Tdl Laan

Types in Logic and Mathematics before 1940

In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whiteheads Principia Mathematica ([71], 1910-1912) and Churchs simply typed ?-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Freges Grundgesetze der Arithmetik for which Russell applied his famous paradox and this led him to introduce the first theory of types, the Ramified Type Theory (RTT). We present RTT formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from RTT leading to the simple theory of types STT. We present STT and Churchs own simply typed ?-calculus (? ? C) and we finish by comparing RTT, STT and ? ?C.

Refining the Barendregt Cube Using Parameters

The Barendregt Cube (introduced in [3]) is a framework in which eight important typed λ-calculi are described in a uniform way. Moreover, many type systems (like Automath [18], LF [11], ML [17], and system F [10]) can be related to one of these eight systems. Furthermore, via the propositions-as-types principle, many logical systems can be described in the Barendregt Cube as well (see for instance [9]). However, there are important systems (including AUTOMATH, LF and ML) that cannot be adequately placed in the Barendregt Cube or in the larger framework of Pure Type Systems. In this paper we add a parameter mechanism to the systems of the Barendregt Cube. In doing so, we obtain a refinement of the Cube. In this refined Barendregt Cube, systems like AUTOMATH, LF, and ML can be described more naturally and accurately than in the original Cube.

A Reflection on Russell's Ramified Types and Kripke's Hierarchy of Truths

Both in Kripkes Theory of Truth KTT [8] and Russells Ramified Type Theory RTT [16, 9] we are confronted with some hierarchy. In RTT, we have a double hierarchy of orders and types. That is, the class of propositions is divided into different orders where a propositional function can only depend on objects of lower orders and types. Kripke on the other hand, has a ladder of languages where the truth of a proposition in language Ln can only be made in Lm where m)n. Kripke finds a fixed point for his hierarchy (something Russell does not attempt to do). We investigate in this paper the similarities of both hierarchies: At level n of KTT the truth or falsehood of all order-n-propositions of RTT can be established. Moreover, there are order-n-propositions that get a truth value at an earlier stage in KTT. Furthermore, we show that wrr is more restrictive than KTT, as some type restrictions are not needed in KTT and more formula., can be expressed in the latter. Looking back at the double hierarchy of Russell, Ramsey [11] and Hilbert and Ackermann [7] considered the orders to cause the restrictiveness, and therefore removed them. This removal resulted in Churchs Simple Type Theory STT [1] We show however that orders in RTT correspond to levels of truth in KTT. Hence, KTT can be regarded as the dual of STT where types have been removed and orders are maintained. As RTT is more restrictive than KTT, we can conclude that it is the combination of types and orders that was the restrictive factor in RTT.

A Correspondence between Martin-Löf Type Theory, the Ramified Theory of Types and Pure Type Systems

In Russells Ramified Theory of Types RTT, two hierarchical concepts dominate:orders and types. The use of orders has as a consequencethat the logic part of RTT is predicative.The concept of order however, is almost deadsince Ramsey eliminated it from RTT. This is whywe find Churchs simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than RTT.Despite the disappearance of orders which have a strong correlation with predicativity, predicative logic still plays an influential role in Computer Science.An important example is the proof checker Nuprl, which is basedon Martin-Löfs Type Theory which uses type universes. Those type universes,and also degrees of expressions in AUTOMATH, are closely related toorders. In this paper, we show that orders have not disappeared frommodern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl.The novelty of our paper lies in: (1) a modest revival of Russellsorders, (2) the placing of the historical system RTTunderlying the famous Principia Mathematica in a context with a modernsystem of computer mathematics (Nuprl) and modern type theories(Martin-Löfs type theory and PTSs), and (3) the presentation of acomplex type system (Nuprl) as a simple and compact PTS.

A modern elaboration of the ramified theory of types

The paper first formalizes the ramified type theory as (informally) described in the Principia Mathematica [32]. This formalization is close to the ideas of the Principia, but also meets contemporary requirements on formality and accuracy, and therefore is a new supply to the known literature on the Principia (like [25], [19], [6] and [7]).As an alternative, notions from the ramified type theory are expressed in a lambda calculus style. This situates the type system of Russell and Whitehead in a modern setting. Both formalizations are inspired by current developments in research on type theory and typed lambda calculus; see [3].

Revisiting the notion of function

Abstract Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both processes were implemented in Frege’s Begriffschrift , Russell’s Ramified Type Theory , and the λ -calculus (originally introduced by Church) showing that the λ -calculus misses a crucial aspect of functionalisation. We then pay attention to some special forms of function abstraction that do not exist in the λ -calculus and we show that various logical constructs (e.g., let expressions and definitions and the use of parameters in mathematics), can be seen as forms of the missing part of functionalisation. Our study of the function concept leads to: (a) an extension of the Barendregt cube [4] with all of definitions, Π -reduction and explicit substitutions giving all their advantages in one system; and (b) a natural refinement of the cube with parameters. We show that in the refined Barendregt cube, systems like A utomath , LF, and ML, can be described more naturally and accurately than in the original cube.

De Bruijn's Automath and pure type systems

We study the position of the Automath systems within the framework of Pure Type Systems (PTSs). In [Barendregt, 1992; Geuvers, 1993], a rough relationship has been given between Automath and PTSs. That relationship ignores three of the most important features of Automath: definitions, parameters and П-reduction, because at the time, formulations of PTSs did not have these features. Since, PTSs have been extended with these features and in view of this, we revisit the correspondence between Automath and PTSs. This paper gives the most accurate description of Automath as a PTS so far.

Parameters in Pure Type Systems

In this paper we study the addition of parameters to typed ?-calculus with definitions. We show that the resulting systems have nice properties and illustrate that parameters allow for a better fine-tuning of the strength of type systems as well as staying closer to type systems used in practice in theorem provers and programming languages.

A History of Types

In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whiteheads Principia Mathematica ([Whitehead and Russell, 1910], 1910–1912) and Churchs simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Freges Grundgesetze der Arithmetik for which Russell applied his famous paradox1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present RTT formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from RTT leading to the simple theory of types STT. We present STT and Churchs own simply typed λ-calculus (λ→C2) and we finish by comparing RTT, STT and λ→C.

Automath and pure type systems

Abstract We study the position of the Automath systems within the framework of Pure Type Systems (PTSs). In [1,15], a rough relationship has been given between Automath and PTSs. That relationship ignores three of the most important features of Automath : definitions, parameters and π-reduction, because at the time, PTSs did not have these features. Since, PTSs have been extended with these features and in view of this, we revisit the correspondence between Automath and PTSs. This paper gives the most accurate description of Automath as a PTS so far.