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Reasoning About Belief Revision

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Belief Revision meets Philosophy of Science

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 21))

Abstract

The theory of belief revision developed by Carlos Alchourrón, Peter Gärdenfors and David Makinson (AGM) is one of the most influential and well-investigated theories of rational belief change; for a comprehensive presentation and references see Hansson (1999) and Rott (2001). This highly successful research program co-exists with another major research program concerned with the belief and knowledge of rational agents, namely doxastic and epistemic logic. With respect to epistemic logic, in Knowledge in Flux (1988), Peter Gärdenfors remarked:

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Notes

  1. 1.

    Another framework of interest in this connection is dynamic epistemic logic, DEL, see van Ditmarsch et al. (2005).

  2. 2.

    We would like to thank Erik Olsson for inviting us to contribute to this volume and an anonymous commentator for her/his useful remarks on an earlier version of this chapter.

  3. 3.

    In a frequently cited paper, Bernard Williams (1973) claimed to have shown that deciding to believe is logically impossible. See also Winters (1979).

  4. 4.

    A sound and complete axiomatization of bdi-stit logic is presented in Semmling and Wansing (2009).

  5. 5.

    Here cl stands for the reflexive and transitive closure of a binary relation.

  6. 6.

    Since we interpret α by α itself and since every situation over \((Tree,\leq)\) corresponds to a situation s on b, it is warrantable that we use the same letters in the tableaux and the notation of \({\cal M} _b\).

  7. 7.

    Choosing v in this way is suitable, since b is open. There is no situation s on b, such that p, s and \(\neg p,s\) occur on b.

  8. 8.

    and the quantifiers, if we want to make explicit the implicit universal quantification over belief sets and formulas from L.

  9. 9.

    Note that we do not translate (K+4’) as \((\alpha\,{bel:}\,\varphi \land \alpha\,{bel:}\, \psi )\supset( \alpha\,{dstit:}\,\alpha\,{bel:}\,\varphi \supset\alpha\,{bel:}\,\psi)\), because \((\alpha\,{bel:}\,\varphi \land \alpha\,{bel:}\, \psi )\supset \alpha\,{bel:}\,(\varphi \land \psi )\) fails to be valid, hence neighbourhoods \(U \in B^{\alpha}_{s}\) represent belief sets.

  10. 10.

    For a discussion of closure principles for belief, see, for instance, Fagin and Halpern (1988) and Semmling and Wansing (2008).

  11. 11.

    An alternative translation would be \(\neg \alpha \ {bel:}\,\psi \supset (\alpha \ {contra:}\,\varphi \supset \neg \alpha \ {dstit:}\,\alpha \ {bel:}\,\psi )\). (“If an agent contracts her beliefs, she does not add new beliefs”.) This formula is also valid.

  12. 12.

    This condition of a not necessary truth refers to the negative condition in the semantics of the dstit-operator.

  13. 13.

    In the following tableau we represent by an arrow the application of exactly one tableau rule, but this rule can be applied to more than one subformula of a given formula.

References

  • Allen, M. 2005. Complexity results for logics of local reasoning and inconsistent belief. In Theoretical aspects of rationality and knowledge, ed. Ron van der Meyden, 92–108. Proceedings of the 10th conference, National University of Singapore, Singapore.

    Google Scholar 

  • Belnap, N.D., and M. Perloff. 1988 Seeing to it that: a canonical form for agentives. Theoria 54: 175–199.

    Article  Google Scholar 

  • Belnap, N.D., M. Perloff, and M. Xu. 2001. Facing the future: Agents and choices in our indeterminist world, New York, NY: Oxford University Press.

    Google Scholar 

  • van Benthem, J. 1995. Logic and the flow of information. In Logic, methodology and philosophy of science IX, eds. D. Prawitz et al., 693–724. Amsterdam: North Holland.

    Google Scholar 

  • Bonanno, G. 2005. A simple modal logic for belief revision. Synthese 147:5–40.

    Google Scholar 

  • Bonanno, G. 2007. Temporal interaction of information and belief. Studia Logica 86:375–401.

    Article  Google Scholar 

  • van Ditmarsch, H., W. van der Hoek, and B. Kooi. 2005. Playing cards with Hintikka. An introduction to dynamic epistemic logic. Australasian Journal of Logic 3:108–134.

    Google Scholar 

  • Fagin, R., and J.Y. Halpern. 1988 Belief, awareness and limited reasoning. Artificial Intelligence 34:39–76.

    Article  Google Scholar 

  • Gärdenfors, P. 1988. Knowledge in flux. Modeling the dynamics of epistemic states. Cambridge: MIT Press.

    Google Scholar 

  • Georgeff, M.P., and A.S. Rao. 1998. Decision procedures for BDI logics. Journal of Logic and Computation 8:293–342.

