Notes on measuring inconsistency in probabilistic logic

Inconsistency measures have recently been put forward to manage inconsistent knowledge bases in the AI community. For conditional probabilistic logics, rationality postulates and computational complexity have driven the formulation of inconsistency measures. Independently, investigations in formal epistemology have used the betting concept of Dutch book to measure an agent’s degree of incoherence. In this paper, we argue for the unsatisfiability of the proposed postulates and put forward alternative ones. Problematic desirable properties are weakened by analyzing the underlying consolidation process. Inconsistency measures suggested in the literature and computable with linear programs are shown to satisfy the postulates. Additionally, it is given a gambling interpretation for these practicable measures, showing they correspond to incoherence measures via Dutch books. Finally, we propose a general linear programming framework, allowing for confidence factors and encompassing measures from both communities that satisfy the reconciled postulates.

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