Non-prioritised kernel contraction

Hansson's kernel contraction is a fundamental operation for contracting information from belief bases, which adheres to the Success postulate, ensuring that all non-tautological beliefs are retractable. In this article, we propose a non-prioritised, weakened variant of kernel contraction. To formalise this operation, we introduce the degree of firmness of arbitrary sentences within a belief base (a measure of epistemic entrenchment), drawing from well-established measures of inconsistency. Instead of necessarily withdrawing beliefs, non-prioritised kernel contraction allows for adjusting their degree of firmness, thereby violating the (controversial) Success postulate, while offering greater flexibility in belief change. As we demonstrate, non-prioritised kernel contraction is also perfectly suitable for mitigating inconsistency within belief bases. This new contraction method is constructed using a special type of incision function that determines which sentences are subject to retraction, and is axiomatically characterised by a relaxation of Hansson's original postulates. Finally, we introduce a refined form of classical kernel contraction designed to mitigate inconsistencies within belief bases.