Abstract
According to the Classical Pessimistic Induction (CPI), otherwise positive features like predictive and explanatory success actually cast doubt on theories that display them. After all, they negatively correlate with truth. From this historical track record, it is inferred that current best theories are probably not (even approximately) true. The CPI is often wielded against realists about science, but others like Hume (Moral philosophy. Geoffrey Sayre McCord (ed), Hackett Publishing Company, 1757), Pareto (The mind and society, vol 1. (Arthur Livingston, Trans.), Harcourt, Brace and Company, New York, 1935), and Hájek (in: Green and Williams (eds) Moore’s paradox: new essays on belief, rationality, and the first person, Oxford University Press, 2007) extend it to philosophical theories more generally. In this paper, I unveil a priori refutations that defang the CPI in all its guises. To date, treatments of this topic typically assume the following: if the historical proportion of false theories increases, then it has increasingly negative bearing against realism. I show that this assumption is false. I identify three types of problems for the CPI, each attaching to some interpretation or other. The first problem is that the CPI violates a universal adequacy condition on the confirmation relation that restricts the relationship between entailment and disconfirmation. The second problem is that the induction’s target hypotheses preclude the track record from instantiating a list of epistemically significant properties that empirical information is supposed to afford. The third problem is that it is constitutively impossible for the CPI to have random samples; this implies that even the use of randomization procedures cannot help. In sum: my refutation is novel, accessible by pure reflection, and quite general.
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Notes
This nomenclature is borrowed from Stanford (2006, ch. 1.2).
The CPI features with varying prominence in many discussions, including Putnam (1978), Newton-Smith (1981), Horwich (1982), Kuhn (1992), Weston (1992), Kitcher (1993), Worrall (1994), Hobbs (1994), Leplin (1984; 1997), Lipton (2000), Lewis (2001), Blackburne (2002), Lange (2002), Kukla and Walmsley (2004), Magnus and Callender (2004), Stanford (2003, 2006, ch. 1.2), Chakravartty (2007), Bird (2007, 2022, p. 232–239), Nola (2008), Magnus (2010), Papineau (2010), Roush (2010), Frost-Arnold (2011), Devitt (2010, ch. 4, 2011), Fahrbach (2011a, 2011b, 2017), Mizrahi (2013), Wray (2013; 2018), Alai (2017), Niiniluoto (2017), Schech (2019), Ladyman (2021), Park (2022), and Vickers (2023). It is immediately worth noting that the CPI is one argument within a broader family of arguments associated with the name ‘Pessimistic Meta-Induction’. Other arguments associated with that moniker merit their own independent treatment. Examples include but needn’t be limited to Stanford’s (2006) ‘New Pessimistic Induction’, Frost-Arnold’s (2019) ‘Misleading Evidence Induction’, and Achinstein’s (2023) version that inductively predicts disconfirmatory evidence against scientific speculations. Each argument poses a different induction that advances different concerns about the epistemic weight afforded by aspects of scientific inquiry. This family of concerns is at least as old as Montaigne’s (1576) ‘Apology for Raymond Sebond’ and its members are sometimes conflated with each other in the literature. Not only are different inductions sometimes conflated, but they are even commonly conflated with Laudan’s (1981) deductive counterexamples to Putnam’s (1978) ‘No Miracles Argument’. For differentiation between Laudan’s argument and the Pessimistic Meta-Induction, see Lyons (2002), Saatsi (2005, 2019), Cartwright et al., (2022, p. 34–35), and Bird (2022, p. 232). In this essay I focus entirely on the CPI, leaving other meta-inductions to receive treatment elsewhere.
This includes the following. First, how one defines the CPI’s reference class. Second, whether we restrict the CPI to have ‘local’ applications to the special sciences only or whether it is less restricted and applied ‘globally’ to science or even philosophy in general. Third, whether the population space from which we sample theories is finite or infinite. Fourth, the failure of the No-Miracles Argument, if indeed it fails. Fifth, whether confirmation holism or non-holism is true. It is a realist custom to advocate for non-holistic ways in which confirmation might distribute throughout a theory. See e.g., Kitcher (1993), Psillos (1999), and Chakravartty (2007). Evidential propagation is thought to be determined in part by the distinction between working and non-working bits of theory. My refutation of the CPI takes no stance on the specifics of evidential propagation throughout a theory. For a more general discussion about holism and associated ideas about the distributive properties of confirmation, see Sober (2000).
Frost-Arnold (2011, 2014) argues that if the realist is correct about contemporary scientific theories, then many mainstream semantic theories imply that terms like “phlogiston” and “miasma” are empty, rendering past theories neither true nor false. In what follows, the reader may use the terms ‘false’ and ‘not true’ interchangeably depending on their favored semantic theory.
