Reasoning under inconsistency: the forgotten connective

In many frameworks for reasoning under inconsistency, it is implicitly assumed that the formulae from the belief base are connected using a weak form of conjunction. When it is consistent, a belief base B = {φ1, . . . , φn}, where the φi are propositional formulae, is logically equivalent to the base {φ1 ∧ . . . ∧ φn}. However, when it is not consistent, both bases typically lead to different conclusions. This illustrates the fact that the comma used in base B has to be considered as an additional, genuine connective, and not as a simple conjunction. In this work we define and investigate a propositional framework with such a “comma connective”. We give it a semantics and show how it generalizes several approaches for reasoning from inconsistent beliefs.