What is the difference between interpretations of the following predicates?

Question

I need to translate 2 sentences into predicate logic using the following definitions:-

A(x, y): x admires y
B(x, y): x attended y
P(x): x is a professor
S(x): x is a student
L(x): x is a lecture
m: mary

Now I want to translate the following 2 sentences into predicate logic:

1. No student attended every lecture.
2. No lecture was attended by every student.

My translations:

1. $ \neg \exists x \space (S(x) \land (\forall y \space L(y) \rightarrow B(x, y) ) ) $
2. $ \neg \exists x \space (L(x) \land (\forall y \space S(y) \rightarrow B(y, x) ) ) $

Problem:

1. First of all, I'm not sure if my translations are correct(although I'm convinced that they are).
2. Second, I am not able to deduce why the following translations are wrong:
   1. $ \neg \exists x \forall y \space (S(x) \land L(y) \rightarrow B(x, y)) $
   2. $ \neg \exists x \forall y \space (L(x) \land S(y) \rightarrow B(y, x)) $
   or,
   1. $ \neg \exists x \space (S(x) \rightarrow (\forall y L(y) \rightarrow B(x, y))) $
   2. $ \neg \exists x \space (L(x) \rightarrow (\forall y S(y) \rightarrow B(y, x))) $
   I know that they are all different, that much is evident by converting each translation into clause form and comparing. But I'm not able to put a finger on what is wrong.
   For the last set of translations, the only reasoning I'm able to use is by translating the predicate into a non-English language and finding that the sentence constructed is not very idiomatic.
3. Lastly, as a result of 3, I find that using an implication in $\exists$ just seems wrong(as I said, the sentence formed are not very idiomatic when translated into a non-English language(I tried 2 other languages)). Are there any cases where an implication makes sense in an $\exists$.

Answer 1

(a) The bracketing is wrong, for a start.

Take the first example in stages, using LogLish (a mix of English and logical symbolism) as a halfway house!

How would you render "x attended every lecture"? $\forall y(Ly \to Bxy)$.

How would you render 'No student is $\varphi$" using a negated existential? $\neg\exists x(Sx \land \varphi x)$

Now you want the case where $\varphi x = x$-attended-every-lecture = $\forall y(Ly \to Bxy)$. So plugging that in, we get

$\neg\exists x(Sx \land \forall y(Ly \to Bxy))$

Do you see why the $Bxy$ needs to be inside the scope of the existential quantifier?

The second example is similar.

(b) On the general issue of translating restricted quantifiers, see e.g. Restricted quantifiers - Logic (and for more, see the reference to Teller's Primer given there).