From $(p\lor q)\to r, s\to\neg p, s\land\neg q$, prove $\neg r$

Question

Need help with this: $$\frac{p\lor q\to r;s\to\neg p;s\land\neg q}{\therefore\neg r}$$ How do I prove that $\neg r$ is in fact true?

Answer 1

The only premise you have involving $r$ is the first one, which has the form $XYZ\to r$. No matter how much you know or can prove about $XYZ$, you will never be able to conclude $\lnot r$ from this; it only tells you a condition under which you can conclude $r$.

Compare: “If you steal, you will be arrested.” Could you ever be certain that you won't be arrested? No, of course not. Whether or not you steal, you might be arrested for tax fraud or for wearing the wrong shirt on Wednesday.