Let me start by describing what I see as two different styles for decision making and exploring issues. Of course, these can be used together and can be disguised, so it is not always easy to draw a clear line between the two. And people can start with logic and when that doesn't work revert to faith, for example, so it can get confusing and feel like you are on shifting sand. But, for the sake of this discussion, let me draw a hard line between them.
The first is the logical or reasoning based approach. In this approach, objectives are used to determine what the outcome should be, for instance minimizing pain, redcuing risk of injury and complications, maximizing sexual potential later as an adult. From these objectives, one can look at the facts, data from other cases, etc and draw conclusions of what the best decision is. One of the striking features of this, for me, is that any reasonable person can see why a particular conclusion is reached, given certain assumptions and common definitions. They may not agree, because they have fundamentally different assumptions of objectives, but they can understand how you go from point A to B and see why you did that.
The second I will call faith based reasoning. In this approach, facts are cherry picked to tell a story that is believed. Instead of carefully looking at all the evidence, the person focuses on just some facts, especially facts that could be called anomolous data, and these anomoles, or "exceptions to the rule" so to speak are give huge amounts of weight. And a ready fall back position is that if you would only open your mind you would see the logic too. This effectivley shuts the dialog down, and sets up wall that is very difficlut to work through.
With logic based reasoning, you can generally work with them to explore the assumptions, objectives, facts, logic, and conlusions. They can be swayed by picking apart the pieces and putting them back together. You may never get to agreement, but you can explore and if assumptions or objectives are different, agree to disagree at least.
With faith based reasoning though, this does not seem to work very well. Assumptions are fixed by them, in their mind. Some objectives can be discussed, but frequently they drift into more assuptions than objectives that can be shared. Facts do not work as they do not seem to accept all facts, just the ones that support their view. Logic does not seem to sway them either, as it frequently violates key assumptions they have.
My experience is that if you want to open their mind, you have to try a more subtle approach that works within their belief system. To do this, you need to have signficiant understanding of their belief system, understand their assumptions and beliefs enough to show them how their core beliefs can fit into the ideas you are presenting. You may be able to demostrate some areas where they already use logic, and expand upon that. One really good approach is if there are others who share their belief system that have come to different conclusions than they have. For example, some Jews are arguing the circumcsion is not necessary and practice an alternative ritual.
I cannot say I have had great success, it is a very difficult thing to do. And most times, it ends up with them getting frustrated and breaking off the dialog in one way or another. But it is valuable to recognize why that is happening, I think. And if you can recognize this pattern up front, you can tailor you approach to best fit the reasoning style used. And you know that at some point, it may be best to disengage.
I hope that is understadabvle and seems reasonable to others. I make no claims that I have expertise in this area, but I have thought about it and experienced it enough to have some half baked thoughts.