×

zbMATH — the first resource for mathematics

Reapproaching Ramsey: conditionals and iterated belief change in the spirit of AGM. (English) Zbl 1233.03015
Summary: According to the Ramsey test, conditionals reflect changes of beliefs: \(\alpha > \beta \) is accepted in a belief state iff \(\beta \) is accepted in the minimal revision of it that is necessary to accommodate \(\alpha \). Since P. Gärdenfors’s seminal paper [“Belief revision and the Ramsey test for conditionals”, Philos. Rev. 95, 81–93 (1986)], a series of impossibility theorems (“triviality theorems”) has seemed to show that the Ramsey test is not a viable analysis of conditionals if it is combined with AGM-type belief revision models. I argue that it is possible to endorse that Ramsey test for conditionals while staying true to the spirit of AGM. A main focus lies on AGM’s condition of preservation according to which the original belief set should be fully retained after a revision by information that is consistent with it. I use concrete representations of belief states and (iterated) revisions of belief states as semantic models for (nested) conditionals. Among the four most natural qualitative models for iterated belief change, two are identified that indeed allow us to combine the Ramsey test with preservation in the language containing only flat conditionals of the form \(\alpha > \beta \). It is shown, however, that preservation for this simple language enforces a violation of preservation for nested conditionals of the form \(\alpha > (\beta > \gamma )\). In such languages, no two belief sets are ordered by strict subset inclusion. I argue that it has been wrong right from the start to expect that preservation holds in languages containing nested conditionals.

