Can there be a rational argument against suicide? Yes, I think, but, like almost all rational arguments, it can be overcome by the facts. (Mathematical arguments are the exception: they do not rely on any fact, and so can not be overcome by any fact.)

The (or a) rational argument against suicide is this: suppose you’ve decided to end your own life because of some problem or pain that seems insurmountable. While it is true that suicide will remove the problem or pain, you must realize that it is also probably true that, given time, the problem/pain will go away anyway, even if you don’t commit suicide. And when that happens, you’ll be glad you didn’t take your own life.

But as I said, facts can overcome this argument. For instance, it could be the case that I have a painful, debilitating, and ultimately fatal disease, for which there is no cure, nor hope for cure. In that case, it’s not true that, given time, the problem/pain will go away, and so there’s no reason to think that I will at some later point in my life be thankful that I didn’t take my life earlier. And in fact, most people agree with the notion that a person in these circumstances should be able to end their life when they want to, in the most painless manner possible.

There may be other rational arguments against suicide. But given the level of suffering that organisms like us are subject to, I don’t think any absolute prohibition against suicide is truly rational.