Abstract
Weight of evidence continues to be a powerful metaphor within formal approaches to epistemology. But attempts to construe the metaphor in precise and useful ways have encountered formidable obstacles. This paper shows that two quite different understandings of evidential weight can be traced back to one 1878 article by C.S. Peirce. One conception, often associated with I.J. Good, measures the balance or net weight of evidence, while the other, generally associated with J.M. Keynes, measures the gross weight of evidence. Conflations of these two notions have contributed to misunderstandings in the literature on weight. This paper shows why Peirce developed each conception of weight, why he distinguished them, and why they are easily mistaken for one another.
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Notes
Levi in his 2011 deserves credit for a clear recognition of the distinction, but he does not pursue the matter.
See W3, pp. 14–109.
For more on the role of deliberately choosing, rather than merely picking, a method of inquiry, see Kasser (2011, pp. 233–234).
For a recent and detailed discussion of the pragmatic maxim, see Burke (2013).
I am grateful to Cheryl Misak for directing me to a more recent paper by Levi in which he seems to deny that Peirce countenaces degrees of belief, even when betting rates can be grounded in statistical information. See Levi (2012, p. 157).
For some details involved in the derivations, see Schum (1994, p. 216).
For a somewhat similar discussion, see Levi (1995, p. 67).
See Joyce (2005, p. 156).
I do not mean to be making the stronger claim that Peirce’s way of thinking about probabilities is fully neutral between conceptualist/Bayesian and materialist/frequentist approaches. As Adler argues, partial belief “points inward toward the inquirer” and one cannot “look through” partial belief to the world, as one arguably can for full belief. So I do not mean to deny that Peirce’s insistence that probability statements must rest on facts puts conceptualists at a disadvantage. I merely insist that it’s not an insurmountable one. See Adler (2002, p. 237).
Unfortunately and surprisingly, Schum’s valuable discussion gets Peirce quite wrong on this point. Schum writes that “one trouble with Peirce’s idea is that logarithms of Pascalian probabilities are always less than or equal to zero.” That’s true enough, but Peirce makes it quite plain that he is taking the logarithms of chances, not probabilities. Schum is misled by Peirce’s use of “chance.” Schum doesn’t see the connection Peirce forges between chances and odds. He then goes on to commend a log-odds approach, not seeing that it is Peirce’s.
Schum (1994, p. 217) Once again, Schum’s valuable discussion is insufficiently charitable to Peirce. Schum claims that “one matter Peirce did not consider is that evidence having any force at all causes a change in belief, which we must determine in order to grade the force of evidence in Pascalian terms” (ibid). Peirce’s discussion is too compressed, but a mere consideration of the fact that “Probability” is part of a serious that continues Peirce’s doubt-belief theory of inquiry as articulated in “Fixation” renders it highly improbable that Peirce ignored the fact that we always set out from a state of belief and that inquiry involves the updating of prior opinions. On the other hand, it must be granted to Schum that Peirce’s distrust of principles of indifference (on which more below) and his insistence that probability judgments must represent a fact (as discussed above) make it difficult to see how, exactly, prior states of belief are to be represented for Peirce. Schum is right that weight of evidence can’t be made to link up to appropriate feelings of belief unless there is some prior belief state involved. I am only gesturing in the direction of an answer here. This matter deserves a paper of its own.
See W3 247 and 263.
Good sometimes credits Turing rather than Peirce with the suggestion of the log-likelihood measure. See Good (1983, pp. x–xi and 36–38). The complications arise because of a dispute about how adequately Peirce characterized what it is for two arguments to be independent of one another. For an exchange between Good and Levi on this topic, see Good (1981) and Levi (2011, p. 40). For Good’s treatment of the issue more generally, see Good (1984). Schum (1994, p. 229), credits Good and Turing with being the first to propose the log-likelihood measure of weight. But this is because Schum has misunderstood Peirce, as we’ve seen.
Schum is an important exception. Cheryl Misak has pointed out that the interpretation of Ramsey on this score is more contentious than is generally recognized.
See, for example, the exchange between Good and Seidenfeld in Good (1985) about whether Good’s conception of weight of evidence captures Popper’s notion of corroboration. For a contemporary survey of Bayesian approaches to incremental confirmation, including the log-likelihood ratio, see Fitelson (2001).
See Good (1985, pp. 253–267).
O’Donnell (1989, p. 70), suggests a qualification. When the probability of a statement is 0 or 1, he suggests, weight is automatically at a maximum.
See Skyrms (1977). The characterization of Joyce’s view involves a simplification. What Joyce actually says is that Keynesian weight “stabilizes not the probabilities of the chance hypotheses themselves, but their probabilities discounted by the distance between X’s chance and its credence” (p. 166).
By “independent,” I don’t mean that the value of Keynesian weight can’t be captured in terms of, say, belief probabilities. Joyce offers an illustration of how that might work. I just have in mind the minimal sense that improvement along one dimension of one’s evidential situation is compatible with weakening along another.
Levi (2011), as noted, is the shining exception. He alone seems to be clear about the distinction between net and gross weight of evidence in Peirce, but he doesn’t seem particularly interested in developing it. That paper, despite its title, does not offer a sustained examination of the two approaches to evidential weight, but instead a comparison of Peirce and Keynes on a number of issues.
See Cottrell (1993) on this point.
See Good (1985, pp. 264–268).
For a bit of discussion of nearness to proof and of the stopping problem, see Levi (2011).
For some discussion, see Adler (2002, p. 252).
For a recent treatment, see Misak (2004).
This forms a crucial component of Peirce’s solution to Peirce’s Puzzle, discussed earlier.
The most extensive discussion of the notion of settled belief in Peirce is in Kasser (2011).
This formulation is merely suggestive and omits many important matters of detail. I elide the distinction Joyce draws between specificity, viz. “the degree to which the data discriminates the truth of the proposition [in question] from that of others” (2005, p. 154) and both force and weight, and I ignore such matters as that a belief can be highly resilient in the face of some kinds of data while remaining highly susceptible to being undermined by other kinds of data.
See Popper (1959, pp. 406 ff).
See Joyce (2005, pp. 170–171).
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Acknowledgments
The author would like to thank Katie McShane, Louis Loeb, James Joyce, Brian Weatherson, David Beisecker, Cheryl Misak, an anonymous referee for this journal, and the audience at the conference in honor of Louis Loeb at The University of Michigan who provided helpful comments on an earlier version of this paper. This work was supported in part by the Endowment Fund of the Department of Philosophy, Colorado State University.
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Kasser, J. Two Conceptions of Weight of Evidence in Peirce’s Illustrations of the Logic of Science . Erkenn 81, 629–648 (2016). https://doi.org/10.1007/s10670-015-9759-5
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DOI: https://doi.org/10.1007/s10670-015-9759-5