The Selective Confirmation Answer to the Paradox of the Ravens

ABSTRACT

Philosophers such as Goodman (1954), Scheffler (1963) and Glymour (1983) aim to answer the Paradox of the Ravens by distinguishing between confirmation simpliciter and selective confirmation. The latter evidential relation occurs when data not only confirms a hypothesis, but also disconfirms one of its ‘rival’ hypotheses. The appearance of paradox is allegedly due to a conflation of valid intuitions about selective confirmation with our intuitions about confirmation simpliciter. Theories of evidence, like the standard Bayesian analysis, should only be understood as explications of confirmation simpliciter; when we disambiguate between selective confirmation and confirmation simpliciter, there is no longer a paradox from these theories. Bandyopadhyay and Brittan (2006) have revived this answer within a sophisticated Bayesian analysis of confirmation and severe testing. I argue that, despite the attractive features of the Selective Confirmation Answer, there is no analysis of this evidential relation that satisfactorily answers the Paradox of the Ravens, and the prospects for any answer along these lines are bleak. We must look elsewhere.

Acknowledgements

I have greatly benefited from feedback from Julian Reiss, Nancy Cartwright, Rune Nyrup, Wendy Parker, Donal Khosrowi, Erin Nash, Richard Williams, and everyone at the Centre for Humanities Engaging Science and Society at Durham University.

Notes

1 In accordance with the tradition in the PR literature, I use these predicates as placeholders for more realistic examples. Since we know that there are white ravens, ‘All ravens are black’ is inconsistent with our background knowledge. I use the example to maintain continuity with the literature and rely on the reader’s sense of charity.

2 I.e. assuming that the selection of the sample, the prior probability of ‘All ¬Y are ¬X’, and any other relevant methodological issues are clement to induction in this case.

3 Unlike Hempel and some others in the PR literature, I shall stick to the formal rather than the material mode of describing confirmation, e.g. reports of black ravens, rather than black ravens, confirm/disconfirm hypotheses.

4 From a Bayesian perspective, Good (1960) argued against the Nicod Criterion for confirmation relative to non-tautological background information, while Maher (2004, section 8) provides a Bayesian counterexample to the Nicod Criterion relative to tautological background information. See also Franklin (2001) (287) for an informal example. While statement (c) in my presentation of the PR instantiates the Nicod Criterion, it does not require that this principle holds in general.

5 This list is adapted from a discussion of the Liar Paradox (Kirkham 1992, 273).

6 However, there is a tradition in the PR literature that denies (1) or (2). For examples, see Stove (1966), Cohen (1987), Sylvan and Nola (1991) and Clarke (2010). I shall not discuss such approaches in this article. I shall assume (1) and (2) for easy dialogue with the Selective Confirmation Approach and the PR literature in general, since the majority of philosophers writing on the PR adopt them.

7 Their use of ‘selective confirmation’ in their 2006 article should not be confused with their use in a 2016 book, written with Mark L. Taper, where they are discussing something different.

8 Note that Mayer’s critique does not presuppose that selective confirmation is generally identifiable on formal grounds alone. Nothing that I say in this article should be read as a criticism of the informal concept of selective confirmation in scientific practice.

9 One might speculate that there is yet another cognitive illusion regarding the logical equivalence of ‘All ravens are black’ and ‘All non-black things are non-ravens’. However, there are at least three problems here. Firstly, if this illusion is widespread in science, then scientists’ beliefs are not deductively closed under even very basic logical operations, which (among other problems) would cause problems for the applicability of Bayesian models to understanding actual scientific reasoning. Secondly, while it might be plausible that even experienced logicians who reject the confirmation of ‘All ravens are black’ by such reports, like Quine (1970), might overlook very small degrees of confirmation, it is implausible that these logicians do not know how to contrapose a basic universal generalisation. Thirdly, assuming that we are Bayesian reasoners in some ways (having degrees of confirmation) but not others (logical blindness) violates the Non-Ad Hoc desideratum, since these assumptions lack a motivation beyond defending this answer to the PR. Again, it is possible that these issues can be addressed, but this resolution is beyond the scope of this article.

10 Their use of selective confirmation is part of a broader analysis of severe testing. Although I disagree with their claim that selective confirmation helps with the PR, I have no objection to their broader project or its details; in fact, their analyses and its defences seem very attractive as a way of incorporating many informal methodological ideas about severe testing into a Bayesian framework.

