It's no secret that the Catholic Church is strongly opposed to abortion in all forms and cases—a stance Francis also takes, having equated it to the death sentence in a 2007 speech, saying that although Argentina doesn't have a death penalty, "a child conceived by the rape of a mentally ill or retarded woman can be condemned to death.”
Birth Control and Pre-Marital Sex
If there's a topic Francis is malleable on, it's contraception. A recent Gallup poll found 82 percent of American Catholics view contraceptive use "morally acceptable," while the new pontiff (along with the Church) believes the use of condoms is permissible in preventing diseases like HIV. However, since he doesn't support the use of contraceptives as forms of birth control—only in prevention of disease—and publicly opposed free distribution of them in Argentina, it's not likely the Church's opinion on the subject will change while he holds papal office. And, although Francis agrees with social doctrine regarding condoms, his stance on premarital sex is very much aligned with that of the Catholic Church, so it'll likely still be strictly verboten.
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As far as divorce is concerned, well, the Catholic Church would rather you stay married. You did, after all, accept each other for better or for worse and so on and so forth. They will, however, give annulments, and that's probably how it will remain while Francis is pope.
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So where might we see change if not on the social issues that receive increasingly progressive opinions from Catholics today? The fact that Pope Francis is the first Jesuit pontiff would seem like a step toward papal reform since the Jesuit order is seen as particularly liberal among Catholics. And while Francis shunned vanity in favor of celebrating Mass with the poor (he famously washed and kissed the feet of AIDS patients in Argentina), he comes from the same conservative tradition as the last two popes.