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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
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 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.
Keywords: Division by zero; Traditional conventions for writing arithmetic; Legal texts; Illegal texts; Proof by contradiction; Bayes-Price Theorem; Common meadows; Signed common meadows
College: Faculty of Science and Engineering
Funders: Swansea University

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