Restricted Rules of Inference and Paraconsistency

In this paper, we study two companions to 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 - Intuitionistic Paraconsistent Weak Kleene logic (IPWK) and Paraconsistent Pre-Rough logic (PPRL) - are presented here as examples of logics of left variable inclusion. IPWK is the left variable inclusion companion of Intuitionistic Propositional logic (IPC) and is also the restricted rules companion of it. PPRL, on the other hand, is the left variable inclusion companion of Pre-Rough logic (PRL) but differs from the restricted rules companion of it. We have discussed algebraic semantics for these logics in terms of P{\l}onka sums. This amounts to introducing a contaminating truth value, intended to denote a state of indeterminacy.
Comment: The final version of this paper has been published online in Logic Journal of the IGPL (https://academic.oup.com/jigpal/advance-article/doi/10.1093/jigpal/jzab019/6299942). Minor typos fixed; Theorem 3.7 has been changed to Remark 3.7; other minor rearrangements of the material done; further justification added in Remark 4.28 (now Remark 4.27)