{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T11:15:16Z","timestamp":1778670916383,"version":"3.51.4"},"reference-count":30,"publisher":"Cambridge University Press (CUP)","issue":"4-5","license":[{"start":{"date-parts":[[2015,9,3]],"date-time":"2015-09-03T00:00:00Z","timestamp":1441238400000},"content-version":"unspecified","delay-in-days":64,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Theory and Practice of Logic Programming"],"published-print":{"date-parts":[[2015,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we provide an alternative semantics for Equilibrium Logic and its monotonic basis, the logic of Here-and-There (also known as G\u00f6del's<jats:italic>G<\/jats:italic><jats:sub>3<\/jats:sub>logic) that relies on the idea of<jats:italic>denotation<\/jats:italic>of a formula, that is, a function that collects the set of models of that formula. Using the three-valued logic<jats:italic>G<\/jats:italic><jats:sub>3<\/jats:sub>as a starting point and an ordering relation (for which equilibrium\/stable models are minimal elements) we provide several elementary operations for sets of interpretations. By analysing structural properties of the denotation of formulas, we show some expressiveness results for<jats:italic>G<\/jats:italic><jats:sub>3<\/jats:sub>such as, for instance, that conjunction is not expressible in terms of the other connectives. Moreover, the denotational semantics allows us to capture the set of equilibrium models of a formula with a simple and compact set expression. We also use this semantics to provide several formal definitions for entailment relations that are usual in the literature, and further introduce a new one called<jats:italic>strong entailment<\/jats:italic>. We say that \u03b1 strongly entails \u03b2 when the equilibrium models of \u03b1 \u2227 \u03b3 are also equilibrium models of \u03b2 \u2227 \u03b3 for any context \u03b3. We also provide a characterisation of strong entailment in terms of the denotational semantics, and give an example of a sufficient condition that can be applied in some cases.<\/jats:p>","DOI":"10.1017\/s1471068415000277","type":"journal-article","created":{"date-parts":[[2015,9,3]],"date-time":"2015-09-03T08:21:21Z","timestamp":1441268481000},"page":"620-634","source":"Crossref","is-referenced-by-count":9,"title":["A denotational semantics for equilibrium logic"],"prefix":"10.1017","volume":"15","author":[{"given":"FELICIDAD","family":"AGUADO","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"PEDRO","family":"CABALAR","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"DAVID","family":"PEARCE","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"GILBERTO","family":"P\u00c9REZ","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"CONCEPCI\u00d3N","family":"VIDAL","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2015,9,3]]},"reference":[{"key":"S1471068415000277_ref26","doi-asserted-by":"crossref","unstructured":"Pearce D. and Valverde A. 2004a. Towards a first order equilibrium logic for nonmonotonic reasoning. 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Conf. on Computational and Mathematical Methods in Science and Engineering CMMSE 2015 (July 3-7)","author":"Aguado","year":"2015"},{"key":"S1471068415000277_ref16","unstructured":"Heyting A. 1930. Die formalen Regeln der intuitionistischen Logik. 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