TITLE: Reductio ad absurdum or the contrapositive?
QUESTION [60 upvotes]: From time to time, when I write proofs, I'll begin with a claim and then prove the contradiction.  However, when I look over the proof afterwards, it appears that my proof was essentially a proof of the contrapositive, and the initial claim was not actually important in the proof.  
Can all claims proven by reductio ad absurdum be reworded into proofs of the contrapositive?  If not, can you give some examples of proofs that don't reduce?  If not all reductio proofs can be reduced, is there any logical reason why not?  Is reductio stronger or weaker than the contrapositive?
Edit: Just another minor question (of course this is optional and will not affect me choosing an answer):  If they are equivalent, then why would you bother using reductio?
And another bonus question (Like the above, does not influence how I choose the answer to accept.)
Are the two techniques intuitionistically equivalent?

REPLY [3 votes]: There is yet another reason for preferring proofs of the contrapositive to proofs by contradiction that has not been mentioned so far: It is harder to come up with wrong arguments that appear to be correct proofs when one tries to prove the contrapositive directly. When one attempts to prove something by contradiction, it is easy to declare victory if one can produce nonsense- even if the nonsense is the result of a mistake.
For an extreme point of view based on this perspective, Halsey L. Royden writes the following in the "Prologue to the Student" in his analysis textbook:

All students are enjoined in the strongest possible terms to eschew
proofs by contradiction! There are two reasons for this prohibition:
First, such proofs are very often fallacious, the contradiction on the
final page arising from an erroneous deduction on an earlier page,
rather than from the incompatibility of $A$ with $\neg B$. Second,
even when correct, such a proof gives little insight into the
connection between $A$ and $B$, whereas both the direct proof and the
proof by contraposition construct a chain of argument connecting $A$
with $B$. One reason that mistakes are so much more likely in proofs
by contradiction than in direct proofs or proofs by contraposition is
that in a direct proof (assuming the hypothesis is not always false)
all deductions from the hypothesis are true in those cases where the
hypothesis holds, and similarly for proofs by contraposition (if the
conclusion is not always true) the deductions from the negation of the
conclusion are true in those cases where the conclusion is false.
Either way, one is dealing with true statements, and one's intuition
and knowledge about what is true help to keep one from making
erroneous statements. In proofs by contradiction, however, you are
(assuming the theorem true) in the unreal world where any statement
can be derived, and so the falsity of a statement is no indication of
an erroneous deduction.

This is of course mostly didactic; Royden is clearly no constructivist. A related didactic point is that students might be able to visualize and imagine the antecedent of a proof of the contrapositive. Since the antecedent of a proof by contradiction is logically impossible, this does not work with proofs by contradiction. At least not if you do not have the skills of the queen in Alice in Wonderland, who in her younger years was able to imagine six impossible things before breakfast.