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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
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.
A Reflection on Russell's Ramified Types and Kripke's Hierarchy of Truths
 Journal of the Interest Group in Pure and Applied Logic 4(2
, 1996
"... Both in Kripke's Theory of Truth ktt [8] and Russell's 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 ..."
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Both in Kripke's Theory of Truth ktt [8] and Russell's 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 ordernpropositions of rtt can be established. Moreover, there are ordernpropositions that get a truth value at an earlier stage in ktt. Furthermore, we show that rtt is more restrictive than ktt, as some type restrictions are not needed in ktt and more formulas can be expressed in the latter. Looking back at the dou...
Journal of the IGPL
"... In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, ..."
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In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logicaloperationsasadjunctions. It offers an answer to the notorious question of the equality of proofs. So we chase diagrams in algebra of proofs. On the basis of these ideas, the present paper investigates proof theory of regular logic: the f; 9gfragment of the first order logic with equality. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes. Regular logic is exactly what is needed to talk about maps, as total and singlevalued relations. However, when enriched with proofsasarrows, this familiar concept must be supplied with an additional conversion rule, conne...
The evolution of types and logic in the 20th century: A journey through Frege, Russell and . . .
 ILLC ALUMNI EVENT, AMSTERDAM 2004
, 2004
"... ..."
From the Foundation of Mathematics to the Birth of Computation
, 2011
"... deduction/Logic was taken as a foundation for Mathematics, computation was also taken throughout as an essential tool in mathematics. • Our ancestors used sandy beaches to compute the circomference of a circle, and to work out approximations/values of numbers like π. • The word algorithm dates back ..."
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deduction/Logic was taken as a foundation for Mathematics, computation was also taken throughout as an essential tool in mathematics. • Our ancestors used sandy beaches to compute the circomference of a circle, and to work out approximations/values of numbers like π. • The word algorithm dates back centuries? Algorithms existed in anciant Egypt at the time of Hypatia. The word is named after AlKhawarizmi. • But even more impressively, the following important 20th century (un)computability result was known to Aristotle. • Assume a problem Π, – If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. HAPOC11: History and Philosophy of Computing 1 • But, this result was already known to Aristotle: • Assume a proposition Φ. – If you give me a proof of Φ, I can check whether this proof really proves Φ. – But, if you ask me to find a proof of Φ, I may go on forever trying but without success. • In fact, programs are proofs: – program = algorithm = computable function = λterm. – By the PAT principle: Proofs are λterms.
A Re ection on Russell's Ramied Types and Kripke's Hierarchy of Truths
"... Both in Kripke's Theory of Truth ktt [8] and Russell's Ramied 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 dierent orders where a propositional function can only dep ..."
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Both in Kripke's Theory of Truth ktt [8] and Russell's Ramied 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 dierent 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 L n can only be made in L m where min. Kripke nds a xed 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 ordernpropositions of rtt can be established. Moreover, there are ordernpropositions that get a truth value at an earlier stage in ktt. Furthermore, we show that rtt is more restrictive than ktt, as some type restrictions are not needed in ktt and more formulas 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 Church's 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 Nuprl and the Ramified Theory of Types
, 1996
"... 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 disappearence 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 MartinLof's Type Theory which uses type universes. Those type universes, and also degrees of expressions in Authomath, are closely related to orders. In this paper, we concentrate on Nuprl. We describe Nuprl as a Pure Type System and relate it to Russell's rtt. To be successful at so doing, we are forced to explicitly typing Nuprl (but this is not res...