The gay parents example jumped out at me as a bad example as well (the two morals stated aren't contradictory in light of a study showing gays to be good parents). The first two examples illustrate Academian's point well though.

Contradictions actually do change peoples minds, I think. Look at birth control in Catholicism. Despite the pope himself saying it is wrong, many Catholics use it and support it (because to do otherwise would contradict other morals/desires they have).