{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T06:39:01Z","timestamp":1757313541253},"reference-count":34,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2021,6,23]],"date-time":"2021-06-23T00:00:00Z","timestamp":1624406400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,5,21]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we study two companions of a logic, viz., the left variable inclusion companion and the restricted rules companion, their nature and interrelations, especially in connection with paraconsistency. A sufficient condition for the two companions to coincide has also been proved. Two new logical systems\u2014intuitionistic paraconsistent weak Kleene logic (IPWK) and paraconsistent pre-rough logic (PPRL)\u2014are presented here as examples of logics of left variable inclusion. IPWK is the left variable inclusion companion of intuitionistic propositional logic and is also the restricted rules companion of it. PPRL, on the other hand, is the left variable inclusion companion of pre-rough logic but differs from the restricted rules companion of it. We have discussed algebraic semantics for these logics in terms of P\u0142onka sums. This amounts to introducing a contaminating truth value, intended to denote a state of indeterminacy.<\/jats:p>","DOI":"10.1093\/jigpal\/jzab019","type":"journal-article","created":{"date-parts":[[2021,5,8]],"date-time":"2021-05-08T08:26:47Z","timestamp":1620462407000},"page":"534-560","source":"Crossref","is-referenced-by-count":5,"title":["Restricted Rules of Inference and Paraconsistency"],"prefix":"10.1093","volume":"30","author":[{"given":"Sankha S","family":"Basu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Indraprastha Institute of Information Technology\u2013Delhi, New Delhi 110020, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mihir K","family":"Chakraborty","sequence":"additional","affiliation":[{"name":"School of Cognitive Science, Jadavpur University, Kolkata 700032, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2021,6,23]]},"reference":[{"volume-title":"Entailment: The Logic of Relevance and Necessity","year":"1975","author":"Anderson","key":"2022052312313464100_ref1"},{"key":"2022052312313464100_ref2","doi-asserted-by":"crossref","DOI":"10.1007\/978-81-322-2719-9_4","article-title":"Three-valued paraconsistent propositional logics","volume-title":"New Directions in Paraconsistent Logic","author":"Arieli","year":"2015"},{"key":"2022052312313464100_ref3","doi-asserted-by":"crossref","first-page":"276","DOI":"10.2307\/2274919","article-title":"Natural 3-valued logics\u2014characterization and proof theory","volume":"56","author":"Avron","year":"1991","journal-title":"The Journal of Symbolic Logic"},{"key":"2022052312313464100_ref4","doi-asserted-by":"crossref","first-page":"211","DOI":"10.3233\/FI-1996-283401","article-title":"Rough sets through algebraic logic","volume":"28","author":"Banerjee","year":"1996","journal-title":"Fundamenta Informaticae"},{"key":"2022052312313464100_ref5","doi-asserted-by":"crossref","DOI":"10.1007\/978-81-322-2719-9_5","article-title":"Strong three-valued paraconsistent logics","volume-title":"New Directions in Paraconsistent Logic","author":"Beziau","year":"2015"},{"key":"2022052312313464100_ref6","first-page":"259","article-title":"On the representation of quasi-Boolean algebras","volume":"5","author":"Bia\u0142ynicki-Birula","year":"1957","journal-title":"Bulletin L\u2019Acad\u00e9mie Polonaise des Science Cl. 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