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A quick answer would be no, i don't think feelings of worthlessness or even actually being "worthless" (to be honest, i don't even know how can one measure that) would give someone the imperative to off themselves.
However,I do understand how someone feel that way, because i have been feeling that way for a long time, and not just being worthless, but also being a burden to those i know. It's bad enough if i go through days feeling that way with minimal interaction with people, but the thing is, the more and more i involve myself in what i do, interacting with people, the more the feeling worsens. I feel like i'm digging a hole for myself. And the people i know, and who i think are great, are getting dragged into that hole with me.
But as a moral imperative? I have to say no, not because of any real certainty on my part, but because i haven't really decided myself.
By your logic most animals should off themselves because they're not really helping matters and a lot of them depend on us to survive. Think for a moment, if you were a common cold virus infecting a brilliant scientists lungs: am I worthless?
Lots of things are worthless in the grand scheme of things. It matters what YOU think.
I do not think that suicide should be a moral imperative for any human being. What makes me different from possibly most people, is that, I strongly hold the view that no human being deserves to die or to execute suicide. Regardless of what anyone has done, said, thought, or what motive anyone have or had; despite what attributes, characteristics, qualities, or traits anyone has or had; regardless of what opinions, views, beliefs, attitudes, personalities, or world views anyone has or had, I do not think absolutely any human being deserves to die. That includes even serial murderers, child molesters, child killers, terrorists, racists, abusers of all types, rapists, and other groups of people who are viewed in contempt by the general populace.
Even providing that the death of someone would be beneficial in some fashion. For example, the death of someone saving hundreds, thousands, millions, or billions of lives, or preventing a disastrous or catastrophic event that would affect the entire world, I still do not think that it would be indispensable for the person to perpetrate suicide or to die. I am the type of person who intensely respect the life of every human being, and that includes every human being who has lived, currently living, and will be living; there are absolutely no exceptions. In other words, even the most contemptible of human beings deserve to live. Adolf Hitler committed suicide, but even I think that he deserves to live.
Although what he had done during his life was despicable and repellent, I was still saddened by the fact that he took his own life. I do not know why I am like that, but for some reason, I cherish my own species, and therefore, I cherish and respect the life of every human being, which insinuates that I am completely against capital punishment and of any human being killing another human being under any circumstances (even in self-defense or to prevent the death of another human being). My views have also frustrated me, because of many moral dilemmas that I have contemplated that induces me to question my own views of human life.
For example, abortion is one of those moral dilemmas of which I am uncertain concerning human life. Further, moral dilemmas that involves choosing the life of one human being over another. Not only do I cherish and respect the life of every human being, I also cherish and respect the life of every human being equally. If I were in a situation where I would have to choose who lives, and the people who were involved in that situation were my mother and my worst enemy, I would not even make a decision. Although most people would choose their mother over their worst enemy. I would also find it laborious to kill another human being intentionally, because the guilt of doing so would be to such a degree severe, that I would have to kill myself to rid myself of the guilt.
I could not live comfortably and happily being cognizant of the reality that I have taken the life of another human being. If the person is "worthless", or does not contribute in any manner, I still believe that person should not die. What I am going to state contradicts everything that I have asserted in regard to my beliefs about human life. I do not believe anyone should be required to die or to take her or his (or other gender) life, but I believe that I deserve to die; that I deserve to take my own life. I do not employ my convictions of human life towards myself, and probably never will.