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

M.Randall Holmes
2003 Electronical Notes in Theoretical Computer Science  
doi:10.1016/s1571-0661(04)80761-8 fatcat:hffzfcwv7vd6xlkb5vpukq7yni

Types in Logic and Mathematics before 1940

Fairouz Kamareddine, Twan Laan, Rob Nederpelt
2002 Bulletin of Symbolic Logic  
1 and this led him to introduce 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.  ... 
doi:10.2307/2693964 fatcat:d4jn6bsgovggnaao77ta7cmv3u

Types in Logic and Mathematics Before 1940

Fairouz Kamareddine, Twan Laan, Rob Nederpelt
2002 Bulletin of Symbolic Logic  
and this led him to introduce 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.  ... 
doi:10.2178/bsl/1182353871 fatcat:h7fifhtu6bfapgzfgarqly6nqu

A formulation of the simple theory of types (for Isabelle) [chapter]

Lawrence C. Paulson
1990 Lecture Notes in Computer Science  
Simple 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.  ... 
doi:10.1007/3-540-52335-9_58 fatcat:hhyrxzj6rnbjjcn2zrmp35uqaa

Types, Sets, and Categories [chapter]

John L. Bell
2012 Handbook of the History of Logic  
This essay is an attempt to sketch 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.  ... 
doi:10.1016/b978-0-444-51621-3.50009-8 fatcat:tbjkroysbfajnfzthut7qbdqba

The history of Standard ML

David MacQueen, Robert Harper, John Reppy
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.  ... 
doi:10.1145/3386336 fatcat:2hrtsaf5azfe3htngsqvxv3kre

Automation of Higher-Order Logic [chapter]

Christoph Benzmüller, Dale Miller
2014 Handbook of the History of Logic  
Besides 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.  ... 
doi:10.1016/b978-0-444-51624-4.50005-8 fatcat:jfcztdvymjfujg3bzb2rq2qyzy

Revisiting the notion of function

Fairouz Kamareddine, Twan Laan, Rob Nederpelt
2003 The Journal of Logic and Algebraic Programming  
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  ...  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] .  ... 
doi:10.1016/s1567-8326(02)00016-4 fatcat:263bn6dy5fawdi7rzswwrgoagi

Foundations of Mathematics from the Perspective of Computer Verification [chapter]

Henk Barendregt
2013 Mathematics, Computer Science and Logic - A Never Ending Story  
We argue that most philosophical views over-emphasize a particular aspect 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.  ... 
doi:10.1007/978-3-319-00966-7_1 fatcat:37urslisnnhabalb6viu4e4j7q

Foundations [article]

Jeremy Avigad
2021 arXiv   pre-print
This is a draft 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  ... 
arXiv:2009.09541v4 fatcat:yatuz32vondo3eburgtj6jmp6e

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 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  ... 
doi:10.1017/bsl.2016.22 fatcat:cm4a5yyvgfajzm4qvlszviblni

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

JEREMY AVIGAD, REBECCA MORRIS
2016 The Review of Symbolic Logic  
Modern presentations 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  ... 
doi:10.1017/s1755020315000398 fatcat:daecga3aynfhvoyhzjnrjzpdxy

The concept of "character" in Dirichlet's theorem on primes in an arithmetic progression [article]

Jeremy Avigad, Rebecca Morris
2013 arXiv   pre-print
We survey implicit and explicit uses 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  ... 
arXiv:1209.3657v3 fatcat:bqc7azwfhrb67gwuyejqzfdj7q

Refinement types for Elm

Lucas Payr
2021
Theorem 1.1: Russell's Paradox To Ąx this problem, Russell added a basic 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] .  ... 
doi:10.35011/risc.21-10 fatcat:xewgamvpwvhfhidfustehd7644
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