{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T07:49:35Z","timestamp":1775461775018,"version":"3.50.1"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12703,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,6]]},"abstract":"<jats:p>In [5] Henkin defined a quantifier, which we shall denote by <jats:italic>Q<jats:sub>H<\/jats:sub><\/jats:italic>: linking four variables in one formula. This quantifier is related to the notion of formulas in which the usual universal and existential quantifiers occur but are not linearly ordered. The original definition of <jats:italic>Q<jats:sub>H<\/jats:sub><\/jats:italic> was<\/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=\"S0022481200048908_eqn0U1\"\/><\/jats:disp-formula><\/jats:p><jats:p>Here (<jats:italic>Q<jats:sub>H<\/jats:sub>x<\/jats:italic><jats:sub>1<\/jats:sub><jats:italic>x<\/jats:italic><jats:sub>2<\/jats:sub><jats:italic>y<\/jats:italic><jats:sub>1<\/jats:sub><jats:italic>y<\/jats:italic><jats:sub>2<\/jats:sub>)\u03c6 is true if for every <jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub> there exists <jats:italic>y<\/jats:italic><jats:sub>1<\/jats:sub> such that for every <jats:italic>x<\/jats:italic><jats:sub>2<\/jats:sub> there exists <jats:italic>y<\/jats:italic><jats:sub>2<\/jats:sub>, whose choice depends only on <jats:italic>x<\/jats:italic><jats:sub>2<\/jats:sub> not on <jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>y<\/jats:italic><jats:sub>1<\/jats:sub> such that \u03c6(<jats:italic>x<\/jats:italic><jats:sup>14<\/jats:sup>, <jats:italic>x<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>y<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>y<\/jats:italic><jats:sub>2<\/jats:sub>). Another way of writing this is<\/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=\"S0022481200048908_eqnU10\"\/><\/jats:disp-formula><\/jats:p><jats:p>In [5] it was observed that the logic <jats:italic>L(Q<jats:sub>H<\/jats:sub>)<\/jats:italic> obtained by adjoining <jats:italic>Q<jats:sub>H<\/jats:sub><\/jats:italic> defined as in (1) is more powerful than first-order logic. In particular, it turned out that the quantifier \u201cthere exist infinitely many\u201d denoted <jats:italic>Q<\/jats:italic><jats:sub>0<\/jats:sub> was definable from <jats:italic>Q<jats:sub>H<\/jats:sub><\/jats:italic> because<\/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=\"S0022481200048908_eqnU2\"\/><\/jats:disp-formula><\/jats:p>","DOI":"10.2307\/2273726","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:50:09Z","timestamp":1146937809000},"page":"184-200","source":"Crossref","is-referenced-by-count":26,"title":["On the semantics of the Henkin quantifier"],"prefix":"10.1017","volume":"44","author":[{"given":"Micha\u0142","family":"Krynicki","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alistair H.","family":"Lachlan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048908_ref008","first-page":"153","article-title":"Quantifiers versus quantification theory","volume":"5","author":"Hintikka","year":"1974","journal-title":"Linguistic Enquiry"},{"key":"S0022481200048908_ref005","first-page":"167","volume-title":"Infinitistic methods","author":"Henkin","year":"1961"},{"key":"S0022481200048908_ref014","first-page":"395","volume-title":"Comptes-Rendus du I Congr\u00e8s des Math\u00e9maticiens des Pays Slaves","author":"Presburger","year":"1929"},{"key":"S0022481200048908_ref018","volume-title":"Recursive function theory and logic","author":"Yasuhara","year":"1971"},{"key":"S0022481200048908_ref004","first-page":"31","volume-title":"Colloquium on the Foundations of Mathematics, Mathematical Machines and their Applications","author":"H\u00e4rtig","year":"1965"},{"key":"S0022481200048908_ref002","volume-title":"Models and ultraproducts","author":"Bell","year":"1969"},{"key":"S0022481200048908_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(74)90016-3"},{"key":"S0022481200048908_ref020","doi-asserted-by":"publisher","DOI":"10.4064\/fm-66-1-147-152"},{"key":"S0022481200048908_ref003","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19700160802"},{"key":"S0022481200048908_ref009","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19690152301"},{"key":"S0022481200048908_ref010","unstructured":"Krynicki M. , O pewnych logikach z dodatkowym kwantyfikatorem, Master's Thesis, University of Warsaw, 1973."},{"key":"S0022481200048908_ref013","first-page":"563","volume":"32","author":"L\u00f6b","year":"1967","journal-title":"Decidability of the monadic predicate calculus with unary function symbols"},{"key":"S0022481200048908_ref015","first-page":"269","volume-title":"Proceedings of the Summer School in Logic, Leeds, 1967, Lecture Notes in Mathematics","author":"Slomson","year":"1968"},{"key":"S0022481200048908_ref016","first-page":"535","volume":"35","author":"Walkoe","year":"1970","journal-title":"Finite partially ordered quantifiers"},{"key":"S0022481200048908_ref007","first-page":"61","article-title":"Reductions in the theory of types","volume":"8","author":"Hintikka","year":"1955","journal-title":"Acta Philosophica Fennica"},{"key":"S0022481200048908_ref019","first-page":"633","volume":"31","author":"Yasuhara","year":"1966","journal-title":"An axiomatic system for first-order languages with an equicardinality quantifier"},{"key":"S0022481200048908_ref017","unstructured":"Weese M. , The undecidability of the theory of well ordering with the quantifier. I (to appear)."},{"key":"S0022481200048908_ref006","unstructured":"Herre H. , Entscheidungsprobleme f\u00fcr Theorien in Logiken mit veralgemeinerten Quantoren, issertation zur Erlangung des akademischen Grades doctor Scientiae naturalis, Humboldt-Universit\u00e4t zu Berlin, 1974."},{"key":"S0022481200048908_ref011","unstructured":"Krynicki M. , O pewnych rozszeneniach logiki L\u03c9\u03c9 , Doctoral Thesis, University of Warsaw, 1976."},{"key":"S0022481200048908_ref012","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1969.tb00356.x"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048908","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T17:20:56Z","timestamp":1558891256000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048908\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,6]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1979,6]]}},"alternative-id":["S0022481200048908"],"URL":"https:\/\/doi.org\/10.2307\/2273726","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,6]]}}}