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I think there are two dimensions to this. The first is: what makes a real-world person irredeemable? A fictional character with the same traits will then, presumably, also be irredeemable. I thin...
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#2: Initial revision
I think there are two dimensions to this. The first is: what makes a real-world person irredeemable? A fictional character with the same traits will then, presumably, also be irredeemable. I think the already-provided answers by Galastel, Amadeus and Francine DeGrood Taylor do a great job of discussing this idea. The second is: how does a character function, within the structure of a story, so that, in terms of storytelling mechanics, the character is irredeemable? At the most basic level, redemption (after extensive evil) requires a deep, challenging character arc. If a character cannot be convincingly given such an arc, the character cannot, mechanically, be redeemed within the story. **And that character will also _feel_ irredeemable to the reader** , because the reader cannot imagine a situation in which the character is (plausibly) redeemed. So, for example, a two-dimensional/cardboard character will be irredeemable because redemption has, effectively, not been foreshadowed--we would need a more rounded character to believe that this is a real person, with flaws and failures and blind spots, who could, therefore, come to understand and regret her failures and, thereafter, _change_. Also, a character we don't invest in, don't feel any empathy towards, will be (mechanically) irredeemable, because a redemption character arc takes us on an emotional journey that is impossible without us being significantly invested. A different way to state this. Consider the _values_ explored by the story. The first way to consider the question is about the _character's_ values--the evil character's values are so perverse and despicable that, once they have been translated into actions, we will never forgive the character. The second way is to focus on _our_ values: the irredeemable character cannot guide us on a journey towards a deeper understanding of our own values, whereas a character may be exceptionally evil and yet lead us on such a journey, and thereby, within the story, be redeemed (to some extent) in our mind\*. A good example of this is American History X: we will never forgive what the protagonist did as a neonazi, but we are confronted by his humanity in a heart-wrenching way that challenges our way of categorising people. Moving on to the Harry Potter examples. I think Umbridge is, simply, an _excruciating_ read. All I want is for her to be the hell out of the story, to stop tormenting Harry and everybody else. Whereas Voldemort is, in the storytelling sense, awesome\*\*: powerful, enigmatic and "fallen"--I want to know more about him, and I enjoy myself imagining his upcoming epic defeat, or, who knows, maybe a more complex resolution to the story (although it turns out Voldemort gets nothing like a redemption arc). Umbridge is, thus, further along the scale towards irredeemable because I, the reader, have no _interest_ in seeing that character developed, I just want her defeated, whereas I _do_ want Voldemort developed as a character. \* perhaps I should drop a reference to Aristotelian "katharsis" here--but my classical education is too many decades in the past for me to do so confidently... \*\* though, personally, I thought he kinda failed as a character late in the series (but maybe this was intentional: when we finally meet him, it turns out he isn't actually that capable at things other than evil magic)