The debate -- more of an argument, really -- has been waged elsewhere and at great, shouting-past-one-another length.
But it is a fundamental one, beginning, as it does, at the basis of all rights: property rights. What's the single most inalienable thing you own? That's easy: it's your own self. That's the most basic thing you've got.
One of the rights that grows from this root is your right to take an early exit. In most cases, it's not the best course and you're best off stopping and rethinking; but it is a right.
However, just as the right to own and carry weapons doesn't give you the right to point 'em any which way without consequences, your property rights over your own person don't mean you get to threaten to off yourself without eliciting unwanted responses from others.
You don't own other people; you don't get to control what they do. When you threaten to harm someone -- even yourself -- you're initiating force, attempting to extort something from the persons to whom you are expressing your threat.
Even in Libertopia, they're justified in trying to stop you, same as they would be if you were making a similar threat to someone other than yourself. (I believe the philosopher Mel Brooks addressed this early on in his monograph, Blazing Saddles). And since we don't live in Libertopia, sometimes the intervention is less justified and more forceful than one would wish.
Do you have the moral right to kill yourself? I think you do;* you even have the right to discuss it. But you don't have the right to colonize other people's minds; if they take your "discussion" as a threat, they'll react. And if you "discuss" the matter in the heat of the moment with an agent of the State -- a policeman, a firefighter, an EMT? Why, at best, you will almost automatically be finding yourself detained and under observation for a day or three while they try to figure out if you're rational or not.
Unfair? Possibly; but it's the world we live in -- a world of dangerous things that will provoke strong reactions if you meddle with them incautiously around others. You don't juggle bottles of rat poison in a crowd; you don't wave firearms around -- and you don't make suicide threats. That doesn't mean you can't use rat poison, defend yourself, or that your right to ownership and control of your person has been abridged. You're a grown-up and your peers grant you a staggering degree of trust; but it is not an unconditional or irrevocable grant.
You're surrounded by persons just as real -- and just as opinionated -- as yourself. Normal life involves interacting with them and accepting that they're not just clones of you. If you want to dictate how they should act and react, you might want to take a closer look at your own philosophical premises: you might not be a libertarian. You might not even be pro-freedom for anyone but yourself.
Update: An excellent discussion in Comments! Thank you. My own "bright line" is that any time threats of suicide are used to attempt to compel others, it's a hostage situation -- and 72 hours to cool off is a truly gentle response. Most of the "gray area" situations aren't that gray when examined closely: few if any serious suicides announce it ahead of time. (I'd like to point out that no ethical suicider would commit the act with unfulfilled obligations, either; you can't leave your messes for others to clean up and claim you're acting responsibly.)
SF writer H. Beam Piper, broke and with no prospects he was aware of, took his own life within days of his agent sending news that he had made a major sale. He never found out. Rational persons contemplating suicide should consider his story before taking irrevocable steps.
* Most religions think you do not. And -- hooray! -- in a free society, you're free to follow the dictates of your faith. You don't get to impose them on other adults.
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