## New foundations for a relational theory of theory-revision.(English)Zbl 1108.03021

This paper has three main purposes. One, addressed in sections 1 through 10, is technical in nature. It is to reformulate the syntactic side of AGM belief change theory in a manner that is essentially equivalent to the familiar presentation, but contrasts with it in several respects. The second, addressed in sections 11 and 12, is to show that the AGM account of belief change, whether in its original form or as reformulated by the author, allows contractions and revisions that are wildly irrational. The third purpose of the paper, in section, 13, is to suggest a possible idea for reconstructing the AGM account to avoid this excessive liberality.
The contrasts between the author’s presentation of revision in the first part of the paper and the usual one are as follows. (a) It is presented as a single three-place relation rather than as a family of two-argument functions. This relation is understood as holding between an initial belief set $$K$$, an input formula $$x$$ and an output belief set $$J$$ iff there is some revision function in the AGM sense such that $$J = K*x$$. (b) The extension of this three-place relation is restricted to what in the AGM account is seen as the principal case – that in which the initial belief set and input formula are jointly inconsistent but separately consistent. (c) The postulates are written as Gentzen-style introduction and elimination rules rather than in the more relaxed metalanguage of AGM. Similar differences apply for contraction.
Reviewer’s comments: Regarding the first goal, it is not initially clear why the author wishes to exclude all limiting cases from the extension of the belief revision relation (and from the domains of the corresponding functions). From remarks made towards the end of the paper (section 12.7) it appears that he believes that such limiting cases may distract attention from his results showing excessive liberality. But the results continue to hold whether or not the limiting cases are admitted, and so there seems little justification for taking the mathematically very unusual step of excluding them from the fundamental definitions.
Concerning the second goal, essentially the same formal results are given in the author’s companion paper [“On the degeneracy of the full AGM-theory of theory revision”, J. Symb. Log. 71, 661–676 (2006; Zbl 1103.03017)], but in the more familiar context of belief change functions rather than relations. The author’s claim (section 1.7) that these results have “not been so much as anticipated or conjectured, let alone rigorously proved”, is somewhat exaggerated. As observed in the Zentralblatt review of the companion paper (and granted in the author’s laconic footnotes 5 and 6), certain of these anomalies (those for maxichoice and full meet operations) were proven by Alchourrón and Makinson in a paper as early as 1982. The author’s contribution is to show that the excessive liberality is not confined to those two cases, but goes further.
Finally, regarding the possible idea for reconstruction, the paper under review undertakes no formal work, although it points to some related material in the author’s [“Theory-contraction is NP-complete”, Log. J. IGPL 11, No. 6, 675–693 (2003; Zbl 1042.03018)]. In this respect, the paper’s title “New foundations for a relational theory of theory-revision” is rather optimistic. It should also be noted that the general idea for reconstruction is close to ideas studied in the theory of defeasible inheritance nets and also in logic programming with default negation. For example, the author’s diagnosis (section 13) of the shortcomings of the Harper identity in belief revision theory is (in effect) that it ignores the existence of “floating conclusions” in the sense of D. Makinson and K. Schlechta in their (uncited) paper [“Floating conclusions and zombie paths: two deep difficulties in the ‘directly skeptical’ approach to defeasible inheritance nets”, Artif. Intell. 48, No. 2, 199–209 (1991)]. Again, the proposal to replace formulae by formulae-with-justifications as the elements of a belief set has been carried out in detail in the neighbouring context of logic programming with default negation (there as a way of recuperating the principle of cumulativity by M. Brewka, D. Makinson and K. Schlechta in the (uncited) paper [“Cumulative inference relations for JTMS and logic programming”, Lect. Notes Comput. Sci. 543, 1–12 (1991; Zbl 0792.68170)]. Existing work in these areas may assist the execution of the author’s programme.

### MSC:

 03B42 Logics of knowledge and belief (including belief change) 03A05 Philosophical and critical aspects of logic and foundations 68T27 Logic in artificial intelligence

### Keywords:

AGM; revision; contraction; belief change

### Citations:

Zbl 1042.03018; Zbl 0792.68170; Zbl 1103.03017
Full Text:

### References:

 [1] Alchourrón, C., Gärdenfors, P. and Makinson, D. (1985): On the logic of theory change: Partial meet contractions and revision functions, J. Symb. Log. 50, 510–530. · Zbl 0578.03011 [2] Alchourrón, C. and Makinson, D. (1982): On the logic of theory change: contraction functions and their associated revision functions, Theoria 48, 14–37. · Zbl 0525.03001 [3] Gärdenfors, P. (1986): Belief revision and the Ramsey test for conditionals, Philosophical Review 95, 81–93. [4] Gärdenfors, P. (1988): Knowledge in Flux, MIT, Cambridge, Massachusetts. [5] Harper, W. (1977): Rational conceptual change, PSA 1976 2, 462–494. [6] Lindström, S. and Rabinowicz, W. (1989): On probabilistic representation of non-probabilistic belief-revision, J. Philos. Logic 18(1), 69–101. · Zbl 0669.03007 [7] Lindström, S. and Rabinowicz, W. (1991): Epistemic entrenchment with incomparabilities and relational belief revision, in A. Fuhrmann and M. Morreau (eds.), The Logic of Theory Change, volume 465 of Lecture Notes in Computer Science, Springer, Berlin Heidelberg New York, pp. 93–126. · Zbl 0925.03128 [8] Makinson, D. (1985): How to give it up: A survey of some formal aspects of the logic of theory change, Synthese 62, 347–363. [9] Makinson, D. (1987): On the status of the postulate of recovery in the logic of theory change, J. Philos. Logic 16, 383–394. · Zbl 0632.03008 [10] Rabinowicz, W. (1996): Stable revision, or is preservation worth preserving? in A. Fuhrmann and H. Rott (eds.), Logic, Action, Information: Essays on Logic in Philosophy and Artificial Intelligence, Walter de Gruyter, Berlin and New York, pp. 101–128. · Zbl 0926.03016 [11] Rott, H. (September 2000): Two dogmas of belief revision, J. Philos. XCVII(9), 503–522. [12] Ryan, M. D. (1996): Belief revision and ordered theory presentations, in A. Fuhrmann and H. Rott (eds.), Logic, Action, Information: Essays on Logic in Philosophy and Artificial Intelligence, Walter de Gruyter, Berlin and New York, pp. 129–151. · Zbl 0922.03036 [13] Tennant, N. (1997): The Taming of The True, Oxford University Press, Oxford. · Zbl 0929.03001 [14] Tennant, N. (2003): Theory-contraction is NP-complete, Log. J. IGPL 11(6), 675–693. · Zbl 1042.03018 [15] Tennant, N. (2005): Contracting intuitionistic theories, Stud. Log. 80, 371–393. · Zbl 1085.03015 [16] Tennant, N. (June 2006): On the degeneracy of the AGM-theory of theory-revision, J. Symb. Log. 71(2), 661–676. · Zbl 1103.03017 [17] Tennant, N. (April 2005): The Gärdenfors impossibility theorem does not show that genuine belief-revision functions cannot be monotonic. Unpublished paper to Princeton Philosophy Colloquium.
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