[FOM] FOM: What is a proof?

[Antonino Drago]

 The questions by John Corcoran about What is a proof? of 22/1/09 and the 
answer by Bill Taylor
lead me to ask in my
 turn the following questions, which in some sense represent a particualr
case of his questions:

 Intuitionist logic ancludes ad absurdum proofs as respectable proofs.
 However their results are not positive sentences, but no more than double
 negated sentences.
 In other terms, intuitiionist logic accepts weak ad absurdum theorems only.
 The passage of these results to the corresponding positive sentences means
 to make use
 of classical logic, because in it the double neagtion law holds true.
 In such a case the previous weak ad absurdum proofs become strong ad
 absurdum proofs.
 I ask:
 1) the notion of a proof allows a change of the kind of logic? If yes, are
 the intuitionistic ad absurdum proofs (i.e. the weak ones) equal to the
 classical ad absurdum proofs (i.e. the strong ad absurdum ones)?. If not, 
is
 the Markoff's principle always invalid?
 2) the status of an ad absurdum proof is the same of a direct proof?
 Best greetings
 Antonino Drago