Confused about proof by contradition In proof by contradiction, I can understand how it works when the hypothesis leads to a clearly false proposition. e.g., if we want to prove $P$, we assume $\neg P$ and show that $\neg P \implies ... \implies Q$, but we know that $\neg Q$ and since we just proved that $\neg Q \implies P$, $P$ is true.
However, I get confused when the hypothesis leads to its own negation. In other words, we have $\neg P \implies ... \implies Q \implies P$. I can't help but feel that $\neg P\implies P$ is just something meaningless that we can't use to conclude anything. We could say that since $Q\implies P$, $\neg P \implies \neg Q \implies P$, but again what this is saying is $\neg P \implies P$.
A good example of this type of proof is Dijkstra's algorithm where the assumption that the selected vertex does not have its shortest path determined leads to the conclusion that it does indeed have its shortest path set. 

Let $Q$ be the statement $P\land\lnot P$. We know that $Q$ is false. 
Then if you have $\lnot P\implies P$ then you have $\lnot P\implies (P\land \lnot P)$, which is $P\implies Q$. But we know $Q$ is false, so $P$ is false.
Indeed, every proof by contradiction can be written as:
$$P\implies(A\land \lnot A)$$
for some predicate $A$. 
In your case, you knew that $\lnot Q$ is true, so you can conclude from $P\implies Q$ that $P\implies (Q\land\lnot Q)$. 
In the case of $P\implies \lnot P$, you have $A$ the same as $P$. But you still reach a contradiction - you know that something is both true and not true.
