After observing that the truth conditions of connectives of non-classical logics are generally defined in terms of formulas of firstorder logic, we introduce 'protologics', a class of logics whose connectives are defined by arbitrary first-order formulas. Then, we introduce atomic and molecular logics, which are two subclasses of protologics that generalize our gaggle logics and which behave particularly well from a theoretical point of view. We also study and introduce a notion of equiexpressivity between two logics based on different classes of models. We prove that, according to that notion, every pure predicate logic with k ≥ 0 free variables and constants is as expressive as a predicate atomic logic, some sort of atomic logic. Then, we prove that the class of protologics is equally expressive as the class of molecular logics. That formally supports our claim that atomic and molecular logics are somehow 'universal'. Finally, we identify a subclass of molecular logics that we call predicate molecular logics and which constitutes its representative core: every molecular logic is as expressive as a predicate molecular logic. N.B.: This version of the paper corrects some typographical errors which are present in the published version, notably those regarding the references in Figures 2 and 3.