I just realized I have a serious problem in properly seeing the logical structure that involves counterexamples.
Here there is an example:
Proposition F: Assume $P$. Then, there is a function $f \in Y^X$ such that $f$ is strictly increasing.
Imagine I have been told that this is wrong (i.e. there is a counterexample).
What does that mean exactly?
Does that mean that, after assuming $P$, I have to show that it is possible to construct a $f \in Y^X$ that is not strictly increasing?
[This doesn't tell us anything on how $f$ really looks like, it just tells us that it is not always in a certain way]Or do I have to negate [F], getting the following: $$ \forall f \in Y^X , \ \exists a, b \in X : a>b , \ f(a) \leq f(b) ? $$ [On the contrary, this tell us exactly how $f$ looks like]
The two are essentially different to me, and I do not really see how to properly address it. Intuitively I am for (1), but if I try to be careful, I reason as follows:
$F$ is false, hence $\neg F$ is true. Thus, apply negation to $F$ following standard quantifiers rules.
Which one is correct?
Any feedback/explanation is most welcome, as always.
Thank you for your time.
Your #1 is certainly not correct, since constructing an $f$ which is not strictly increasing doesn't mean that some other $f$ couldn't be constructed which is strictly increasing. #2 is the correct interpretation.
I understand the issue you're having, though. To give a counterexample to a statement $F$ means to give an 'example' showing that $F$ is false, i.e., an 'example' showing that $\neg F$ is true. When $F$ is an existentially quantified statement, then $\neg F$ will be a universally quantified statement, and it is not always clear what kind of 'example' can demonstrate a universally quantified statement to be true. An existentially quantified statement is not easily disproved by giving a 'counterexample' in the same way that a universally quantified statement can be disproved.
Unless there were something special in the assumptions $P$, if you want to disprove $F$, that would involve finding a situation where the assumptions $P$ hold and then proving that there is no function $f \in Y^X$ such that $f$ is strictly increasing.