RESUMO 40 | COMUNICAÇÕES

On the Impredicativity and Circularity of Frege’s definition of the Ancestral

SCHMIDT, João Vitor | UNICAMP, Brasil

Frege’s definition of the Ancestral, as given in Begriffsschrift, is a second-order definition for the transitive closure of any first-order relation. For any first-order relation R and objects a and b, we say that Frege’s Ancestral, denoted by R*(a,b), holds if, and only if, b has all the hereditary properties that a has and that all ancestral R-successors of a also have. Formally, this is

with Her (F,R) meaning “F is hereditary in R”, formally

The name “Ancestral” is due to the fact that, if we consider R as the parent relation, then R* is the ordinary ancestral relation. It is also considered that Frege’s definition is a logical reduction of this ordinary notion. This is an important step for logicism, since it provides the conditions for transforming the predecessor relation into a linear-ordered series, being both transitive and trichotomous. It is also Frege’s first major result, both for his logic and for his philosophical goals, already in 1879, i.e. an argument against the necessity of Kantian intuitions for ordered series and the informativity of | second-order logic. But since the definition uses quantification over properties, it was recognized as impredicative, since it quantifies over a domain that contains itself. Consider the property R*(a,x) which says that “x is descendent of a”. Clearly, this is an hereditary property in R, given that, using the analogous relation, my ancestrals are also ancestrals of my children. Then, R*(a,b) holds if, and only if, b has the hereditary properties that a pass through R, including the property above. But this is R*(a,b) again. We have, then, a circular definition. This was first pointed out by Benno Kerry in 1887 and more recently, by Ignacio Angelelli in 2012. Angelelli not only restate Kerry’s objection, but add his own, arguing that Frege’s logical reduction of the ordinary notion was undermined by this problem. In this talk, I aim to clarify this problem and to point out some mistakes made by Angelelli in his paper. Specifically, I want to argue that, although Frege’s definition is in fact impredicative, it is not circular as Angelelli supposes. In such, I’ll state that Angelelli’s argument is solid only by making violence to Frege’s logical and philosophical motivations. A similar response can be made to Kerry’s, one that Russell did in 1902 in defense of Frege’s definition. Finally, we shall point that the impredicativity of the Ancestral is not as problematic as both Kerry and Angelelli considered.