Interpreting logical forms involving quantifiers I have been trying to translate these two logical form into English
statements without using any quantifier laws:
(a) ∃x∀y ¬L(x,y)
(b) ¬(∃x∀y L(x,y))

where L(x,y) means x likes y
I have translated the (b) part like this:


*

*¬(Someone likes everyone)

*Someone doesn't like everyone.


And the (a) part like this:


*

*There exists some x such that for all y, x doesn't like y.

*Someone doesn't like anyone.


Are the translations correct ? Is my reasoning correct ? What are the general guidelines while trying to reason out the these things ?
 A: Your translation for (a) is entirely correct.
However, for (b), you made an error. Let us investigate closely what your translation amounts to when we make a formula of it once again. The following is not formal notation, but should get the point across:


*

*Someone doesn't like everyone

*$\exists x: x$ doesn't like everyone

*$\exists x: \neg$ ($x$ likes everyone)

*$\exists x: \neg (\forall y: L(x,y))$


Conversely, if we start from the given expression $\neg(\exists x: \forall y: L(x,y))$:


*

*$\neg(\exists x: \forall y: L(x,y))$

*$\neg(\exists x: x$ likes everyone$)$

*$\neg($ someone likes everyone $)$

*Nobody likes everyone


As you can see, the part where you went wrong is that in our informal hybrid language, "someone" and $\neg$ cannot be exchanged. The proper interpretation of "$\neg$ someone" is "nobody".
If you find it hard to perform such translations, it is better to just start with "someone" and "everyone", translate e.g. the quantifier laws, and see what they amount to.
For example, we could consider $\forall x: \neg P(x) \iff \neg \exists x: P(x)$: "Everyone is not $P$" $\iff$ "Not someone is $P$", from which it becomes clear immediately that "not someone" is the same as "nobody".
On the other hand, the importance of these translations to natural language should not be overestimated -- after all, it was the inadequacy of natural language in expressing logical theorems that inspired the creation of the symbolism in the first place; so it shouldn't be a surprise that it does a better job of avoiding ambiguity, misunderstanding and other problems than logical reasoning performed in natural language.
