I don't know a lot about ILP algorithms, but my understanding of them is that they basically search the space of logical rules for rules that "explain" the exam

I don't know a lot about ILP algorithms, but my understanding of them is that they basically search the space of logical rules for rules that "explain" the examples they are given. More formally, they look for rules from which you could then prove the given examples. So in an ideal world you could give an ILP system a bunch of formulas and it would give you back the Peano axioms or something like that I'm not sure if state of the art ILP systems are good enough to do that in practice or not.