What is the correct way to use the implication in first-order logic?

Question

I know the implication symbol, $\rightarrow$, is used for conditions like

If $A$ is true, then $B$ will be true.

which can be written as

$$ A \rightarrow B $$

However, sometimes the implication symbol is also used in other contexts. For example, if we want to say that

All $A$'s are $B$.

We could write

$$\forall X (A(X) \rightarrow B(X))$$

I don't understand why the implication is used here? And if the implication is necessary to use here, then why isn't the implication used in the example written below?

Some $A$'s are $B$'s.

$$\exists X (A(X) \land B(X))$$

Answer 1

When we state in English that "All As are Bs", this means that we gain information as soon as we observe an A, we can immediately deduce that it must also be a B. These are the kinds of situations where we use an implication. So, this would be written in formal logic as:

$$\forall X \left( A(X) \rightarrow B(X) \right)$$

When we state in English that "Some As are Bs", we do not gain any new information just from observing that something is an A, we cannot deduce anything about that A. It might happen to be one of the As that simultaneously is a B, but it also might happen not to be one of those examples. So, it would be wrong to use an implication here. The only information that the English sentence gives us is that there is at least one thing somewhere that happens to be an A as well as a B, which is written formally as:

$$\exists X (A(X) \land B(X))$$

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Suppose that we would have written the following in logic:

$$\exists X (A(X) \rightarrow B(X))$$

This would be translated to English as follows:

There exists some $X$ such that, if it is an $A$, it is also a $B$.

That bolded part is very important there. Note that this logical statement is also true as soon as I find one example $X$ that is not an $A$. For example, the following statement is true in the real world:

There exists some human $X$ such that, if $X$ can fly, $X$ can also shoot fireballs from his or her hands.

(this is true in the real world, because I can come up with many examples of humans who cannot fly)