Before I begin, i'll admit this isn't a maths question, however it is intrinsically a part of maths, and systems such as logic, set theory, and of course mathematics, will have been studied by many serious mathematicians. So this seems like the most appropriate(though not perfect) place to ask this.
If there's a more appropriate place, please notify me, and I shall mitigate it immediately.
Ok, so by system, I mean a completely independent structure that has it's own sets(i.e classes), operations, and objects(i.e data types), where proofs and the rules of syntax are then derived as a result of the nature in which the operations are applied to objects, and the classes they belong to.
I figure that there must be some universal traits of a good and effective system, not so much in how it can be applied, but to how it's built. For e.g, you can't have a system without any objects (such as the number in maths, the proposition in logic, or the set in, well, set theory). You obviously need something to work with, and thus something to apply operations to.