{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T23:48:14Z","timestamp":1771544894955,"version":"3.50.1"},"reference-count":8,"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":4210,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A propositional system of modal logic is<jats:italic>second-order<\/jats:italic>if it contains quantifiers \u2200<jats:sub>p<\/jats:sub>and \u2203<jats:sub>p<\/jats:sub>which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.<\/jats:p><jats:p>In this paper we generalize this framework by allowing multiple modalities. While this does not affect the undecidability of K, B, T, K4 and S4, poly-modal second-order S5 is dramatically more expressive than its mono-modal counterpart. As an example, we establish the definability of the transitive closure of finitely many modal operators. We also take up the decidability issue, and, using a novel encoding of sets of unordered pairs by partitions of the leaves of certain graphs, we show that the second-order propositional logic of two S5 modalitities is also equivalent to full second-order logic.<\/jats:p>","DOI":"10.2178\/jsl\/1190150147","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:14:26Z","timestamp":1197573266000},"page":"1039-1054","source":"Crossref","is-referenced-by-count":12,"title":["Representability in second-order propositional poly-modal logic"],"prefix":"10.1017","volume":"67","author":[{"given":"G. Aldo","family":"Antonelli","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Richmond H.","family":"Thomason","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200009403_ref004","first-page":"355","volume":"35","author":"Kaplan","year":"1970","journal-title":"S5 with quantifiable propositional variables"},{"key":"S0022481200009403_ref002","doi-asserted-by":"crossref","DOI":"10.7551\/mitpress\/5803.001.0001","volume-title":"Reasoning about knowledge","author":"Fagin","year":"1995"},{"key":"S0022481200009403_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71260-6"},{"key":"S0022481200009403_ref007","first-page":"58","volume-title":"Logic, methodology and philosophy of science. Proceedings of the 1964 international congress","author":"Rabin","year":"1965"},{"key":"S0022481200009403_ref001","volume-title":"Solvable cases of the decision problem","author":"Ackermann","year":"1954"},{"key":"S0022481200009403_ref003","first-page":"336","volume-title":"Theoria","author":"Fine","year":"1970"},{"key":"S0022481200009403_ref008","volume-title":"Undecidable theories","author":"Tarski","year":"1953"},{"key":"S0022481200009403_ref005","first-page":"529","volume":"62","author":"Kremer","year":"1997","journal-title":"On the complexity of propositional quantification in intuitionistic logic"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200009403","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T17:51:52Z","timestamp":1737654712000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200009403\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,9]]},"references-count":8,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2002,9]]}},"alternative-id":["S0022481200009403"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150147","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,9]]}}}