, Volume 41, Issue 2, pp 505-517
Date: 19 Apr 2011

If Logic, Definitions and the Vicious Circle Principle

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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.