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A normatively adequate credal reductivism

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It is a prevalent, if not popular, thesis in the metaphysics of belief that facts about an agent’s beliefs depend entirely upon facts about that agent’s underlying credal state. Call this thesis ‘credal reductivism’ and any view that endorses this thesis a ‘credal reductivist view’. An adequate credal reductivist view will accurately predict both when belief occurs and which beliefs are held appropriately, on the basis of credal facts alone. Several well-known—and some lesser known—objections to credal reductivism turn on the inability of standard credal reductivist views to get the latter, normative, results right. This paper presents and defends a novel credal reductivist view according to which belief is a type of “imprecise credence” that escapes these objections by including an extreme credence of 1.

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  1. 1.

    I adopt the term ‘credal reductivism’ and its cognates from Ross and Schroeder (2012)—rather than ‘lockeanism’, ‘thresholdism’ or one of the other various labels appearing in the literature, since those terms are more appropriately thought of as picking out specific types of reduction rather than reductions of the general kind.

  2. 2.

    Where this threshold is set will be discussed in Sect. 3.

  3. 3.

    While the exact view on offer makes novel predictions in many of the cases discussed below, it draws inspiration from, and bears some structural resemblance to other accounts. In particular, it can be seen as relative of the “stability centric” views of credal norms espoused in Fraassen (1995), Leitgeb (2013) and, more recently, Arló-Costa and Pedersen (2012). One notable difference that makes the comparison imperfect is that the view being advanced attempts a metaphysical reduction of belief to credences rather than being a specification of the norms governing beliefs in credal terms. The account also draws on “imprecise credence” views, like Levi (1974) and Sturgeon (2008).

  4. 4.

    A convincing argument for the stronger claim that this normative fact is partially constitutive of the concept of belief can be found in Shah (2003).

  5. 5.

    This principle finds many defenders in the literature. It is argued for in Fantl and McGrath (2009, Chap. 5), from whom ((Ross and Schroeder 2012, p. 17)) borrow the principle. A principle entailing Correctness Footnote 5 continued is also advanced and defended in Wedgwood (2002), amongst other places. For a dissenting view when Correctness is understood as a truth norm to which one can be held accountable—as we are understanding it here—see Feldman and Conee (1985) and Feldman (2000).

  6. 6.

    Since it makes sense to conditionalize on probability zero events in transfinite cases, like that of Archimedes and Reid, rational credence functions should be understood in terms of fundamentally dyadic probability functions—the standard Kolmogorov definition of conditional probability as the ratio of monadic probabilities \( {\mathbb {P}}\left[ {A} \big | {B} \right] = \frac{ {\mathbb {P}}\left[ {A \wedge B} \right] }{ {\mathbb {P}}\left[ {B} \right] } \) will be undefined in some of the relevant cases. I consequently take the notion of probability at issue in the account given to be fundamentally two-placed, as in Popper (2002). Good arguments for the conclusion that conditional probability is the more basic notion can be found in Hájek (2003). I bring these issues up only to highlight them and set them aside for the remainder of the paper. I am also assuming that we do not make so-called ‘infinitesimal credence assignments’. Good reasons for making this assumption can be found in Easwaran (2014).

  7. 7.

    This is not to say that one gets everything epistemicially right when one has an extreme credence 1 in the truth. For instance, it is plausible that one should also “respect one’s evidence” in a way that can come into conflict with truth norms. It is also possible that considerations of probabilistic coherence, the causal antecedents of one’s credences, or other considerations matter epistemically.

  8. 8.

    Interestingly, “extreme” but non-1 credences, in the sense of being a high credence that, conditional on any compatible proposition, remains above a given threshold might also be able to capture Correctness since there is a sense in which such credences are close to being “fixed-points” in reasoning. Thus, for instance, the Arló-Costa and Pedersen (2012) view might be able to secure something like Correctness. I set aside this strategy—though it certainly merits further examination—on the grounds that (i) such views need to restrict either the range of the propositions conditionalized over, or the algebra over which credences are defined, in a substantial, and as of yet unexplained, way in order to avoid the conclusion that we never have beliefs; and (ii) it is unclear that these views have the resources to explain the relationship between belief and epistemic possibility that are unpacked in the following paragraphs.

  9. 9.

    See, for example, Holton (2013) and Ross and Schroede (2012, pp. 27–8). Notably, while these authors take this observation to weigh against credal reductivism, it is a point that weighs in favor of the credal reductivism proposed here.

  10. 10.

    I thank an anonymous referee for encouraging me to expand on, and clarify, this point.

  11. 11.

    For example, Foley (1993), Kaplan (1996, p. 91 ff), and Ganson (2008, p. 451).

  12. 12.

    If being belief-level-committed to the truth of \(p\) is a vague matter, then the constraint that \(\gamma \) places on \(\tau \) will be a vague matter as well. If the level of \(\gamma \) overridingly fixes the lower bound of \(\tau \), then the value of \(\tau \) will be vague too.

  13. 13.

    This is also welcome given that belief feels coarse “from the inside”, as has been noted in Sturgeon (2008).

  14. 14.

    Of course, nothing hangs on this being the correct value for \(\tau \). The argument could be reformulated with any threshold value less than 1.

  15. 15.

    Principles that attempt to capture the truism are often referred to as “direct inference” principles in the literature. See Pollock (1990) for further discussion. For a discussion of the truism under the heading of the “Principal Principle”, see Lewis (1986).

  16. 16.

    See, for example, Harman (1967) and Nelkin (2000).

  17. 17.

    A recent exemplar is Hawthorne (2004, pp. 15–20).

  18. 18.

    Perhaps it might be rational for a subject to have credences of this sort if one thought that the lottery was fair, but suspected that there was a chance that it was rigged.

  19. 19.

    For an example of a credal reductivist who does, see Sturgeon (2008).

  20. 20.

    It is worth pointing out that certain intuitions tell against this treatment of the lottery case. In particular, contra NACR, one might have the intuition that it is possible to believe a lottery proposition while having a precise credence less than 1 in that proposition. To those worried by this possibility, I offer two responses. First, the motivations for the Extreme Credence Thesis given in Sect. 2 tell against this possibility. Second, it should be noted that recent philosophical and psychological research concerning how we reason with credences suggests that much credal reasoning, though perhaps not credal reasoning that constitutes reasoning with belief on the present account, occurs below the conscious level (Staffel 2012). Thus, we should not expect our access to our credal levels to be particularly transparent. I thank an anonymous referee for pressing me to address this point.


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I would like to thank John Hawthorne, Mark Schroeder, and Julia Staffel for valuable comments on an earlier draft of this paper. Special thanks are due to Kenny Easwaran who provided comments on several drafts of this paper. Of course, any mistakes in the current paper are my own. I would also like to thank the Social Sciences and Humanities Research Council of Canada for financial support during the writing of this paper in the form of a doctoral award, number 752-2010-0298.

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Correspondence to Justin M. Dallmann.

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Dallmann, J.M. A normatively adequate credal reductivism. Synthese 191, 2301–2313 (2014). https://doi.org/10.1007/s11229-014-0402-9

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  • Belief
  • Credal reductivism
  • Imprecise credences
  • Correctness
  • The lottery paradox
  • The preface paradox