Is it acceptable in formal logic to achieve proof by contradiction by obtaining the negation of the assumption made? I am (re-)working through the Gensler logic book to refresh my command of formal logic. For the most part, he is using proof by contradiction to achieve results. I noticed that the proofs I am writing are somewhat different from his own. Typically, for a set of premises S and a conclusion Q, he assumes ~Q and then from there derives both A and ~A for some A, thereby proving Q. I realized that I have been assuming ~Q just as he has, but instead of finding a complementary A and ~A from there to make my contradiction, I have instead proved that Q is true from S and ~Q. Is this also contradictory, or am I missing something?
 A: That's fine: you're proving $\neg Q\implies Q$, which implies $\neg Q\implies Q\wedge \neg Q$, so that $\neg Q$ again implies a contradiction.
A: I would like to add some comments on this question from a formal point of view.
In mathematical logic, one way to formalize the law of contradiction is through the propositional law :

$(A → B) → ((A → ¬B) → ¬A)$ --- (*).

This form is intuitionistically valid; it is enough to change it a little bit, to have classical logic :

$(\lnot A → B) → ((\lnot A → \lnot B) → A)$ --- (§).

This law is easily derivable from (*) through Double Negation : $\lnot \lnot A \rightarrow A$, which is not intuitionistically valid.
Having said that, the law we are discussing :


$(\lnot Q \rightarrow Q) \rightarrow Q$


is itself easily derivable from the law of contradiction :
i) $\lnot Q \rightarrow Q$ --- assumed
ii) $(\lnot Q → Q) → ((\lnot Q → \lnot Q) → Q)$ --- from (§) with $Q$ in place of both $A,B$ [this step amounts to applying Henning's comment above : "just taking $A$ to be $Q$"]  
iii) $(\lnot Q → \lnot Q) → Q$ --- from i) and ii) by modus ponens
iv) $Q$ --- from iii) and the logic law : $A \rightarrow A$, by modus ponens

v) $(\lnot Q \rightarrow Q) \rightarrow Q$ --- from i) and iv) by Deduction Theorem.

Thus, we are licensed to say that it is also a form of the law of contradiction.
