{"id":"https://openalex.org/W2071369407","doi":"https://doi.org/10.2307/2273727","title":"Failures of the interpolation lemma in quantified modal logic","display_name":"Failures of the interpolation lemma in quantified modal logic","publication_year":1979,"publication_date":"1979-06-01","ids":{"openalex":"https://openalex.org/W2071369407","doi":"https://doi.org/10.2307/2273727","mag":"2071369407"},"language":"en","primary_location":{"id":"doi:10.2307/2273727","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273727","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/A5016451627","display_name":"Kit Fine","orcid":"https://orcid.org/0000-0002-9438-4900"},"institutions":[{"id":"https://openalex.org/I204250578","display_name":"University of California, Irvine","ror":"https://ror.org/04gyf1771","country_code":"US","type":"education","lineage":["https://openalex.org/I204250578"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Kit Fine","raw_affiliation_strings":["University of California, Irvine, California 92664"],"affiliations":[{"raw_affiliation_string":"University of California, Irvine, California 92664","institution_ids":["https://openalex.org/I204250578"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5016451627"],"corresponding_institution_ids":["https://openalex.org/I204250578"],"apc_list":null,"apc_paid":null,"fwci":6.8692,"has_fulltext":false,"cited_by_count":58,"citation_normalized_percentile":{"value":0.96464646,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"44","issue":"2","first_page":"201","last_page":"206"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9948999881744385,"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":0.9948999881744385,"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":0.9936000108718872,"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":0.9850000143051147,"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/modal-logic","display_name":"Modal logic","score":0.7225227952003479},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.6893801689147949},{"id":"https://openalex.org/keywords/lemma","display_name":"Lemma (botany)","score":0.5937947034835815},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5800472497940063},{"id":"https://openalex.org/keywords/unary-operation","display_name":"Unary operation","score":0.5599798560142517},{"id":"https://openalex.org/keywords/interpolation","display_name":"Interpolation (computer graphics)","score":0.5567509531974792},{"id":"https://openalex.org/keywords/accessibility-relation","display_name":"Accessibility relation","score":0.550236165523529},{"id":"https://openalex.org/keywords/multimodal-logic","display_name":"Multimodal logic","score":0.477192759513855},{"id":"https://openalex.org/keywords/modal","display_name":"Modal","score":0.4720899760723114},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4617123603820801},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.4555266499519348},{"id":"https://openalex.org/keywords/normal-modal-logic","display_name":"Normal modal logic","score":0.45186880230903625},{"id":"https://openalex.org/keywords/intermediate-logic","display_name":"Intermediate logic","score":0.43954014778137207},{"id":"https://openalex.org/keywords/dynamic-logic","display_name":"Dynamic logic (digital electronics)","score":0.43588605523109436},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.39594385027885437},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.37026143074035645},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2654293179512024},{"id":"https://openalex.org/keywords/description-logic","display_name":"Description logic","score":0.15535369515419006},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.1441488265991211},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.13031089305877686}],"concepts":[{"id":"https://openalex.org/C27508121","wikidata":"https://www.wikidata.org/wiki/Q210841","display_name":"Modal logic","level":3,"score":0.7225227952003479},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.6893801689147949},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.5937947034835815},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5800472497940063},{"id":"https://openalex.org/C78023250","wikidata":"https://www.wikidata.org/wiki/Q657596","display_name":"Unary operation","level":2,"score":0.5599798560142517},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.5567509531974792},{"id":"https://openalex.org/C11191006","wikidata":"https://www.wikidata.org/wiki/Q2663311","display_name":"Accessibility relation","level":4,"score":0.550236165523529},{"id":"https://openalex.org/C203659156","wikidata":"https://www.wikidata.org/wiki/Q6934805","display_name":"Multimodal logic","level":3,"score":0.477192759513855},{"id":"https://openalex.org/C71139939","wikidata":"https://www.wikidata.org/wiki/Q910194","display_name":"Modal","level":2,"score":0.4720899760723114},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4617123603820801},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4555266499519348},{"id":"https://openalex.org/C114092440","wikidata":"https://www.wikidata.org/wiki/Q840226","display_name":"Normal