If Logic, Definitions and the Vicious Circle Principle

In a definition (x)((xr)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 (ba) or (ba). 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. 1. Correcting Frege " s mistake Gottlob Frege is usually credited with having constructed (discovered and codified) the central part of our common logic, the logic of quantifiers (plus propositional connectives). If so, he did not finish the job. He left at least ones serious flaw in his construct. This flaw was not a random mistake. It is due to Frege's failure to understand fully the nature of the most important ingredients of his edifice, the (existential and universal) quantifiers. This mistake did not only affect Frege's own work. Subsequent logicians have until recently repeated his mistake. This paper will show through case studies how logicians' continued failure to recognize the mistake has led to serious gaps in the contemporary logical theory and misguided the development of both logical theory and set theory. What was the aspect of the meaning of quantifiers that Frege did not appreciate? What is the semantical job description of quantifiers, anyway? The answer that one is likely to receive from any logician or philosopher is that they range over a class of values. An existentially quantified sentence is like a disjunction of its substitution-instances and a universally quantified sentence is like the analogous conjunction. This " ranging over " is obviously part of what quantifiers do. But is it their whole job? Frege thought so, and expressed his thought in the form of his interpretation of quantifiers as higher-order predicates expressing the nonemptyness and the exceptionlessness of lower-order