As one of fundamental properties to characterize inconsistency measures for knowledge bases, the property of free formula independence captures well 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 notion of Bi-free formula to describe formulas that are free from inconsistency in both syntactic characterization and paraconsistent models in the framework of Priest's minimally inconsistent LP. 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.