More in the spirit than the letter of the question, some of Alan Turing's work can be seen as applications of contradiction. The development of the Turing Machine and using it to resolve Hilbert's Entscheidungsproblem can be seen as a successful use of the Liar Paradox, and elsewhere I read a quote which suggested that Turing took a result from mathematical logic (from a contradiction one can prove anything) and used it to help build one of the codebreaking systems used in World War II.
Gerhard "Ask Me About Me Unasking" Paseman, 2012.12.15