限制性的推断和不一致规则 (Restricted Rules of Inference and Paraconsistency)

from arxiv, 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)

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.

翻译：在本文中,我们研究了逻辑的两个相伴因素,即左边的可变包容伴侣和受限制的规则伴侣,其性质和相互关系,尤其是与偏执性有关的情况。两个伴侣的足够条件也得到了证明。两个新的逻辑体系――自闭式的自相矛盾逻辑(IPWK)和偏执式的预留式逻辑(PPRL)――在这里作为左变量融入的逻辑示例加以介绍。IPWK是自闭式先行逻辑(IPC)的可变包容配方,也是其受限制的规则伴侣。另一方面,PPRL是预留式逻辑(PRL)的可变包容伴方,但不同于其受限制的规则伴方。我们讨论了这些逻辑的代数语义语义语义,这相当于引入一种污染性的事实价值,意在表示非决定性状态。