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Polymorphic type-checking for the ramified theory of types of Principia Mathematica

2003
*
Electronical Notes in Theoretical Computer Science
*

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Types in Logic and Mathematics before 1940

2002
*
Bulletin of Symbolic Logic
*

1 and this led him to introduce

doi:10.2307/2693964
fatcat:d4jn6bsgovggnaao77ta7cmv3u
*the*first*theory**of**types*,*the**Ramified**Type**Theory*(rtt). ... 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*-calculus*of*1940 ... Russell's*ramified**theory**of**types*in*Principia**Mathematica*applied*the*vicious circle principle, assuming all*the*elements*of**the*set before constructing it. ...##
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Types in Logic and Mathematics Before 1940

2002
*
Bulletin of Symbolic Logic
*

and this led him to introduce

doi:10.2178/bsl/1182353871
fatcat:h7fifhtu6bfapgzfgarqly6nqu
*the*first*theory**of**types*,*the**Ramified**Type**Theory*(RTT). ... 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*λ-calculus*of*1940 ... Russell's*ramified**theory**of**types*in*Principia**Mathematica*applied*the*vicious circle principle, assuming all*the*elements*of**the*set before constructing it. ...##
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A formulation of the simple theory of types (for Isabelle)
[chapter]

1990
*
Lecture Notes in Computer Science
*

Simple

doi:10.1007/3-540-52335-9_58
fatcat:hhyrxzj6rnbjjcn2zrmp35uqaa
*type**theory*is formulated*for*use with*the*generic theorem prover Isabelle. This requires explicit*type*inference rules. There are function, product, and subset*types*, which may be empty. ... Descriptions (*the*η-operator) introduce*the*Axiom*of*Choice. Higher-order logic is obtained through reflection between formulae and terms*of**type*bool . ...*Polymorphism**The*'typical ambiguity' in*Principia*is a form*of**polymorphism*where*type*symbols in expressions are simply not shown. ...##
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Types, Sets, and Categories
[chapter]

2012
*
Handbook of the History of Logic
*

This essay is an attempt to sketch

doi:10.1016/b978-0-444-51621-3.50009-8
fatcat:tbjkroysbfajnfzthut7qbdqba
*the*evolution*of**type**theory*from its beginnings early in*the*last century to*the*present day. ... Central to*the*development*of**the**type*concept has been its close relationship with set*theory*to begin with and later its even more intimate relationship with category*theory*. ... It is this feature*of*Russell's*theory**of**types*which led to its being called*the**ramified**theory**of**types*. ...##
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The history of Standard ML

2020
*
Proceedings of the ACM on Programming Languages (PACMPL)
*

*The*use

*of*parametric

*polymorphism*in its

*type*system, together with

*the*automatic inference

*of*such

*types*, has influenced a wide variety

*of*modern languages (where

*polymorphism*is often referred to as ... Standard ML, and

*the*ML family

*of*languages, have had enormous influence on

*the*world

*of*programming language design and

*theory*. ... There was only an incomplete definition

*of*when two terms łhave

*the*same

*type*. ž 24 This

*ramified*

*theory*

*of*

*types*was both complex and not fully defined in

*Principia*

*Mathematica*. ...

##
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Automation of Higher-Order Logic
[chapter]

2014
*
Handbook of the History of Logic
*

Besides

doi:10.1016/b978-0-444-51624-4.50005-8
fatcat:jfcztdvymjfujg3bzb2rq2qyzy
*the*readers*of*this chapter, we thank Zakaria Chihani, Julian Röder, Leon Weber, and Max Wisnieswki*for*proofreading*the*document. ... We thank Chad Brown*for*sharing notes that he has written related to*the*material in this chapter. ... Simple*type**theory*corresponds to*the**ramified**theory**of**type*plus*the*axiom*of*reducibility. ...##
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Revisiting the notion of function

2003
*
The Journal of Logic and Algebraic Programming
*

We explain how both processes were implemented in Frege's Begriffschrift, Russell's

doi:10.1016/s1567-8326(02)00016-4
fatcat:263bn6dy5fawdi7rzswwrgoagi
*Ramified**Type**Theory*, and*the*λ-calculus (originally introduced by Church) showing that*the*λ-calculus misses a crucial ... Functions play a central role in*type**theory*, logic and computation. ... However, both parts*of**the*functionalisation process are present in*Principia**Mathematica*by Whitehead and Russell [42] . ...##
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Foundations of Mathematics from the Perspective of Computer Verification
[chapter]

