{"id":"https://openalex.org/W2575470425","doi":"https://doi.org/10.2307/2586859","title":"Frege proof system and TNC\u00b0","display_name":"Frege proof system and TNC\u00b0","publication_year":1998,"publication_date":"1998-06-01","ids":{"openalex":"https://openalex.org/W2575470425","doi":"https://doi.org/10.2307/2586859","mag":"2575470425"},"language":"en","primary_location":{"id":"doi:10.2307/2586859","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2586859","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":"The 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/A5081729792","display_name":"Gaisi Takeuti","orcid":null},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]},{"id":"https://openalex.org/I2801919071","display_name":"University of Illinois System","ror":"https://ror.org/05e94g991","country_code":"US","type":"education","lineage":["https://openalex.org/I2801919071"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Gaisi Takeuti","raw_affiliation_strings":["Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA, E-mail:","Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA, E-mail: takeuti@saul.cis.upenn.edu"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA, E-mail:","institution_ids":["https://openalex.org/I2801919071"]},{"raw_affiliation_string":"Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA, E-mail: takeuti@saul.cis.upenn.edu","institution_ids":["https://openalex.org/I157725225"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5081729792"],"corresponding_institution_ids":["https://openalex.org/I157725225","https://openalex.org/I2801919071"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.26590563,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"63","issue":"2","first_page":"709","last_page":"738"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9991000294685364,"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":0.9991000294685364,"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/T10126","display_name":"Logic, programming, and type systems","score":0.9973999857902527,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9948999881744385,"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":0.6756473779678345},{"id":"https://openalex.org/keywords/modus-ponens","display_name":"Modus ponens","score":0.5846519470214844},{"id":"https://openalex.org/keywords/soundness","display_name":"Soundness","score":0.5425269603729248},{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.5306115746498108},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5164666175842285},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.4604620933532715},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.45343154668807983},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.44304850697517395},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3355298638343811},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.15373331308364868},{"id":"https://openalex.org/keywords/inference","display_name":"Inference","score":0.13881096243858337},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.0697106420993805}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6756473779678345},{"id":"https://openalex.org/C193885149","wikidata":"https://www.wikidata.org/wiki/Q655742","display_name":"Modus ponens","level":3,"score":0.5846519470214844},{"id":"https://openalex.org/C39920170","wikidata":"https://www.wikidata.org/wiki/Q693083","display_name":"Soundness","level":2,"score":0.5425269603729248},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.5306115746498108},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5164666175842285},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.4604620933532715},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45343154668807983},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.44304850697517395},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3355298638343811},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.15373331308364868},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.13881096243858337},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0697106420993805},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2586859","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2586859","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":"The 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":17,"referenced_works":["https://openalex.org/W49401050","https://openalex.org/W169256259","https://openalex.org/W1531368347","https://openalex.org/W1543891730","https://openalex.org/W1607926272","https://openalex.org/W1968584500","https://openalex.org/W1971856509","https://openalex.org/W1988083342","https://openalex.org/W2010055185","https://openalex.org/W2068638452","https://openalex.org/W2074624468","https://openalex.org/W2081469776","https://openalex.org/W2136868774","https://openalex.org/W2171353564","https://openalex.org/W3157713018","https://openalex.org/W4255155072","https://openalex.org/W4388289526"],"related_works":["https://openalex.org/W2029563506","https://openalex.org/W1588067599","https://openalex.org/W2611198522","https://openalex.org/W2389239788","https://openalex.org/W4246940373","https://openalex.org/W2615257120","https://openalex.org/W2288267523","https://openalex.org/W2166691384","https://openalex.org/W2290698615","https://openalex.org/W10327757"],"abstract_inverted_index":{"A":[0],"Frege":[1,28],"proof":[2],"system":[3,8,16,29,212,260,279],"F":[4,34,60,167,181,195,234],"is":[5,31,40,55,89,110,124,264],"any":[6,98,125,134,203],"standard":[7],"of":[9,43,59,81,99,192,233,245,269],"prepositional":[10],"calculus,":[11],"e.g.,":[12],"a":[13,41,117,152,215,266,276],"Hilbert":[14],"style":[15],"based":[17],"on":[18],"finitely":[19],"many":[20],"axiom":[21,58,114],"schemes":[22],"and":[23,67,115,142,158,196,228,235,242,261],"inference":[24],"rules.":[25],"An":[26,37,121],"Extended":[27],"EF":[30,38,122,126,156,199,236,246],"obtained":[32],"from":[33,63],"as":[35],"follows.":[36],"-sequence":[39,127],"sequence":[42],"formulas":[44],"\u03c8":[45,49,65,68,71,74,100],"1":[46,101],",":[47,61,214],"\u2026,":[48],"\u03ba":[50],"such":[51],"that":[52,146,160,230,263],"each\u03c8":[53],"i":[54,75,103],"either":[56],"an":[57,90,112],"inferred":[62],"previous":[64],"u":[66,72],"v":[69],"(=":[70],"\u2192":[73],")":[76],"by":[77,177],"modus":[78],"ponens":[79],"or":[80],"the":[82,147,188,211,231,243,257],"form":[83],"q":[84,88,107,116],"\u2194":[85,108],"\u03c6,":[86,109],"where":[87],"atom":[91],"occurring":[92],"neither":[93],"in":[94,97,240,248],"\u03c6":[95],"nor":[96],",\u2026,\u03c8":[102],"\u22121":[104],".":[105,185,250,271],"Such":[106],"called":[111],"extension":[113,119,135,268],"new":[118],"atom.":[120,136],"-proof":[123,157,182],"whose":[128,220],"last":[129],"formula":[130],"does":[131,162],"not":[132,163,201],"contain":[133],"In":[137,169,205,251],"[12],":[138],"S.":[139,171,207,253],"A.":[140,208],"Cook":[141,209],"R.":[143,172,254],"Reckhow":[144],"proved":[145,247],"pigeonhole":[148],"principle":[149],"PHP":[150,161,184],"has":[151,200],"simple":[153],"polynomial":[154,165,179,193,197],"size":[155,166,180,194,198],"conjectured":[159],"admit":[164],"-proof.":[168],"[5],":[170],"Buss":[173,255],"refuted":[174],"this":[175],"conjecture":[176],"furnishing":[178],"for":[183],"Since":[186],"then":[187],"important":[189],"separation":[190],"problem":[191],"shown":[202],"progress.":[204],"[11],":[206],"introduced":[210,256,275],"PV":[213,241,249,270],"free":[216],"variable":[217],"equational":[218],"logic":[219],"provable":[221],"functional":[222],"equalities":[223],"are":[224],"\u2018polynomial":[225],"time":[226],"verifiable\u2019":[227],"showed":[229,262],"metamathematics":[232],"can":[237],"be":[238],"developed":[239],"soundness":[244],"[3],":[252],"first":[258],"order":[259,278],"essentially":[265],"conservative":[267],"There":[272],"he":[273],"also":[274],"second":[277],"(":[280],"BD":[281],").":[282]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}