    Article  Google Scholar 

  • Hansson, S.O. 1999. A textbook of belief dynamics: theory change and database updating. Dordrecht: Kluwer.

    Google Scholar 

  • Kripke, S.A. 1963. Semantical analysis of modal logic I: Normal propositional calculi. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 9:6796.

    Article  Google Scholar 

  • Leitgeb, H., and K. Segerberg. (2007). Dynamic doxastic logic: Why, how, and where to? Synthese 155:167–190.

    Article  Google Scholar 

  • Nottelmann, N. 2007. Blameworthy belief: A study in epistemic deontologism. Berlin: Springer.

    Google Scholar 

  • Priest, G. 2001. An introduction to non-classical logic. Cambridge: Cambridge University Press.

    Google Scholar 

  • de Rijke, M. 1994. Meeting some neighbours. In Logic and information flow eds. J. van Eijk, and A. Visser. Cambridge, MA: MIT.

    Google Scholar 

  • Rott, H. 2001. Change, choice and inference. A study of belief revision and non-monotonic reasoning. Oxford: Oxford University Press.

    Google Scholar 

  • Segerberg, K. 1999. Two traditions in the logic of belief: Bringing them together. In Logic, language and reasoning. Essays in honour of Dov Gabbay, eds. H.-J. Ohlbach, and U. Reyle, 135–147. Dordrecht: Kluwer.

    Google Scholar 

  • Semmling, C., and H. Wansing. 2008. From BDI and stit to bdi-stit logic. Logic and Logical Philosophy 17:185–207.

    Google Scholar 

  • Semmling, C., and H. Wansing. 2009. A sound and complete axiomatic system of bdi-stit logic, In Logica Yearbook 2008, ed. M. Pelis, 193–210. London: College Publications.

    Google Scholar 

  • Wansing, H. 2006a. Tableaux for multi-agent deliberative-stit logic. In Advances in modal logic, vol. 6, eds. G. Governatori, I. Hodkinson, and Y. Venema, 503–520. London: College Publications.

    Google Scholar 

  • Wansing, H. 2006b. Doxastic decisions, epistemic justification, and the logic of agency. Philosophical Studies 128:201–227.

    Article  Google Scholar 

  • Williams, B. 1973. Deciding to believe. In Problems of the self, chapter 9, 136–151. New York, NY: Cambridge University Press.

    Google Scholar 

  • Winters, B. 1979. Believing at will. Journal of Philosophy 76:243–256.

    Article  Google Scholar 

  • Wooldridge, M. 2000. Reasoning about rational agents. Cambridge, MA: MIT Press.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Philosophy, Dresden University of Technology, Dresden, Germany

    Caroline Semmling & Heinrich Wansing

Authors
  1. Caroline Semmling
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  2. Heinrich Wansing
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Correspondence to Caroline Semmling .

Editor information

Editors and Affiliations

  1. Dept. Philosophy, University of Lund, Kungshuset, Lund, 222 22, Sweden

    Erik J. Olsson

  2. Dept. Philosophy, University of Lund, Kungshuset, Lund, 222 22, Sweden

    Sebastian Enqvist

Appendix: A Tableau Proof of (tK*8)

Appendix: A Tableau Proof of (tK*8)

Before applying the tableau calculus, we formulate the negation of (tK*8):

$$\begin{array}{cc} & \neg(\alpha \ {rev:}\,(\varphi \wedge \neg \psi) \supset (\alpha \ {dstit:}\,\alpha \ {bel:}\,\psi \supset \alpha \ {rev:}\,(\varphi \wedge \psi)))\\ \equiv & \alpha \ {rev:}\,(\varphi \wedge \neg \psi) \land (\alpha \ {dstit:}\,\alpha \ {bel:}\,\psi \land \neg \alpha \ {rev:}\,(\varphi \wedge \psi)) \end{array}$$

Note that \(\alpha \ {rev:}\,\varphi\equiv (\alpha \ {dstit:}\,\neg\alpha \ {bel:}\,\neg \varphi \land \alpha \ {dstit:}\,\alpha \ {bel:}\,\varphi)\) and that \(\neg \alpha \ {rev:}\,\varphi\equiv (\neg\alpha \ {dstit:}\,\neg\alpha \ {bel:}\,\neg \varphi \lor \neg \alpha \ {dstit:}\,\alpha \ {bel:}\,\varphi)\). Now, we show that it is not possible to construe a counter model for the negation of (tK*8) by demonstrating that the finite tableau in Table 13.13 is closed.Footnote 13

Table 13.13 Another tableau proof
Full size table

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Semmling, C., Wansing, H. (2010). Reasoning About Belief Revision. In: Olsson, E., Enqvist, S. (eds) Belief Revision meets Philosophy of Science. Logic, Epistemology, and the Unity of Science, vol 21. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9609-8_13

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