Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-25T23:42:39.022Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALIZED PARTIAL MEET AND KERNEL CONTRACTIONS

Part of: General logic

Published online by Cambridge University Press:  24 November 2022

MARCO GARAPA  and
MAURÍCIO D. L. REIS
MARCO GARAPA
Affiliation:
FACULDADE DE CIÊNCIAS EXATAS E DA ENGENHARIA UNIVERSIDADE DA MADEIRA, CAMPUS UNIVERSITÁRIO DA PENTEADA 9020-105 FUNCHAL, PORTUGAL and CIMA—CENTRO DE INVESTIGAÇÃO EM MATEMÁTICA E APLICAÇÕES ÉVORA, PORTUGAL E-mail: [email protected] E-mail: [email protected]
MAURÍCIO D. L. REIS
Affiliation:
FACULDADE DE CIÊNCIAS EXATAS E DA ENGENHARIA UNIVERSIDADE DA MADEIRA, CAMPUS UNIVERSITÁRIO DA PENTEADA 9020-105 FUNCHAL, PORTUGAL and CIMA—CENTRO DE INVESTIGAÇÃO EM MATEMÁTICA E APLICAÇÕES ÉVORA, PORTUGAL E-mail: [email protected] E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Two of the most well-known belief contraction operators are partial meet contractions (PMCs) and kernel contractions (KCs). In this paper we propose two new classes of contraction operators, namely the class of generalized partial meet contractions (GPMC) and the class of generalized kernel contractions (GKC), which strictly contain the classes of PMCs and of KCs, respectively. We identify some extra conditions that can be added to the definitions of GPMCs and of GKCs, which give rise to some interesting subclasses of those classes of functions, namely the classes of extensional and of uniform GPMCs/GKCs. In the context of contractions on belief sets the classes of partial meet contractions, uniform GPMCs and extensional GPMCs are all identical. Nevertheless, when considered as operations on belief bases, the class of uniform GPMCs coincides with the class of partial meet contractions, but the extensional GPMCs constitute a new kind of belief base contraction functions whose characterizing postulate of irrelevance of syntax is extensionality—the same postulate of irrelevance of syntax which occurs in the classical axiomatic characterization of partial meet contractions for belief sets—rather than the postulate of uniformity—which is the irrelevance of syntax postulate used in the axiomatic characterization for partial meet contractions on belief bases. Analogous results are obtained regarding the classes of extensional and of uniform GKCs. We present the interrelations in the sense of inclusion among all the new classes of operators presented in this paper and several well known classes of PMCs and of KCs.

Keywords

belief changecontractiongeneralized partial meet contractionsgeneralized kernel contractionssentence based selection and incision functions

MSC classification

Primary: 03B42: Logics of knowledge and belief (including belief change)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 17 , Issue 2 , June 2024 , pp. 366 - 394
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Alchourrón, C., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet contraction and revision functions. The Journal of Symbolic Logic, 50, 510–530.CrossRefGoogle Scholar
Alchourrón, C. & Makinson, D. (1981). Hierarchies of regulations and their logic. In Hilpinen, R., editor. New Studies in Deontic Logic: Norms, Actions, and the Foundations of Ethics, Dordrecht: Springer Netherlands, pp. 125–148.CrossRefGoogle Scholar
Alchourrón, C. & Makinson, D. (1985). On the logic of theory change: Safe contraction. Studia Logica, 44, 405–422.CrossRefGoogle Scholar
Alchourrón, C. & Makinson, D. (1986). Maps between some different kinds of contraction functions: The finite case. Studia Logica, 45, 187–198.CrossRefGoogle Scholar
Dalal, M. (1988). Investigations into a theory of knowledge base revision: Preliminary report. In Seventh National Conference on Artificial Intelligence (AAAI-88) (St. Paul), USA: AAAI Press, pp. 475–479.Google Scholar
Fermé, E. & Hansson, S. O. (2011). AGM 25 years: Twenty-five years of research in belief change. Journal of Philosophical Logic, 40, 295–331.CrossRefGoogle Scholar
Gärdenfors, P. (1988). Knowledge in Flux: Modeling the Dynamics of Epistemic States. Cambridge: The MIT Press.Google Scholar
Hansson, S. O. (1991). Belief Base Dynamics. Ph.D. Thesis, Uppsala University.Google Scholar
Hansson, S. O. (1992). A dyadic representation of belief . In Gärdenfors, P., editor. Belief Revision. Cambridge Tracts in Theoretical Computer Science, Vol. 29. Cambridge: Cambridge University Press, pp. 89–121.CrossRefGoogle Scholar
Hansson, S. O. (1993). Reversing the Levi identity. Journal of Philosophical Logic, 22, 637–669.CrossRefGoogle Scholar
Hansson, S. O. (1994). Kernel contraction. The Journal of Symbolic Logic, 59, 845–859.CrossRefGoogle Scholar
Hansson, S. O. (1999). A Textbook of Belief Dynamics: Theory Change and Database Updating. Applied Logic Series. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
Rott, H. (1992). On the logic of theory change: More maps between different kinds of contraction functions . In Gärdenfors, P., editor. Belief Revision. Cambridge Tracts in Theoretical Computer Science, Vol. 29. Cambridge: Cambridge University Press, pp. 122–141.CrossRefGoogle Scholar
Rott, H. (2000). Two dogmas of belief revision. Journal of Philosophy, 97(9), 503–522.CrossRefGoogle Scholar
Rott, H. & Hansson, S. O. (2014). Safe contraction revisited . In Hansson, S. O., editor. David Makinson on Classical Methods for Non-Classical Problems. Outstanding Contributions to Logic, Vol. 3. Dordrecht: Springer, pp. 35–70 (in English).CrossRefGoogle Scholar
Tarski, A. (1956). Logic, Semantics, Metamathematics. Oxford: Clarendon Press. Papers from 1923 to 1938. Translated by J. H. Woodger.Google Scholar