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A Variable Occurrence-Centric Framework for Inconsistency Handling

AAAI 2025 · Probabilistic and scientific modeling › Causal discovery · approach: Counterfactual identification and bounds · 0 citations (OpenAlex)

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Proposes a syntactic, variable-occurrence-based framework for propositional inconsistency (MIRs and MCRs with dual properties), using them to define non-explosive inference relations and an occurrence-based semantics.

Problem

Classical propositional bases become uninformative under inconsistency; existing syntactic approaches (MISes, MCSes) ignore the internal syntactic structure of formulas, missing conflicts involving specific variable occurrences.

Approach

Treat conflicts as equivalence relations on variable occurrences: minimal inconsistency relations (MIRs) capture conflicts, maximal consistency relations (MCRs) describe ways to restore consistency by renaming occurrences rather than dropping formulas; MCR-based inference relations retain all formulas and variables, and an occurrence-based semantics assigns truth values to occurrences.

Key results

• Proves a duality theorem: MCRs are maximal hitting-set complements of MIRs and vice versa, analogous to MIS/MCS duality.
• Shows the MCR-based inference relations are strictly ordered in cautiousness and that the cautious relation of MCRs is contained in LPm consequence (Theorem 4: K |-o1 phi implies K |-LPm phi).
• Shows that for consistent bases all four MCR-based inference relations coincide with classical inference (Proposition 7).

Surprising findings

• MIRs can capture conflicts that MISes miss; e.g., in {p∧q, ¬p∧r, ¬q∨¬r} there is one MIS but two O-MISes.
• The MCR-based inference relation |-o1 is not contained in LPm consequence for some cases, i.e., the frameworks are incomparable in one direction.

Limitations

• Computational complexity of MIRs, MCRs and the inference relations is not yet studied.
• Framework restricted to classical propositional logic; extension to non-classical logics is future work.

Experimental setup

Baselines: MCS-based inference relations LPm (Minimally Inconsistent Logic of Paradox) Forgetting-based inconsistency measure (Besnard 2016)

Builds on: Maximal consistent subsets (Rescher and Manor inference) Minimal inconsistent subsets / minimal hitting set duality (Reiter) Forgetting-based repair (Besnard 2016) Logic of Paradox (LPm) (Priest 1991)

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