What is the meaning of these first order predicate statements?

Question

Statement is "Some boys like every girl." I'm trying to figure out which first order predicate fits the statement.

1. $\forall(x)\ [\ \text{girl}(x) \Rightarrow \exists(y)\ [\ \text{boy}(y) \land \text{likes} (y,x)]]$
2. $\forall(x)\ [\ \text{girl}(x) \land \exists(y)\ [\ \text{boy}(y) \Rightarrow \text{likes} (y,x)]]$
3. $\forall(x)\ [\ \text{girl}(x) \Rightarrow \exists(y) [\ \text{boy}(y) \Rightarrow \text{likes} (y,x)]]$
4. $\exists(y)\ [\ \text{boy}(x) \land \left(\ \forall(y)\ \text{girl} (y) \Rightarrow \text{likes} (x,y)\right)]$

Statement 1. If every x satisfies "girl", then some y satisfy "boy" and those particular instances of y like all instances of x.

Statement 2. Every x satisfies "girl" and there exists some y which if satisfies "boy" likes all instances of x.

Statement 3. If every x satisfies "girl" then there exists some y which if satisfies "boy" likes all instances of x.

Statement 4. There exists some x which satisfies "boy" and if every y satisfy "girl", then all those y are liked by those particular instances of x.

Statement 1 says that there should be some "boy" atleast. Statement 2 says that "everything is girl" Statement 3 says "if everything is girl", and "if something is "boy", then that instance of "boy" likes every instance of "girl". Statement 4 seems to be the same as statement 3 but written in a different way. Statement 3 and 4 seem correct to me. Statement 1 is wrong because it says that there should be some boy atleast (conjuction instead of implication; in implication, we can get the first clause to be false). Statement 2 seems wrong to me because everything can't be a girl. It doesn't seem so elegant.

Another easier way of interpreting statement 2 could be that universal quantifier is distributive over conjunction, so the statement roughly becomes "everything is girl and if "boy" exists, then it likes everything." This also implies a overlap between "girl" and "boy". And if overlap is not allowed then it would become "everything is girl, "boy" can't exist." Now that also seems like correct statement to me.

I'm having difficulty understanding this topic. So, please try to give me a detailed answer that can help me understand this.

Answer 1

One statement at a time.

First statement

$\forall x(\text{Girl}(x)\to \exists y(\text{Boy}(y)\land \text{Likes}(y,x))$

Statement 1. If every x satisfies "girl", then some y satisfy "boy" and those particular instances of y like all instances of x.

This is incorrect. What you wrote would be modeled by something like $$\underbrace{\forall x\text{Girl}(x)\to}_{\text{If every }x{ \text{ satisfies "girl"}}} \text{Something},$$ more specifically it would be modeled by $$\forall x\text{Girl}(x)\to \exists y(\text{Boy}(y)\land \underbrace{\color{blue}{\forall x\text{Likes}(y,x)}}_{y\text{ likes all instances of }x})$$ and this isn't at all what you want, nor is it equivalent (the universal quantifier over $x$ doesn't distribute over $\to$). I noted that you made the mistake highlighted in blue on all statements, so I won't mention it again.

In plain English the first statement could be written as "For every girl there's a boy who likes her" or maybe "Every girl has a boy who likes her". I'm not entirely comfortable with the second one because "has" sort of implies ownership, but I'm not sure.

Then you claim that "statement 1 says that there should be some "boy" at least". This isn't right, at best the statement would imply that there exists a boy, which isn't the same as saying that there's a boy. This is probably wrong, it depends on the universe you're considering. In a universe with only boys the statement is true. Assuming the universe is the people of Earth, I don't think anyone can tell you the truth value of this statement.

Second statement

$\forall x(\text{Girl}(x) \land \exists y(\text{Boy}(y)\to \text{Likes}(y,x)))$

"Statement 2. Every x satisfies "girl" and there exists some y which if satisfies "boy" likes all instances of x."

What you wrote would be modeled by $\forall x\text{Girl}(x)\land \exists y(\text{Boy}(y)\to\forall x\text{Likes}(y,x))$ or $\forall x(\text{Girl}(x)\land \exists y(\text{Boy}(y)\to \forall x\text{Likes}(y,x)))$, depending on interpretation. It doesn't really matter which it is because they are equivalent in most FOLs and they're both wrong as mentioned in the analysis of the first statement.

Correct would be "Given any one person, that person is a girl and there exists someone who if it happens to be a boy, than the latter likes the former". I couldn't find a more natural sounding way of putting this statement in English and I don't think a 100% correct translation can be put in natural sounding English.

"Statement 2 says that "everything is girl" - again, this isn't what it says, even though it doesn't imply this.

"Another easier way of interpreting statement 2 could be that universal quantifier is distributive over conjunction, so the statement roughly becomes "everything is girl and if "boy" exists, then it likes everything." This also implies a overlap between "girl" and "boy". And if overlap is not allowed then it would become "everything is girl, "boy" can't exist."

Very good analysis, except for the part where the boy likes everything, which is under the assumption that your translation is correct and it isn't because of what was already mentioned.

Again, the truth value depends on the universe. In a blessed paradise in which only girls exist, the statement would be true. Assuming the universe is the people of Earth, the statement is false because there are boys.