{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T22:20:38Z","timestamp":1786054838802,"version":"3.56.0"},"reference-count":4,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":3479,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,9]]},"abstract":"<jats:p>We begin with a prepositional language<jats:italic>L<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub>containing conjunction (\u039b), a class of sentence names {<jats:italic>S<\/jats:italic><jats:italic>\u03b1<\/jats:italic>}<jats:sub><jats:italic>\u03b1\u03f5A<\/jats:italic><\/jats:sub>, and a falsity predicate<jats:italic>F<\/jats:italic>. We (only) allow unrestricted infinite conjunctions, i.e., given any non-empty class of sentence names {<jats:italic>S<\/jats:italic><jats:sub><jats:italic>\u03b2<\/jats:italic><\/jats:sub>}<jats:sub><jats:italic>\u03b2\u03f5B<\/jats:italic><\/jats:sub>,<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200007611_Uequ1\"\/><\/jats:disp-formula><\/jats:p><jats:p>is a well-formed formula (we will use<jats:italic>WFF<\/jats:italic>to denote the set of well-formed formulae).<\/jats:p><jats:p>The language, as it stands, is unproblematic. Whether various paradoxes are produced depends on which names are assigned to which sentences. What is needed is a denotation function:<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200007611_Uequ2\"\/><\/jats:disp-formula><\/jats:p><jats:p>For example, the<jats:italic>L<\/jats:italic><jats:sub><jats:italic>P<\/jats:italic><\/jats:sub>sentence \u201c<jats:italic>F<\/jats:italic>(<jats:italic>S<\/jats:italic><jats:sub>1<\/jats:sub>)\u201d (i.e.,<jats:italic>\u039b<\/jats:italic>{<jats:italic>F<\/jats:italic>(<jats:italic>S<\/jats:italic><jats:sub>1<\/jats:sub>)}), combined with a denotation function<jats:italic>\u03b4<\/jats:italic>such that<jats:italic>\u03b4<\/jats:italic>(<jats:italic>S<\/jats:italic><jats:sub>1<\/jats:sub>)\u201c<jats:italic>F<\/jats:italic>(<jats:italic>S<\/jats:italic><jats:sub>1<\/jats:sub>)\u201d, provides the (or, in this context, a)<jats:italic>Liar Paradox<\/jats:italic>.<\/jats:p><jats:p>To give a more interesting example,<jats:italic>Yablo's Paradox<\/jats:italic>[4] can be reconstructed within this framework.<jats:italic>Yablo's Paradox<\/jats:italic>consists of an \u03c9-sequence of sentences {<jats:italic>S<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>}<jats:sub><jats:italic>k\u03f5\u03c9<\/jats:italic><\/jats:sub>where, for each<jats:italic>n<\/jats:italic>\u03f5<jats:italic>\u03c9<\/jats:italic>:<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200007611_Uequ3\"\/><\/jats:disp-formula><\/jats:p><jats:p>Within<jats:italic>L<\/jats:italic><jats:sub><jats:italic>P<\/jats:italic><\/jats:sub>an equivalent construction can be obtained using infinite conjunction in place of universal quantification - the sentence names are {<jats:italic>S<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>}<jats:sub><jats:italic>i<\/jats:italic><jats:italic>\u03f5\u03c9<\/jats:italic><\/jats:sub>and the denotation function is given by:<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200007611_Uequ4\"\/><\/jats:disp-formula><\/jats:p><jats:p>We can express this in more familiar terms as:<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200007611_Uequ5\"\/><\/jats:disp-formula><\/jats:p><jats:p>etc.<\/jats:p>","DOI":"10.2178\/jsl\/1096901765","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:42:34Z","timestamp":1109799754000},"page":"767-774","source":"Crossref","is-referenced-by-count":34,"title":["Patterns of paradox"],"prefix":"10.1017","volume":"69","author":[{"given":"Roy T.","family":"Cook","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200007611_ref001","volume-title":"Graphs","author":"Berge","year":"1985"},{"key":"S0022481200007611_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/2027167"},{"key":"S0022481200007611_ref003","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534501.001.0001","volume-title":"Diagonalization and Self-reference","author":"Smullyan","year":"1994"},{"key":"S0022481200007611_ref004","doi-asserted-by":"publisher","DOI":"10.1093\/analys\/53.4.251"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200007611","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,23]],"date-time":"2024-01-23T22:45:59Z","timestamp":1706049959000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200007611\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,9]]},"references-count":4,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2004,9]]}},"alternative-id":["S0022481200007611"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1096901765","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,9]]}}}