  Russell introduced another hierarchy, that was not motivated by any formal paradoxes expressed in a formal system, but rather by the fear of “circularity” and by informal paradoxes similar to the paradox of the liar. If a man says “I am lying”, then we have a situation reminiscent of Russell’s paradox: a proposition which is equivalent to its own negation. Another informal such paradoxical situation is obtained if we define an integer to be the “least integer not definable in less than 100 words”. In order to avoid such informal paradoxes, Russell thought it necessary to introduce another kind of hierarchy, the so-called “ramified hierarchy”. The need for such a hierarchy is hinted in Appendix B of Russell 1903. Interestingly this is connected there to the question of the identity of equivalent propositions and of the logical product of a class of propositions. A thorough discussion can be found in Chapter 10 of Russell 1959. Since this notion of ramified hierarchy has been extremely influential in logic and especially proof theory, we describe it in some details. 