The issue of how a semantics should deal with self-attacking arguments was always a subject of debate among argumentation scholars. A consensus exists for extension-based semantics because those arguments are always rejected (as soon as the semantics in question respects conflict-freeness). In case of gradual semantics, the question is more complex, since other criteria are taken into account. In this paper, we check the impact of those arguments by using a principle-based approach. Principles like self-contradiction and strong self-contradiction prescribe how to deal with self-attacking arguments. We show that they are incompatible with the well-known equivalence principle (which is satisfied by almost all the existing gradual semantics), as well as with some other principles (e.g. counting). This incompatibility was not studied until now and the class of semantics satisfying self-contradiction is under-explored. In the present paper, we explore that class of semantics. We show links and incompatibilities between several principles. We define a new general oriented argumentation semantics that satisfies (strong) self-contradiction and a maximal number of compatible principles. We introduce an iterative algorithm to calculate our semantics and prove that it always converges. We also provide a characterization of our semantics. Finally, we experimentally show that our semantics is computationally efficient.