{"id":"https://openalex.org/W1990478150","doi":"https://doi.org/10.1305/ndjfl/1093893726","title":"A reduction procedure for Sheffer stroke formulas.","display_name":"A reduction procedure for Sheffer stroke formulas.","publication_year":1969,"publication_date":"1969-07-01","ids":{"openalex":"https://openalex.org/W1990478150","doi":"https://doi.org/10.1305/ndjfl/1093893726","mag":"1990478150"},"language":"en","primary_location":{"id":"doi:10.1305/ndjfl/1093893726","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093893726","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-10/issue-3/A-reduction-procedure-for-Sheffer-stroke-formulas/10.1305/ndjfl/1093893726.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-10/issue-3/A-reduction-procedure-for-Sheffer-stroke-formulas/10.1305/ndjfl/1093893726.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5022564707","display_name":"Robert D. Carnes","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Robert D. Carnes","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5022564707"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.12897775,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"10","issue":"3","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":0.21410000324249268,"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":0.21410000324249268,"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/reduction","display_name":"Reduction (mathematics)","score":0.532682478427887},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5037164092063904},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.3943861126899719},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3440372347831726},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.34213075041770935},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.13529366254806519},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.04367348551750183}],"concepts":[{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.532682478427887},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5037164092063904},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3943861126899719},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3440372347831726},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.34213075041770935},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.13529366254806519},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.04367348551750183},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1305/ndjfl/1093893726","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093893726","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-10/issue-3/A-reduction-procedure-for-Sheffer-stroke-formulas/10.1305/ndjfl/1093893726.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/1093893726","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.ndjfl/1093893726","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/1093893726","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093893726","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-10/issue-3/A-reduction-procedure-for-Sheffer-stroke-formulas/10.1305/ndjfl/1093893726.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":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1990478150.pdf","grobid_xml":"https://content.openalex.org/works/W1990478150.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":{"In":[0],"this":[1,23],"paper,":[2],"I":[3,18],"shall":[4,19],"present":[5],"a":[6,10,36,122,183,231,275,333,340],"reduction":[7],"procedure":[8,53],"for":[9,61,127],"propositional":[11],"calculus":[12],"employing":[13,70],"the":[14,33,51,58,71,97,114,132,154,159,176,188,191,202,207,212,221,227,239,295,300,305,308,313,337,349],"Sheffer":[15,28,123],"stroke.":[16],"1":[17],"show":[20],"that":[21],"when":[22],"technique":[24],"is":[25,35,105,257,269,272,274,310,319,332,345],"applied":[26],"to":[27,56,217,294,312,321],"stroke":[29,72,98,116],"formulas":[30],"(hereafter,":[31,41],"'strokeformulas'),":[32],"result":[34],"finite":[37,249],"set":[38],"of":[39,44,110,130,156,178,190,251,253,266,285,292,302,307,315,329,339,348],"sentences":[40,45],"'FSS')":[42],"consisting":[43],"which":[46,151,173,197,210,229,244],"contain":[47,74],"fewer":[48],"strokes":[49],"than":[50],"original.This":[52],"enables":[54],"one":[55,265,283],"test":[57],"original":[59],"formula":[60],"tautologousness,":[62],"self-contradictoriness,":[63],"or":[64,92,117,147,169,336],"contingency.1.":[65],"Preparatory":[66],"Considerations":[67],"Object":[68],"languages":[69],"usually":[73],"sentence":[75,296,334,341],"variables":[76,134,162,297],"(for":[77],"example,":[78],"'/>',":[79],"'q":[80],"9":[81,84,90,138,141,167,242],",":[82,85,139,142,243,351],"'r":[83],"(":[86,136],"s',":[87],"and":[88,96,144,165,237,260,278,325,355],"'t":[89],"with":[91],"without":[93,148,170],"numerical":[94,149,171],"subscripts)":[95],"('I')":[99],"as":[100,220],"primitive":[101],"signs.Punctuation,":[102],"where":[103],"necessary,":[104],"achieved":[106],"by":[107],"symmetrical":[108],"groups":[109],"dots":[111],"2":[112],"flanking":[113],"appropriate":[115],"strokes.Let":[118],"SC":[119,128],"be":[120,218,322,357],"such":[121],"calculus.The":[124],"metalanguage":[125],"M":[126],"consists":[129],"(1)":[131],"syntactical":[133],"P',":[135],"Q":[137],"<R":[140],"S',":[143],"'T'":[145],"(with":[146,168],"subscripts),":[150,172],"range":[152,174],"over":[153,175,199,246],"wffs":[155,180],"SC;":[157],"(2)":[158],"quasi-syntactical":[160],"3":[161],"T',":[163],"'>,":[164],"'":[166],"FSSs":[177],"SC.The":[179],"occurring":[181,298],"in":[182,206,299],"non-empty":[184],"FSS":[185,256,318,347],"are":[186],"called":[187],"members":[189,268,301,309,331],"FSS.(3)":[192],"The":[193],"negation":[194,338],"bar":[195],"'--',":[196],"occurs":[198],"wffs;":[200],"(4)":[201],"arrow":[203],"'->'":[204],"(used":[205],"rewrite":[208,254],"rules),":[209],"permits":[211],"expression":[213,222],"on":[214,223,289],"its":[215,224,267,330],"left":[216],"rewritten":[219],"right;":[225],"(5)":[226],"'#',":[228],"indicates":[230],"split":[232],"into":[233],"two":[234],"separate":[235],"FSSs;":[236],"(6)":[238],"variable":[240,335],"'R":[241],"ranges":[245],"reductions,":[247],"i.e.,":[248],"sequences":[250],"applications":[252],"rules.An":[255],"true":[258,288],"if":[259,262,277,280,324,327],"only":[261,279,326],"at":[263,281],"least":[264,282],"true;":[270],"otherwise,":[271],"false.":[273],"tautology":[276,344],"member":[284],"comes":[286],"out":[287],"every":[290],"assignment":[291],"truthvalues":[293],".":[303],"Obviously,":[304],"order":[306],"irrelevant":[311],"truth-value":[314],"an":[316],"FSS.An":[317],"said":[320],"basic":[323],"each":[328],"variable.A":[342],"sequence":[343],"any":[346],"form":[350],"P,":[352],"F,":[353],"(where":[354],"may":[356],"empty).":[358]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
