{"id":"https://openalex.org/W2084199183","doi":"https://doi.org/10.1305/ndjfl/1093893523","title":"The shortest axioms of the implicational calculus.","display_name":"The shortest axioms of the implicational calculus.","publication_year":1968,"publication_date":"1968-10-01","ids":{"openalex":"https://openalex.org/W2084199183","doi":"https://doi.org/10.1305/ndjfl/1093893523","mag":"2084199183"},"language":"en","primary_location":{"id":"doi:10.1305/ndjfl/1093893523","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093893523","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-9/issue-4/The-shortest-axioms-of-the-implicational-calculus/10.1305/ndjfl/1093893523.pdf","source":{"id":"https://openalex.org/S78544662","display_name":"Notre Dame Journal of Formal Logic","issn_l":"0029-4527","issn":["0029-4527","1939-0726"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315698","host_organization_name":"Duke University Press","host_organization_lineage":["https://openalex.org/P4310315698","https://openalex.org/P4310315572"],"host_organization_lineage_names":["Duke University Press","Duke University"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Notre Dame Journal of Formal Logic","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-9/issue-4/The-shortest-axioms-of-the-implicational-calculus/10.1305/ndjfl/1093893523.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5041570546","display_name":"Richard Tursman","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Richard Tursman","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5041570546"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.7097,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.89163288,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"9","issue":"4","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.987500011920929,"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":0.987500011920929,"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":0.9621000289916992,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9106000065803528,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.6788904070854187},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.6689918041229248},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4080788791179657},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.390184223651886},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3431304097175598},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.18446430563926697},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.1381167471408844},{"id":"https://openalex.org/keywords/orthodontics","display_name":"Orthodontics","score":0.07864373922348022},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.06581041216850281}],"concepts":[{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.6788904070854187},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.6689918041229248},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4080788791179657},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.390184223651886},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3431304097175598},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.18446430563926697},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.1381167471408844},{"id":"https://openalex.org/C29694066","wikidata":"https://www.wikidata.org/wiki/Q118301","display_name":"Orthodontics","level":1,"score":0.07864373922348022},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.06581041216850281}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1305/ndjfl/1093893523","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093893523","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-9/issue-4/The-shortest-axioms-of-the-implicational-calculus/10.1305/ndjfl/1093893523.pdf","source":{"id":"https://openalex.org/S78544662","display_name":"Notre Dame Journal of Formal Logic","issn_l":"0029-4527","issn":["0029-4527","1939-0726"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315698","host_organization_name":"Duke University Press","host_organization_lineage":["https://openalex.org/P4310315698","https://openalex.org/P4310315572"],"host_organization_lineage_names":["Duke University Press","Duke University"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Notre Dame Journal of Formal Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.ndjfl/1093893523","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.ndjfl/1093893523","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1305/ndjfl/1093893523","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093893523","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-9/issue-4/The-shortest-axioms-of-the-implicational-calculus/10.1305/ndjfl/1093893523.pdf","source":{"id":"https://openalex.org/S78544662","display_name":"Notre Dame Journal of Formal Logic","issn_l":"0029-4527","issn":["0029-4527","1939-0726"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315698","host_organization_name":"Duke University Press","host_organization_lineage":["https://openalex.org/P4310315698","https://openalex.org/P4310315572"],"host_organization_lineage_names":["Duke University Press","Duke University"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Notre Dame Journal of Formal Logic","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.550000011920929,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2084199183.pdf","grobid_xml":"https://content.openalex.org/works/W2084199183.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2748952813","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W4225152035","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694","https://openalex.org/W2911598644"],"abstract_inverted_index":{"Jan":[0],"Lukasiewicz":[1],"has":[2,21],"proved":[3,71],"that":[4,28,72],"all":[5,37],"theses":[6,38],"of":[7,11,58,85,93,111,134,151,161,207,211,217,224,231,233,245,256,267,284],"the":[8,16,26,31,59,75,82,91,124,180,195,208,218,264],"Implicational":[9],"Calculus":[10],"Propositions":[12],"are":[13,39,45,108,131,145,155],"derivable":[14],"from":[15,35,251],"axiom":[17,34,80,172],"(I)":[18,29,51,73,122],"CCCpqrCCrpCsp":[19],"and":[20,142,149,222,238],"outlined":[22],"a":[23,101,109,168,205,215,229,276],"proof":[24],"to":[25,178,193,199,243,262],"effect":[27],"is":[30,64,70,74,123,164,177,198,204,214,228,270,278,281],"shortest":[32],"single":[33,56,79,102,115,128,171],"which":[36,52,163,232],"derivable.1":[40],"Whether":[41],"or":[42,95],"not":[43,65,96],"there":[44,107,130,144,154],"13":[46,77,113,126,138,158],"letter":[47,78,114,127,139,159],"axioms":[48,57],"other":[49],"than":[50],"could":[53],"serve":[54,99],"as":[55,100],"Calculus,":[60],"\u03afiUkasiewicz":[61],"noted,":[62],"2":[63],"known.In":[66],"this":[67],"article":[68,88],"it":[69,104,118],"only":[76,125,282],"with":[81],"possible":[83],"exception":[84],"(II)":[86,97],"CCrpCCCpqrCsp.This":[87],"leaves":[89],"unanswered":[90],"question":[92],"whether":[94],"does":[98,119],"axiom.If":[103],"does,":[105],"then":[106,121],"total":[110],"two":[112,257],"axioms;":[116],"if":[117],"not,":[120],"axiom.Since":[129],"132":[132,209],"forms":[133,210],"punctuation":[135],"for":[136,170],"these":[137,162,234],"wff":[140],"'s,":[141],"since":[143,272],"877":[146,219,235],"unique":[147,157,220,236],"combinations":[148,221,237],"permutations":[150,223,239],"seven":[152,225,260],"variables,":[153,261],"115,764":[156,186],"wff's.Each":[160],"analytic":[165,181,196],"is,":[166],"initially,":[167],"candidate":[169],"status.The":[173],"first":[174],"practical":[175],"problem":[176],"determine":[179,194],"wff's":[182,197],"without":[183],"writing":[184],"out":[185],"128":[187,253],"row":[188],"truth":[189],"tables.The":[190],"easiest":[191],"way":[192],"use":[200],"three":[201],"charts:Chart":[202],"I":[203],"list":[206,216,230],"punctuation.Chart":[212],"II":[213],"variables.Chart":[226],"III":[227],"can":[240],"be":[241],"assigned":[242],"each":[244,268],"64":[246],"symmetrical":[247],"couplets.(These":[248],"couplets":[249],"result":[250],"reducing":[252],"standard":[254],"assignments":[255],"values,":[258],"involving":[259],"64;":[263],"second":[265],"member":[266],"couplet":[269],"superfluous,":[271],"anything":[273],"ending":[274],"in":[275],"'1'":[277],"analytic.)(What":[279],"follows":[280],"part":[283],"Chart":[285],"I\u03a0.)":[286]},"counts_by_year":[{"year":2016,"cited_by_count":1}],"updated_date":"2026-08-06T08:24:18.245995","created_date":"2025-10-10T00:00:00"}