{"id":"https://openalex.org/W2572204993","doi":"https://doi.org/10.2307/2272818","title":"Formalisations of further \u2135<sub>0</sub>-valued \u0141ukasiewicz propositional calculi","display_name":"Formalisations of further \u2135<sub>0</sub>-valued \u0141ukasiewicz propositional calculi","publication_year":1978,"publication_date":"1978-06-01","ids":{"openalex":"https://openalex.org/W2572204993","doi":"https://doi.org/10.2307/2272818","mag":"2572204993"},"language":"en","primary_location":{"id":"doi:10.2307/2272818","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272818","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5110467009","display_name":"Alan Rose","orcid":null},"institutions":[{"id":"https://openalex.org/I142263535","display_name":"University of Nottingham","ror":"https://ror.org/01ee9ar58","country_code":"GB","type":"education","lineage":["https://openalex.org/I142263535"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Alan Rose","raw_affiliation_strings":["University of Nottingham, Nottingham, NG7 2RD, England"],"affiliations":[{"raw_affiliation_string":"University of Nottingham, Nottingham, NG7 2RD, England","institution_ids":["https://openalex.org/I142263535"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5110467009"],"corresponding_institution_ids":["https://openalex.org/I142263535"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.18772967,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"43","issue":"2","first_page":"207","last_page":"210"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9296000003814697,"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"}},"topics":[{"id":"https://openalex.org/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9296000003814697,"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"}},{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9276999831199646,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13500","display_name":"Probability and Statistical Research","score":0.9139999747276306,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.792375922203064},{"id":"https://openalex.org/keywords/functor","display_name":"Functor","score":0.7152470350265503},{"id":"https://openalex.org/keywords/modus-ponens","display_name":"Modus ponens","score":0.6429644227027893},{"id":"https://openalex.org/keywords/propositional-calculus","display_name":"Propositional calculus","score":0.5123015642166138},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.5118738412857056},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4840051829814911},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.429458349943161},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.4263717532157898},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09136238694190979},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.07447871565818787}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.792375922203064},{"id":"https://openalex.org/C156772000","wikidata":"https://www.wikidata.org/wiki/Q864475","display_name":"Functor","level":2,"score":0.7152470350265503},{"id":"https://openalex.org/C193885149","wikidata":"https://www.wikidata.org/wiki/Q655742","display_name":"Modus ponens","level":3,"score":0.6429644227027893},{"id":"https://openalex.org/C69562738","wikidata":"https://www.wikidata.org/wiki/Q200694","display_name":"Propositional calculus","level":2,"score":0.5123015642166138},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.5118738412857056},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4840051829814911},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.429458349943161},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.4263717532157898},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09136238694190979},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.07447871565818787},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2272818","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272818","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":12,"referenced_works":["https://openalex.org/W1975627271","https://openalex.org/W1989632075","https://openalex.org/W2003976385","https://openalex.org/W2080126541","https://openalex.org/W2126160338","https://openalex.org/W2324223338","https://openalex.org/W2331720955","https://openalex.org/W2331855851","https://openalex.org/W4291439952","https://openalex.org/W6644446173","https://openalex.org/W6651285518","https://openalex.org/W6678934100"],"related_works":["https://openalex.org/W2393619947","https://openalex.org/W2000439261","https://openalex.org/W1500440713","https://openalex.org/W143006200","https://openalex.org/W320130745","https://openalex.org/W4388432581","https://openalex.org/W2352522588","https://openalex.org/W2573300539","https://openalex.org/W155209255","https://openalex.org/W2359568481"],"abstract_inverted_index":{"It":[0,177],"has":[1],"been":[2],"shown":[3],"that,":[4,82],"for":[5],"all":[6],"rational":[7],"numbers":[8],"r":[9,13,32,60,64,76,91,173,204],"such":[10,30,74,149],"that":[11,31,75,150,186,214,253,255],"0\u2264":[12],"\u2264":[14,33,35,77,217],"1,":[15,93],"the":[16,57,67,84,87,100,107,116,122,138,145,156,187,209,236],"\u2135":[17],"0":[18,206,265],"-valued":[19],"\u0141ukasiewicz":[20],"propositional":[21],"calculus":[22],"whose":[23],"designated":[24,68],"truth-values":[25,28,69,72,157],"are":[26,70,168],"those":[27,71],"x":[29,34,73,78,158,163,216,223,242,258,261,272,283],"1":[36],"may":[37],"be":[38,132,135],"formalised":[39],"completely":[40],"by":[41,99,248],"means":[42],"of":[43,51,86,109,184,195],"finitely":[44],"many":[45],"axiom":[46,103],"schemes":[47,104],"and":[48,66,172,197,254],"primitive":[49],"rules":[50],"procedure.":[52],"We":[53,80,211],"shall":[54],"consider":[55,143],"now":[56,144,234],"case":[58,89],"where":[59,90],"is":[61,97,191,259],"rational,":[62],"0\u2265":[63],"\u22641":[65],"\u22641.":[79],"note":[81,212],"in":[83,121,137,193,199],"subcase":[85],"previous":[88],"=":[92,174,205,229,246],"a":[94,182],"complete":[95],"formalisation":[96],"given":[98],"following":[101],"four":[102],"together":[105],"with":[106],"rule":[108],"modus":[110],"ponens":[111],"(with":[112],"respect":[113],"to":[114,134],"C),":[115],"functor":[117,146,188],"A":[118],"being":[119],"defined":[120,136],"usual":[123,139],"way.":[124,140,202],"The":[125],"functors":[126],"B,":[127],"K,":[128],"L":[129],"will":[130],"also":[131],"considered":[133],"Let":[141,232],"us":[142,233],"D":[147,153,189],"\u03b1\u03b2":[148,154,161,190,221,240],"if":[151,215,257,267],"P,":[152],"take":[155],",":[159],"d":[160,220,238],"(":[162,222,241,244,282],")":[164,243],"respectively,":[165],"\u03b1,":[166],"\u03b2":[167],"relatively":[169],"prime":[170],"integers":[171],"\u03b1/\u03b2":[175,218,263],"then":[176,219,277],"follows":[178,251],"at":[179],"once":[180],"from":[181],"theorem":[183],"McNaughton":[185],"definable":[192],"terms":[194],"C":[196],"N":[198],"an":[200],"effective":[201],"If":[203],"we":[207],"make":[208],"definition":[210],"first":[213],")\u2264(\u03b2":[224],"+":[225,275,279],"1)\u03b1/\u03b2":[226],"\u2212":[227,262,270,273],"\u03b1":[228],"\u03b1/\u03b2.":[230],"Hence":[231],"define":[235],"functions":[237],"n":[239,245,268,281],"0,1,\u2026)":[247],"Since":[249],"it":[250],"easily":[252],"Thus,":[256],"designated,":[260],"&gt;":[264,269,285],"and,":[266],"log(":[271],"\u03b1/\u03b2)/log(\u03b2":[274],"1),":[276],"(\u03b2":[278],"1)":[280],"\u2212\u03b1/\u03b2)":[284],"1.":[286]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}