I'm trying to derive conclusions from premisies in a propositional logic class, and I believe the below question has a typo, if not, I have no idea where to even begin on a problem like this:
$P\vee (Q\wedge R), -P\wedge -Q \vdash S$
Is there something I'm missing, or is this a sequent that's impossible to derive?
I am using the following rule sets:
First 7:
Last 4:
Let me know if there is anything I can do to pose this question in a better way.
On
You should notice that that $\{P\vee(Q\wedge R), \neg P\wedge \neg Q\}$ is an inconsistent set of propositions. That is that they contradict.
Assume $P\vee (Q\wedge R)$. ...blahblahblah... Therefore $P\vee(Q\wedge R)$ contradicts $\neg P\wedge\neg Q$.
Recall that a contradiction entails anything, vaccuously. $S$ is something.
$\therefore\quad P\vee(Q\wedge R), \neg P\wedge \neg Q~\vdash~S$
$\blacksquare$
I can't see all of the rules from those links you gave ... But:
Do an $\lor$ Elim on $P \lor (Q \land R)$:
assuming $P$ will contradict with the $\neg P$, and assuming $Q \land R$ gives you $Q$ which will contradict the $\neg Q$. So, either way you get a contradiction, and from a contradiction you can infer anything you want, and what you want is of course $S$
So yes, this is valid, and can be derived, exactly because the premises are inconsistent. No typo!
OK, guessing a your rules ...
\begin{array}{llll} 1&(1)&P\lor (Q \land R)&Assumption\\ 2&(2)&\neg P \land \neg Q&Assumption\\ 2&(3)&\neg P& \land \ E \ 2\\ 2&(4)&\neg Q& \land \ E \ 2\\ 5&(5)&P&Assumption\\ 6&(6)&\neg S&Assumption\\ 5,6&(7)&\neg S\land P&\land \ I \ 5,6\\ 5,6&(8)&P&\land\ E \ 7\\ 2,5,6&(9)&P\land \neg P&\land \ I \ 3,8\\ 2,5&(10)&\neg \neg S&\neg \ I \ 9\\ 2,5&(11)&S&\neg \ E \ 10\\ 12&(12)&Q \land R&Assumption\\ 12&(13)&Q & \land \ E \ 12\\ 6,12&(14)&Q\land \neg S&\land \ I \ 6,13\\ 6,12&(15)&Q&\land \ E \ 14\\ 2,6,12&(16)&Q \land \neg Q&\land \ I \ 4,15\\ 2,12&(17)&\neg \neg S&\neg \ I \ 16\\ 2,12&(18)&S&\neg \ E \ 17\\ 1,2&(19)&S&\lor \ E \ 1,11,18\\ \end{array}