## A New Paradox in Type Theory (1994)

Venue: | Logic, Methodology and Philosophy of Science IX : Proceedings of the Ninth International Congress of Logic, Methodology, and Philosophy of Science |

Citations: | 7 - 0 self |

### BibTeX

@INPROCEEDINGS{Coquand94anew,

author = {Thierry Coquand},

title = {A New Paradox in Type Theory},

booktitle = {Logic, Methodology and Philosophy of Science IX : Proceedings of the Ninth International Congress of Logic, Methodology, and Philosophy of Science},

year = {1994},

pages = {7--14},

publisher = {Elsevier}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper is to present a new paradox for Type Theory, which is a type-theoretic refinement of Reynolds' result [24] that there is no set-theoretic model of polymorphism. We discuss then one application of this paradox, which shows unexpected connections between the principle of excluded middle and the axiom of description in impredicative Type Theories. 1 Minimal and Polymorphic Higher-Order Logic

### Citations

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Citation Context ...redicative Type Theories. 1 Minimal and Polymorphic Higher-Order Logic 1.1 Minimal Higher-Order Logic 1.1.1 A presentation of the system We assume known simply typed lambda calculus (see for instance =-=[3]-=-.) The lambda-terms will always be considered up to fi-conversion. The types of minimal higher-order logic consist of one basic type o and function types of the form ff ! fi: The terms of minimal high... |

425 |
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Citation Context ...the value F, and hence cannot get a proof. Such a semantics is not faithful to the intuitionistic character of minimal higher-order logic. Topos theory provides various intuitionistic interpretations =-=[13]-=-, which fail however to reflect the definitional equality on propositions. 1.2 Polymorphic Higher-Order Logic 1.2.1 Second-Order Lambda Calculus Second-order lambda-calculus has been introduced indepe... |

238 |
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Citation Context ...cription operator (another version contains an extensionality axiom and the axiom of choice). It is possible to interpret classical higher-order propositional logic in minimal higher-order logic (see =-=[11]-=-). 1.1.4 Semantics Minimal higher-order logic has a direct set-theoretic semantics. Each type denotes a finite set: the type o a set with two elements f T,F g, and the function type operator is interp... |

229 |
Une extension de l’interprétation de Gödel à l’analyse et son application à l’élimination de coupures dans l’analyse et la théorie des types
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Citation Context ...er to reflect the definitional equality on propositions. 1.2 Polymorphic Higher-Order Logic 1.2.1 Second-Order Lambda Calculus Second-order lambda-calculus has been introduced independently by Girard =-=[10]-=- and Reynolds [22]. One motivation is to provide a syntax for polymorphic (or generic, or uniform) procedure. Typically, the identity operation is of type ff ! ff; where ff is arbitrary, and such an o... |

157 | An intuitionistic theory of types: Predicative part - Martin-Löf - 1975 |

141 |
Intuitionistic Type Theory. Bibliopolis
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Citation Context ...ed from A-translation in polymorphic higher-order logic. In [8], it is shown that it is possible to build such a fixed-point operator in the presence of a the well-founded type operator of Martin-Lof =-=[15]-=-. 4.2 Strong existence The results about excluded middle in impredicative theory were first expressed as consequence of the inconsistency of the system U of Girard, which extends polymorphic higher-or... |

135 |
The Principles of Mathematics
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Citation Context ... does not appear free in any proposition of \Gamma: 1.1.2 Definition of other logical connectives It is possible to define other logical connectives. This fact was in essence already known to Russell =-=[25]-=-, at least for negation and conjunction. ? = 8OE o :OE : o : = OE o :OE )? : o!o T = 8OE o :OE ) OE : os= OE o ; / o :8ffi o :(OE )/) ffi )) ffi : o!o!os= OE o ; / o :8ffi o :(OE ) ffi) )(/) ffi) ) ff... |

72 |
Constructions: A Higher Order Proof System for Mechanizing Mathematics
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Citation Context ...was possible to avoid the quantification over type variables used in [11, 5]. 3 Application to Impredicative Type Theory 3.1 Impredicative Type Theory Impredicative type theory has been introduced in =-=[4] and -=-is analysed in [6]. We will not present in detail this type theory, but limit ourselves to a short description. Impredicative type theory is a direct expression of the principle of "propositions-... |

