If you allow someone to die when you are capable of saving they life, but do not kill them directly, are you a murderer?

In both the legal and the familiar sense of the word "murderer," the answer is no. You certainly wouldn't be charged with murder in a case like this, and if you were, successfully arguing that you didn't actually kill the person but merely allowed them to die would lead to a "not guilty" verdict. Murder, as it's usually understood, is unlawful killing or, in the non-legal sense, morally unjustified killing.

That said, someone might argue that if you're in a position to save someone's life and you don't, then you're guilty of something just as bad as murder. No doubt we can come up with hypothetical cases where this might be so. For instance: Alex intends to kill Bob; he's got the means and the will. But on his way to do the deed, he discovers Bob unconscious and bleeding by the side of the road. Suppose it's clear that Alex could save him; calling 911 and staunching his wound until help arrives would do. But Alex does nothing except wait for Bob to breathe his last. In this case, we might say that Alex has really just taken advantage of a twist of fate to accomplish what he would have done by himself anyway. We might well think: Alex is as morally guilty as he would be if he'd shot Bob to death. But whatever we say about this case, it's hard to draw general conclusions from it. In general, both the law and common-sense morality distinguish between intentionally harming (or killing) someone and simply not helping them and the fact that there are some cases where the distinction seems untenable doesn't show that it's untenable in general. After all, there might be many explanations for the fact that someone doesn't act to save someone else: shock or fear or confusion, for example. None of those add up to malice, let alone murder.

Just to give you something more to chew on: suppose you could save Rob and could also save Bob, but no way could you save both. You flip a coin and save Rob, letting Bob die. You were capable of saving Bob, but you didn't, but pretty clearly this doesn't make you a murderer. It doesn't even mean that you did anything even slightly wrong. Not the case you had in mind, but a clear example of letting die that's not the moral equivalent of murder.

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