{"id":"https://openalex.org/W4411440366","doi":"https://doi.org/10.1007/s10849-025-09438-8","title":"Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts","display_name":"Logical models of mathematical texts II: Legality conventions for division by zero in inconsistent contexts","publication_year":2025,"publication_date":"2025-06-20","ids":{"openalex":"https://openalex.org/W4411440366","doi":"https://doi.org/10.1007/s10849-025-09438-8"},"language":"en","primary_location":{"id":"doi:10.1007/s10849-025-09438-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10849-025-09438-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10849-025-09438-8.pdf","source":{"id":"https://openalex.org/S197419112","display_name":"Journal of Logic Language and Information","issn_l":"0925-8531","issn":["0925-8531","1572-9583"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Logic, Language and Information","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10849-025-09438-8.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5032380170","display_name":"J.A. Bergstra","orcid":"https://orcid.org/0000-0003-2492-506X"},"institutions":[{"id":"https://openalex.org/I887064364","display_name":"University of Amsterdam","ror":"https://ror.org/04dkp9463","country_code":"NL","type":"education","lineage":["https://openalex.org/I887064364"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Jan A. Bergstra","raw_affiliation_strings":["Informatics Institute, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Informatics Institute, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands","institution_ids":["https://openalex.org/I887064364"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5103271292","display_name":"John V. Tucker","orcid":"https://orcid.org/0000-0003-4689-8760"},"institutions":[{"id":"https://openalex.org/I39586589","display_name":"Swansea University","ror":"https://ror.org/053fq8t95","country_code":"GB","type":"education","lineage":["https://openalex.org/I39586589"]},{"id":"https://openalex.org/I4210113874","display_name":"Energy Safety Research Institute","ror":"https://ror.org/026znxe07","country_code":"GB","type":"facility","lineage":["https://openalex.org/I39586589","https://openalex.org/I4210113874"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"John V. Tucker","raw_affiliation_strings":["Department of Computer Science, Swansea University, Bay Campus, Fabian Way, SA1 8EN, Swansea, United Kingdom"],"raw_orcid":"https://orcid.org/0000-0003-4689-8760","affiliations":[{"raw_affiliation_string":"Department of Computer Science, Swansea University, Bay Campus, Fabian Way, SA1 8EN, Swansea, United Kingdom","institution_ids":["https://openalex.org/I39586589","https://openalex.org/I4210113874"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5103271292"],"corresponding_institution_ids":["https://openalex.org/I39586589","https://openalex.org/I4210113874"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.20557656,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9365000128746033,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9365000128746033,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9078999757766724,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/principle-of-legality","display_name":"Principle of legality","score":0.7643250226974487},{"id":"https://openalex.org/keywords/contradiction","display_name":"Contradiction","score":0.6553117036819458},{"id":"https://openalex.org/keywords/arithmetic-function","display_name":"Arithmetic function","score":0.5903551578521729},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.49826478958129883},{"id":"https://openalex.org/keywords/division","display_name":"Division (mathematics)","score":0.4938550591468811},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4918649196624756},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.454524427652359},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.4370880126953125},{"id":"https://openalex.org/keywords/commutative-property","display_name":"Commutative property","score":0.43490517139434814},{"id":"https://openalex.org/keywords/proof-theory","display_name":"Proof theory","score":0.41061273217201233},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.38233059644699097},{"id":"https://openalex.org/keywords/epistemology","display_name":"Epistemology","score":0.375198632478714},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.36062294244766235},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.3382267653942108},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.33371907472610474},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.31321871280670166},{"id":"https://openalex.org/keywords/law","display_name":"Law","score":0.2654728293418884},{"id":"https://openalex.org/keywords/linguistics","display_name":"Linguistics","score":0.2400619387626648},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.2340298295021057},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.20137885212898254},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.18615993857383728},{"id":"https://openalex.org/keywords/political-science","display_name":"Political science","score":0.13162416219711304}],"concepts":[{"id":"https://openalex.org/C42027317","wikidata":"https://www.wikidata.org/wiki/Q2307054","display_name":"Principle of legality","level":2,"score":0.7643250226974487},{"id":"https://openalex.org/C2776728590","wikidata":"https://www.wikidata.org/wiki/Q363948","display_name":"Contradiction","level":2,"score":0.6553117036819458},{"id":"https://openalex.org/C112343008","wikidata":"https://www.wikidata.org/wiki/Q140744","display_name":"Arithmetic function","level":2,"score":0.5903551578521729},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.49826478958129883},{"id":"https://openalex.org/C60798267","wikidata":"https://www.wikidata.org/wiki/Q1226939","display_name":"Division (mathematics)","level":2,"score":0.4938550591468811},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4918649196624756},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.454524427652359},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.4370880126953125},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.43490517139434814},{"id":"https://openalex.org/C2318724","wikidata":"https://www.wikidata.org/wiki/Q852732","display_name":"Proof theory","level":3,"score":0.41061273217201233},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.38233059644699097},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.375198632478714},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.36062294244766235},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.3382267653942108},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.33371907472610474},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.31321871280670166},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.2654728293418884},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.2400619387626648},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.2340298295021057},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.20137885212898254},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.18615993857383728},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.13162416219711304},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10849-025-09438-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10849-025-09438-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10849-025-09438-8.pdf","source":{"id":"https://openalex.org/S197419112","display_name":"Journal of Logic Language and Information","issn_l":"0925-8531","issn":["0925-8531","1572-9583"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Logic, Language and Information","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10849-025-09438-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10849-025-09438-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10849-025-09438-8.pdf","source":{"id":"https://openalex.org/S197419112","display_name":"Journal of Logic Language and Information","issn_l":"0925-8531","issn":["0925-8531","1572-9583"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Logic, Language and Information","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16","score":0.7699999809265137}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4411440366.pdf","grobid_xml":"https://content.openalex.org/works/W4411440366.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W1801045077","https://openalex.org/W2024366700","https://openalex.org/W2049352524","https://openalex.org/W2099937335","https://openalex.org/W2171361542","https://openalex.org/W2269907218","https://openalex.org/W2314261248","https://openalex.org/W3022570832","https://openalex.org/W3104006239","https://openalex.org/W3105499720","https://openalex.org/W3105604967","https://openalex.org/W3165895558","https://openalex.org/W4212959354","https://openalex.org/W4220950893","https://openalex.org/W4230900352","https://openalex.org/W4231911220","https://openalex.org/W4316813701","https://openalex.org/W4386927940","https://openalex.org/W4392982669","https://openalex.org/W4399670193","https://openalex.org/W4399752369","https://openalex.org/W4400205773","https://openalex.org/W4400606801","https://openalex.org/W4401665823","https://openalex.org/W6666064338","https://openalex.org/W6834379202"],"related_works":["https://openalex.org/W3122346207","https://openalex.org/W4280488862","https://openalex.org/W2361823710","https://openalex.org/W2382763736","https://openalex.org/W4280647041","https://openalex.org/W2381825231","https://openalex.org/W2378357372","https://openalex.org/W2888591139","https://openalex.org/W2358425177","https://openalex.org/W2385114175"],"abstract_inverted_index":{"Abstract":[0],"To":[1],"avoid":[2],"the":[3,87,136,155,161],"risk":[4],"of":[5,39,51,86,94,105,124,160],"problems":[6],"to":[7,47,76,134,157],"do":[8],"with":[9,117],"division":[10,17],"by":[11,109],"zero":[12],"(DbZ),":[13],"arithmetical":[14,52],"texts":[15,58],"involving":[16],"use":[18],"what":[19],"may":[20],"be":[21,60],"called":[22,69],"traditional":[23],"conventions":[24,35,139],"on":[25,143],"DbZ.":[26,118],"Earlier,":[27],"we":[28,99,146],"developed":[29],"a":[30,81,101,131,144,158],"method":[31,156],"for":[32,91,126,150],"exploring":[33],"these":[34,57,95],"using":[36],"informal":[37],"notions":[38],"legal":[40],"and":[41],"illegal":[42],"texts,":[43],"which":[44,112],"are":[45,74],"used":[46],"analyse":[48],"simple":[49],"fragments":[50],"texts.":[53],"We":[54,119,129,153],"showed":[55],"how":[56],"can":[59,115],"transformed":[61],"into":[62],"logical":[63],"formulae":[64],"over":[65],"special":[66],"total":[67,82],"algebras,":[68],"common":[70],"meadows":[71],",":[72],"that":[73,140],"able":[75],"approximate":[77],"partiality":[78],"but":[79],"in":[80,111,164],"world.":[83],"The":[84],"subtleties":[85],"legal/illegal":[88],"distinction":[89],"call":[90,147],"further":[92],"development":[93],"mathematical":[96],"methods.":[97],"Here":[98],"examine":[100],"more":[102,121],"complex":[103],"type":[104],"text,":[106],"namely":[107],"proof":[108,159,168],"contradiction,":[110],"inconsistent":[113],"assumptions":[114],"coexist":[116],"formulate":[120],"advanced":[122],"criteria":[123],"legality":[125],"this":[127],"case.":[128],"introduce":[130],"three-valued":[132],"logic":[133],"capture":[135],"resulting":[137],"semiformal":[138],"is":[141],"based":[142],"notion":[145],"frugal":[148],"equality":[149],"partial":[151],"operators.":[152],"apply":[154],"Bayes-Price":[162],"Theorem":[163],"probability":[165],"theory,":[166],"whose":[167],"has":[169],"DbZ":[170],"issues.":[171]},"counts_by_year":[],"updated_date":"2026-06-13T06:13:01.061226","created_date":"2025-10-10T00:00:00"}
