The word ’predicative ’ first appeared in Russell’s note On Some Difficulties in the Theory of Transfinite Numbers and Order Types [Rus07]. Paradoxes such

as Russell’s Paradox show that we cannot form the class {x | φ(x)} for all propositional functions φ(x). Russell proposed we call φ(x) predicative if it defines a class and non-predicative otherwise, but did not offer a criterion by which we could decide which propositional functions are which. A first such criterion was offered by Poincare ́ in the third part of his paper Les Mathématiques et la Logique [Poi06]. He proposed the vicious circle principle: “The definitions which ought to be regarded as non-predicative are those which contain a vicious circle. ” (p. 1063) He indicated by example what he meant by ‘vicious circle’: in both the Richard paradox and the Burali-Forti paradox, we define an aggregate E, and make use of E within its own definition. Poincare ́ proposed that definitions involving such a ‘vicious circle ’ are il-legitimate: “A definition containing a vicious circle defines nothing. ” [Poi06, p. 1065] Poincare ́ justified this as follows. For a definition to be legitimate, it must be possible to substitute the definiens for the defined term. Recursive definition