© 1998 by Oxford University Press
Original Articles |
Updates and Counterfactuals
Department of Computer Science PO Box 26, FIN-00014 University of Helsinki, Finland. E-mail: grahne{at}cshelsinki.fi
We study the problem of combining updates—a special instance of theory change—and counterfactual conditionals in propositional knowledge bases. Intuitively, an update means that the world described by the knowledge base has changed. This is opposed to revisions—anotherinstance of theory change—where our knowledge about a static world changes. A countcrfactual 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 update and counterfactual implication as connectives 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 VCU2 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 Gärdenfors' 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