proof by contradiction - making more than one assumption I have a pet peeve with a proof by contradiction. Such proofs begin by an assumption which is shown to lead to a contradiction thereby leading us to conclude that our initial assumption must have been wrong.
However, it often happens that we make more than one assumption in the course of the proof by contradiction. My question is: How can we know which assumption is responsible for the contradiction?
See amWhys answer in deducing $\lnot B \implies \lnot A$ from $A \implies B$ - she makes two assumptions: not B and A.

If you assume $A$, $B$, $C$, and $D$ and manage to derive a contradiction, then you have a proof by contradiction of each of the following theorems (which are all logically equivalent):
$$ A\land B \land C \implies \neg D $$
$$ A\land B \land D \implies \neg C $$
$$ A\land C \land D \implies \neg B $$
$$ B\land C \land D \implies \neg A $$
Often it will only be one of these you're actually interested in, but you're free to choose to state any or all of them as a conclusion once you've done the work.
