Here, Terence Tao presents a collection of similar mathematical arguments that he calls "no self-defeating object" (examples are Euclids proof of the infinitude of the primes and Cantor's theorem). In the second post, he remarks that one can reformulate these "no-self defeating object" arguments to get a "every object can be defeated"-version.
The simplest example "no self-defeating object" goes as follows:
Proposition 1 (No largest natural number). There does not exist a natural number N that is larger than all the other natural numbers.
Proof: Suppose for contradiction that there was such a largest natural number N. Then N+1 is also a natural number which is strictly larger than N, contradicting the hypothesis that N is the largest natural number.
The corresponding "every object can be defeated"-version is:
Proposition 1′. Given any natural number N, one can find another natural number N' which is larger than N.
Proof. Take N' := N+1.
Terence Tao also remarks:
"This is done by converting the “no self-defeating object” argument into a logically equivalent “any object can be defeated” argument, with the former then being viewed as an immediate corollary of the latter. This change is almost trivial to enact (it is often little more than just taking the contrapositive of the original statement), but it does offer a slightly different “non-counterfactual” (or more precisely, “not necessarily counterfactual”) perspective on these arguments which may assist in understanding how they work."
My question: What has the contrapositive to do with the change from "no self-defeating object" to "every object can be defeated"?
As I understand it, the "no self-defeating object"-version is of the form "$\neg\exists x: P(x)$" and the "every object can be defeated"-version is "$\forall x:\neg P(x)$". That these are equivalent is de Morgan for quantifiers, what does it have to do with contrapositives?