In propositional logic, for example: $$\neg p \vee q.$$
If $p$ is true at the outset, does that mean it must be considered false when comparing with q in the disjunction?
P.S. I am unsure about tags for this question.
If $p$ is true, then $\neg p$ is false. To evaluate $\neg p \vee q$, you must evaluate $\neg p$ and you must evaluate $q$. If either $\neg p$ is true or $q$ is true, then $\neg p\vee q$ is true.
In other words, you really need to figure out $(\neg p)\vee q$, performing first the operation inside the parentheses, then the disjunction.
If $p$ is true, then $\lnot p \lor q \Leftrightarrow q$.
In general, $p$ and $\lnot p$ have the opposite value: if one is true then the other is false, and vice versa.
You can think of $p$ as some proposition, say "today is Sunday". Then $\lnot p$ stands for "today is not Sunday".
I'm not sure I understand your question, but this may help.
Truth Table for ~p v q:
~ p v q
If p is true, and ~p v q is true (first line only), then q is true.
Note that ~p v q is logically equivalent to p => q.