Russell's Orders in Kripke's Theory of Truth and Computational Type Theory

Abstract

In Russell's Ramified Theory of Types rttas 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 rttpredicative. Ramsey [1926]and Hilbert and Ackermann [1928]considered the orders to be too restrictive and therefore removed them. This

The Ramified Theory of Types RTT

The basic aim of rttis to exclude the logical paradoxes from logic by eliminating all self-references. An extended philosophical motivation for rttcan be found in Principia Mathematica [Whitehead and Russell, 1910, 1927], pages 38–55. In this section we review a formal description of rttthat is both faithful to Russell's original informal presentation and compatible with the present formulations of type theories (see [[Laan and Nederpelt, 1996], [Kamareddine and Laan, 1996]]).

In Subsection 2a

Kripke's Theory of Truth KTT

In this section, we describe Kripke's Theory of Truth KTT ([Kripke, 1975]). Kripke expresses higher-order formulas within a first-order language, using the fact that many interesting languages are rich enough to reflect their own syntax (for instance, via a Gödel numbering).

Let us assume a first-order language L, with variables ranging over a domain D, and primitive predicates interpreted by (totally defined) relations on D. Let us also assume two subsets S₁and S₂of Dsuch that S₁∩S₂= ∅. Kripke

RTT in KTT

Both in rttand in kttwe are confronted with a hierarchy. Russell constructs a hierarchy by dividing propositions and propositional functions into different orders, taking care that a propositional function fcan only depend on objects of a lower order than the order of f.

Kripke does not make this distinction beforehand. He has only one truth-predicate (T), but decisions about truth of propositions are split into different levels: At the first level only decisions about propositions that do not

The Nuprl And Martin-Löf Type Theories

Martin-Löf's type theory itt(see [[Martin-Lof, 1975], [Martin-Löf,1982]] and also [Nordström et al., 1990]) was originally developed as a foundation of constructive mathematics, and the cttrelative of it was developed and implemented in Nuprl [[Jackson, 1995], [Constable et al, 1986]] to formalize and unify many basic concepts in computer science as well as constructive mathematics. A fundamental idea common to both theories is the interpretation of logic within type theory through the

Computational Type Theory CTT

In this section we look at Computational Type Theory as implemented in Nuprl and MetaPRL, the implicitly typed extensional theory ctt[[Constable et al, 1986], [Allen et al, 2006]] which supports proofs as programs. We will show how the implicitly typed theory offers expressive advantages that compensate for its predicative character. We point out how results from Section 5apply to this theory. Implicit typing also supports polymorphism, and we show the advantages of polymorphism by defining

Conclusions

At the beginning of the 20th century, the paradoxes led to many new formulations of logical systems and an amazing variety of ideas and approaches. Later on, some of these ideas were abandoned when they should not have been. Even more, some of the ideas proposed were found later to contribute nothing to the solution of the paradoxes. For example, even though ZF set theory uses the foundation axiom, it is clear now that it is the separation rather than the foundation axiom which was responsible