I encountered earlier today a question "Is the proof by contradiction same as that $A \rightarrow B$ is true when $A$ is false?" continued by "Are they related, then? How?"
I think the answer is "no, they´re not the same thing", and demonstrates some confusion on the subject. However, as I am still more than a little confused about mathematical logic myself, too, I couldn't explain properly why this felt wrong to me. I pointed that proving statement $P$ by contradiction can be formulated symbolically as a $\neg P \rightarrow Q \wedge \neg Q$, and if $P = A \rightarrow B$ (because statements like that were the topic of discussion), we get something ...quite different. (I think I lost my trail of thought here.) However, it (the method of proof by contradiction) does feel like a different thing than the existence of a line in the truth table for $A \rightarrow B$ where $A$ is false and $B$ is true and thus the implication is true. Was I correct in this?
However, afterwards I remembered that $A \rightarrow B = \neg B \rightarrow \neg A$, where the right side is known as proof by contraposition. And the proof by contraposition is "in practice" usually done by assuming that $\neg B$ is true and then showing that $\neg A$ follows. (Because of the equality above, this is same as showing that "$A \rightarrow B$".)
Drawing the truth tables for both $A \rightarrow B$ and $\neg B \rightarrow \neg A$, this line (where $\neg B$ and $\neg A$ are true) does coincide with with $A$ being false and $B$ being true. So maybe one could say that $A \rightarrow B$ is true when $A$ is false is related to (or even "same thing as"?) the proof by contraposition.
Maybe we're both (or just me) confused about the relation of truth tables of statements like $A\rightarrow B$ to actually proving statements like "if A, then B".
Enlighten me, please.