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Constructive natural deduction and its ‘ω-set’ interpretation†

Published online by Cambridge University Press:  04 March 2009

Giuseppe Longo  and
Eugenio Moggi
Giuseppe Longo
Affiliation:
Laboratoire d'Informatique, CNRS Ecole Normale Superieure, Paris, France
Eugenio Moggi
Affiliation:
LFCS, Computer Science Dept., University of Edinburgh, Edinburgh, UK
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Extract

Various Theories of Types are introduced, by stressing the analogy ‘propositions-as-types’: from propositional to higher order types (and Logic). In accordance with this, proofs are described as terms of various calculi, in particular of polymorphic (second order) λ-calculus. A semantic explanation is then given by interpreting individual types and the collection of all types in two simple categories built out of the natural numbers (the modest sets and the universe of ω-sets). The first part of this paper (syntax) may be viewed as a short tutorial with a constructive understanding of the deduction theorem and some work on the expressive power of first and second order quantification. Also in the second part (semantics, §§6–7) the presentation is meant to be elementary, even though we introduce some new facts on types as quotient sets in order to interpret ‘explicit polymorphism’. (The experienced reader in Type Theory may directly go, at first reading, to §§6–8).

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Research Article
Information
Mathematical Structures in Computer Science , Volume 1 , Issue 2 , July 1991 , pp. 215 - 254
Copyright
Copyright © Cambridge University Press 1991

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