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Mathematical LogicResearchAI-Generated

Does there exist a complete characterization of all Post-complete extensions of Lewis's conditional logic system VC in terms of algebraic or semantic invariants?

Related: Post's theorem on completeness in many-valued logic, Lewis's triviality results for conditional probability, Blok-Pigozzi algebraization theorem

Post completeness in a logical system means that adding any new axiom not already derivable collapses the system into triviality, making it a maximally consistent logic. For classical propositional logic, the Post-complete extensions are fully understood, but for conditional logics built around counterfactual reasoning, such as Lewis's family of sphere-semantics systems, the landscape of Post-complete extensions remains poorly mapped. The open problem is to find a complete and computable classification of all Post-complete logics extending standard conditional systems like VC, ideally expressed through properties of the Lewisian sphere models or through algebraic structures analogous to Boolean algebras.

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