We propose a general framework to investigate semantics of Dung-style argumentation frameworks (AFs) by means of a generic defeat notion formalized by refute operators. After establishing the technical foundations, we propose natural generic versions of Dung's classical semantics. We demonstrate how classical as well as recent proposals can be captured by our approach when utilizing suitable notions of refutal. We perform an investigation of basic properties which semantics inherit from the underlying refute operator. In particular, we show under which conditions a counterpart to Dung's fundamental lemma can be inferred and how it ensures the existence of the generalized version of complete extensions. We contribute to a principle-based study of AF semantics by discussing properties tailored to compare different refute operators. Finally, we report computational complexity results for basic reasoning tasks which hold in our general framework.