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Against Grue Mysteries


The paper develops an inductive extension of AGM-style belief base revision theory with the aim of formally implementing Freitag’s (Philos Q 65(259):254–267, 2015, Dialectica 70(2):185–200, 2016) solution to Goodman’s paradox. It shows that the paradox dissolves once belief revision takes place on inductively closed belief bases, rather than on belief sets.

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  1. See, e.g., Gärdenfors (1990) for a discussion of coherentist versus foundational belief revision theory.

  2. AGM theory can also be adpated to non-classical consequence relations. See, e.g., Rott (2001) for a discussion and Wassermann (2011) for an overview.

  3. Partial meet contraction was developed by Alchourrón et al. (1985).

  4. For Hansson’s original examples, see Hansson (1992, 89–90, 1999, sec. 1.7).

  5. Of course, in any even only remotely plausible scenario, Ann and Cen have numerous further beliefs. We here solely concentrate on those beliefs that are relevant for the present case.

  6. There are various further suggestions on how to model the notion of doxastic dependence, which also account for the interplay between dependence and belief change. However, these proposals aim at notions different from our notion of dependence. For example, Parikh (1999) and Makinson and Kourousias (2006) try to capture some notion of syntactic dependence, and Fariñas del Cerro and Herzig (1996); Hansson and Wassermann (2002), and Oveisi et al. (2014) discuss notions of purely inferential dependence. Obviously, the justification operator of Haas (2005) is not meant to capture the relevant notion of doxastic dependence either.

  7. Let me remark that this framework is not committed to the view that there are ‘absolutely basic’ beliefs. In the discussed examples, we never consider everything an agent believes, but only concentrate on those aspects of her doxastic state that are relevant for the case. With respect to the situation described in Hansson’s case, we will say that Ann’s belief base contains the belief that Marry goes to the party and the belief that Harry goes to the party. This does not mean that these beliefs form the foundation of Ann’s doxastic system, but only that they play the role of basic beliefs in the very restricted scenario under consideration.

  8. To serve readability, we use the same symbols for expansion, revision, and contraction, respectively, on beliefs sets and belief bases. For the following definitions, see, e.g., Hansson (1999). For a discussion of different versions of basic belief change, see Rott (2001, sec. 3.4.1. and ch. 5). It has been proved that the contraction operation defined below satisfies the Postulates Success, Inclusion, Relevance, and Uniformity, and, furthermore, that every contraction operation for belief bases that satisfies these postulates is a partial meet contraction (see Hansson 1999, Ch.2). Contraction on belief bases does not satisfy Gärdenfors’ recovery postulate. However, this is the most contested basic AGM postulate anyway.

  9. Fuhrmann (1991, 184). For a similar remark, see Makinson (1997, 475). See especially Rott (2000) for a discussion of filtering conditions and believing ’just because’.

  10. This inference principle is rather simplistic. For example, it seems to violate the principle of total evidence. Again, my aim is not to develop a theory of (enumerative) induction, but to provide a formal reconstruction of a recent solution strategy to Goodman’s paradox.

  11. One might want to formulate certain constraints on \(\delta \). However, in the examples discussed here, there will always only be exactly one inductive closure of the belief base, so that nothing hinges on the particular choice of \(\delta \).

  12. Belief bases were defined as non-empty, consistent sets of sentences (see Sect. 2). It directly follows from Definition 3 that I-closures are inductive closures of belief bases, and it follows directly from Definition 1 that inductive closures of belief bases are belief bases.

  13. Contracted I-closures are not necessarily inductively closed. However, this doesn’t cause any problems as contraction and expansion are defined for all belief bases, not just inductively closed ones.

  14. The above definitions allow that an inductively inferred belief is still an element of the I-closure after one has contracted the evidence from which it is inductively inferred. From a foundationalist perspective, this seems undesirable. We could prevent this by revising, not the I-closure of a belief base, but the belief base itself, close under induction in a second step, and close under deduction in a third step. However, as this is unnecessarily complicated for our present purposes, we stick to the above definitions.

  15. This is a simplification of Goodman’s original definition, which goes as follows: x is grue iff (x is green and has been observed before t) or (x is blue and is observed after t), where t is some point of time in the near future (Goodman 1983). We have substituted ‘not green’ for ‘blue’, and omitted the explicit reference to a future point in time, but assumed t to be now. This yields ‘x is grue iff (x is green and has been observed) or (x is not green and has not been observed yet)’, which is logically equivalent to the above formulation.

