Assume I have formulas $H$, $P$ and $Q$. Assume further that I can show in classical logic that $P$ follows from $H$:
$$H \vdash P$$
And that the negation of $Q$ follows from $H$:
$$H \vdash \neg Q$$
Can I then jump to the conclusion that $Q$ does not follow from $P$:
$$P\not\vdash Q \text{ ?} $$
Bye