inference rules application (introduction / elimination): two examples Got stuck while trying out how to apply inference rules (introduction and elimination) for the following examples:


*

*From $\lnot(P\land Q)$ and $P$ infer $\lnot Q$

*From $P\lor Q$ and $Q$ infer $\lnot P$
Could you please demonstrate how this can be done?
Thanks a lot in advance!
 A: For the first question, recall DeMorgan's laws.  Thus, the statement becomes $¬P\vee¬Q$.  Then, taking $P$ as true, you have $¬P$ false. So, then $¬Q$ must be true for the statement to be true.
For the second I don't believe you can infer $¬P$.  $P\vee Q$ reads as 'At least one of P or Q is true'.  If Q is true by assumption, $P\vee Q$ is always true regardless of the truth value of $P$.
A: The second isn't valid, and almost surely not proveable in your system.
For the first, you want the negation of Q.  So, you might want to assume Q and see if you can then infer a contradiciton.  Does that help enough?
A: For 1.
Claim: Let $\Gamma =\{\neg(P \wedge Q), P\}$. Then, $\Gamma \vdash \neg Q$.
Proof:
By DeMorgann's Law, $\neg(P \wedge Q) \equiv \neg P \vee \neg Q$. Then, by disjunctive syllogism, since we know $P$, we may conclude $\neg Q$.
For 2. Claim: Let $\Gamma = \{P \vee Q, Q\}$. Then, $\Gamma \vdash \neg P$.
Disproof: Let $Q$ be true and $P$ be true, then $P \vee Q$ holds, $Q$ holds, and $P$ holds. Therefore, $\Gamma \nvdash \neg P$.
