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Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts

Jan A. Bergstra, John Tucker Orcid Logo

Journal of Logic, Language and Information

Swansea University Author: John Tucker Orcid Logo

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DOI (Published version): 10.1007/s10849-025-09438-8

Abstract

To avoid the risk of problems to do with division by zero (DbZ), arithmetical texts involving division use what may be called traditional conventions on DbZ. Earlier, we developed a method for exploring these conventions using informal notions of legal and illegal texts, which are used to analyse si...

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Published in: Journal of Logic, Language and Information
ISSN: 0925-8531 1572-9583
Published: Springer Nature 2025
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URI: https://cronfa.swan.ac.uk/Record/cronfa69593
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last_indexed 2025-08-02T04:59:59Z
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spelling 2025-08-01T14:25:13.5612146 v2 69593 2025-05-30 Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts 431b3060563ed44cc68c7056ece2f85e 0000-0003-4689-8760 John Tucker John Tucker true false 2025-05-30 MACS To avoid the risk of problems to do with division by zero (DbZ), arithmetical texts involving division use what may be called traditional conventions on DbZ. Earlier, we developed a method for exploring these conventions using informal notions of legal and illegal texts, which are used to analyse simple fragments of arithmetical texts. We showed how these texts can be transformed into logical formulae over special total algebras, called common meadows, that are able to approximate partiality but in a total world. The subtleties of the legal/illegal distinction call for further development of these mathematical methods. Here we examine a more complex type of text, namely proof by contradiction, in which inconsistent assumptions can coexist with DbZ. We formulate more advanced criteria of legality for this case. We introduce a three-valued logic to capture the resulting semiformal conventions that is based on a notion we call frugal equality for partial operators. We apply the method to a proof of the Bayes-Price Theorem in probability theory, whose proof has DbZ issues. Journal Article Journal of Logic, Language and Information 0 Springer Nature 0925-8531 1572-9583 Division by zero; Traditional conventions for writing arithmetic; Legal texts; Illegal texts; Proof by contradiction; Bayes-Price Theorem; Common meadows; Signed common meadows 20 6 2025 2025-06-20 10.1007/s10849-025-09438-8 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University SU Library paid the OA fee (TA Institutional Deal) Swansea University 2025-08-01T14:25:13.5612146 2025-05-30T11:36:31.2277430 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Jan A. Bergstra 1 John Tucker 0000-0003-4689-8760 2 69593__34562__41ec5abeb7e6499abf870ccc85977e8c.pdf 69593.VOR.pdf 2025-06-24T14:35:08.8738274 Output 402411 application/pdf Version of Record true © The Author(s) 2025. This article is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). true eng http://creativecommons.org/licenses/by/4.0/
title Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts
spellingShingle Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts
John Tucker
title_short Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts
title_full Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts
title_fullStr Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts
title_full_unstemmed Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts
title_sort Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts
author_id_str_mv 431b3060563ed44cc68c7056ece2f85e
author_id_fullname_str_mv 431b3060563ed44cc68c7056ece2f85e_***_John Tucker
author John Tucker
author2 Jan A. Bergstra
John Tucker
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institution Swansea University
issn 0925-8531
1572-9583
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publisher Springer Nature
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hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science
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description To avoid the risk of problems to do with division by zero (DbZ), arithmetical texts involving division use what may be called traditional conventions on DbZ. Earlier, we developed a method for exploring these conventions using informal notions of legal and illegal texts, which are used to analyse simple fragments of arithmetical texts. We showed how these texts can be transformed into logical formulae over special total algebras, called common meadows, that are able to approximate partiality but in a total world. The subtleties of the legal/illegal distinction call for further development of these mathematical methods. Here we examine a more complex type of text, namely proof by contradiction, in which inconsistent assumptions can coexist with DbZ. We formulate more advanced criteria of legality for this case. We introduce a three-valued logic to capture the resulting semiformal conventions that is based on a notion we call frugal equality for partial operators. We apply the method to a proof of the Bayes-Price Theorem in probability theory, whose proof has DbZ issues.
published_date 2025-06-20T05:28:36Z
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