The key to having opposing or contradictory ideas in your head is to realize that every idea has a probability of being right. There are very very few ideas that are 100% correct.
Cromwell’s rule – Wikipedia states that you should never assign a 100% probability to any event.
Once you understand this you can have an idea that you believe strongly in and an idea that you only have a little bit of belief in even if they are contradictory. For both ideas, you can have the reasons why you believe it and the reasons why you don’t.
In this way you will find for a lot of beliefs there are times where either belief could apply. The world is not always black and white.
When we think this way it is far easier to have discussions and make compromises with people who have different beliefs than we do.