(X) be X is dedicated

G(X) be X can play the game well

W(X) be X will win his/her game

Axiom Form :

1. VX [P(X) --> C(X) ]

2. [ ( C(X) ^ D(X) ) --> G(X) ]

3. VX [ G(X) --> W(X) ]

4. P(Diana) --> D(X)

Clausal Form :

1. ~P(X) v C(X)

2. ~C(X) v ~D(X) v G(X)

3. ~G(X) v W(X)

4. ~P(Diana) v D(X)

Can anyone please explain how to prove that "Diana will win the game" , using above clausal forms.

Thanks!

Answer 1

Prove it by contradiction...

Suppose Diana will LOSE the game, so ~W

Then clausal statement 3 implies that Diana is not a good player.

Clausal statement 2 implies that Diana is not Clever OR Diana is not Dedicated.

Clausal Statement #1 then shows that Diana is not a player since she is not clever

OR

Clausal Statement #4 then shows that Diana is not a player at all since she is not dedicated.

Both statements are contradictions, especially the latter