Cognitive Dissonance
It's not just two opposite ideas
Cognitive dissonance is often defined as a kind of discomfort in the mind when someone has two beliefs that contradict each other.
However, to really understand cognitive dissonance, you need to know that at least one of the ideas has to be something that a person considers to be defining of their character.
If you believed all crows are black, and then were confronted with evidence of a white crow, you would probably not experience any cognitive dissonance. The white crow is just a new fact that changes what you know about crows, and there's nothing about what kind of birds there are in the world that really matters to you. It's all just arbitrary facts.
The classic example of cognitive dissonance is with police who later discover that they have put an innocent person in jail.
Even when the evidence that overturns the conviction is absolutely incontrovertible, such as rock solid DNA evidence, studies have shown that most, if not all, police and prosecutors will continue to believe they did the right thing by putting that person in jail.
Putting criminals in prison is part of how these law enforcement agents define themselves as people. Discovering that they put good people as well as bad people in jail changes their self identity from agents of justice to cogs in an oppressive state machine.
The harsher the punishment, the more clearly innocent the accused turned out the be, the more the law enforcement officer will cling to the idea that they did the right thing in spite of the evidence. They will turn to alternate justifications that preserve their sense of self, such as saying that the suspect was probably guilty of something else anyway.
The definitive book on cognitive dissonance, and where I derive my definition from, is Mistakes Were Made (but Not By Me), by Carol Tavris and Eliot Aronson .