I'm trying to translate English into Predicate Logic but I am not sure if I did it correctly. So I want to check it before I attempt to prove it.

Question 1

The sentence is : Everyone who loves Bella also loves either Claire or Daisy, and none of them speak French, but at least one person who loves Daisy speaks German.

L(x, y) = x loves y,

b = Bell, c = Claire, d = Daisy

F(x) = speaks French

G(x) = speaks German

My first translation is

$$\forall x[L(x,b) \to (L(x,c) \lor L(x,d)] \land \neg\exists x[F(x)] \lor \exists x[L(x,d) \land G(x)]$$

For this one, I am unsure whether to put ¬∃x[F(x)] inside or outside the universal quantifier. Would it make a different if I put it out?

My second translation attempt :

$$\forall x\neg\exists y[ [(L(x,b) \to (L(x,c) \lor L(x,c)) \land \neg F(y) \land \neg(x = y)] ] \land \exists x[L(x,d) \land G(x)]$$

Which of these are correct? (If any)

Question 2

Sentence is : if everyone either speaks French or loves Daisy, and Bella speaks neither French nor German, then somebody must love Daisy.

In this one I translate it as :

$$\forall x[F(x) \lor L(x,d)] \land \neg(F(b) \land G(b)) \to \exists x[L(x,d)]$$

Is the correct? and should I add rule like ¬(x = d) saying that daisy cannot love herself? is it necessary?


For 1: this is a bit ambiguous, but it seems the 'them' in 'none of them' is referring to the people who love Bella, and so you do need to put $\neg F(x)$ inside the scope of the first quantifier. Also, the $\lor$ you have near the end should be an $\land$

For 2: Bella speaking neither French nor German translates as $\neg (F(b) \lor G(b))$


I would interpret "either ... or" as exclusive or which means you have to exclude the case that they love all three of them. Further not speaking french is an attribute of the people who love daisy, while your first try reintroduces x with a new quantifier (which means it is a different variable). So you are saying $\neg\exists x: F(x)$ which is equivalent to $\forall x:\neg F(x)$ which means no person speaks French at all, compared to people who love daisy don't speak French.

Oh and I think the "but" would be a logical and. So there is someone who loves daisy and speaks german, AND the other stuff. Not or the other stuff.

My attempt (Spoiler):

$$\forall x [L(x,b)\rightarrow \left((L(x,c)\vee L(x,d))\wedge\neg(L(x,c)\wedge L(x,d)\right) \wedge \neg F(x)]\wedge \exists y[L(y,d)\wedge G(y)]$$

The second one looks good besides the either or again. Why can daisy not love herself? That is not in the statement.


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