{"id":"https://openalex.org/W1982687715","doi":"https://doi.org/10.1305/ndjfl/1093887829","title":"Negation, material equivalence, and conditioned nonconjunction: completeness and duality.","display_name":"Negation, material equivalence, and conditioned nonconjunction: completeness and duality.","publication_year":1977,"publication_date":"1977-01-01","ids":{"openalex":"https://openalex.org/W1982687715","doi":"https://doi.org/10.1305/ndjfl/1093887829","mag":"1982687715"},"language":"en","primary_location":{"id":"doi:10.1305/ndjfl/1093887829","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093887829","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-18/issue-1/Negation-material-equivalence-and-conditioned-nonconjunction-completeness-and-duality/10.1305/ndjfl/1093887829.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-18/issue-1/Negation-material-equivalence-and-conditioned-nonconjunction-completeness-and-duality/10.1305/ndjfl/1093887829.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5017543831","display_name":"Gerald J. Massey","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Gerald J. Massey","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5017543831"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.12156078,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":93,"max":97},"biblio":{"volume":"18","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9291999936103821,"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":0.9291999936103821,"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/negation","display_name":"Negation","score":0.8797434568405151},{"id":"https://openalex.org/keywords/completeness","display_name":"Completeness (order theory)","score":0.6017357110977173},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.5934654474258423},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5927227735519409},{"id":"https://openalex.org/keywords/negation-as-failure","display_name":"Negation as failure","score":0.43756601214408875},{"id":"https://openalex.org/keywords/linguistics","display_name":"Linguistics","score":0.36692994832992554},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.356416255235672},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3521125912666321},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32833972573280334},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.19979986548423767},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.18420076370239258},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.13540402054786682},{"id":"https://openalex.org/keywords/semantics","display_name":"Semantics (computer science)","score":0.09976071119308472},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.09850871562957764},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09581038355827332},{"id":"https://openalex.org/keywords/operational-semantics","display_name":"Operational semantics","score":0.04118800163269043}],"concepts":[{"id":"https://openalex.org/C2185349","wikidata":"https://www.wikidata.org/wiki/Q190558","display_name":"Negation","level":2,"score":0.8797434568405151},{"id":"https://openalex.org/C17231256","wikidata":"https://www.wikidata.org/wiki/Q5156540","display_name":"Completeness (order theory)","level":2,"score":0.6017357110977173},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5934654474258423},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5927227735519409},{"id":"https://openalex.org/C194146651","wikidata":"https://www.wikidata.org/wiki/Q2119400","display_name":"Negation as failure","level":5,"score":0.43756601214408875},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.36692994832992554},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.356416255235672},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3521125912666321},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32833972573280334},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.19979986548423767},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.18420076370239258},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.13540402054786682},{"id":"https://openalex.org/C184337299","wikidata":"https://www.wikidata.org/wiki/Q1437428","display_name":"Semantics (computer science)","level":2,"score":0.09976071119308472},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.09850871562957764},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09581038355827332},{"id":"https://openalex.org/C156325763","wikidata":"https://www.wikidata.org/wiki/Q1930895","display_name":"Operational semantics","level":3,"score":0.04118800163269043},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C127001435","wikidata":"https://www.wikidata.org/wiki/Q7595770","display_name":"Stable model semantics","level":4,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1305/ndjfl/1093887829","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093887829","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-18/issue-1/Negation-material-equivalence-and-conditioned-nonconjunction-completeness-and-duality/10.1305/ndjfl/1093887829.