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The Procedures for Belief Revision

  • Piotr ŁukowskiEmail author
Part of the Trends in Logic book series (TREN, volume 28)

Abstract

The idea of belief revision is strictly connected with such notions as revision and contraction given by two sets of postulates formulated by Alchourrón, Gärdenfors and Makinson in (Theoria, 48: 14–17, 1982; Journal of Symbolic Logic, 50: 510–530, 1985; Philosophical Essays Dedicated to Lennard Aqvist on His Fiftieth Birthday, pp. 88–101, 1982). In the paper expansion, contraction and revision are defined probably in the most orthodox way, i.e. by Tarski’s consequence operation (e.g. Compt. Rend. Séances Soc. Sci. Lett. Varsovie, cl.III, 23, 22–29) and Tarski-like elimination operation (see Łukowski, P., Logic and Logical Philosophy, 10: 59–78, 2002). In our approach nonmonotonicity appears as a final result of alternate using of two steps of our reasoning: “step forward” and “step backward”. Step forward extends the set of our beliefs and it is used when some new belief appears. Step backward reduces the set of our beliefs and it is used when we reject some previously accepted belief. A decision of adding or rejecting of some sentences is arbitrary and depends on our wish only. Thus, this decision cannot be logical and logic cannot justify it. In our approach logic is a tool for faultless and precise realization of extension or reduction of the set of our beliefs. But why some sentences should be added or refused depends on extralogical reasons. Such understood nonmonotonicity can be considered also on logics other than classical. A reconstruction of a given logic in its deductive-reductive form is here the basis of nonmonotonicity. It means that nonmonotonic reasoning can be formalized on the base of every logic for which its deductive-reductive form can be reconstructed. Firstly our approach is tested on the ground of the classical logic. Next it is confronted with the intuitionism represented by the Heyting-Brouwer logic. Procedures of contraction and revision are verified by using of the AGM postulates. We limit our considerations to first six conditions for contraction and revision, because of the well known relation between contraction satisfying first four conditions together with the sixth one and revision defined by this contraction and consequence operation. Satisfaction of almost every postulate is for us a good sign that our approach is reasonable. The only exception we make for the fifth postulate and for Harper identity. Analyzing the reasoning alternately extending and contracting the set of beliefs it is difficult to accept the postulate that adding previously rejected sentence we obtain again the same set of beliefs as before the contraction. This problem is deeper considered in the paper. The original AGM settles the relations between contraction and revision with, the well known Levi and Harper identities. In all cases of our approach, revision is defined on the basis of contraction which with respect to revision is prime.

Keywords

belief revision expansion contraction revision consequence operation elimination operation logic of truth logic of falsehood nonmonotonic reasoning classical logic Heyting-Brouwer logic intuitionistic logic 

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References

  1. [1]
    Alchourrón, C., Makinson, D., ‘On the logic of theory change: contraction functions and their associated revision functions’, Theoria, 48: 14–37, 1982. MathSciNetCrossRefGoogle Scholar
  2. [2]
    Alchourrón, C., Gärdenfors, P., Makinson, D., ‘On the logic of theory change: Partial meet contraction revision functions’, Journal of Symbolic Logic, 50: 510–530, 1985. zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Bryll, G., Metody Odrzucania Wyrażeń, Akademicka Oficyna Wydawnicza PLJ, Warszawa, 1996. Google Scholar
  4. [4]
    Dutkiewicz, R., ‘The method of axiomatic rejection for the intuitionistic propositional logic’, Studia Logica, 48: 449–460, 1989. zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Gärdenfors, P., ‘Rules for rational changes of belief’, in Pauli, T. (ed.), Philosophical Essays Dedicated to Lennart Aqvist on His Fiftieth Birthday, University of Uppsala Philosophical Studies 34, 1982, pp. 88–101. Google Scholar
  6. [6]
    Mc Kinsey, J.C.C., Tarski, A., ‘On closed elements in closure algebras’, Ann. of Math., 47: 122–162, 1946. CrossRefMathSciNetGoogle Scholar
  7. [7]
    Łukowski, P., ‘A deductive-reductive form of logic: general theory and intuitionistic case’, Logic and Logical Philosophy, 10: 59–78, 2002. zbMATHGoogle Scholar
  8. [8]
    Rauszer, C., ‘Semi-Boolean algebras and their applications to intuitionistic logic with dual operations’, Fundamenta Mathematicae, LXXXIII: 219–249, 1974. MathSciNetGoogle Scholar
  9. [9]
    Rauszer, C., An algebraic and Kripke-style approach to a certain extension of intuitionistic logic, Dissertationes Mathematicae, vol. CLXVII, PWN, Warszawa, 1980. Google Scholar
  10. [10]
    Skura, T., ‘A complete syntactical characterization of the intuitionistic logic’, Reports on the Mathematical Logic, 23: 75–80, 1990. MathSciNetGoogle Scholar
  11. [11]
    Tarski, A., ‘Über einige fundamentale Begriffe der Metamathematik’, Compt. Rend. Séances Soc. Sci. Lett. Varsovie, cl.III, 23, pp. 22–29. Google Scholar
  12. [12]
    Wójcicki, R., ‘Dual counterparts of consequence operation’, Bulletin of the Section of Logic, 2(1): 54–57, 1973. Google Scholar
  13. [13]
    Wójcicki, R., Lectures on Propositional Calculi, The Polish Academy of Sciences, Institute of Philosophy and Sociology, Ossolineum, 1984. zbMATHGoogle Scholar
  14. [14]
    Wójcicki, R., Theory of Logical Calculi: Basic Theory of Consequence Operations, Synthese Library vol. 199, Kluwer Academic Publishers, 1988. Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of Logic and Methodology of ScienceŁódź UniversityŁódźPoland

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