Three quick points. First, by ‘best’ I just mean those theories that at some time displayed the realist’s positive epistemic markers to the highest degree. Second, I’m assuming on behalf of the CPI that there is a useful theory/observation distinction ready to hand. For some discussion on this, see Hobbs (1994). Third, we don’t yet need to distinguish between truth and approximate truth until Sect. 3. Until then, for brevity’s sake, I simply use ‘not (even approximately) true’ and ‘untrue’ interchangeably, and the same for ‘approximately true’ and ‘true’.
I consider another way to interpret it in Sect. 5.
I use the term ‘data’ to refer to the relatum in the confirmation relation that plays the role of foreground information. The term ‘evidence’ is typically used to refer to this relatum. I opted not to use that term since an aim of this paper is to show that such foreground information cannot amount to genuine evidence in the first place. While ‘data’ is used differently in other contexts, it is best suited for my purposes here. It conveys foreground information that is taken for granted but leaves open the question of whether it can make a difference to hypotheses.
For proofs, see Appendices A and B. Note also that it isn’t just our best quantitative theories of evidence that satisfy the Converse Entailment Restriction. Popperian confirmation skeptics likewise accept the Converse Entailment Restriction since they hold that H is disconfirmed by D exactly if H and D are logically inconsistent and D has been empirically established. To Popperians, the only way to get disconfirmation is full falsification. So, the condition specified in the antecedent above would be just another way that disconfirmation fails in Popperian eyes.
For a proof, see Appendix C.
A similar issue is anticipated by Augustine (1995) in his objection to Ciceronian Skepticism. In his Against the Academicians, Augustine writes: ‘Please pay the closest attention…If a man unacquainted with your father were to see your brother and assert that he is like your father, won’t he seem to you crazy or simple minded?’ [2.7.16.20]. Augustine’s point is that one needs a standard of comparison to make proper ascriptions of truth-likeness.
For examples of these types of conditionals in use, see Lyons (2002, 2006), Vickers (2017), Hricko (2021), and Tulodziecki (2021). For a method that is supposed to epistemically justify these conditionals—for more than one version of the pessimistic induction—see Stanford (2018). Lastly, these two conditionals are worded in a somewhat cumbersome manner. This is intentional since it makes the conditionals either fully true or false and thus amenable to standard probabilistic treatment if one so desired. This trick for transforming verisimilitude claims into truth claims is used by Bird (2007, p. 76–78).
Possible objection: the CPI shouldn’t bring Bn and Pn to bear period. Reply: that is surely incorrect. Insofar as the CPI is meant to put realism to the test, realists must be able to use the full ambit of their commitments to see if the putative structural problem with realism is genuine.
Objection: there is plenty of mystery in scientific reasoning (e.g., the confirmational value of peer review). Reply: to defuse the analogy I need only identify a difference in kind such that the negative aspects of the society can plausibly serve as the basis for my ‘guilt by association’ explanation for why it seemed that Jones’s final rejoinder was illicit, without having those same negative aspects map onto genuine science. The aspects of science that eliminate error suffice for this. Even if there are remaining questions about scientific reasoning, we can at least say that error identification is possible and even common there. Not so for the black-box method. We have no positive conception of the box’s inner circuitry, so we can’t identify any such bits as sources of error. Cumulative error elimination makes it doubtful that successive iterations of scientific theorizing are mutually independent. The black-box, however, straightforwardly conforms to toy models of induction—e.g., coin-flipping—with independent successive trials. Note though, that I do not appeal to these dynamic aspects of science to refute the CPI proper, like Leplin (1997, p. 141), Lipton (2000, p. 204), Doppelt (2007, p. 111), Devitt (2010, p. 96–98), Roush (2010, p. 34), Fahrbach (2011a, 2011b, 2017), Mizrahi (2013), and Park (2022, p. 32). My appeal aims simply to break the analogy between the marble society and science.
Reflection on these four dubious features reveals that they are properties of the track record itself which is conceptually prior to any confirmational role it plays. This implies that these same issues manifest whether one considers the disconfirmation or base rate interpretations of the CPI that use a true/false binary.
We do not know which historical theories will be shaved off ~ Hn and relocated to K prior to historical investigation. That varies from world to world. What we do know a priori is that whatever theories get shaved off from ~ Hn and relocated to K in a well-defined induction, the resulting ~ Hn* will imply whatever historical series that world contains. The inevitability of implication regardless of world-specific detail is my point.