28 |
Type Dependence and Constructive Mathematics
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Citation Context ...retic argument that the axiom of description, and hence the axiom of choice, is not provable in impredicative Type Theory (personal communication.) A "syntactic" version of this model is des=-=cribed in [1]-=-. It is similar to the proof irrelevance model, but the inhabited sets are interpreted instead by the set of all untyped lambda terms. This also models the principle of excluded middle, but not the pr... |

24 |
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Citation Context ... some comments in order to motivate the argument of the next section. Since there is no set of all sets, there is a problem in interpreting set-theoretically second-order lambda-calculus. However, in =-=[23]-=-, Reynolds conjectured that there is a non trivial set-theoretic model where the function operator is interpreted as set-theoretic exponentiation. The idea was that, in interpreting a product of a fam... |

16 | On functors expressible in the polymorphic typed lambda calculus
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- 1990
(Show Context)
Citation Context ... is provable. This proposition expresses the fourth Peano axiom for arithmetic. 2 A Type-Theoretic Refinement of Reynolds' Theorem 2.1 An heuristic presentation of Reynolds' Theorem Reynolds' theorem =-=[24, 21] states th-=-at there is no set-theoretic model of second-order lambdacalculus. We do not need here to detail the notion of "set-theoretic model" required in order to make this statement precise. But we ... |

14 |
The computational behaviour of Girard’s paradox
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(Show Context)
Citation Context ...consistency of polymorphic higher-order logic, or even of the system U of [11], entails, by direct translation, the existence of a non normalisable term in a type system with a type of all types (see =-=[16, 5, 12]-=-). The existence of a fixed-point combinator in such a type system is an open problem since [16]. The article [12] contains a proof, using computers in an essential way, that shows the existence of a ... |

11 |
M.B.Reinhold: 'Type' is not a type
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Citation Context ...consistency of polymorphic higher-order logic, or even of the system U of [11], entails, by direct translation, the existence of a non normalisable term in a type system with a type of all types (see =-=[16, 5, 12]-=-). The existence of a fixed-point combinator in such a type system is an open problem since [16]. The article [12] contains a proof, using computers in an essential way, that shows the existence of a ... |

6 | Non-trivial power types can’t be subtypes of polymorphic types
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Citation Context ...ternatively, ? is provable. This follows directly from the lemmas, and the usual intuitionistic proof of Cantor's theorem, that there cannot be onto maps from a set to its power set (see for instance =-=[19]-=-). This argument has been checked and found using a computer, and the formal proof is presented in [7]. 2.3 Connection with Girard's paradox In [11], Girard considers essentially the extension of poly... |

6 |
Definite descriptions and excluded middle in the theory of constructions, TYPES mailing list
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Citation Context ...y S. Berardi, and checked in the proof checker LEGO of R. Pollack. The present result, which concerns the principle of definite descriptions, generalises and was motivated by a result of G. Pottinger =-=[20]-=-. 4.3 Consistency and Independence Results S. Berardi has shown by a model theoretic argument that the axiom of description, and hence the axiom of choice, is not provable in impredicative Type Theory... |

5 |
On a Nonconstructive Type Theory and Program Derivation. To appear
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Citation Context ...ed Middle The last principle we shall consider is the principle of excluded middle. (\PiA : Set)As:(A): The extension of Martin-Lof's set theory with this principle has been considered by J. Smith in =-=[29]-=-. It is direct to check that this principle is equivalent to (\PiA : Set):(:(A)))A: 3.3 An application We can now state the application of the inconsistency of polymorphic higher-order logic. Lemma: T... |

4 |
Inductive definitions in the calculus of constructions
- Paulin-Mohring
- 1989
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Citation Context ... seems to be closely connected to the well-known "mismatch" in the representation of destructors for recursively defined types represented in secondorder lambda-calculus (as presented for in=-=stance in [17]-=-). But the author has not been able to make this connection precise. The existence of a family of looping combinators entails the undecidability of type checking for a type system with a type of all t... |

3 |
Excluded middle without definite descriptions in the theory of constructions
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Citation Context ...ped lambda terms. This also models the principle of excluded middle, but not the principle of proof irrelevance. It shows that the principle of proof irrelevance is independent of excluded middle. In =-=[28]-=-, a purely proof theoretic argument shows the consistency of a context implying classical arithmetic, where the set of integers is interpreted as a small type. 4.4 Related results in Category Theory T... |

