First Order Logic Question: How to interpret ∀x~∃y when L(x,y) means ___x loves ___y?

Question

I am given a sentence and my professor wishes for me to convert it to FOL. The problem is that I do not know what the form of this quantifier order would look like. For instance, if L(x,y) means ___x loves ___y.This is not the real question, as I want to learn it on my own. But, How would I interpret it through the aforementioned quantifier sequence. My initial thought is that it would mean something like, 'Everyone does not love somebody." Really I'm completely lost. Thanks.

Answer 1

A fairly literal translation of

∀x~∃yL(x,y)

would be

For every x, there is no y, such that x loves y.

And a less literal version:

Everthing loves nothing.

If we assume the domain of objects are persons instead of ojects in general, this means:

Everyone loves no-one.

The latter is equivalent to

Nobody loves anyone.

Which would be formalized as

~∃x∃yL(x,y)

This is logically equivalent to the first expression above.