{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T06:41:50Z","timestamp":1757313710373,"version":"3.37.3"},"reference-count":19,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2021,2,22]],"date-time":"2021-02-22T00:00:00Z","timestamp":1613952000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100003977","name":"The Israel Science Foundation","doi-asserted-by":"publisher","award":["grant no. 550\/18"],"award-info":[{"award-number":["grant no. 550\/18"]}],"id":[{"id":"10.13039\/501100003977","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,5,21]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The logic $G3^{&amp;lt;}_{{{}^{\\scriptsize{-}}}\\!\\!\\textrm{L}}$ was introduced in Robles and Mend\u00e9z (2014, Logic Journal of the IGPL, 22, 515\u2013538) as a paraconsistent logic which is based on G\u00f6del\u2019s 3-valued matrix, except that Kleene\u2013\u0141ukasiewicz\u2019s negation is added to the language and is used as the main negation connective. We show that $G3^{&amp;lt;}_{{{}^{\\scriptsize{-}}}\\!\\!\\textrm{L}}$ is exactly the intersection of $G3^{\\{1\\}}_{{{}^{\\scriptsize{-}}}\\!\\!\\textrm{L}}$ and $G3^{\\{1,0.5\\}}_{{{}^{\\scriptsize{-}}}\\!\\!\\textrm{L}}$, the two truth-preserving 3-valued logics which are based on the same truth tables. (In $G3^{\\{1\\}}_{{{}^{\\scriptsize{-}}}\\!\\!\\textrm{L}}$ the set ${\\cal D}$ of designated elements is $\\{1\\}$, while in $G3^{\\{1,0.5\\}}_{{{}^{\\scriptsize{-}}}\\!\\!\\textrm{L}}$ \u00a0${\\cal D}=\\{1,0.5\\}$.) We then construct a Hilbert-type system which has (MP) for $\\to $ as its sole rule of inference, and is strongly sound and complete for $G3^{&amp;lt;}_{{{}^{\\scriptsize{-}}}\\!\\!\\textrm{L}}$. Then we show how, by adding one axiom (in the case of $G3^{\\{1\\}}_{{{}^{\\scriptsize{-}}}\\!\\!\\textrm{L}}$) or one new rule of inference (in the case of $G3^{\\{1,0.5\\}}_{{{}^{\\scriptsize{-}}}\\!\\!\\textrm{L}}$), we get strongly sound and complete systems for $G3^{\\{1\\}}_{{{}^{\\scriptsize{-}}}\\!\\!\\textrm{L}}$ and $G3^{\\{1,0.5\\}}_{{{}^{\\scriptsize{-}}}\\!\\!\\textrm{L}}$. Finally, we provide quasi-canonical Gentzen-type systems which are sound and complete for those logics and show that they are all analytic, by proving the cut-elimination theorem for them.<\/jats:p>","DOI":"10.1093\/jigpal\/jzab013","type":"journal-article","created":{"date-parts":[[2021,1,31]],"date-time":"2021-01-31T12:06:34Z","timestamp":1612094794000},"page":"437-453","source":"Crossref","is-referenced-by-count":3,"title":["Proof Systems for 3-valued Logics Based on G\u00f6del\u2019s Implication"],"prefix":"10.1093","volume":"30","author":[{"given":"Arnon","family":"Avron","sequence":"first","affiliation":[{"name":"School of Computer Science, Tel-Aviv University, Israel"}]}],"member":"286","published-online":{"date-parts":[[2021,2,22]]},"reference":[{"volume-title":"Entailment","year":"1975","author":"Anderson","key":"2022052312303611100_ref1"},{"key":"2022052312303611100_ref2","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1093\/logcom\/9.6.977","article-title":"On the expressive power of three-valued and four-valued languages","volume":"9","author":"Avron","year":"1999","journal-title":"Journal of Logic and Computation"},{"key":"2022052312303611100_ref3","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1080\/11663081.2014.911515","article-title":"Paraconsistency, paracompleteness, Gentzen systems and trivalent semantics","volume":"24","author":"Avron","year":"2014","journal-title":"Journal of Applied Non-classical Logics"},{"key":"2022052312303611100_ref4","article-title":"Quasi-canonical systems and their semantics","author":"Avron","year":"2018","journal-title":"Synthese"},{"volume-title":"Theory of Effective Propositional Paraconsitent Logics","year":"2018","author":"Avron","key":"2022052312303611100_ref5"},{"key":"2022052312303611100_ref6","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1093\/logcom\/exi001","article-title":"Non-deterministic multi-valued structures","volume":"15","author":"Avron","year":"2005","journal-title":"Journal of Logic and Computation"},{"key":"2022052312303611100_ref7","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1093\/jigpal\/jzy064","article-title":"Paraconsistency, self-extensionality, modality","volume":"28","author":"Avron","year":"2020","journal-title":"The Logic Journal of the IGPL"},{"key":"2022052312303611100_ref8","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/978-3-642-35722-0_9","article-title":"Automated support for the investigation of paraconsistent and other logics","volume-title":"Logical Foundations of Computer Science","author":"Ciabattoni","year":"2013"},{"key":"2022052312303611100_ref9","first-page":"31","article-title":"The completeness and compactness of a three-valued first-order logic","volume":"XIX","author":"D\u2019Ottaviano","year":"1985","journal-title":"Revista Colombiana de Matematicas"},{"key":"2022052312303611100_ref10","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-0525-2","volume-title":"The Semantic Foundation of Logic. 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