What is a word to describe holding two opposites at the same time as truth. To believe and hold opposing truths or doctrines both as true at the same time, though they are completely in opposition. Example. "I believe that women are equal to men even though they are inferior to men."
I think the term you're looking for is "cognitive dissonance," which is "the mental discomfort (psychological stress) experienced by a person who simultaneously holds two or more contradictory beliefs, ideas, or values." LINK It's often misused to describe a person's skepticism or disbelief when presented with something he or she disagrees with.
Cognitive dissidence is a general term for mental conflicts in belief, but I don’t think it’s exactly what you’re looking for for two reasons. Firstly, as long as it doesn’t bother you it’s not cognitive dissidence. Secondly, cognitive dissidence applies to many contexts that aren’t what you’ve described in the question.
The term that describes precisely the belief in two contradictory things is doublethink. Doublethink comes from the book Nineteen Eighty-Four, a dystopian novel by George Orwell. The Wikipedia article for doublethink reads
Doublethink is the act of simultaneously accepting two mutually contradictory beliefs as correct, often in distinct social contexts.
the novel explicitly shows people learning doublethink and newspeak due to peer pressure and a desire to "fit in", or gain status within the Party—to be seen as a loyal Party Member.