Transcending the theory of types

The intensional paradoxes present a continuing challenge to any theory of concepts (properties, attributes, propositional functions). In his seminal paper on Russell, Gödel expressed sympathy for the strategy of limited ranges of significance, which is derived from but logically independent of Russell’s theory of types. According to this strategy, every predicate determines a concept, but applying a concept to certain arguments may take us outside the range of meaningfulness. Gödel’s idea is most naturally implemented in a logic that admits truth-value gaps. Unfortunately, attempts in this direction often result in deductively weak theories. Although Gödel rejected the theory of types, one can make a case that a satisfactory type-free system needs to be able to recover the deductive strength of classical simple type theory. Ordinary fixed-point theories à la Kripke fail to satisfy this desideratum. Based on Gödel’s ideas, we present a naïve theory of concepts, formulated over Weak Kleene logic, which preserves the deductive strength of classical simple type theory. Our language contains some novel restricted quantifiers, but no additional conditional or recapture operators.