Why most online arguments about politics and religion end in a stalemate

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Scenario: Two people are engaged in mortal ideological combat on Facebook.  Each person argues their point passionately, genuinely baffled that what is manifestly obvious to them does not register at all with the other.  Since The Truth™ is so obvious, each combatant starts wondering why their opponent refuses to concede the point, eventually attributing such incorrigibility to bad character or, even worse, to mental illness. 

What is actually going on here?  Why do most skirmishes like these end with nobody persuaded and with bad feelings all around?

I believe this happens because the actual conflict usually does not lie in the topic itself, but in the long chain of presuppositions that lead rationally and inexorably to each person's view on the topic.  Within the closed system of a person's worldview, the point they're arguing makes complete sense to them, flowing logically from the worldview's foundational principles (i.e. the collection of assumptions forming the filter through which reality is interpreted), many or most of which are accepted as givens.

You won't get them to budge on the topic being debated unless either (a) you can convince them that the point they're arguing is inconsistent with the foundational principles of their worldview, or (b) you can somehow erode their confidence in the foundational principles themselves.

Strategy (a) is possible, but only those who embrace the same worldview are likely to see any success.  If someone of an opposing worldview tries this strategy, the natural reaction is to question the aggressor's motives (thus allowing the target to ignore the argument being made).

Even if strategy (b) has any chance of eventual success, the foundational principles aren't even part of the current debate, so forget about that.

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Imagine a worldview as a tree.  The trunk and roots are the foundational principles; the opinion being argued at the moment is more akin to a cluster of leaves at the end of one of the tree's branches.  Everything between the cluster and the trunk is the chain of presuppositions connecting the opinion to the foundational principles.  Even if someone succeeds in damaging or lopping off that cluster, it'll grow back in due time -- nothing really gained or lost.

Facebook is where I'm most likely to encounter people of significantly differing worldviews.  I've developed an instinct for recognizing when, for my potential opponent, an opinion represents a leafy cluster or something closer to the trunk.  If the former, I'm more likely to refrain from joining the fray, because the most likely outcome is bad feelings and nothing else.  If the latter, I might cautiously engage, in the hope that something worthwhile might transpire.

With the leafy-cluster topics, I will occasionally come across a viewpoint that baffles me so much that I step back and ask myself, How could someone possibly hold such an opinion?  Then, over time, through observation and sometimes through tactfully-worded questioning, I will attempt to discern that opinion's chain of presuppositions.  As I analyze the presuppositions, I usually am forced to reanalyze and reaffirm the presuppositions of my own worldview.

While very rare, the process does occasionally lead me to soften or change my opinions (or at least stop thinking of my opponent as insane).  But -- and this is the point -- I came to this conclusion on my own, not as the result of being outmaneuvered in an online argument.