The status of first-order logic (FO) as the single most important logic has been questioned by a number of researchers in the past, perhaps most notably by Hintikka. His independence-friendly (IF) logic extends first-order logic by independence declaration operators. These operators are relatively natural and do occur in the everyday practice of mathematics together with the usual quantifiers and connectives. However, IF logic has its shortcomings. For example, it is not clear why the expressive power of IF logic should be regarded as fundamental. In this talk we show how to extend FO into a Turing-complete framework by considering two classes of simple operators that are omnipresent in everyday research mathematics. We also discuss the interface between natural language and the novel logic.