What's the difference between $∀x\,∃y\,L(x, y)$ and $∃y\,∀x\,L(x, y)$? 
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*Everybody loves somebody. $∀x\,∃y\,L(x, y)$

*There is somebody whom everybody loves. $∃y\,∀x\,L(x, y)$
What's the difference between these two sentences? If they are same, can I switch $\exists y$ and $\forall x$?

That's a great example of why quantifiers don't commute! For the sake of simplicity, assume everybody in the world is married, and everybody loves his spouse.  Then the first formula is satisfied.  However, there is no reason to think that the second formula is satisfied; in fact, it could be that people only love their spouses, so that there is nobody in the world who is loved by everybody.
