Ask them why

That many people tend to be irrational and inconsistent is due to the fact we as humans are primarily driven by emotion. This means that protection of the psyche will override critical thinking. It is why you will sometimes witness a very partisan voter caught on camera having a hypocritical stance on an issue for which they attack the opposition for but then justify their own side for the very same thing.

Clearing our minds of emotional bias is very difficult but I am having increasing success as I get older and remove emotional attachment. Rather than protect myself behind a barrier, I want to examine carefully why I believe something by challenging my own thought process.

More ethereal issues like religion are less easily dissected but even there we can examine whether our beliefs are consistent with the ethos of what our faith is trying to accomplish. Some so called "Christians" for example are fine to take exceptions to love of neighbor if it contradicts some deeply held bias about another.

I no longer argue with irrational people because it becomes pointless. The more they are glued to a fixed orthodoxy the less success you will have in having them assess their viewpoint with a critical eye. But if you want to catch someone off guard  simply have them explain to you why they believe something. 

The inconsistencies are often fascinating and wildly amusing.