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Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
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Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus 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 Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 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 Church’s own simply typed λcalculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced
Refining the Barendregt Cube using Parameters
, 2001
"... The Barendregt Cube (introduced in [3]) is a framework in which eight important typed 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 propositionsastype ..."
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The Barendregt Cube (introduced in [3]) is a framework in which eight important typed 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 propositionsastypes 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 re nement of the Cube. In this re ned Barendregt Cube, systems like Automath, LF, and ML can be described more naturally and accurately than in the original Cube.
Revisiting the Notion of Function
"... 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 pro ..."
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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 [17], Russell's Ramified Type Theory [42] and the lambdacalculus (originally introduced by Church [12, 13]) showing that the lambdacalculus misses a crucial aspect of functionalisation. We then pay attention to some special forms of function abstraction that do not exist in the lambdacalculus 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...
Reviewing the classical and the de Bruijn notation for λcalculus and pure type systems
 Logic and Computation
, 2001
"... This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentat ..."
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This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λcalculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λcalculus is introduced and some of its advantages are outlined.
A Correspondence between MartinLöf Type Theory, the Ramified Theory of Types and Pure Type Systems
 Journal of Logic, Language and Information
, 2001
"... In Russell's Ramified Theory of Types rtt, two hierarchical concepts dominate: orders and types. The use of orders has as a consequence that the logic part of rtt is predicative. The concept of order however, is almost dead since Ramsey eliminated it from rtt. This is why we find Church's ..."
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In Russell's Ramified Theory of Types rtt, two hierarchical concepts dominate: orders and types. The use of orders has as a consequence that the logic part of rtt is predicative. The concept of order however, is almost dead since Ramsey eliminated it from rtt. This is why we find Church's 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 based on MartinLöf's Type Theory which uses type universes. Those type universes, and also degrees of expressions in Automath, are closely related to orders. In this paper, we show that orders have not disappeared from modern 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 Russell's orders, 1 2) the placing of the historical system rtt underlying the famous Principia Mathematica in a context with a modern system of computer mathematics (Nuprl) and modern type theories (MartinLöf's type theory and PTSs), and 3) the presentation of a complex type system (Nuprl) as a simple and compact PTS.
An Abstract Syntax for a Formal Language of Mathematics
, 2001
"... This paper provides an abstract syntax for a formal language of mathematics. We call our language Weak Type Theory (abbreviated WTT ). WTT will be as faithful as possible to the mathematician 's language yet will be formal and will not allow ambiguities. WTT can be used as an intermediary be ..."
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This paper provides an abstract syntax for a formal language of mathematics. We call our language Weak Type Theory (abbreviated WTT ). WTT will be as faithful as possible to the mathematician 's language yet will be formal and will not allow ambiguities. WTT can be used as an intermediary between the natural language of the mathematician and the formal language of the logician. As far as we know, this is the rst extensive formalization of an abstract syntax of a formal language of mathematics. We compare our work with existing formalizations of languages of mathematics. 1
The evolution of types and logic in the 20th century: A journey through Frege, Russell and . . .
 ILLC ALUMNI EVENT, AMSTERDAM 2004
, 2004
"... ..."
Supervisors:
, 2009
"... A proof assistant is a computer program that is used for proving theorems in an interactive way. Many proof assistants are based on the theory of pure type systems and the propositions as types principle. During this master project a proof assistant has been developed that has such a theoretic found ..."
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A proof assistant is a computer program that is used for proving theorems in an interactive way. Many proof assistants are based on the theory of pure type systems and the propositions as types principle. During this master project a proof assistant has been developed that has such a theoretic foundation, but instead of using the ordinary pure type system framework, it uses a pure type system framework that is extended with additional type constructors. Initially, it was supposed to be implemented by using the FoolProof components to test their usability. One of FoolProof main features is its ability to manipulate terms with binding structures. When it turned out those components were not available in time, the decision was made to use the infrastructure of Cocktail instead, which manipulates terms with binding structures in the same way as FoolProof. Acknowledgements I am grateful to my supervisor Kees Hemerik for letting me start immediately on my master project when I asked him to be my supervisor. It did not take him long to come up with a research subject that I liked to investigate. When the progress of the project stalled due to
Parameters in Pure Type Systems
"... Abstract. 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 finetuning of the strength of type systems as well as staying closer to type systems used in pra ..."
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Abstract. 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 finetuning of the strength of type systems as well as staying closer to type systems used in practice in theorem provers and programming languages. 1 What are parameters? Parameters occur when functions are only allowed to occur when provided with arguments. As we will show below, both in mathematics and in programming languages the use of parameters is abundant and closely connected to the use of constants and definitions. If we want to be able to use type systems in accordance with practice and yet described in a precise manner, we therefore need parameters, constants, and definitions in type theory as well. Parameters, constants and and���� � ��������� definitions in theorem proving It is interesting to note that
ALLIGATOR: THEOREM PROVING FOR DEPENDENT TYPE SYSTEMS WITH SIGMA TYPES
, 2006
"... Abstract This paper describes a theorem prover for Dependent Type Systems. We start with an introduction to Dependent Type Systems and highlight the properties that make them specifically suited for computational semantics. We proceed with a brief description of the ALLIGATOR system, including its a ..."
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Abstract This paper describes a theorem prover for Dependent Type Systems. We start with an introduction to Dependent Type Systems and highlight the properties that make them specifically suited for computational semantics. We proceed with a brief description of the ALLIGATOR system, including its architecture and some implementation issues. Alligator works with a specific generalization of Dependent Type Systems: Pure Type Systems extended with Sigma Types. The paper concludes with examples of proofs constructed by ALLIGATOR. These have been selected to illustrate how certain problems in anaphora/presupposition resolution can be addressed with the current system. 1