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As one of fundamental properties to characterize inconsistency measures for knowledge bases, the property of free formula independence well captures the intuition that free formulas are independent of the amount of inconsistency in a knowledge base for cases where inconsistency is characterized in terms of minimal inconsistent subsets. But it has been argued that not all the free formulas are independent of inconsistency in some other contexts of inconsistency characterization. In this paper, we propose a characterization of formulas independent of inconsistency in the framework of Priest's minimally inconsistent LP. Based on an atom-based counterpart of the notion of free formula, we propose a notion of Bi-free formula to describe formulas that are free from inconsistency in both syntax and paraconsistent models in this logic. Then we propose the property of Bi-free formula independence, which is more suitable for characterizing the role of formulas free from inconsistency in measuring inconsistency from both syntactic and semantic perspectives.