Assume it is given $2$ predicates as below:
$A(x):$ $x$ is a horse;
$B(x,y):$ $x$ is a tail of $y$.
Then, translate the following sentence into predicate logic formula: "$w$ is a tail of horse" - where $w$ is arbitrary variable
My ideas: I am struggling on which variable out of $w$ and horse, we should pick up as existential, and universal? Would appreciate to know your ideas.
Update: Is the formula for above sentence expressed as $\forall(y)(A(y) \Rightarrow B(w,y))$?
Given the instructions, $w$ is supposed to be a free variable that remains unquantified. It acts as a placeholder to insert arbitrary names for, like in the definitions of the two predicates you gave, where $x$ and $y$ are not bound either.
Your proposal is incorrect. Your formula expresses "$w$ is the tail of every horse". But that's not the meaning of "a tail's horse". "horse's tail" means "a horse's tail" -- that particular tail is the tail of one horse, not all the horses in the world -- so the horse variable will be existentially, not universally quantified.
Also, as explained in the answer to your other post, "a horse's tail" means that $w$ is the tail of $y$ and $y$ is a horse -- so you need conjunction, not implication. As a rule of thumb, existential quantifiers go with conjunction and universal quantifiers go with implication.
Can you figure out the correct solution given these hints?