Consider also that the so-called reference class problem for induction (and its corresponding candidate solutions) is entirely motivated by the fact that the specific choice of reference class is a substantive matter since it can alter whether the track record has positive or negative bearing on the inductive hypothesis. Pessimists like Hobbs (1994, p. 185–186) and Wray (2013) take care to specify the right reference class, and their adversaries display no shortage of attempts to ‘limn’ the historical list of theories by advocating for ever more specific formulations of historical realist commitments.
Possible retort: the logical relationships within the argument are a feature rather than a bug once the argument is shorn of any pretense to empiricality. My reply: this innovation would cease to even resemble the argument of interest that has been discussed since the time of Montaigne. Consideration of an argument which is that different would take us beyond the remit of this paper.
For discussions on different philosophically significant notions of statistical ‘randomness’, see Kyburg (1974, p. 216–246), Levi (1980, p. 392–398), Campbell and Franklin (2004), and McGrew and McGrew (2007, p. 148–153). The point that I make in what follows is not affected by the question of whether randomness is a property of sampling procedures or a property of the samplers’ knowledge states.
Mizrahi (2013) applies a randomization device to the historical record after noticing that few seem to be concerned with proper procedure. But the epistemic role of his procedure is downstream from the problem source.
This example is borrowed from Titelbaum (2022, p. 198).
This diagnosis is restricted to the CPI proper. In the Golden Marble Society’s meta-induction, the problem arose from the property of uniqueness.
For other versions of the Pessimistic Meta-Induction, see footnote 2.
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Acknowledgements
Thanks to Alan Sidelle, Peter Vranas, Elliott Sober, Hayley Clatterbuck, Meryem Keskin, Christopher Pincock, Keshav Singh, Peter Tan, Paul Kelley, Nate Lauffer, and two anonymous referees at Erkenntnis for helpful feedback on this essay. Thanks also to Lindsey Brainard and Kenneth Boyce for their helpful comments at the 2022 APA Central Division Meeting’s symposium on this paper. Lastly, special thanks to Mike Titelbaum for his wisdom throughout this essay’s development.
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Appendices
Appendix A
The following is a proof that Bayesianism satisfies the Converse Entailment Restriction. Here, PrK(-) and PrK(- | -) are probability functions with the background corpus encoded into them.
1. Suppose that there is a well-defined probability distribution D1 containing logically consistent data D, hypothesis H, and total background corpus K | |
where H&K╞ D | Assumption |
2. D disconfirms H relative to K iff PrK(H) > PrK(H|D) | Bayesian Assumption |
3. PrK(H|D) = PrK(H)PrK(D|H)/PrK(D) | Bayes’s Theorem |
4. PrK(D) = PrK(H)PrK(D|H) + PrK(~ H)PrK(D|~ H) | Law of Total Probability |
5. Either PrK(H) < PrK(D|H), PrK(H) = PrK(D|H), or PrK(H) > PrK(D|H) | Tautology |
6. PrK(D|H) = 1 | 1 Math |
7. PrK(H) > PrK(D|H) | Assume for Reductio |
8. PrK(H) > 1 | 6, 7 Identity Substitution |
9. It is false that PrK(H) > 1 | Maximality Theorem |
10. PrK(H) > 1 and it is false that PrK(H) > 1 | 8, 9 Conjunction Introduction |
11. It is false that PrK(H) > PrK(D|H) | 7–10 Reductio |
12. Either PrK(H) < PrK(D|H) or PrK(H) = PrK(D|H) | 5, 11 Disjunctive Syllogism |
13. PrK(H) = PrK(D|H) | Assume for Conditional Proof |
14. PrK(H) = 1 | 6, 13 Identity Substitution |
15. PrK(H|D) = 1 × PrK(D|H)/PrK(D) | 3, 14 Identity Substitution |
16. PrK(H|D) = 1 × 1/PrK(D) | 6, 15 Identity Substitution |
17. PrK(H|D) = 1/PrK(D) | 16 Math |
18. PrK(H|D) = 1/PrK(H)PrK(D|H) + PrK(~ H)PrK(D|~ H) | 4, 17 Identity Substitution |
19. PrK(H|D) = 1/1 × PrK(D|H) + PrK(~ H)PrK(D|~ H) | 16, 18 Identity Substitution |
20. PrK(H|D) = 1/1 × 1 + PrK(~ H)PrK(D|~ H) | 6, 19 Identity Substitution |
21. PrK(~ H) = 1-PrK(H) | Negation Theorem |
