{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T00:02:05Z","timestamp":1649030525079},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10968,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,3]]},"abstract":"<jats:p>In this paper, we establish an extended joint consistency theorem for an infinite family of free modal logics with equality. The extended joint consistency theorem incorporates the Craig and Lyndon interpolation lemmas and the Robinson joint consistency theorem. In part, the theorem states that two theories which are jointly unsatisfiable are separated by a sentence in the vocabulary common to both theories.<\/jats:p><jats:p>Our family of free modal logics includes the free versions of <jats:italic><jats:bold>I, M<\/jats:bold><\/jats:italic>, and <jats:bold><jats:italic>S4<\/jats:italic><\/jats:bold> studied by Leblanc [5, Chapters 8 and 9], supplemented with equality as in [3]. In the relational semantics for these logics, there is no restriction on the accessibility relation in <jats:italic><jats:bold>I<\/jats:bold><\/jats:italic>, while in <jats:italic>M(S4)<\/jats:italic> the restriction is reflexivity (refiexivity and transitivity). We say that a restriction on the accessibility relation <jats:italic>countenances backward-looping<\/jats:italic> if it implies a sentence of the form \u2200<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub> \u2026<jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub><jats:italic>Rx<\/jats:italic><jats:sub>2<\/jats:sub> &amp;\u2026&amp;<jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic> \u2283 <jats:italic>x<jats:sub>k<\/jats:sub>Rx<jats:sub>j<\/jats:sub><\/jats:italic>) (1 \u2264 <jats:italic>j<\/jats:italic> &lt; <jats:italic>k<\/jats:italic> \u2264 <jats:italic>n<\/jats:italic> \u2265 2), where the <jats:italic>x<jats:sub>i<\/jats:sub><\/jats:italic> (1 \u2264 <jats:italic>i<\/jats:italic> \u2264 <jats:italic>n<\/jats:italic>) are distinct individual variables. Just as reflexivity and transitivity do not countenance backward-looping, neither do any of the restrictions in our family of free modal logics. (The above terminology is derived from the effect of such restrictions on Kripke tableaux constructions.) The Barcan formula, its converse, the Fitch formula, and the formula <jats:italic>T<\/jats:italic> \u2260 <jats:italic>T<\/jats:italic>\u2032 \u2283 \u25a1<jats:italic>T<\/jats:italic> \u2260 <jats:italic>T<\/jats:italic>\u2032 do not hold in our logics.<\/jats:p>","DOI":"10.2307\/2274100","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:07:02Z","timestamp":1146938822000},"page":"174-183","source":"Crossref","is-referenced-by-count":3,"title":["An extended joint consistency theorem for a family of free modal logics with equality"],"prefix":"10.1017","volume":"49","author":[{"given":"Raymond D.","family":"Gumb","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200034034_ref006","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19820280102"},{"key":"S0022481200034034_ref005","volume-title":"Truth-value semantics","author":"Leblanc","year":"1976"},{"key":"S0022481200034034_ref001","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19790251302"},{"key":"S0022481200034034_ref002","first-page":"201","volume":"44","author":"Fine","year":"1979","journal-title":"Failures of the interpolation lemma in quantified modal logic"},{"key":"S0022481200034034_ref003","volume-title":"Evolving theories","author":"Gumb","year":"1979"},{"key":"S0022481200034034_ref004","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093882539"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200034034","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T17:26:43Z","timestamp":1558632403000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200034034\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,3]]},"references-count":6,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1984,3]]}},"alternative-id":["S0022481200034034"],"URL":"https:\/\/doi.org\/10.2307\/2274100","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1984,3]]}}}