{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T23:53:39Z","timestamp":1770508419320,"version":"3.49.0"},"reference-count":19,"publisher":"Wiley","issue":"1-2","license":[{"start":{"date-parts":[[2015,2,2]],"date-time":"2015-02-02T00:00:00Z","timestamp":1422835200000},"content-version":"vor","delay-in-days":1,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2015,2]]},"abstract":"<jats:p>We study a real valued propositional logic with unbounded positive and negative truth values that we call <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300068-math-0003.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300068:malq201300068-math-0003\"\/><jats:italic>\u2010valued logic<\/jats:italic>. Such a logic is semantically equivalent to continuous propositional logic, with a different choice of connectives. After presenting the deduction machinery and the semantics of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300068-math-0004.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300068:malq201300068-math-0004\"\/>\u2010valued logic, we prove a completeness theorem for finite theories. Then we define unital and Archimedean theories, in accordance with the theory of Riesz spaces. In the unital setting, we prove the equivalence of consistency and satisfiability and an approximated completeness theorem similar to the one that holds for continuous propositional logic. Eventually, among unital theories, we characterize Archimedean theories as those for which strong completeness holds. We also point out that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201300068-math-0005.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201300068:malq201300068-math-0005\"\/>\u2010valued logic provides alternative calculi for \u0141ukasiewicz logic and for propositional continuous logic.<\/jats:p>","DOI":"10.1002\/malq.201300068","type":"journal-article","created":{"date-parts":[[2015,2,2]],"date-time":"2015-02-02T09:50:59Z","timestamp":1422870659000},"page":"32-44","source":"Crossref","is-referenced-by-count":2,"title":["The eal truth"],"prefix":"10.1002","volume":"61","author":[{"given":"Stefano","family":"Baratella","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica Universit\u00e0 di Trento via Sommarive 14 38123 Povo Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Domenico","family":"Zambella","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica Universit\u00e0 di Torino  via Carlo Alberto 10 10123 Torino Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2015,2,2]]},"reference":[{"key":"e_1_2_7_2_1","volume-title":"Positive Operators, Pure and Applied Mathematics Vol. 119","author":"Aliprantis C. 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