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Categories with families and first-order logic with dependent sorts
Stockholm University, Faculty of Science, Department of Mathematics. (Matematisk logik)
2019 (English)In: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 170, no 12, article id 102715Article in journal (Refereed) Published
Abstract [en]

First-order logic with dependent sorts, such as Makkai's first-order logic with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL), may be regarded as logic enriched dependent type theories. Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-Löf type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and DFOL fit in this semantical framework. A soundness and completeness theorem is proved for DFOL. The semantics is functorial in the sense of Lawvere, and uses a dependent version of the Lindenbaum-Tarski algebra for a DFOL theory. Agreement with standard first-order semantics is established. Applications of DFOL to constructive mathematics and categorical foundations are given. A key feature is a local propositions-as-types principle.

Place, publisher, year, edition, pages
Elsevier, 2019. Vol. 170, no 12, article id 102715
Keywords [en]
Intuitionistic first-order logic, Dependent types, Categorical logic, Models of type theory
National Category
Natural Sciences
Research subject
Mathematical Logic
Identifiers
URN: urn:nbn:se:su:diva-174770DOI: 10.1016/j.apal.2019.102715OAI: oai:DiVA.org:su-174770DiVA, id: diva2:1359661
Funder
Swedish Research Council, 2015-03835Available from: 2019-10-09 Created: 2019-10-09 Last updated: 2019-10-10Bibliographically approved

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CiteExportLink to record
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  • apa
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