Everyone, at some time or another, has shaken his head at the tragedy of an untimely death. Deep down inside we know that each person's life is meant for something. It's not just that they have personal aspirations. A very young child dying of leukemia doesn't have aspirations. By contrast, we look at someone like Mother Theresa or Jonas Salk, and we applaud them for having accomplished something wonderful, having fulfilled something of their purpose in life.
Human beings ought to fulfill their potential.
If someone kills himself, he's wasting his life. If there's no purpose, then there's no tragedy, no waste. Yet we intuitively know these things are tragic losses. Therefore, our lives must have some purpose waiting to be discovered and lived. This is one extended example of all kinds of things we're aware of, things we know intuitively.