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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 <jats:italic>common meadows<\/jats:italic>, 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 <jats:italic>frugal equality<\/jats:italic> for partial operators. We apply the method to a proof of the Bayes-Price Theorem in probability theory, whose proof has DbZ issues.<\/jats:p>","DOI":"10.1007\/s10849-025-09438-8","type":"journal-article","created":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T21:24:31Z","timestamp":1750368271000},"page":"371-393","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts"],"prefix":"10.1007","volume":"34","author":[{"given":"Jan A.","family":"Bergstra","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4689-8760","authenticated-orcid":false,"given":"John V.","family":"Tucker","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,6,20]]},"reference":[{"key":"9438_CR1","doi-asserted-by":"publisher","unstructured":"Anderson, J.A., & Bergstra, J.A. (2021). Review of Suppes\u2019s 1957 proposals for division by zero. Transmathematica. https:\/\/doi.org\/10.36285\/tm.53","DOI":"10.36285\/tm.53"},{"key":"9438_CR2","doi-asserted-by":"publisher","unstructured":"Bayes, T., & Price, R. (1763). An essay towards solving a problem in the doctrine of chance. By the late Rev. Mr. Bayes, communicated by Mr. Price, in a letter to John Canton, F.R.S. Philosophical Transactions of the Royal Society 53, 370-418. https:\/\/doi.org\/10.1098\/rstl.1763.0053","DOI":"10.1098\/rstl.1763.0053"},{"key":"9438_CR3","doi-asserted-by":"publisher","unstructured":"Bergstra, J.A. (2020). Arithmetical datatypes, fracterms, and the fraction definition problem. Transmathematica. https:\/\/doi.org\/10.36285\/tm.33","DOI":"10.36285\/tm.33"},{"key":"9438_CR4","doi-asserted-by":"crossref","unstructured":"Bergstra, J.A., & Middelburg, C.A. (2015). Transformation of fractions into simple fractions in divisive meadows. Journal of Applied Logic, 16 (2015), 92\u2013110. arXiv:1510.06233","DOI":"10.1016\/j.jal.2016.03.001"},{"key":"9438_CR5","doi-asserted-by":"publisher","unstructured":"Bergstra, J. A., & Ponse, A. (2011). Proposition algebra. ACM Transactions on Computational Logic 12 (2) (2011). Article, 21, 1\u201336. https:\/\/doi.org\/10.1145\/1929954.1929958","DOI":"10.1145\/1929954.1929958"},{"key":"9438_CR6","doi-asserted-by":"crossref","unstructured":"Bergstra, J.A. & Ponse, A. (2015). Division by zero in common meadows. In R. de Nicola and R. Hennicker (editors), Software, Services, and Systems (Wirsing Festschrift), Lecture Notes in Computer Science 8950, 46-61, Springer.","DOI":"10.1007\/978-3-319-15545-6_6"},{"key":"9438_CR7","doi-asserted-by":"publisher","unstructured":"Bergstra, J.A., & Ponse, A. (2016). Fracpairs and fractions over a reduced commutative ring. 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