Journal of Logic, Language and Information

, Volume 6, Issue 2, pp 123-146

First online:

Counterfactuals and Updates as Inverse Modalities

  • Mark RyanAffiliated withSchool of Computer Science, University of Birmingham
  • , Pierre-Yves SchobbensAffiliated withInstitut d‘Informatique, Facultées Universitaires de Namur

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We point out a simple but hitherto ignored link between the theoryof updates, the theory of counterfactuals, and classical modal logic: update is a classicalexistential modality, counterfactual is a classical universalmodality, and the accessibility relations corresponding to these modalities are inverses. The Ramsey Rule (often thought esoteric) is simply an axiomatisation of this inverse relationship.

We use this fact to translate between rules for updates andrules for counterfactuals. Thus, Katsuno and Mendelzon‘s postulatesU1--U8 are translated into counterfactual rules C1--C8(Table VII), and many of the familiar counterfactual rulesare translated into rules for updates (Table VIII). Ourconclusions are summarised in Table V.

From known properties of inverse modalities we deduce that notall rules for updates may be translatedinto rules for counterfactuals, and vice versa. We present asyntactic condition which is sufficient to guarantee that atranslation from update to counterfactual (or vice versa) is possible.

Updates counterfactuals conditional logic belief revision inverse modalities multi-modal logic correspondence theory Kripke semantics