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Journal of Logic and Computation, Volume 8, Issue 1: February 1998.
Updates and counterfactuals
Department of Computer Science, PO Box 26, Department of Computer Science, PO Box 26, FIN-0014 University of Helsinki, Finland. E-mail: firstname.lastname@example.org
We study the problem of combining updates - a special instance of theory change - and counterfactual conditionals in prepositional knowledge bases. Intuitively, an update means that the world described by the knowledge base has changed. This is opposed to revisions - another instance of theory change - where our knowledge about a static world changes. A counterfactual implication is a statement of the form 'If A were the case, then B would also be the case'. Where the negation of A may be derivable from our current knowledge. We present a decidable logic, called VCU2, that has both the update and counterfactual in the object language. Our update operator is a generalization of operators previously proposed and studied in the literature. We show that our operator satisfies certain postulates set forth for any reasonable update. The logic VCU2 is an extension of D. K. Lewis' logic VCU for counterfactual conditionals. The semantics of VCU 2 is that of a multimodal prepositional calculus, and is based on possible worlds. The infamous Ramsey Rule becomes a derivation rule in our sound and complete axiomatization. We then show that Gardenfors' Triviality Theorem about the impossibility to combine theory change and counterfactual conditionals via the Ramsey Rule, does not hold in our logic. It is thus seen that the Triviality Theorem applies only to revision operators, not to updates.
Keywords: Belief revision, updates, conditional logic, hypothetical reasoning, theory change.