I am having a hard time on a homework problem which involves converting a given English sentence into first order logic and then converting that into Conjunctive Normal Form.
The sentences are (EDIT: Along with the entire problem statement)
Translate the following into First Order Logic and then convert to Conjunctive Normal Form (CNF):
According to the Pidgeon: If little girls eat eggs, then they are a kind of serpent. Alice (who is a little girl) eats eggs. Therefore, she is a kind of serpent.
I think I have done well so far below but I seem to be stuck trying to reduce the sentences any further, I am not sure what rule to apply from here.
Hypothesis where L(x) = LittleGirl(x), E(x) = EatEggs(x), S(x) = Serpent(x)
{[∀x ((L(x) ∧ E(x)) ⇒ S(x))] ∧ [L(Alice) ∧ E(Alice)]} ⇒ S(Alice)
Implication: [A ⇒ B ≡ ¬A ∨ B] Applied to: L and E
{[∀x (¬(L(x) ∧ E(x)) ∨ S(x))] ∧ [L(Alice) ∧ E(Alice)]} ⇒ S(Alice)
Implication: [A ⇒ B ≡ ¬A ∨ B] Applied to: Pidgeon's final implication
¬{[∀x (¬(L(x) ∧ E(x)) ∨ S(x))] ∧ [L(Alice) ∧ E(Alice)]} ∨ S(Alice)
Universal Instantiation: [∀x P(x) ⇒ P(a/x)] Applied to: The only occurance
¬{[(¬(L(a/x) ∧ E(a/x)) ∨ S(a/x))] ∧ [L(Alice) ∧ E(Alice)]} ∨ S(Alice)
DeMorgan's Law: [¬(A ∧ B) ≡ ¬A ∨ ¬B, ¬(A ∨ B) ≡ ¬A ∧ ¬B] Applied to: Statement in brackets
¬[(¬(L(a/x) ∧ E(a/x)) ∨ S(a/x))] ∨ ¬[L(Alice) ∧ E(Alice)] ∨ S(Alice)
DeMorgan's Law: [¬(A ∧ B) ≡ ¬A ∨ ¬B, ¬(A ∨ B) ≡ ¬A ∧ ¬B] Applied to: a/x statement
(¬¬(L(a/x) ∧ E(a/x)) ∧ ¬S(a/x)) ∨ ¬[L(Alice) ∧ E(Alice)] ∨ S(Alice)
DeMorgan's Law: [¬(A ∧ B) ≡ ¬A ∨ ¬B, ¬(A ∨ B) ≡ ¬A ∧ ¬B] Applied to: Middle statement
(¬¬(L(a/x) ∧ E(a/x)) ∧ ¬S(a/x)) ∨ ¬L(Alice) ∨ ¬E(Alice) ∨ S(Alice)
Double negation elimination: [¬¬A ≡ A] Applied to: Left statement
((L(a/x) ∧ E(a/x)) ∧ ¬S(a/x)) ∨ ¬L(Alice) ∨ ¬E(Alice) ∨ S(Alice)
Though after doing some reading I think that I might have done this wrong. After reviewing this webpage describing Universal Instantiation I feel like I could apply its example problem directly to mine. I could have initially applied Universal Instantiation and then Modus ponens to have simply ended with Serpent(Alice)
which is in CNF. But this feels like I am "cheating" the problem, or is this most likely the solution the professor seeks for the question?
Am I on the right track above or should I try applying the example from the website to my problem? If I am on the right track what rule might I apply next (specifically to the top statement) to continue?