{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/first-order-logic-with-isomorphism","title":"First-Order Logic with Isomorphism","arxiv_id":"1603.03092","date":"2016-03-09","proceeding":null,"authors":["Dimitris Tsementzis"],"abstract":"The Univalent Foundations requires a logic that allows us to define structures on homotopy types, similar to how first-order logic with equality ($\\text{FOL}_=$) allows us to define structures on sets. We develop the syntax, semantics and deductive system for such a logic, which we call first-order logic with isomorphism ($\\text{FOL}_{\\cong}$). The syntax of $\\text{FOL}_{\\cong}$ extends $\\text{FOL}_{=}$ in two ways. First, by incorporating into its signatures a notion of dependent sorts along the lines of Makkai's FOLDS as well as a notion of an $h$-level of each sort. Second, by specifying three new logical sorts within these signatures: isomorphism sorts, reflexivity predicates and transport structure. The semantics for $\\text{FOL}_{\\cong}$ are then defined in homotopy type theory with the isomorphism sorts interpreted as identity types, reflexivity predicates as relations picking out the trivial path, and transport structure as transport along a path. We then define a deductive system $\\mathcal{D}_{\\cong}$ for $\\text{FOL}_{\\cong}$ that encodes the sense in which the inhabitants of isomorphism sorts really do behave like isomorphisms and prove soundness of the rules of $\\mathcal{D}_{\\cong}$ with respect to its homotopy semantics. Finally, as an application, we prove that precategories, strict categories and univalent categories are axiomatizable in $\\text{FOL}_{\\cong}$.","url_abs":"http://arxiv.org/abs/1603.03092v2","url_pdf":"http://arxiv.org/pdf/1603.03092v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"first-order-logic-with-isomorphism","repo_url":"https://github.com/dimtsem/FOLiso","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}