{"id":"https://openalex.org/W2075609053","doi":"https://doi.org/10.2307/2270373","title":"A note on S5","display_name":"A note on S5","publication_year":1964,"publication_date":"1964-12-01","ids":{"openalex":"https://openalex.org/W2075609053","doi":"https://doi.org/10.2307/2270373","mag":"2075609053"},"language":"en","primary_location":{"id":"doi:10.2307/2270373","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2270373","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/A5040916739","display_name":"H\u00e9ctor-Neri Casta\u00f1eda","orcid":null},"institutions":[{"id":"https://openalex.org/I185443292","display_name":"Wayne State University","ror":"https://ror.org/01070mq45","country_code":"US","type":"education","lineage":["https://openalex.org/I185443292"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Hector-Neri Castaneda","raw_affiliation_strings":["Wayne State University"],"affiliations":[{"raw_affiliation_string":"Wayne State University","institution_ids":["https://openalex.org/I185443292"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5040916739"],"corresponding_institution_ids":["https://openalex.org/I185443292"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.21183491,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"29","issue":"4","first_page":"191","last_page":"192"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9957000017166138,"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"}},"topics":[{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9957000017166138,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9936000108718872,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9918000102043152,"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/mathematics","display_name":"Mathematics","score":0.777449369430542},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4970238506793976},{"id":"https://openalex.org/keywords/completeness","display_name":"Completeness (order theory)","score":0.4836355447769165},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.43395406007766724},{"id":"https://openalex.org/keywords/propositional-variable","display_name":"Propositional variable","score":0.41885894536972046},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.38334548473358154},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12720182538032532},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10035049915313721},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.06266719102859497},{"id":"https://openalex.org/keywords/intermediate-logic","display_name":"Intermediate logic","score":0.05958184599876404}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.777449369430542},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4970238506793976},{"id":"https://openalex.org/C17231256","wikidata":"https://www.wikidata.org/wiki/Q5156540","display_name":"Completeness (order theory)","level":2,"score":0.4836355447769165},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.43395406007766724},{"id":"https://openalex.org/C14523651","wikidata":"https://www.wikidata.org/wiki/Q3492863","display_name":"Propositional variable","level":4,"score":0.41885894536972046},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.38334548473358154},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12720182538032532},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10035049915313721},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.06266719102859497},{"id":"https://openalex.org/C33203268","wikidata":"https://www.wikidata.org/wiki/Q5361594","display_name":"Intermediate logic","level":3,"score":0.05958184599876404},{"id":"https://openalex.org/C102993220","wikidata":"https://www.wikidata.org/wiki/Q387196","display_name":"Description logic","level":2,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2270373","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2270373","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W22110506","https://openalex.org/W2041833446","https://openalex.org/W2080823411","https://openalex.org/W2093157821","https://openalex.org/W2320524668"],"related_works":["https://openalex.org/W4237277701","https://openalex.org/W2121424666","https://openalex.org/W4301250310","https://openalex.org/W2055420561","https://openalex.org/W2067700950","https://openalex.org/W1669998927","https://openalex.org/W3162039195","https://openalex.org/W2071225893","https://openalex.org/W4250321999","https://openalex.org/W2498652051"],"abstract_inverted_index":{"This":[0],"is":[1,25,39,45,64,75,81,128,140,146],"really":[2],"a":[3,26,65,95,107,129,149],"note":[4],"to":[5],"Wajsberg's":[6,118],"[4].":[7],"In":[8],"[3]":[9],"J.":[10],"C.":[11,12],"McKinsey":[13],"proved":[14],"the":[15],"following":[16],"two":[17],"completeness":[18],"theorems:":[19],"(1)":[20],"theorem":[21,60,97,111],"6:":[22],"If":[23,126],"A":[24,38,44,63,74,80,127,139,145],"wff":[27,66,130],"of":[28,67,131],"S2":[29,42],"with":[30,51,69,87,133,152],"just":[31,70,134],"r":[32,71],"(proper":[33],"or":[34],"improper)":[35],"sub-wffs,":[36,72],"then":[37,73,138],"provable":[40,76,141],"in":[41,77,122,142],"iff":[43,79,144],"satisfied":[46,82,147],"by":[47,83,148],"every":[48,84],"normal":[49,85,150],"S2-matrix":[50],"no":[52,88],"more":[53,89],"than":[54,90],"2":[55,91,153],"2r+1":[56],"elements":[57,155],";":[58],"(2)":[59],"13:":[61],"if":[62],"S4":[68,78],"S4-matrix":[86],"2r":[92],"elements.":[93],"Now,":[94],"similar":[96],"has":[98,112],"not":[99],"been":[100,113],"explicitly":[101],"formulated":[102],"for":[103],"S5,":[104],"even":[105,109],"though":[106],"similar,":[108],"simpler,":[110],"almost":[114],"at":[115],"hand":[116],"since":[117],"[4]":[119],"was":[120],"published":[121],"1933,":[123],"namely:":[124],"Theorem.":[125],"S5":[132,143],"n":[135],"propositional":[136],"variables,":[137],"SS-matrix":[151],"2n":[154],".":[156]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2016-06-24T00:00:00"}
