Journal of Philosophical Logic

, Volume 41, Issue 2, pp 505–517

If Logic, Definitions and the Vicious Circle Principle

Authors

    • Department of PhilosophyBoston University
    • Collegium for Advanced StudiesUniversity of Helsinki
Article

DOI: 10.1007/s10992-011-9184-4

Cite this article as:
Hintikka, J. J Philos Logic (2012) 41: 505. doi:10.1007/s10992-011-9184-4

Abstract

In a definition (∀x)((xєr)↔D[x]) of the set r, the definiens D[x] must not depend on the definiendum r. This implies that all quantifiers in D[x] are independent of r and of (∀x). This cannot be implemented in the traditional first-order logic, but can be expressed in IF logic. Violations of such independence requirements are what created the typical paradoxes of set theory. Poincaré’s Vicious Circle Principle was intended to bar such violations. Russell nevertheless misunderstood the principle; for him a set a can depend on another set b only if (bєa) or (b ⊆ a). Likewise, the truth of an ordinary first-order sentence with the Gödel number of r is undefinable in Tarki’s sense because the quantifiers of the definiens depend unavoidably on r.

Keywords

(In)dependence IF logic Definitions Vicious circle principle Truth-definition

Copyright information

© Springer Science+Business Media B.V. 2011