MSC:
03A05 Philosophical and critical aspects of logic and foundations
03B42 Logics of knowledge and belief (including belief change)
03B48 Probability and inductive logic
68T27 Logic in artificial intelligence
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alchourrón, C. E., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50, 510–530. · Zbl 0578.03011 · doi:10.2307/2274239
[2] Alchourrón, C. E., & Makinson, D. (1981). Hierarchies of regulations and their logic. In R. Hilpinen (Ed.), New studies in deontic logic (pp. 125–148). Dordrecht: Reidel.
[3] Alchourrón, C. E., & Makinson, D. (1982). On the logic of theory change: Contraction functions and their associated revision functions. Theoria, 48, 14–37. · Zbl 0525.03001 · doi:10.1111/j.1755-2567.1982.tb00480.x
[4] Booth, R., & Meyer, T. (2006). Admissible and restrained revision. Journal of Artificial Intelligence Research, 26, 127–151. · Zbl 1182.68284
[5] Boutilier, C. (1996). Iterated revision and minimal change of conditional beliefs. Journal of Philosophical Logic, 25, 263–305. · Zbl 0858.03030 · doi:10.1007/BF00248151
[6] Boutilier, C., & Goldszmidt, M. (1995). On the revision of conditional belief sets. In G. Crocco, L. Fariñas del Cerro, & A. Herzig (Eds.), Conditionals. From Philosophy to Computer Science (pp. 267–300). Oxford University Press.
[7] Bradley, R. (2007). A defence of the Ramsey test. Mind, 116(461), 1–21. · doi:10.1093/mind/fzm001
[8] Carnap, R. (1952). The continuum of inductive methods. Chicago: University of Chicago Press. · Zbl 0047.37210
[9] Crocco, G., & Herzig, A. (2002). Les opérations de changement basées sur le test de Ramsey’. In P. Livet (Ed.), Révision des croyances (pp. 21–41). Paris: Hermes Science Publications.
[10] Darwiche, A., & Pearl, J. (1997). On the logic of iterated belief revision’. Artificial Intelligence, 89, 1–29. · Zbl 1018.03012 · doi:10.1016/S0004-3702(96)00038-0
[11] Gärdenfors, P. (1978). Conditionals and changes of belief’. In I. Niiniluoto, & R. Tuomela (Eds.), The logic and epistemology of scientific change. Acta philosophica Fennica (Vol. 30, Nos. 2–4, pp. 381–404). Amsterdam: North Holland.
[12] Gärdenfors, P. (1986). Belief revisions and the Ramsey test for conditionals. Philosophical Review, 95, 81–93. · doi:10.2307/2185133
[13] Gärdenfors, P. (1987). Variations on the Ramsey test: More triviality results. Studia Logica, 46, 321–327. · Zbl 0639.03024 · doi:10.1007/BF00370643
[14] Gärdenfors, P. (1988). Knowledge in flux: Modeling the dynamics of epistemic states. Cambridge: Bradford Books, MIT Press. · Zbl 1229.03008
[15] Grahne, G. (1998). Updates and counterfactuals. Journal of Logic and Computation, 8, 87–117. · Zbl 0901.03025 · doi:10.1093/logcom/8.1.87
[16] Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17, 157–170. · Zbl 0639.03025 · doi:10.1007/BF00247909
[17] Hansson, S. O. (1999). A textbook on belief dynamics. Dordrecht: Kluwer. · Zbl 0947.03023
[18] Katsuno, H., & Mendelzon, A. O. (1991). Propositional knowledge base revision and minimal change. Artificial Intelligence, 52, 263–294. · Zbl 0792.68182 · doi:10.1016/0004-3702(91)90069-V
[19] Katsuno, H., & Mendelzon, A. O. (1992). On the difference between updating a knowledge base and revising it. In P. Gärdenfors (Ed.), Belief revision (pp. 183–203). Cambridge: Cambridge University Press. · Zbl 0765.68197
[20] Kern-Isberner, G. (1999). Postulates for conditional belief revision. In T. Dean (Ed.), Proceedings of the sixteenth international joint conference on artificial intelligence, IJCAI’99 (pp. 186–191). Morgan Kaufmann.
[21] Konieczny, S., & Pino Pérez, R. (2000). A framework for iterated revision. Journal of Applied Non-Classical Logics, 10, 339–367. · Zbl 1033.03506 · doi:10.1080/11663081.2000.10511003
[22] Leitgeb, H. (2010). On the Ramsey test without triviality. Notre Dame Journal of Formal Logic, 51, 21–54. · Zbl 1207.03012 · doi:10.1215/00294527-2010-003
[23] Levesque, H. J. (1990). All I know: A study in autoepistemic logic. Artifcial Intelligence, 42, 263–309. · Zbl 0724.03019 · doi:10.1016/0004-3702(90)90056-6
[24] Levi, I. (1967). Gambling with truth. New York: Alfred Knopf, New York.
[25] Levi, I. (1996). For the sake of argument: Ramsey test conditionals, inductive inference and nonmonotonic reasoning. Cambridge University Press. · Zbl 0935.03001
[26] Lewis, D. (1973). Counterfactuals (2nd edn.). Oxford: Blackwell 1986.
[27] Lewis, D. (1976). Probabilities of conditionals and conditional probabilities. Philosophical Review, 85, 297–315. Reprinted, with a postscript. In D. Lewis (Ed.), Philosophical papers (Vol. 2, pp. 133–156). Oxford University Press, Oxford 1986.
[28] Lindström, S., & Rabinowicz, W. (1998). Conditionals and the Ramsey test’. In D. M. Gabbay, & P. Smets (Eds.), Handbook of defeasible reasoning and uncertainty management systems (Vol. 3, pp. 147–188, Belief Change). Kluwer. · Zbl 0937.03025
[29] Makinson, D. (1990). The Gärdenfors impossibility theorem in non-monotonic contexts. Studia Logica, 49, 1–6. · Zbl 0705.03011 · doi:10.1007/BF00401549
[30] McGee, V. (1985). A counterexample to modus ponens. Journal of Philosophy, 82, 462–471. · doi:10.2307/2026276
[31] Nayak, A. (1994). Iterated belief change based on epistemic entrenchment. Erkenntnis, 41, 353–390. · doi:10.1007/BF01130759
[32] Nayak, A., Pagnucco, M., Foo, N. Y., & Peppas, P. (1996). Learning from conditionals: Judy Benjamin’s other problems. In W. Wahlster (Ed.), Proceedings of 12th European conference on artificial intelligence, Budapest (pp. 75–79). New York: Wiley.
[33] Nute, D., & Cross, C. B. (2001). Conditional logic. In D. M. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 4, pp. 1–98, 2nd edn.). Dordrecht: Kluwer. · Zbl 1003.03500
[34] Ramsey, F. P. (1931). General propositions and causality. In J. B. Braithwaite (Ed.), The foundations of mathematics and other logical essays (pp. 237–255). London: Kegan Paul. 4th imprint 1965.
[35] Rott, H. (1986). Ifs, though and because. Erkenntnis, 25, 345–370. · doi:10.1007/BF00175348
[36] Rott, H. (1991). A nonmonotonic conditional logic for belief revision I. In A. Fuhrmann, & M. Morreau (Eds.), The logic of theory change. LNAI (Vol. 465, pp. 135–181). Berlin: Springer-Verlag. · Zbl 0925.03134
[37] Rott, H. (2001). Change, choice and inference: A study in belief revision and nonmonotonic reasoning. Oxford: Oxford University Press. · Zbl 1018.03004
[38] Rott, H. (2003). Coherence and conservatism in the dynamics of belief. Part II: Iterated belief change without dispositional coherence. Journal of Logic and Computation, 13, 111–145. · Zbl 1020.03015 · doi:10.1093/logcom/13.1.111
[39] Rott, H. (2009). Shifting priorities: Simple representations for twenty-seven iterated theory change operators. In D. Makinson, J. Malinowski, & H. Wansing (Eds.), Towards mathematical philosophy (pp. 269–296). Trends in Logic. Berlin: Springer Verlag. · Zbl 1159.03012
[40] Ryan, M. D., & Schobbens, P.-Y. (1997). Counterfactuals and updates as inverse modalities. Journal of Logic, Language and Information, 6, 123–146. · Zbl 0921.03035 · doi:10.1023/A:1008218502162
[41] Schwitzgebel, E. (2006). Belief. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy.
[42] Segerberg, K. (1998). Irrevocable belief revision in dynamic doxastic logic. Notre Dame Journal of Formal Logic, 39, 287–306. · Zbl 0972.03015 · doi:10.1305/ndjfl/1039182247
[43] Stalnaker, R. (1968). A theory of conditionals. In N. Rescher (Ed.), Studies in logical theory. APQ monograph series (Vol. 2, pp. 41–55). Oxford: Blackwell. Reprinted in W. L. Harper, R. Stalnaker, & G. Pearce (Eds.), Ifs (pp. 98–112). Dordrecht 1980.
[44] Stalnaker, R. (2009). Iterated belief revision. Erkenntnis, 70, 189–209. · Zbl 1178.03028 · doi:10.1007/s10670-008-9147-5
[45] Veltman, F. (1996). Defaults in update semantics. Journal of Philosophical Logic, 25, 221–261. (Reprinted in Philosopher’s Annual 19, 1996). · Zbl 0860.03025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.