{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,4]],"date-time":"2023-05-04T04:07:42Z","timestamp":1683173262090},"reference-count":3,"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":13798,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1976,6]]},"abstract":"<jats:p>This note describes a simple interpretation * of modal first-order languages <jats:italic>K<\/jats:italic> with but finitely many predicates in derived classical second-order languages <jats:italic>L<\/jats:italic>(<jats:italic>K<\/jats:italic>) such that if \u0393 is a set of <jats:italic>K<\/jats:italic>-formulae, \u0393 is satisfiable (according to Kripke's 55 semantics) iff \u0393* is satisfiable (according to standard (or nonstandard) second-order semantics).<\/jats:p><jats:p>The motivation for the interpretation is roughly as follows. Consider the \u201ctrue\u201d modal semantics, in which the relative possibility relation is universal. Here the necessity operator can be considered a universal quantifier over possible worlds. A possible world itself can be identified with an assignment of extensions to the predicates and of a range to the quantifiers; if the quantifiers are first relativized to an existence predicate, a possible world becomes simply an assignment of extensions to the predicates. Thus the necessity operator can be taken to be a universal quantifier over a class of assignments of extensions to the predicates. So if these predicates are regarded as naming functions from extensions to extensions, the necessity operator can be taken as a string of universal quantifiers over extensions.<\/jats:p><jats:p>The alphabet of a \u201cfinite\u201d modal first-order language <jats:italic>K<\/jats:italic> shall consist of a non-empty countable set Var of individual variables, a nonempty finite set Pred of predicates, the logical symbols \u2018\u00ac\u2019 \u2018\u2227\u2019, and \u2018\u2227\u2019, and the operator \u2018\u25ca\u2019. The formation rules of <jats:italic>K<\/jats:italic> generate the usual Polish notations as <jats:italic>K<\/jats:italic>-formulae. \u2018\u03bd\u2019, \u2018\u03bd<jats:sub>1<\/jats:sub>\u2019, \u2026 range over Var, \u2018<jats:italic>P<\/jats:italic>\u2019 over Pred, \u2018<jats:italic>A<\/jats:italic>\u2019 over <jats:italic>K<\/jats:italic>-formulae, and \u2018\u0393\u2019 over sets of <jats:italic>K<\/jats:italic>-formulae.<\/jats:p>","DOI":"10.1017\/s0022481200051392","type":"journal-article","created":{"date-parts":[[2014,3,13]],"date-time":"2014-03-13T12:40:02Z","timestamp":1394714402000},"page":"337-340","source":"Crossref","is-referenced-by-count":0,"title":["An interpretation of \u201cfinite\u201d modal first-order languages in classical second-order languages"],"prefix":"10.1017","volume":"41","author":[{"given":"Scott K.","family":"Lehmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200051392_ref002","first-page":"716","volume":"37","author":"Thomason","year":"1972","journal-title":"Noncompactness in prepositional modal logic"},{"key":"S0022481200051392_ref003","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19660120131"},{"key":"S0022481200051392_ref001","first-page":"83","article-title":"Semantical considerations on modal logic","volume":"16","author":"Kripke","year":"1963","journal-title":"Acta Philosophica Fennica"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200051392","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,3]],"date-time":"2023-05-03T06:00:59Z","timestamp":1683093659000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200051392\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1976,6]]},"references-count":3,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1976,6]]}},"alternative-id":["S0022481200051392"],"URL":"https:\/\/doi.org\/10.1017\/s0022481200051392","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1976,6]]}}}