Another clue we get about the narrator's unstable mentality in this story is that his reason for killing the old man is that he does not like the old man's eye. When the narrator ponders his reason for committing murder, he says,
There was no reason for what I did [....]. I think it was his eye [....]. When the old man looked at me with his vulture eye a cold feeling went up and down my back; even my blood became cold. And so, I finally decided I had to kill the old man and close that eye forever!
Now, there is probably never a great reason to kill another person, but some reasons are certainly more understandable, or more "sane," than others: self-defense, a person had wronged you terribly, a person poses a major danger to you or a loved one, and so forth. However, the narrator says that he literally has no reason to kill the old man other than that he hates the man's eye. This is a pretty unacceptable reason to kill a person, at least for a sane person.
Further, when he...
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