Can you change someone's mind? Can you change your own?
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Socrates is the hero of mind-change in the Western tradition.
The celebrated elenchus, or Socratic method, involves probing questions that generate counter-examples, expose special pleading, and force self-contradictions into the open.
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Particulars are not universals!

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Exceptions destroy rules!

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Rational standards are everywhere hidden in plain sight!

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The cases are instructive. Thrasymachus is a member of the species known as political realists, the kind of people who think might makes right.
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Euthyphro represents an even larger sect, the fideists: he is a man who believes because he has prior faith in the gods.
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Consider, then, a combination of these two meta-rational figures: a politician with no regard for truth or argument, who passionately believes in greatness and power (especially his own), and holds himself to no standards except the ones generated by his own megalomania.

You do not change the mind of a man like that. He’s already done all the changing, and it is, ironically, along the lines attributed to Mohammed Atta:
“The world changes first in the mind of the man who decides to change the world.”
Confirmation bias, recency bias, cognitive dissonance, post facto rationalization, and layer upon layer of ideological distortion allow all of us to believe we are engaged in rational debate when, in fact, we are simply piling sandbags against the base of our long-held positions. We pursue self-defeating desires; we vote against our interest; we dig in and die.
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Diving deeper, we get even more primitive: fear, anger, anxiety, and associated pheromonal networks. Otherness of all kinds generates a hackle-raising primate response. Exposed inner contradiction, once thought to spawn change, instead prompts frenzied efforts to scaffold the problem -- the mental equivalent of insane astronomical elaborations meant to preserve the geocentric model of the universe.

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Moral panic sets in, and the response is feral retrenchment. The gates of our fortress of belief are drawn up, even as heralds on the battlements keep bellowing their slogans of reasoned discourse. We make ourselves easy prey for hucksters and ideologues, for the siren call of authority. Yes, we honor epistemic consistency; but how much of that is driven by how we want to see ourselves: sensitive soul, staunch protector of the realm, social justice warrior, piratical outsider.

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Good luck changing anyone’s mind once those various kinds of battlements have been erected.
Click cards to shuffle
Meanwhile, the tenets of emotional conjugation show that we always
(irrationally) grant more moral flexibility to ourselves than to others.

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When it comes to other people, the best we can hope for is a degree of understanding and, maybe, sympathy for why others believe the things they do.
Can one perform the mind-change operation internally, then, cleansing one’s mental mechanism of prejudicial sludge and pseudo-rational evasion?
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Maybe.
But recall another part of Plato’s Socratic message: the mind is not one coherent thing, nor is it obviously ruled by reason. Imagine instead a sort of lame-duck executive, charged with carrying out directives emanating from a mysterious shadow bureaucracy, spinning them as coherent and responsive to norms of reason.

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Maybe the mind is lacking a central executive altogether, being instead a sort of squabbling, fractious, borderline insane committee of voices as depicted in Inside Out (2015), with routine power struggles and multiple internal coups. More often than not, our most basic desires carry the day.
It is hard enough to resist the siren call of ice cream; how in the world is change in basic belief possible?
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Results of a
now-canonical psychological study: When people are confronted by things they want to believe, the question is “Can I?” When they are confronted by things they don’t want to believe, the question is “Must I?”
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We can see why dramatic revelation is, historically, the preferred method of mind-change.

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I was struck from my horse, and arose a Christian! The word was spoken to me, and I turned my back on Babylon!
For the rest of us – well, we go on, twiddling our internal dials, trying to find stable platforms in the shifting sands of mental life, and working out our beliefs as best we can.
And while it is true that emotion often drives reason, Plato knew that the mind is ever a complex matter of feedback loops, desires for stability, and a powerful (if fragile) alliance between reason, appetite, and will.