Some university-level courses have no specific prerequisites, yet are mathematically involved to the extent that someone with little to no experience in math will probably find themselves in over their heads. Such courses are often cross-listed and appeal to students in disciplines outside of math like philosophy, computer science, economics, psychology, etc. For example, certain courses in logic or decision theory often fit this bill.
One standard way of "warning" the students about the mathematical nature of the course is by saying something like, "The only prerequisite is mathematical maturity."
To me, this seems like an apt characterization: they're going to be asked to prove things on the homework, to absorb formal definitions fairly rapidly, etc. But just because I have an intuitive grasp on the meaning of "mathematical maturity" doesn't mean they do. I can't help but feel that this warning falls short with some and goes too far with others. I know from experience that some students underestimate this warning, and remain in the course despite, e.g., having no idea what it means to rigourously prove something. Unsurprisingly, these students tend to struggle greatly. On the other hand, I like to view such courses as good alternatives to more traditional courses like, say, calculus, for helping to strengthen and deepen general mathematical facility. It would be a shame if some students who would otherwise find the course quite useful overestimate the meaning of "mathematical maturity" and shy away.
How can I adequately convey to students the meaning of "mathematical maturity"? In some sense, it's one of those, "If you don't know it, you haven't got it" type things, but I'd like to have something more tangible than this, perhaps a method of assessment or a representative class of examples I could offer to students who are unsure.