  16. Of course, Ann never entertains the belief that emerald \(a_{n+1}\) has already been observed, and never actually revises her belief base with \(a_{n+1}\)has not been observed. She believes this to be the case all along. However, we do not intend to provide a psychologically or procedurally adequate description of the case, but aim at displaying its logical structure.

  17. The objection has actually been put forward in Dorst (2017), but see Freitag (forthcoming) for a reply.


  • 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.

    Article  Google Scholar 

  • Dorst, C. (2017). Bet accepted—A reply to Freitag. Philosophical Quaterly, 68, 175–183.

    Google Scholar 

  • Fariñas del Cerro, L., & Herzig, A. (1996). Belief chance and dependence. In Shoham, Y. (Ed)Proceedings of the 6th conference on theoretical aspects of rationality and knowledge (pp. 147–161).

  • Freitag, W. (forthcoming). Why doxastic dependence defeats grue. Reply to Dorst. Philosophical Quaterly.

  • Freitag, W. (2015). I bet you’ll solve Goodman’s riddle. Philosophical Quarterly, 65(259), 254–267.

    Article  Google Scholar 

  • Freitag, W. (2016). The disjunctive riddle and the grue paradox. Dialectica, 70(2), 185–200.

    Article  Google Scholar 

  • Fuhrmann, A. (1991). Theory contraction through base contraction. Journal of Philosophical Logic, 20, 175–203.

    Article  Google Scholar 

  • Gärdenfors, P. (1990). The dynamics of belief systems: Foundations vs. coherence theories. Artificial Intelligence, 44(172), 24–46.

    Google Scholar 

  • Goodman, N. (1983). Fact, fiction, and forecast. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Haas, G. (2005). Revision und Rechtfertigung: Eine Theorie der Theorieänderung. Heidelberg: Synchron.

    Google Scholar 

  • Hansson, S. O. (1992). A dyadic representation of belief. In P. Gärdenfors (Ed.), Belief revision (pp. 89–121). Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • Hansson, S. O. (1999). A textbook of belief dynamics: Theory change and database updating. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Hansson, S. O., & Wassermann, R. (2002). Local change. Studia Logica, 70, 49–76.

    Article  Google Scholar 

  • Makinson, D. (1997). On the force of some apparent counterexamples to recovery. In E. G. Valdés, W. Krawietz, G. H. von Wright, & R. Zimmerling (Eds.), Normative systems in legal and moral theory (pp. 475–481). Berlin: Duncker & Humblot.

    Google Scholar 

  • Makinson, D., & Kourousias, G. (2006). Respecting relevance in belief change. Análisis Filosófico, 26, 53–61.

    Google Scholar 

  • Oveisi, M., Delgrande, J. P., Pelletier, F. J., & Popowich, F. (2014). Belief change and base dependence. In: 14th international conference on principles of knowledge representation and reasoning (pp. 151–159).

  • Parikh, R. (1999). Belief, belief revision, and splitting languages. Logic, Language, and Computation, 2, 266–278.

    Google Scholar 

  • Rott, H. (2000). ‘Just Because’: Taking belief bases seriously. In Buss, S. R., Hájek, P., & Pudlák, P. (eds.), Logic colloquium ’98—Proceedings of the annual European summer meeting of the association for symbolic logic held in Prague (pp. 387–408). Urbana III: Association for Symbolic Logic.

  • Rott, H. (2001). Chance, choice, and inference: A study of belief revision and nonmonotonic reasoning. Oxford: Clarendon Press.

    Google Scholar 

  • Wassermann, R. (2011). On AGM for non-classical logics. Journal of Philosophical Logic, 40, 271–294.

    Article  Google Scholar 

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An early version of this paper was presented at the MCMP, Munich. I thank the audience there, especially Hannes Leitgeb and Berna Kilinc, for discussion. I am also indebted to two anonymous referees of this journal and to Wolfgang Freitag for comments on an earlier draft of this paper.

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Correspondence to Alexandra Zinke.

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Zinke, A. Against Grue Mysteries. Erkenn 85, 1023–1033 (2020).

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