Predicativity

A special way of forming concepts, characterized by the absence of a "vicious circle" in the definitions: the object to be defined should not participate in its own definition. If the language in which the definitions are stated is formalized, then predicativity means, as a rule, that the defining formula should not contain a bound variable whose domain of variation includes the object to be defined.

A non-predicative definition, on the other hand, is distinguished by the presence of a "vicious circle" within it. The phenomenon of non-predicativity can also be encountered in certain reasoning wherein the process of substantiating a certain part of the reasoning under consideration is itself considered as an object of reasoning. It is exactly the use of this kind of reasoning that is a ground for the appearance of semantic antinomies (cf. Antinomy). A typical example is the contradiction in the liar's paradox: If someone states "I am lying" , then this statement can be neither true nor false.

Comments

For an enriched set theory that does admit circularity see [a1]; for an account of "the liar" based on this cf. [a2].

References

[a1]	P. Aczel, "Non well-founded sets" , Centre Study of Language and Inform., Stanford Univ. (1987)
[a2]	J. Barwise, J. Etchemendy, "The liar. An essay on truth and circularity" , Oxford Univ. Press (1987)