Difficulty understanding proof by contradiction Here's my understanding of proof by contradiction based on what I've read and I've been taught.
We show $\neg P \implies (c \land \neg c)$ is always true. This is done by assuming $\neg P$ is true. Then, we realize that $(c \land \neg c)$ is logically equivalent to $F$ so we've just shown that $\neg P \implies F$ is always true. But, examining the $\implies$ truth table, we realize that $\neg P$ must be false to ensure $\neg P \implies F$ is always true. Therefore, we conclude that $\neg P$ is false, and thus $P$ is true. 
The conundrum I have is that our analysis is based on the assumption that $\neg P$ is true. Then later we conclude that $\neg P$ is false. So why do we accept that $\neg P$ is false even though we assume $\neg P$ is true?


If my dog is in the bathroom, I will hear him bark there.
I do not hear him bark there.
Hence he is not in the bathroom.

Nothing more profound than that.
