Russell's Orders in Kripke's Theory of Truth and Computational Type Theory
by Robert L. Constable, Fairouz Kamareddine, Twan Laan
Published in Handbook of the History of Logic: Sets and Extensions in the Twentieth Century. Vol. 6
available from Elsevier
In Russell’s Ramified Theory of Types RTT as presented in Principia Mathematica by Whitehead and Russell [1910, 1927], two hierarchical concepts dominate: orders and types. The class of propositions over types is divided into different orders where a propositional function can only depend on objects of lower orders. The use of orders renders the logic part of RTT predicative. Ramsey  and Hilbert and Ackermann  considered the orders to be too restrictive and therefore removed them. This led to the development of Church’s Simple Type Theory STT  which uses types without orders. Since, numerous type systems abandoned the hierarchy of orders. For example, all the eight influential Pure Type Systems(PTSs) of the Barendregt cube  avoid orders. Despite this lack of explicit use of orders in some modern type systems, orders still play an influential role in understanding hierarchy in modern type systems. In this chapter, we reflect on the use of orders in providing an adequate foundation for basic concepts in computer science and computational mathematics as expressed in Computational Type Theory CTT [Constable et al., 1986; Allen et al., 2006; Kamareddine and Laan, 2001] and in explaining the truth levels in Kripke’s Theory of Truth KTT [Kripke, 1975; Kamareddine and Laan, 2001].
bibTex ref: KLC12