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i'm new to first order logic and i'm having some confusion with translating the sentence above.

My solution for the sentence is :

  • ∀x ∀y ∀z ((DOG(y) ∧ LOVES(x, y) ∧ ARMADILLO(z)) → ¬LOVES(x, z)) (1)

But when i search on the Internet, there is a solution that says:

  • ¬ ∃ x ((∀ y (DOG(y) → LOVES(x,y))) ∧ ∃ z (ARMADILLO(z) ∧ LOVES(x,z))) (2)

The conversion into CNF of (1) and (2) are different. Can anyone let me know which one is correct and give me a brief explanation? Thanks.

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The second of these is correct. Observe that the first formulation is equivalent to $$\forall x\forall y\forall z(\neg Dog(y)\vee\neg Loves(x,y)\vee\neg Armadillo(z)\vee\neg Loves(x,z)).$$ The negation of which is $$\exists x\exists y\exists z(Dog(y)\wedge Loves(x,y)\wedge Armadillo(z)\wedge Loves(x,z));$$ that is, "there is a being, a dog, and an armadillo such that the being loves the dog and the armadillo." But this doesn't contradict the statement you were trying to formulate; we don't know that our being who loves this dog loves all dogs, and our statement is about beings who love all dogs.

If the situation you're trying to describe is not true, then someone both loves all dogs, and loves some armadillo. And the second formula is precisely the negation of this scenario.

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