Categories with families and first-order logic with dependent sorts

TL;DR

This paper develops a categorical semantics for first-order logic with dependent sorts, integrating it into categories with families, and proves soundness and completeness results for the logic.

Contribution

It introduces a hyperdoctrine over cwfs framework for dependent first-order logics and demonstrates how FOLDS and DFOL fit into this structure.

Findings

Proves soundness and completeness for DFOL

Establishes functorial semantics using a dependent Lindenbaum-Tarski algebra

Shows agreement with standard first-order semantics

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-L\"of 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.