  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.
