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A Twofold Tension in Schurz’s Meta-Inductive Solution to Hume’s Problem of Induction


This paper examines a twofold tension in Gerhard Schurz’s (2019) recent proposal to solve Hume’s problem of induction. Schurz concedes to the skeptic that there is no non-circular epistemic justification of the reliability of induction, but then argues for the optimality of meta-induction so that if any prediction method is reliable, then meta-induction is. There is a tension in this proposal between meta-induction and our inductive practice: Are we supposed to abandon our inductive practice in favor of meta-induction? Schurz claims that given the actual way the world has been, the meta-inductive method supports the standard inductive method. There then arises a second tension: The challenges Schurz cites against the reliability of induction—such as the anti-inductive hypothesis and the state-uniform probability distribution—cast doubt on Schurz’s claim that meta-induction supports induction. The reasoning in this paper moves in the opposite direction: There are ways of answering these challenges in defense of Schurz’s claim that meta-induction supports induction, but these ways also point to ways of defending induction directly without the mediation of meta-induction.

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  1. Schurz himself argues against Reichenbach’s reasoning (Sect. 5.3). Hereafter, section numbers refer to those in Schurz (2019) unless noted otherwise.

  2. This is not quite right. If we can only justify our inductive practice by its optimality and fail to show that it is reliable, then we still need to change some part of our inductive practice. We may continue to use the inductive method for the reason of optimality, but we should not count on it as much as we usually do, and be prepared for frequent prediction failures with fallback plans in place. Justification by optimality also leaves us with an explanatory puzzle. If there is no good reason to think the inductive method is reliable, why has it been so reliable in so many applications? See Shogenji (2019) for these points.

  3. There is one technical issue here, viz. Schurz imposes the restriction that the set of prediction methods to consult is finite (cf. Section 9.2). We can meet the restriction by considering a series that starts with tn−m and ends with tn+m for some m. The points below are unaffected as long as m is sufficiently large.

  4. The earliest and the best-known among those formulas is Akaike Information Criterion (AIC) due to Akaike (1974). See Burnham and Anderson (2002) for an overview of various formulas. In case some people worry about the reliance on complexity because of the grue-bleen challenge (Goodman 1955) that complexity is relative to the language of choice, Schurz distinguishes the problem of language-dependence from Hume’s problem of induction (Sect. 1.2, 4.2), and his meta-inductive approach is intended for solving Hume’s problem, which remains a challenge even if the problem of language-dependence is solved. That is also my focus in this paper.

  5. See Shogenji (2018, Ch. 6–7) for more on this approach to the problem of induction.

  6. It is due to Carnap (1950, 564–566) but Schurz also relates it to the No-Free-Lunch theorem (Wolpert 1996) in computer science.

  7. Horwich (1982, Ch. 5) uses the coin toss case of this kind to question the common analysis of surprise by probability—that we are surprised at an outcome if (and only if) its prior probability is very low. Against this analysis, Horwich points out that 100 consecutive heads (out of 100 coin tosses) is surprising, while an irregular sequence of about 50 heads and 50 tails is not, though they are equally probable.

  8. The state-descriptions cover all of the non-probabilistic predictions, but many predictions are probabilistic.

  9. Note that this idea is not applicable to the series of anti-inductive methods discussed in Sect. 3. There is no single prediction method that makes the same prediction as the series of anti-inductive methods (as a group) at any point in the prediction game and regardless of any stream of events.

  10. Hindsight bias (also known as the knew-it-all-along phenomenon) is the tendency of people overestimating the prior predictability of an event after the information is given that the event actually occurred. The locus classicus on the subject is Fischhoff (1975). See Roese and Vohs (2012) for a review of the literature.

  11. A “scoring rule” SR(p, i) measures the degree of inaccuracy of the probability distribution p over the partition {x1, …, xn} when the actual outcome is xi. For example, SRL(p, i) = – log p(xi) by the logarithmic scoring rule. See Gneiting and Raftery (2007) for a review of the literature on scoring rules, and Winkler and Jose (2010) for an accessible overview.

  12. The expected inaccuracy of the probability distribution p over the partition {x1, …, xn} is the weighted average \({\sum }_{i}p\left({x}_{i}\right)\text{S}\text{R}(p,i)\) of inaccuracies, where SR(p, i) is the degree of inaccuracy of p given the outcome xi. See Shogenji (2021) for more on the expected inaccuracy and its role in the measure of surprise.

  13. To clarify the point, it is not that XH/10 = {x0, x1, x2, …, x10} is the only legitimate partition in the case. The original partition of the possible outcomes into 1024 exact sequences is also a legitimate partition. The point here is that we should question the default assumption if the actual outcome makes the probability distribution much more inaccurate than expected for at least one of the legitimate partitions.

  14. See Gelman and Shalizi (2013, Sect. 4) on the role of prior probabilities for stabilizing the estimates. There are other ways of stabilizing the estimates, such as Carnap’s (1952) λ rule.


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I would like to thank two anonymous referees for valuable comments that improved the paper significantly.

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Correspondence to Tomoji Shogenji.

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Shogenji, T. A Twofold Tension in Schurz’s Meta-Inductive Solution to Hume’s Problem of Induction. J Gen Philos Sci (2023).

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  • Schurz
  • Induction
  • meta-induction
  • Reliability
  • Optimality
  • Anti-induction
  • State-uniform probability distribution
  • Probabilistic model
  • Adjustable parameter
  • Prior probability
  • Explanatory demand