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—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 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 Płonka sums. This amounts to introducing a contaminating truth value, intended to denote a state of indeterminacy.

中文翻译：

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推理和平行一致性的限制规则

在本文中，我们研究了逻辑的两个同伴，即左变量包含同伴和受限规则同伴，它们的性质和相互关系，特别是与副一致性相关的关系。两个同伴重合的充分条件也被证明了。这里介绍了两个新的逻辑系统——直觉的平行一致弱 Kleene 逻辑 (IPWK) 和平行一致的预粗逻辑 (PPRL)——作为左变量包含逻辑的示例。IPWK是直觉命题逻辑的左变量包含伴侣，也是它的受限规则伴侣。另一方面，PPRL 是 pre-rough 逻辑的左变量包含伴侣，但不同于它的受限规则伴侣。我们已经根据 Płonka 和讨论了这些逻辑的代数语义。