MHB Understanding Ke Logic Rules & Finding Contradictions

[lyd123]

Hi, the question and Ke logic rules are attached.

This is my attempt at the question.

$1. P \land (R\implies Q) $ Premise
$2. ( P \land Q ) \implies \lnot S) $ Premise
$3. ( P \land S) \implies R) $ Premise
$4. \lnot S $ Conclusion
$5. P \land Q$ $ \beta 2,4$
$6. P $ $ \alpha 5$
$7. Q$ $ \alpha 5$
$8. R\implies Q $ $ \alpha 1$

I don't think the lines I wrote after this make a lot of sense. Usually a contradiction would be found, but in this case I don't seem to find a contradiction. I think maybe I have to negate the conclusion, I thought it was already negated because of the \lnot. But how do I know when the argument form is valid (invalid being if there is a contradiction).Thank you for any help. :)

View attachment 8735
View attachment 8736

 

[Evgeny.Makarov]

I assume that $\neg S$ is the original conclusion, not its negation.

You cannot derive $P\land Q$ from $P\land Q\implies \neg S$ and $\neg S$.

To prove $\neg S$, one must use the law of excluded middle, or the branching rule. From premise 1 we have $P$ and $R\implies Q$. If $S$, then we get $R$ from premise 3, $Q$ from premise 1 and finally $\neg S$ from premise 2. If $\neg S$, then nothing is left to do.

 

[lyd123]

Thank you, I understand now.If an argument was valid, how would we know? For example, in this case if the the original was S and the negated conclusion is ¬S ?

1.P∧(R⟹Q) Premise
2.(P∧Q)⟹¬S Premise
3.(P∧S)⟹R Premise
4.¬S Negated Conclusion

 

[Evgeny.Makarov]

These premises do not imply $S$. The easiest way to see this is to find a counterexample, i.e., an assignment of truth values to variables that makes all premises true and the conclusion false. In this case it is $R=Q=S=F$ and $P=T$.

 

[solakis1]

To prove $\neg S$, one must use the law of excluded middle, or the branching rule

I suppose you mean : $ S\vee\neg S$

If $S$, then we get $R$ from premise 3

I suppose you mean from P and S we get : $P\wedge S$ and then using premise 3 and α rules (Modus Ponens) we get $R$

If yes, there is no rule in α or β rules to account for: $P$,$S$ $\Rightarrow P\wedge Q$