Paper Info
Title
Categories with families and first-order logic with dependent sorts.
Abstract
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.
Year
DOI
Venue
2019
10.1016/j.apal.2019.102715
Annals of Pure and Applied Logic
Keywords
DocType
Volume
03B15,03F50,03G30,18C10,68Q55
Journal
170
Issue
ISSN
Citations 
12
0168-0072
0
PageRank 
References 
Authors
0.34
0
1
Name
Order
Citations
PageRank
Erik Palmgren123343.17