2 |
An Analysis of Girard's Paradox
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Citation Context ... quantification over set variables. The idea of replacing this quantification by an "induction principle" appears also, independently, in the framework of topos theory in a paper of A. Pitts=-= [19]. In [5], a slight-=- simplification of Girard's argument is presented. We have not been able however to formulate a "Burali-Forti" like paradox in polymorphic higher-order logic, that is, we have not seen if it... |

2 |
Programming in Martin-Lof Type Theory
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Citation Context ...hows that it is impossible to think of the present sets as sets in the sense of Zermelo-Skolem-Fraenkel. This basic feature is the main difference with Martin-Lof's logical framework, as presented in =-=[18]-=-. Otherwise, these systems are quite similar. In particular, a fundamental role is played by the notion of context, which is a finite set of typed variables declaration. This notion is also a basic no... |

2 |
Towards a Theory of Type Structure." Programming Symposium
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Citation Context ...definitional equality on propositions. 1.2 Polymorphic Higher-Order Logic 1.2.1 Second-Order Lambda Calculus Second-order lambda-calculus has been introduced independently by Girard [10] and Reynolds =-=[22]. One-=- motivation is to provide a syntax for polymorphic (or generic, or uniform) procedure. Typically, the identity operation is of type ff ! ff; where ff is arbitrary, and such an operation behaves "... |

2 |
Polymorphism is not set-theoretic," in Semantics of Data Types
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Citation Context ... of Goteborg/Chalmers S-412 96 Goteborg, Sweden April 13, 1994 Introduction The aim of this paper is to present a new paradox for Type Theory, which is a type-theoretic refinement of Reynolds' result =-=[24]-=- that there is no set-theoretic model of polymorphism. We discuss then one application of this paradox, which shows unexpected connections between the principle of excluded middle and the axiom of des... |

1 |
A survey of the project Automath." In: To H.B. Curry: Essays in combinatory logic, lambda calculus and formalism
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Citation Context ...In particular, a fundamental role is played by the notion of context, which is a finite set of typed variables declaration. This notion is also a basic notion of Automath, and we refer to the article =-=[2]-=- for an intuitive description of contexts. If A and B are types, we let A ! B be the product of the constant type family B over the type A. In the case where A and B are small types or sets, we write ... |

1 |
Metamathematical Investigations of a Calculus of Constructions
- Th
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(Show Context)
Citation Context ...he quantification over type variables used in [11, 5]. 3 Application to Impredicative Type Theory 3.1 Impredicative Type Theory Impredicative type theory has been introduced in [4] and is analysed in =-=[6]. We will not p-=-resent in detail this type theory, but limit ourselves to a short description. Impredicative type theory is a direct expression of the principle of "propositions-as-types" and "proof-as... |

1 |
Reynolds' paradox, with the Type:Type axiom
- Th
- 1989
(Show Context)
Citation Context ...f Cantor's theorem, that there cannot be onto maps from a set to its power set (see for instance [19]). This argument has been checked and found using a computer, and the formal proof is presented in =-=[7]. 2.3 Conn-=-ection with Girard's paradox In [11], Girard considers essentially the extension of polymorphic higher-order logic with quantification over type variables (called "system U ") and proves tha... |

1 |
The paradox of trees in Type Theory
- Th
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(Show Context)
Citation Context ...ndecidability of type checking for a type system with a type of all types. In [9], the existence of a family of looping combinators is derived from A-translation in polymorphic higher-order logic. In =-=[8]-=-, it is shown that it is possible to build such a fixed-point operator in the presence of a the well-founded type operator of Martin-Lof [15]. 4.2 Strong existence The results about excluded middle in... |

1 |
An application of A-translation to the existence of families of looping combinators." Submitted to the Journal of Functional Programing
- Herbelin
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(Show Context)
Citation Context ...e author has not been able to make this connection precise. The existence of a family of looping combinators entails the undecidability of type checking for a type system with a type of all types. In =-=[9]-=-, the existence of a family of looping combinators is derived from A-translation in polymorphic higher-order logic. In [8], it is shown that it is possible to build such a fixed-point operator in the ... |

1 |
On Denoting." Mind
- Russell
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(Show Context)
Citation Context ... : B)R(x; y)] ) [(9f : A!B)(\Pix : A)R(x; f(x))] This principle appears in the system of Church [3], in the form of a description operator ': The motivation comes from Russell's work on denoting (see =-=[27, 26]-=-). 3.2.3 Excluded Middle The last principle we shall consider is the principle of excluded middle. (\PiA : Set)As:(A): The extension of Martin-Lof's set theory with this principle has been considered ... |