Implication vs Conjunction (Natural Language to Predicate Logic)

Question

So im confused when should i use implication and when should i use conjunction. Let me give an example.

"All parrots like fruits."

I converted this sentence into 2 predicates.

P(x) = "x" is a parrot.
F(x) = "x" likes fruits.
where Universe of Discourse is all the birds.

Now applying logic, i got 2 possible answers.
$\forall x$ ( P(x) -> F(x) )
OR
$\exists x$ ( P(x) ^ F(x) )

Now i am confused between the two. I read the first one as " For all the birds it is true that if they are parrots then they like fruits."

I read the second one as "There exist some birds that are parrots and they like fruits."

When i think about it both the statements seem to fulfill the english meaning of the sentence.

Answer 1

These two statements are just completely different. The first one indeed means that "all parrots like fruits", but the second one means "there is at least one parrot that likes fruits".

This is not equivalent at all : consider the fact that I have a good friend who likes brocolis, but god knows that doesn't imply I do.

Answer 2

Sentence "All parrots like fruits" is better written as $$\forall x\ \big(P(x) \to F(x)\big),$$ because "all" corresponds to the universal quantifier $\forall$.

If you have trouble with implication v.s. conjugation with respect to quantifiers, then this is just how you translate "such that" (i.e., conditions imposed on the quantified variable) using quantifiers, with $\forall$ you use implication, and with $\exists$ you use conjunction, that is,

\begin{align} &\forall x \in P\ \big(F(x)\big) &\text{ corresponds to }& &&\forall x\ \big(x \in P \to F(x)\big) \\ &\exists x \in P\ \big(F(x)\big) &\text{ corresponds to }& &&\exists x\ \big(x \in P \land F(x)\big) \end{align}

I hope this helps $\ddot\smile$