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A New Paradox in Type Theory
 Logic, Methodology and Philosophy of Science IX : Proceedings of the Ninth International Congress of Logic, Methodology, and Philosophy of Science
, 1994
"... this paper is to present a new paradox for Type Theory, which is a typetheoretic refinement of Reynolds' result [24] that there is no settheoretic model of polymorphism. We discuss then one application of this paradox, which shows unexpected connections between the principle of excluded middle and ..."
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this paper is to present a new paradox for Type Theory, which is a typetheoretic refinement of Reynolds' result [24] that there is no settheoretic 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 HigherOrder Logic
The Calculus of Constructions and Higher Order Logic
 In preparation
, 1992
"... The Calculus of Constructions (CC) ([Coquand 1985]) is a typed lambda calculus for higher order intuitionistic logic: proofs of the higher order logic are interpreted as lambda terms and formulas as types. It is also the union of Girard's system F! ([Girard 1972]), a higher order typed lambda calcul ..."
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The Calculus of Constructions (CC) ([Coquand 1985]) is a typed lambda calculus for higher order intuitionistic logic: proofs of the higher order logic are interpreted as lambda terms and formulas as types. It is also the union of Girard's system F! ([Girard 1972]), a higher order typed lambda calculus, and a first order dependent typed lambda calculus in the style of de Bruijn's Automath ([de Bruijn 1980]) or MartinLof's intuitionistic theory of types ([MartinLof 1984]). Using the impredicative coding of data types in F! , the Calculus of Constructions thus becomes a higher order language for the typing of functional programs. We shall introduce and try to explain CC by exploiting especially the first point of view, by introducing a typed lambda calculus that faithfully represent higher order predicate logic (so for this system the CurryHoward `formulasastypes isomorphism' is really an isomorphism.) Then we discuss some propositions that are provable in CC but not in the higher or...
Formalising mathematics in UTT: fundamentals and case studies
, 1994
"... We give a detailed account of the use of type theory as a foundational language to formalise mathematics. We develop in the type system UTT a coherent approach to naive set theory and elementary mathematical notions. In the second part of the paper, we present a fullychecked example based on our re ..."
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We give a detailed account of the use of type theory as a foundational language to formalise mathematics. We develop in the type system UTT a coherent approach to naive set theory and elementary mathematical notions. In the second part of the paper, we present a fullychecked example based on our representation of naive set theory. Contents 1 Introduction 1 2 Fundamentals 3 2.1 Naive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.2 Discrete sets . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.3 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.4 The category of sets . . . . . . . . . . . . . . . . . . . . . 5 2.1.5 Multivariate maps . . . . . . . . . . . . . . . . . . . . . . 6 2.1.6 Predicates and relations . . . . . . . . . . . . . . . . . . . 7 2.1.7 Subsets and powerset . . . . . . . . . . . . . . . . . . . . 7 2.1.8 Quotients . . . . . . . . . . . . . . . ...
Excluded Middle without Definite Descriptions in the Theory of Constructions
 MWPLT91 Spring, Proceedings of The First Montreal Workshop on Proglramming Language Theory, M. Okada and P.J. Scot eds
, 1990
"... this document was sponsored in part by the U.S. Air Force Systems Command, Rome Air Development Center, Gri#ss AFB, New York 134415700 under Contract No. F3060285C0098, in part by grant EQ41840 from the program Fonds pour la Formation de Chercheurs et l'aide a la Recherche (F.C.A.R.) of the Queb ..."
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this document was sponsored in part by the U.S. Air Force Systems Command, Rome Air Development Center, Gri#ss AFB, New York 134415700 under Contract No. F3060285C0098, in part by grant EQ41840 from the program Fonds pour la Formation de Chercheurs et l'aide a la Recherche (F.C.A.R.) of the Quebec Ministry of Education, and in part by grant OGP0023391 from the Natural Sciences and Engineering Research Council of Canada. 1 classical arithmetic is consistent. Since one of the Peano axioms asserts that zero is not a successor, proof degeneracy would imply the inconsistency of this environment. Coquand [1] also proved that excluded middle and strong sums imply proof degeneracy. Garrel Pottinger has pointed out to me (private communication) that the strong sums, when interpreted under the CurryHoward isomorphism, have the disjunction property. Since this property is known to be characteristic of constructive logic and incompatible with classical logic, this result of Coquand is really a confirmation of what we should expect of classical logic. The result of Pottinger [3], on the other hand, is unwelcome, since both excluded middle and definite descriptions are desirable in some circumstances. The result proved here shows that we are more likely to have to give up definite descriptions than excluded middle. An ASCII version of this document was circulated on the TYPES mailing list in September 1990. A L a T E X version was prepared as the result of a request from some readers of TYPES in November 1990. I intend to incorporate this material in [6]. I would like to thank Garrel Pottinger for his helpful comments and suggestions. 2 TOC