Beyond Consistency: A Closer Look at Free Formulas

In this paper, we introduce novel methods for drawing conclusions from inconsistent information in a highly cautious manner. While standard paraconsistent approaches typically rely on the formulas in the intersection of maximally consistent subsets, known as the free formulas, we argue that not all these formulas share the same degree of reliability. Our refined reasoning frameworks distinguish between free formulas, based on their actual involvement in the inconsistency, an so enabling inference only when conclusions are robustly supported. These methods are particularly valuable in high-stakes contexts where decisions carry irreversible or far-reaching consequences. We present several implementation techniques grounded in multi-valued semantics and syntactic independence, analyze their fundamental logical properties, and establish a hierarchy of their inferential strength.