2013
*
Mathematics, Computer Science and Logic - A Never Ending Story
*

We argue that most philosophical views over-emphasize a particular aspect

doi:10.1007/978-3-319-00966-7_1
fatcat:37urslisnnhabalb6viu4e4j7q
*of**the*mathematical endeavor. ... In*the*philosophy*of*mathematics one speaks about Formalism, Logicism, Platonism and Intuitionism. Actually one should add also Calculism. ...*The*impact*of**Principia**Mathematica*was not mathematical, but metamathematical. ...##
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Foundations
[article]

2021
*
arXiv
*
pre-print

This is a draft

arXiv:2009.09541v4
fatcat:yatuz32vondo3eburgtj6jmp6e
*of*a chapter on mathematical logic and foundations*for*an upcoming handbook*of*computational proof assistants. ... I am grateful to Guillaume Dubach, Randy Pollack, and Pedro Sánchez Terraf*for*comments and corrections. ... Russell and Whitehead presented a system*of**ramified**type**theory*in their landmark three-volume work,*Principia**Mathematica*,*of*1910*Mathematica*,*of*-1913 In 1940, Church introduced simple*type**theory*...##
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2015 EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC LOGIC COLLOQUIUM '15 Helsinki, Finland August 3–8, 2015

2016
*
Bulletin of Symbolic Logic
*

(joint with CLMPS) Steve Awodey (University

doi:10.1017/bsl.2016.22
fatcat:cm4a5yyvgfajzm4qvlszviblni
*of*Pittsburgh), Cubical homotopy*type**theory*and univalence. ... We shall design model-*checking*games*for*logics with team semantics in a general and systematic way, based on a notion*of*second-order reachability games. ... We prove that*for*some sequence*of*tautologies ϕn*the*proof steps and*the*proof sizes in ...##
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Page 64 of Mathematical Reviews Vol. , Issue Index
[page]

*
Mathematical Reviews
*

03B15
Laan, Twan (with Nederpelt, Rob) A modern elaboration

*of**the**ramified**theory**of**types*. ... ., (971:68005) Kumar, Ramayya sce Eisenbiegler, Dirk, (97k:68007) Landini, Gregory Will*the*real*Principia**mathematica*please stand up? Reflections on*the*formal logic*of**the**Principia*. ...##
###
CHARACTER AND OBJECT

2016
*
The Review of Symbolic Logic
*

Modern presentations

doi:10.1017/s1755020315000398
fatcat:daecga3aynfhvoyhzjnrjzpdxy
*of**the*proof are explicitly*of*higher-order, in that they involve quantifying over and summing overDirichlet characters, which are certain*types**of*functions. ...*The*notion*of*a character is only implicit in Dirichlet's original proof, and*the*subsequent history shows a very gradual transition to*the*modern mode*of*presentation.In this essay, we study*the*history ... Simple*type**theory*can be viewed as a descendant*of**the**ramified**type**the*-ory*of*Russell and Whitehead's*Principia**Mathematica*[61] , which, in turn, was inspired by*the*formal system*of*Frege's Grungesetze ...##
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The concept of "character" in Dirichlet's theorem on primes in an arithmetic progression
[article]

2013
*
arXiv
*
pre-print

We survey implicit and explicit uses

arXiv:1209.3657v3
fatcat:bqc7azwfhrb67gwuyejqzfdj7q
*of*Dirichlet characters in presentations*of*Dirichlet's proof in*the*nineteenth and early twentieth centuries, with an eye towards understanding some*of**the*pragmatic ... pressures that shaped*the*evolution*of*modern mathematical method. ... Simple*type**theory*can be viewed as a descendent*of**the**ramified**type**theory**of*Russell and Whitehead's*Principia**mathematica*[92] , which, in turn, was inspired by*the*formal system*of*Frege's Grungesetze ...##
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Refinement types for Elm

2021

Theorem 1.1: Russell's Paradox To Ąx this problem, Russell added a basic

doi:10.35011/risc.21-10
fatcat:xewgamvpwvhfhidfustehd7644
*theory**of**types*in*the*appendix*of**Principia**Mathematica*[WR27] , which at*the*time was already in*the*printer. ... As an example, we consider*the*simple*type**theory*extended by*polymorphic**types*, as used in*the*Hindley-Milner*type*system. 1. Let T be a*type*. Then ∀a.T is a*polymorphic**type*. ... It is a research language and a system*of*tools*for*designing and formalizing programming languages. These include tools*for*parsing, execution,*type**checking*and program veriĄcation [Ste+16] . ...
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