modal logic","level":4,"score":0.45186880230903625},{"id":"https://openalex.org/C33203268","wikidata":"https://www.wikidata.org/wiki/Q5361594","display_name":"Intermediate logic","level":3,"score":0.43954014778137207},{"id":"https://openalex.org/C2777796570","wikidata":"https://www.wikidata.org/wiki/Q2351326","display_name":"Dynamic logic (digital electronics)","level":4,"score":0.43588605523109436},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.39594385027885437},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.37026143074035645},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2654293179512024},{"id":"https://openalex.org/C102993220","wikidata":"https://www.wikidata.org/wiki/Q387196","display_name":"Description logic","level":2,"score":0.15535369515419006},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.1441488265991211},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.13031089305877686},{"id":"https://openalex.org/C46757340","wikidata":"https://www.wikidata.org/wiki/Q43238","display_name":"Poaceae","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C172385210","wikidata":"https://www.wikidata.org/wiki/Q5339","display_name":"Transistor","level":3,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C188027245","wikidata":"https://www.wikidata.org/wiki/Q750446","display_name":"Polymer chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C165801399","wikidata":"https://www.wikidata.org/wiki/Q25428","display_name":"Voltage","level":2,"score":0.0},{"id":"https://openalex.org/C104114177","wikidata":"https://www.wikidata.org/wiki/Q79782","display_name":"Motion (physics)","level":2,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2273727","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273727","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"},{"id":"pmh:oai:CULeuclid:euclid.jsl/1183740386","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183740386","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":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W1970175046","https://openalex.org/W2029598760","https://openalex.org/W2063167657","https://openalex.org/W2495091526","https://openalex.org/W3144889436","https://openalex.org/W6793045953"],"related_works":["https://openalex.org/W3108798275","https://openalex.org/W2795425183","https://openalex.org/W176661328","https://openalex.org/W3126675290","https://openalex.org/W2952939288","https://openalex.org/W2260113151","https://openalex.org/W2901238796","https://openalex.org/W2963628981","https://openalex.org/W4310201982","https://openalex.org/W4233435269"],"abstract_inverted_index":{"Beth's":[0],"Definability":[1],"Theorem,":[2],"and":[3,84,115,123,188],"consequently":[4],"the":[5,10,25,73,78,92,96,133,155],"Interpolation":[6,74],"Lemma,":[7],"fail":[8],"for":[9,77,85,95,137,157,169,182],"version":[11],"of":[12,46,87,121],"quantified":[13,64,138],"S5":[14,39],"that":[15],"is":[16,36,60,127,152,177],"presented":[17],"in":[18,61,146],"Kripke's":[19],"[6].":[20],"These":[21],"failures":[22],"persist":[23],"when":[24],"constant":[26,82],"domain":[27],"axiom-scheme":[28],"\u2200":[29],"x":[30,34,186,194],"\u25a1\u03c6":[31],"\u2261":[32],"\u25a1\u2200":[33],"\u03c6":[35],"added":[37],"to":[38,42,63],"or,":[40],"indeed,":[41],"any":[43],"weaker":[44],"extension":[45],"quantificational":[47,56,125],"K":[48],".":[49],"\u00a71":[50],"reviews":[51],"some":[52,112],"standard":[53],"material":[54],"on":[55],"modal":[57,124,139],"logic.":[58,140],"This":[59],"contrast":[62],"intermediate":[65],"logics":[66],"for,":[67],"as":[68,179,192],"Gabbay":[69],"[6]":[70],"has":[71],"shown,":[72],"Lemma":[75],"holds":[76],"logic":[79,126,160],"CD":[80],"with":[81,161],"domains":[83],"several":[86],"its":[88],"extensions.":[89],"\u00a7\u00a72\u20144":[90],"establish":[91],"negative":[93,106],"results":[94,114],"systems":[97],"based":[98],"upon":[99],"S5.":[100],"\u00a75":[101],"establishes":[102],"a":[103,165],"more":[104],"general":[105],"result":[107],"and,":[108],"finally,":[109],"\u00a76":[110],"considers":[111],"positive":[113],"open":[116],"problems.":[117],"A":[118],"basic":[119],"knowledge":[120],"classical":[122,158],"presupposed.":[128],"Let":[129],"me":[130],"briefly":[131],"review":[132],"relevant":[134],"model":[135],"theory":[136],"Further":[141],"details":[142],"can":[143],"be":[144,190],"found":[145],"[3]":[147],"or":[148],"[7].":[149],"The":[150,171],"language":[151,156],"obtained":[153],"from":[154],"first-order":[159],"identity":[162],"by":[163],"adding":[164],"unary":[166],"operator":[167],"\u25a1":[168],"necessity.":[170],"atomic":[172],"formula":[173],"\u2018":[174,193],"Ex":[175],"\u2019":[176],"used":[178],"an":[180],"abbreviation":[181],"\u2018\u2203":[183],"y(y":[184],"=":[185],")\u2019":[187],"may":[189],"read":[191],"exists\u2019.":[195]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":3},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
