The paper discusses the strengths and weaknesses of arguments of proper self-reference, arguments of self-application and arguments of iterative application. A formalization of the underlying logical structure of these arguments helps to identify the implicit premises on which these arguments rest. If the premises are plausible, the conclusions reached by these arguments must be taken seriously. In particular, all the types of argument discussed, when sound, show that certain theories that purport to be universally applicable are not tenable. The argumentative power of such arguments then depends on how devastating it is for the theories in question to give up their claim of universal applicability.