22. PrK(~ H) = 0 | 14, 21 Math |
23. PrK(H|D) = 1/1 × 1 + [0 × PrK(D|~ H)] | 20, 22 Identity Substitution |
24. PrK(H|D) = 1 | 23 Math |
25. PrK(H) = PrK(H|D) | 14, 24 Identity Transitivity |
26. It is false that D disconfirms H relative to K | 2, 25 Biconditional Exchange |
27. If PrK(H) = PrK(D|H), then it is false that D disconfirms H relative to K | 13–26 Conditional Proof |
28. It is false that PrK(H) < PrK(D|H) only if it is false that D disconfirms H relative to K | Assume for Reductio |
29. PrK(H) < PrK(D|H) and it is not false that D disconfirms H relative to K | 28 Negated Conditional |
30. It is not false that D disconfirms H relative to K | 29 Conjunction Elimination |
31. D disconfirms H relative to K | 30 Double Negation |
32. PrK(H|D) = PrK(H) × 1/PrK(D) | 3, 6 Identity Substitution |
33. PrK(H|D) = PrK(H)/PrK(D) | 32 Math |
34. PrK(H|D) = PrK(H&D)/PrK(D) | Ratio Formula |
35. PrK(H&D)/PrK(D) = PrK(H)/PrK(D) | 33, 34 Identity Transitivity |
36. PrK(H&D) = PrK(H) | 35 Math |
37. PrK(H) = PrK(H&D) + PrK(H& ~ D) | Decomposition Theorem |
38. PrK(H& ~ D) = 0 | 36, 37 Math |
39. PrK(D) = PrK(D&H) + PrK(D& ~ H) | Decomposition Theorem |
40. PrK(H|D) = PrK(H&D) + PrK(H& ~ D)/PK(D) | 33, 37 Identity Substitution |
41. PrK(H|D) = PrK(H&D) + PrK(H& ~ D)/PrK(D&H) + PrK(D& ~ H) | 39, 40 Identity Substitution |
42. PrK(H|D) = PrK(H&D) + 0/PrK(D&H) + PrK(D& ~ H) | 38, 41 Identity Substitution |
43. PrK(H|D) = PrK(H&D) + 0/PrK(H&D) + PrK(D& ~ H) | 42 Commutation |
44. PrK(H|D) = 0/PrK(D& ~ H) | 43 Math |
45. PrK(H&D)/PrK(D) = 0/PrK(D& ~ H) | 34, 44 Identity Substitution |
46. PrK(H&D)PrK(D& ~ H) = PrK(D) × 0 | 45 Math |
47. PrK(H&D)PrK(D& ~ H) = 0 | 46 Math |
48. PrK(D& ~ H) = 0/PrK(H&D) | 47 Math |
49. PrK(H&D) ≥ 0 | Tautology |
50. PrK(H&D) = 0 | Assume for Reductio |
51. PrK(D& ~ H) = 0/0 | 48, 50 Identity Substitution |
52. PrK(D& ~ H) is not well-defined | 51 Math |
53. PrK(D& ~ H) is well-defined and PrK(D& ~ H) is not well-defined | 1, 52 Conjunction Introduction |
54. PrK(H&D) ≠ 0 | 50–53 Reductio |
55. PrK(H&D) > 0 | 49, 54 Disjunctive Syllogism |
56. PrK(D& ~ H) = 1 | 48, 55 Math |
57. PrK(H|D) = 0/1 | 44, 56 Identity Substitution |
58. PrK(H|D) = 0 | 57 Math |
59. It is false that PrK(H) < 0 | Non-Negativity Axiom |
60. PrK(D) < PrK(D|H) iff PrK(H) < PrK(H|D) | Confirmation Symmetry |
61. PrK(D) < PrK(D|H) iff PrK(H) < 0 | 58, 60 Identity Substitution |
62. Either PrK(D) < PrK(D|H), PrK(D) = PrK(D|H), or PrK(D) > PrK(D|H) | Tautology |
63. Either PrK(D) < 1, PrK(D) = 1, or PrK(D) > 1 | 6, 62 Identity Substitution |
64. It is false that PrK(D) > 1 | Maximality Theorem |
65. Either PrK(D) < 1, PrK(D) = 1 | 63, 64 Disjunctive Syllogism |
66. PrK(D) = 1 | Assume for Reductio |
67. PrK(H|D) = PrK(H)/1 | 33, 66 Identity Substitution |
68. PrK(H|D) = PrK(H) | 67 Math |
69. It is false that D disconfirms H relative to K | 2, 68 Biconditional Exchange |
70. D disconfirms H relative to K and It is false that D disconfirms H relative to K | 31, 69 Conjunction Introduction |
71. PrK(D) ≠ 1 | 66–70 Reductio |
72. PrK(D) < 1 | 65, 71 Disjunctive Syllogism |
73. PrK(D) < 1 iff PK(H) < 0 | 61, 72 Identity Substitution |
74. PrK(H) < 0 | 72, 73 Biconditional Exchange |
75. PrK(H) < 0 and it is false that PrK(H) < 0 | 59, 74 Conjunction Introduction |
76. PrK(H) < PrK(D|H) only if it is false that D disconfirms H relative to K | 28–75 Reductio |
77. PrK(H) = PrK(D|H) only if it is false that D disconfirms H relative to K, and PrK(H) < PrK(D|H) only if it is false that D disconfirms H relative to K | 27, 76 Conjunction Introduction |
78. Either it is false that D disconfirms H relative to K, or it is false that D disconfirms H relative to K | |
12, 77 Disjunctive Dilemma | |
79. It is false that D disconfirms H relative to K | 78 Tautology |
80. For any well-defined probability distribution D containing logically consistent data D, hypothesis H, and background knowledge K: if H&K╞ D, then | |
D does not disconfirm H relative to K. | ■ |
Appendix B
The following is a proof that Likelihoodism satisfies the Converse Entailment Restriction.
