{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T13:13:54Z","timestamp":1778764434316,"version":"3.51.4"},"reference-count":6,"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":10784,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,9]]},"abstract":"<jats:p>In 1965 Zadeh introduced the concept of fuzzy sets. The characteristic of fuzzy sets is that the range of truth value of the membership relation is the closed interval [0, 1] of real numbers. The logical operations \u2283, \u223c on [0, 1] which are used for Zadeh's fuzzy sets seem to be \u0141ukasiewciz's logic, where <jats:italic>p<\/jats:italic> \u2283 <jats:italic>q<\/jats:italic> = min(1, 1 \u2212 <jats:italic>p<\/jats:italic> + <jats:italic>q<\/jats:italic>), \u223c <jats:italic>p<\/jats:italic> = 1 \u2212 <jats:italic>p<\/jats:italic>. L. S. Hay extended in [4] \u0141ukasiewicz's logic to a predicate logic and proved its weak completeness theorem: if <jats:italic>P<\/jats:italic> is valid then <jats:italic>P<\/jats:italic> + <jats:italic>P<jats:sup>n<\/jats:sup><\/jats:italic> is provable for each positive integer <jats:italic>n<\/jats:italic>. She also showed that one can without losing consistency obtain completeness of the system by use of additional infinitary rule.<\/jats:p><jats:p>Now, from a logical standpoint, each logic has its corresponding set theory in which each logical operation is translated into a basic operation for set theory; namely, the relation \u2286 and = on sets are translation of the logical operations \u2192 and \u2194. For \u0141ukasiewicz's logic, <jats:italic>P<\/jats:italic> \u039b (<jats:italic>P<\/jats:italic> \u2283 <jats:italic>Q<\/jats:italic>). \u2283 <jats:italic>Q<\/jats:italic> is not valid. Translating it to the set version, it follows that the axiom of extensionality does not hold. Thus this very basic principle of set theory is not valid in the corresponding set theory.<\/jats:p>","DOI":"10.2307\/2274139","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:09:38Z","timestamp":1146953378000},"page":"851-866","source":"Crossref","is-referenced-by-count":207,"title":["Intuitionistic fuzzy logic and intuitionistic fuzzy set theory"],"prefix":"10.1017","volume":"49","author":[{"given":"Gaisi","family":"Takeuti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Satoko","family":"Titani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200043000_ref004","first-page":"77","volume":"28","author":"Hay","year":"1963","journal-title":"Axiomatization of the infinite-valued predicate calculus"},{"key":"S0022481200043000_ref003","first-page":"402","volume-title":"Applications of sheaves (Proceedings of the research symposium, Durham, 1981)","volume":"753","author":"Grayson","year":"1979"},{"key":"S0022481200043000_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01201353"},{"key":"S0022481200043000_ref002","unstructured":"Grayson R. J. , A sheaf approach to models of set theory, M.Sc. Thesis, Oxford, 1975."},{"key":"S0022481200043000_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090986"},{"key":"S0022481200043000_ref005","first-page":"221","volume":"40","author":"Powell","year":"1975","journal-title":"Extending G\u00f6del's negative interpretation to ZF"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200043000","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T19:49:14Z","timestamp":1558640954000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200043000\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,9]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1984,9]]}},"alternative-id":["S0022481200043000"],"URL":"https:\/\/doi.org\/10.2307\/2274139","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1984,9]]}}}