Abstract
In this paper we study AGM contraction and revision of rules using input/output logical theories. We replace propositional formulas in the AGM framework of theory change by pairs of propositional formulas, representing the rule based character of theories, and we replace the classical consequence operator Cn by an input/output logic. The results in this paper suggest that, in general, results from belief base dynamics can be transferred to rule base dynamics, but that a similar transfer of AGM theory change to rule change is much more problematic. First, we generalise belief base contraction to rule base contraction, and show that two representation results of Hansson still hold for rule base contraction. Second, we show that the six so-called basic postulates of AGM contraction are consistent only for some input/output logics, but not for others. In particular, we show that the notorious recovery postulate can be satisfied only by basic output, but not by simple-minded output. Third, we show how AGM rule revision can be defined in terms of AGM rule contraction using the Levi identity. We highlight various topics for further research.
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Notes
K-7 and K-8 can be formulated on contraction with a set of formulas, so, the contraction of \(a\wedge b\) can be understood as the contraction of \(\{a,b\}\) see, e.g., Billington and Antoniou (1999).
This follows directly from Proposition 1, see also Makinson and Torre (2000).
However, there may be special cases where it holds, for example for \({ out}_2\).
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Acknowledgments
Thanks to David Makinson and Jörg Hansen for discussions on the issues raised in this paper. We also thank the two anonymous referees for their valuable comments and suggestions that helped us improving the content and readability of the paper. Gabriella Pigozzi benefited from the support of the project AMANDE ANR-13-BS02-0004 of the French National Research Agency (ANR).
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Boella, G., Pigozzi, G. & van der Torre, L. AGM Contraction and Revision of Rules. J of Log Lang and Inf 25, 273–297 (2016). https://doi.org/10.1007/s10849-016-9244-9
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DOI: https://doi.org/10.1007/s10849-016-9244-9