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An Adaptive Logic for Rational Closure

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Adaptive Logics for Defeasible Reasoning

Part of the book series: Trends in Logic ((TREN,volume 38))

Abstract

Lehmann and Magidor have proposed a powerful approach to model defeasible conditional knowledge: Rational Closure. Their account significantly strengthens previous proposals based on the so-called KLM-properties. In this chapter a dynamic proof theory for Rational Closure is presented.

This is a substantially revised version of a paper that has been published under the name “An adaptive logic for Rational Closure” in “The Many Sides of Logic” in “Studies in Logic” Series, College Publications, London, 2009, [1]

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Notes

  1. 1.

    For representational reasons it will be more transparent to stick with the “\(\mathop {{\mathop {\mid \!\sim }}}\)”-notation of Lehmann and Magidor instead of using “\(\leadsto \)”.

  2. 2.

    This is the characterization presented in [12]. In [9] the authors give an alternative (equivalent) characterization replacing (And) with (Cut).

  3. 3.

    Proofs can be found in Appendix E.

  4. 4.

    Recall that we used a similar condition for our threshold functions in the generalized standard format for ALs in Chap. 5. This ensures the strong reassurance property for ALs.

  5. 5.

    In [12] the authors use the “\(\prec \)” symbol for this relation. In order to disambiguate the usage of the various symbols for ordering relations in this chapter and therefore to serve readability, in this chapter “\(\sqsubset \)” is being used exclusively for the preferability relation of Definition 7.1.4.

  6. 6.

    See for instance the states \(s_1\) and \(s_4\) in Fig. 7.1.

  7. 7.

    Since in this chapter we only deal with a finite language, the set of worlds \(W\) is finite as well. Hence, we don’t need to add the smoothness condition that we used in the definition of preferential models.

  8. 8.

    In order to extend \({\mathbf {R}}\) such that its consequence relation maps formulas in the Boolean closure of \(\mathcal {V}_{\mathop {{\mathop {\mid \!\sim }}}} \times \mathcal {V}_p\) to formulas in the Boolean closure of \(\mathcal {V}_{\mathop {{\mathop {\mid \!\sim }}}} \times \mathcal {V}_p\) we can add an actual world \(@\) to rational models and define \(M \models p_i\) iff \(@ \models p_i\). Conditional assertions and complex formulas get their truth values as before. However, here we will keeps things simple and only focus on Boolean combinations of conditional assertions.

  9. 9.

    The interested reader can find the proofs for these fact in Appendix E.

  10. 10.

    The proofs can be found in Appendix E.

  11. 11.

    The proof can be found in Appendix E.

  12. 12.

    This has been presented e.g. in [12].

  13. 13.

    The well-foundedness of a preferential consequence relation \(\mathop {{\mathop {\mid \!\sim }}}\) should not be mistaken for well-foundedness of preferential models, e.g. a model defining \(\mathop {{\mathop {\mid \!\sim }}}\).

References

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Acknowledgments

I am thankful to Diderik Batens and Joke Meheus for many constructive comments which helped to improve this paper.

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    Christian Straßer

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Straßer, C. (2014). An Adaptive Logic for Rational Closure. In: Adaptive Logics for Defeasible Reasoning. Trends in Logic, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-00792-2_7

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