Ben Reiniger wrote on Sat, Oct 6, 2012 11:33 PM UTC:
Rodrigo, I think the idea is that if after white's move, black's only response will put white in (non-mate) check, then that black move is illegal, so black is in fact in checkmate, and white's move was legal.
Then, what happens if black's only response is to put white in a similar position? This is the "paradox" that is referenced.