Problem understanding multiple quantifiers in Predicate Logic So, my problem is that I have trouble knowing when and why I should write two quantifiers in the front of the whole proposition, rather than one in the front and one within ...
If I have "Everyone likes everyone", I have seen it written (where Lxy is x likes y; P is x is a person):
$$ ∀x∀y((Px \land Py) \to Lxy)$$
but can I also write it like:
$$ ∀x(Px \to ∀y(Py \to Lxy))$$
or anything similar ...?
or can I write it like:
$$ ∀x∀y((Px \to (Py \to Lxy)$$
If so, why do we sometimes write both in the front and sometimes not?
Does the same go for cases with different quantifiers? Like "Everyone likes someone":
$$ ∀x∃y((Px \land Py) \to Lxy)$$
is equivalent to
$$ ∀x(Px \to ∃y(Py \to Lxy)).$$
Are there any cases where the two quanifiers must be in the front? Or must be separated?
Sorry for a perhaps stupid question, but my lecturer did not discuss it very much, he just wrote some sentences with two quantifiers in the front and others without it - and now I've gotten a bunch of exercises to do.
 A: In a way this is just a matter of style. Of course we encounter situations where quantifiers are spread across the statement especially during formal proofs, e.g. from $\forall x (Ax)$ and $\forall y (By)$ we immediately get $(\forall x(Ax))\land(\forall y(By))$ formally, and only after a bit rearrangement $\forall x\forall y(Ax\land By)$ or even $\forall x(Ax\land Bx)$. It is sometimes desireable to have all (or at least most) quantifiers in the beginning as it can simplify using the staement in a proof (per specialization etc.).
For example from $\forall x\forall y ((Px\land Py)\to Lxy)$ we can immediately (allright, in two specialization steps) infer $$ (P\,{\rm Jack}\land P\,{\rm Jill})\to L\,\rm{Jack\,Jill}.$$
To get a similar result from $\forall x(Px\to \forall y(Py\to Lxy))$ we first obtain
$$ P{\rm Jack}\to \forall y(Py\to L\,{\rm Jack}\,y),$$
then verify that jack is a person (or make a more complex deduction to obtain the more general result), employ modus ponens to find
$$\forall y(Py\to L\,{\rm Jack}\,y),$$
whence
$$P{\rm Jill}\to L\,{\rm Jack}\,{\rm Jill}).$$
Beyond that, the situation is the way it always is with equivalent formulations (e.g. $(A\land B)\to C$ versus $A\to(B\to C)$) and may be a matter of personal taste even.
