https://hal-ens.archives-ouvertes.fr/hal-03316282Longo, GiuseppeGiuseppeLongoLIENS - Laboratoire d'informatique de l'école normale supérieure - DI-ENS - Département d'informatique - ENS Paris - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris sciences et lettres - Inria - Institut National de Recherche en Informatique et en Automatique - CNRS - Centre National de la Recherche Scientifique - CNRS - Centre National de la Recherche ScientifiqueMoggi, EugenioEugenioMoggiLFCS - Laboratory for the Foundations of Computer Science [Edinburgh] - University of EdinburghConstructive natural deduction and its "omega-set" interpretationHAL CCSD1991[INFO.INFO-LO] Computer Science [cs]/Logic in Computer Science [cs.LO]République des savoirs, UAR 36082021-08-12 10:55:182023-02-07 14:45:152021-08-12 11:19:43enJournal articleshttps://hal-ens.archives-ouvertes.fr/hal-03316282/document10.1017/S0960129500001298application/pdf1Various 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).