1. Suppose that there is a well-defined probability distribution D1 containing logically consistent data D, competing hypotheses H1 and H2, and total | |
background corpus K where H1&K╞ D | Assumption |
2. D disfavors hypothesis H1 to H2 iff PrK(D|H1) < PrK(D|H2) | Likelihoodist Assumption |
3. PrK(D|H1) = 1 | 1 Math |
4. Either PrK(D|H2) < 1, PrK(D|H2) = 1, or PrK(D|H2) > 1 | Tautology |
5. It is false that PrK(D|H2) > 1 | Maximality Theorem |
6. PrK(D|H2) < 1 or PrK(D|H2) = 1 | 4, 5 Disjunctive Syllogism |
7. PrK(D|H2) = 1 | Assume for Conditional Proof |
8. PrK(D|H1) = PrK(D|H2) | 3, 7 Math |
9. It is false that D disfavors H1 to H2 | 2, 8 Biconditional Exchange |
10. If PrK(D|H2) = 1, then it is false that D disfavors H1 to H2 | 7–9 Conditional Proof |
11. PrK(D|H2) < 1 | Assume for Conditional Proof |
12. PrK(D|H1) > PrK(D|H2) | 3, 11 Math |
13. It is false that D disfavors H1 to H2 | 2, 12 Biconditional Exchange |
14. If PrK(D|H2) < 1, then it is false that D disfavors H1 to H2 | 11–13 Conditional Proof |
15. PrK(D|H2) = 1 only if it is false that D disfavors H1 to H2, and | |
PrK(D|H2) < 1 only if it is false that D disfavors H1 to H2 | 10, 14 Conjunction Introduction |
16. It is false that D disfavors H1 to H2 or it is false that | |
D disfavors H1 to H2 | 6, 15 Disjunctive Dilemma |
17. It is false that D disfavors H1 to H2 | 16 Tautology |
18. For any well-defined probability distribution D containing logically consistent data D, competing hypotheses H1 and H2, and total | |
background corpus K where H1&K╞ D, then D does not disfavor H1 to H2 relative to K. | ■ |
Appendix C
Here is a derivation of GER from the Converse Entailment Restriction. For any hypothesis H, data D, total background corpus K and its proper subset K*:
1. If H&K╞ D, then D does not disconfirm H relative to K | Assumption |
2. K╞ K* | Tautology |
3. H&K*╞ D | Assume for Conditional Proof |
4. H&K | Assume for Conditional Proof |
5. K | 4 Conjunction Elimination |
6. K* | 2, 5 Modus Ponens |
7. H | 4 Conjunction Elimination |
8. H&K* | 6, 7 Conjunction Introduction |
9. D | 3, 8 Modus Ponens |
10. H&K╞ D | 5–9 Conditional Proof |
11. D does not disconfirm H relative to K | 1, 10 Modus Ponens |
12. If H&K*╞ D, then D does not disconfirm H relative to K | 3–11 Conditional Proof |
13. For any hypothesis H, data D, total background corpus K and its subset K*: If H&K*╞ D, then D does not disconfirm H relative to K | ■ |
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Cronin, P. An A Priori Refutation of the Classical Pessimistic Induction. Erkenn (2024). https://doi.org/10.1007/s10670-024-00832-5
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DOI: https://doi.org/10.1007/s10670-024-00832-5