Ive associated suicide as correlated with too much logic. It takes a litle bit of delusion deep in the core of your being to assign value, meaning and purpose to the process of moving molecules from a high energy state to lower energy states. Which is all we are doing. Convincing a logical person that delusion is needed is a tough sell.
Ive had the experience of trying to talk some normality into a suicidal person. Its like two kids on a carnival ride and one is queasy and hates it and wants to get off. Every turn and moment is filled with a desire to get off. what does the kid having fun on the ride say to the one who wants off? It comes down to "stop feeling the way you do". the misrable one has to figure out the source of the pain and hijack a feedback loop happening in the subconscious mind.
For the record, logical, analytic, probabilistically correct thinking is only a way of reaching goals. The goals themselves cannot be established by means of logic, only deconflicted/clarified/etc. To think otherwise is illogical.
On the latter point, that's a pretty major dispute in philosophy and mathematics. The axiomatic Hilbert-style approach probably has the upper hand in mathematics, but a Russell-style logicist approach that sees foundations themselves as subject to rational analysis is still a major position, and among philosophers, the idea that axioms are arbitrary and rationality only applies past that is probably the minority view (though existentialists hold a view somewhat like that). Not that that's necessarily practical advice.
According to Camus, one's freedom – and the opportunity to give life meaning – lies in the recognition of absurdity. In acknowledging the absurdity of seeking any inherent meaning, but continuing this search regardless, one can be happy, gradually developing his or her own meaning from the search alone.
It's what keeps me going. I can't control what happens after I die but I can control what happens in this life. So I'm going to make my purpose in this life ---making an impact now. Even if it's ultimately meaningless in the universe's timeframe, so be it.