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/1093887829","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.ndjfl/1093887829","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/1093887829","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093887829","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-18/issue-1/Negation-material-equivalence-and-conditioned-nonconjunction-completeness-and-duality/10.1305/ndjfl/1093887829.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":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.49000000953674316}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1982687715.pdf","grobid_xml":"https://content.openalex.org/works/W1982687715.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W2041833446"],"related_works":["https://openalex.org/W1596105581","https://openalex.org/W4214701509","https://openalex.org/W4212824748","https://openalex.org/W2077585551","https://openalex.org/W2015627330","https://openalex.org/W4388593575","https://openalex.org/W1605488371","https://openalex.org/W3104092934","https://openalex.org/W286008560","https://openalex.org/W3197254143"],"abstract_inverted_index":{"Object":[0],"of":[1,4,14,23,38,41,93,131,158,175],"paperThe":[2],"object":[3],"this":[5],"paper":[6],"is":[7,100,138,160,176,189,199,218,227,243,254,289,299,324,328,343],"threefold:":[8],"to":[9,19,25,49,102,106,245,249,272,291,295,345],"prove":[10],"the":[11,31,39,86,129,156,168],"functional":[12],"incompleteness":[13,62],"{~,":[15,46,136,221,332],"=}":[16,64,137],"without":[17],"appeal":[18],"a":[20,27,70,143,145,146,150,161,177,188,190,200,228,255,301,325,347],"tedious":[21],"analysis":[22],"cases;":[24],"give":[26],"proof":[28],"simpler":[29],"than":[30,77,214,236],"one":[32],"in":[33,95,154,186],"[2],":[34],"p.":[35],"284":[36],"f.,":[37],"nonexistence":[40],"indigenous":[42],"Sheffer":[43,56,337],"connectives":[44,57,75,212,234,338],"for":[45,58,149,180,193,203,206,258,304],"=};":[47],"and":[48,79,81,116,171,205,263],"furnish":[50],"self-dual":[51],"(in":[52],"Church's":[53],"sense)":[54],"ternary":[55],"propositional":[59,71],"logic.2":[60],"Functional":[61],"of{~,":[63],"Lemma":[65,125,224,284],"1":[66,84,104,126,135,225,285,334],"Let":[67,109],"A":[68,83,96,99,103],"be":[69,85,270,346],"wff":[72,87,151,208,229],"containing":[73,232],"no":[74,211,233],"other":[76,213,235],"~":[78,94],"=,":[80],"let":[82],"that":[88,174,197,209,216,239],"results":[89],"when":[90,162],"all":[91,163],"occurrences":[92],"are":[97,166],"deleted.Then":[98],"equivalent":[101,244,271,290],"or":[105,248,294],"~Ar":[107],".Proof:":[108],"<#^>":[110],"signify":[111],"(semantic)":[112],"equivalence.Since":[113],"B":[114,152,159],"<^$>~~B":[115],"since":[117,252],"~(B":[118],"=":[119,121,123,204,237],"C)#=>~J3":[120],"C\u03a6\u03a6B":[122],"~C,":[124],"follows":[127],"from":[128,220,331],"substitutivity":[130],"equivalents":[132],"by":[133,223],"induction.Theorem":[134],"functionally":[139],"incomplete.Proof:":[140],"We":[141],"call":[142],"truth-value":[144],"fixed":[147,178,191,201,256,302],"point":[148,179,192,202,257,303],"just":[153,185],"case":[155,187],"value":[157,169],"its":[164],"variables":[165],"assigned":[167],"a,":[170],"we":[172,308],"say":[173],"an":[181],"ra-ary":[182],"connective":[183,341],"\u00ae":[184,342],"\u0398(/>i,":[194],".":[195],"..,/>\").Notice":[196],"t":[198,253,298],"any":[207],"contains":[210],"=.Suppose":[215],"&":[217,241,261,267,277,281,306,313,327],"definable":[219,330],"=}.Then":[222],"there":[226],"A(p,":[230,246,287,292,310],"a)":[231,247],"such":[238],"p":[240,260,266,276,280,305,312],"q":[242,262,268],"~A(p,":[250,296],"a).But":[251],"both":[259],"A{p,":[264],"a),":[265],"cannot":[269],"~A":[273],"(p,":[274],"q).So,":[275,315],"q<^>A(p,":[278],"a).Therefore,":[279],"~q<\u03a6A(p,":[282],"~q).By":[283],"again,":[286],"~q)":[288,311],"q)":[293],"q).Since":[297],"not":[300,329],"~q9":[307],"have":[309],"~q<#=>~A(p,":[314],"~(P":[316],"8\u03b9~q)":[317],"<":[318],"^\u0386(p,":[319],"q)\u03a6\u03a6p":[320],"&,":[321],"q,":[322],"which":[323],"contradiction.So":[326],"=}.Theorem":[333],"follows.3":[335],"Indigenous":[336],"An":[339],"n-ary":[340],"said":[344]},"counts_by_year":[{"year":2023,"cited_by_count":3},{"year":2021,"cited_by_count":2}],"updated_date":"2026-08-19T15:00:24.278416","created_date":"2025-10-10T00:00:00"}
