{"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":1771544894960,"version":"3.50.1"},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7681,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1993,3]]},"abstract":"<jats:p>A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take a<jats:italic>model<\/jats:italic>to be one of these structures, \u03b1, together with some function or relation which associates with every formula<jats:italic>A<\/jats:italic>a subset of<jats:italic>\u03b1<\/jats:italic>. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of \u03b1 with which a formula can be associated: for example, in the semantics of Routley and Meyer [1973], a formula can only be associated with subsets which are closed upwards. It is natural to take a<jats:italic>proposition<\/jats:italic>of \u03b1 to be such a subset of \u03b1, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers the<jats:italic>primary<\/jats:italic>interpretation associated with the semantics.<\/jats:p>","DOI":"10.2307\/2275341","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:47:31Z","timestamp":1146955651000},"page":"334-349","source":"Crossref","is-referenced-by-count":20,"title":["Quantifying over propositions in relevance logic: nonaxiomatisability of primary interpretations of \u2200<i>p<\/i>and \u2203<i>p<\/i>"],"prefix":"10.1017","volume":"58","author":[{"given":"Philip","family":"Kremer","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200021800_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/BF00649991"},{"key":"S0022481200021800_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71260-6"},{"key":"S0022481200021800_ref001","volume-title":"Entailment, the logic of relevance and necessity","volume":"1","author":"Anderson","year":"1975"},{"key":"S0022481200021800_ref020","first-page":"159","volume":"37","author":"Urquhart","year":"1972","journal-title":"Semantics for relevant logics"},{"key":"S0022481200021800_ref003","volume-title":"Introduction to mathematical logic","author":"Church","year":"1956"},{"key":"S0022481200021800_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12898-5"},{"key":"S0022481200021800_ref004","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1111\/j.1755-2567.1970.tb00432.x","article-title":"Propositional quantifiers in modal logic","volume":"36","author":"Fine","year":"1970","journal-title":"Theoria"},{"key":"S0022481200021800_ref019","volume-title":"Relevant logics and their rivals 1. The basic philosophical and semantical theory","author":"Routley","year":"1982"},{"key":"S0022481200021800_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/BF00650498"},{"key":"S0022481200021800_ref002","volume-title":"Entailment, the logic of relevance and necessity","volume":"2","author":"Anderson","year":"1992"},{"key":"S0022481200021800_ref018","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/S0049-237X(08)71541-6","volume-title":"Truth, syntax and modality. Proceedings of the Temple University conference on alternative semantics","author":"Routley","year":"1973"},{"key":"S0022481200021800_ref012","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19550010205"},{"key":"S0022481200021800_ref005","first-page":"347","author":"Fine","year":"1974","journal-title":"Models for entailment"},{"key":"S0022481200021800_ref010","first-page":"1","volume":"24","author":"Kripke","year":"1959","journal-title":"A completeness theorem in modal logic"},{"key":"S0022481200021800_ref021","unstructured":"Urquhart Alasdair , [1973], The semantics of entailment, Doctoral dissertation, University of Pittsburg, University Microfilms, Ann Arbor, MI."},{"key":"S0022481200021800_ref022","volume-title":"Entailment, the logic of relevance and necessity","volume":"2","author":"Urquhart","year":"1992"},{"key":"S0022481200021800_ref014","volume-title":"Classical recursion theory: The theory of functions and sets of natural numbers","author":"Odifreddi","year":"1989"},{"key":"S0022481200021800_ref008","volume-title":"An introduction to modal logic","author":"Hughes","year":"1968"},{"key":"S0022481200021800_ref006","first-page":"81","volume":"15","author":"Henkin","journal-title":"Completeness in the theory of types"},{"key":"S0022481200021800_ref011","first-page":"92","volume-title":"Formal systems and recursive functions. Proceedings of the eighth logic colloquium","author":"Kripke","year":"1963"},{"key":"S0022481200021800_ref015","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1944-08111-1"},{"key":"S0022481200021800_ref009","first-page":"355","volume":"35","author":"Kaplan","year":"1970","journal-title":"S5 with quantifiable propositional variables"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200021800","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T07:27:36Z","timestamp":1707031656000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200021800\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,3]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1993,3]]}},"alternative-id":["S0022481200021800"],"URL":"https:\/\/doi.org\/10.2307\/2275341","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,3]]}}}