Modus ponens reasoning: truth of p implies q and why can we say q follows? I've found Isn't the modus ponens just the definition of what 'if' means? but I don't feel like it is particularly relevant. I also found  Why is modus ponens a q-implication? which I understand if I don't dig too deep ($p$'s truth and $q$'s truth) but I don't feel like I fully understand modus ponents.
Modus ponens states
$$ p \rightarrow q $$
Given $p$
Therefore $q$

I'm trying to approach this by looking at the truth table but it isn't helping me.




$p$
$q$
$p\rightarrow q$




1
1
1


1
0
0


0
1
1


0
0
1




What is modus ponens saying exactly? I'm having a problem interpreting it.
Is it saying that given some $p$ (can this be true or false or must this only be true) and given some implication (can THIS be true or false or must this be true)?
We can determine that $q$ is true?
Assuming $p$ must be true and $p\rightarrow q$ must be true that DOES imply that $q$ must be true but in any other case we cannot determine $q$?
 A: Abstract answer. An inference rule is something of the form (for some $n \geq 0$)
$$\tag{*}
\frac{H_1 \quad \dots \quad H_n}{T}
$$
where $H_1, \dots, H_n, T$ are formulas; more precisely, $H_1, \dots, H_n$ are the hypotheses or premises of $(*)$ and $T$ is the thesis or conclusion of $(*)$.
The inference rule $(*)$ is read as follows: "Given $H_1, \dots, H_n$, then $T$ follows" (or similar expressions). This means that whenever $H_1, \dots, H_n$ hold simultaneously true, then by necessity $T$ holds true too.
Said differently, it is impossible that $H_1, \dots, H_n$ are all simultaneously true and $T$ is not.

Concrete answer. Modus ponens is the following inference rule:
$$
\frac{p \to q \qquad p}{q}
$$
According to the abstract explanation above, modus ponens says that whenever $p \to q$ and $p$ are simultaneously true, then $q$ is necessarily true.
Indeed, this is exactly what happens in the truth table you wrote.
Check the rows where both $p\to q$ and $p$ are simultaneously true (this happens only on the first row): in each of them (i.e. in the first row) it turns out that $q$ is true too.
Said differently, in the truth table you wrote, it is impossible to find a row where $p \to q$ and $p$ are both true and $q$ is false.