11 More widely, the notion of ‘comparable generality’ for hypotheses seems to depend on two things: (1) the quantifiers used in a particular hypothesis and (2) the scope of those quantifiers. These are fairly easy to compare for simple hypotheses over a particular domain D, e.g. ∀x(Rx → Bx) is more general than ∃x(Rx → Bx), since the former implies non-trivial conclusions for every member of D, whereas the latter does not. I do not know how Glymour and other selective confirmationists would compare hypotheses with multiple and distinct quantifiers like ∀x∃x(Rx → Bx) and ∃x∀x(Rx → Bx). Intuitively, the presence of at least one universal quantifier is sufficient to make a hypothesis completely general (because they have non-trivial implications for each member of D) so that both of these hypotheses are equally general, and as general as ∀x(Rx → Bx).

12 An anonymous reviewer suggests that a selective confirmationist could address this issue by requiring that the evidence is deductively inconsistent with the putatively rival hypothesis. In this case, ∃x(Rx ^ ¬Bx) is consistent with (¬Ra ^ ¬Ba), so this response would address the particular issue of excess breadth. However, this new restriction leads to problems with Completeness, because in general our evidence will not be deductively inconsistent with rival hypotheses: such crucial experiments (or observations) are at best exceptions in science, especially for interesting hypotheses. Often, we only have indirect evidence, such as measurements from scientific instruments that a particle or microbe satisfies a generalisation. Therefore, if we modify the example away from ‘ravens’ and ‘black’, we end up in conflict with the intuition that indirect evidence in favour of ‘This X is a Y’ will selective confirm ‘All X are Y’ in many cases.

13 The literature on the Selective Confirmation Answer to the PR is replete with the use of ‘contraries’ and ‘incompatible’ to describe ∀x(Rx → Bx) and ∀x(Rx → ¬Bx). While there is definitely a non-logical sense in which they are inconsistent (in particular, they cannot both be true if something satisfies the antecedent) we must be on our guard against forgetting that, in the strict logical sense of these terms and given the standard contemporary interpretation of their truth conditions, these hypotheses are logically independent. I shall simply avoid talking this way and instead use terms like ‘rival’ rather than terms like ‘consistent’, as I hope that the former has less of a tendency to suggest that the hypotheses are logical contraries.

14 I am grateful to an anonymous reviewer for help on specifying the form of this predicate.

15 Throughout this article, I assume that none of the hypotheses I discuss has a prior probability of zero unless they are inconsistent with our background knowledge.

16 Given the Scientific Laws Condition and the Equivalence Condition, this also entails that (¬Ra ^ ¬Ba) does not selectively confirm ∀x(¬Bx → ¬Rx). This is because, given their logical equivalence and the Equivalence Condition, the class of rivals for two hypotheses are identical. For example, if there are ravens, then both ∀x(¬Bx → ¬Rx) and ∀x(Rx → ¬Bx) cannot both be true, while both have the same quantifiers and the same (non-logical) terms. I have not found an instance of a defender of Glymour’s version of the Selective Confirmation Answer who explores this consequence.

17 To be precise, our background knowledge minus any knowledge of white ravens we might have. If this is hard to imagine, consider a realistic example.

18 One might argue that our intuition here does not use selective confirmation, but confirmation simpliciter. (I thank an anonymous referee for this suggestion.) However, unless we can justify why we switch our concepts between considering whether (¬Ra ^ ¬Ba) confirms ∀x(Rx → Bx) and whether (Ra ^ Ba) does so, this response violates the Non-Ad Hoc desideratum, since we have no apparent reason beyond the PR to draw an asymmetry between two cases.

19 I thank an anonymous referee for this suggestion.

20 It does not seem to be the typical case: for example, with ‘All samples of alkaline metals react with water’ we know that there are such samples, but also that at least some of them react with water.

21 (∀x(¬Bx → Rx) ^ ∀x(Rx → ¬Bx)) does not violate the comparable generality clause (at least in the sense that I have been able to interpret it) because its quantifiers have the same scope as those of ∀x(Rx → Bx). It might be possible to determine a sense of ‘comparable generality’ which is relative to the scope of the antecedent terms, but the details and broader implications of this would need to be explored.

22 ‘All ravens are black’ and ‘All ravens are not black’ might have different pragmatics even if we disregard our background knowledge, but the current versions of the Selective Confirmation Answer all define the notion in terms of semantic rather than pragmatic features of the hypotheses. Perhaps there could be an answer that is like the existing versions of the Selective Confirmation Answer, but which uses pragmatic features of universal generalisations to draw an asymmetry between ∀x(Rx → Bx) and ∀x(Rx → ¬Bx) when we disregard our background knowledge that there are ravens. However, such an answer is outside the scope of this article. I have not located any such a position in the literature.

23 It is insufficient to formalise them as □∀x(Rx → Bx) and □∀x(Rx → ¬Bx), because both would be true if it was impossible (for the type of modality in question, e.g. physical possibility) that anything was a raven. Arguably, the existential conjuncts are part of our background knowledge, but recall that at this point I am discussing confirmation without reference to background knowledge.

24 This formalisation of universal generalisations would have problems with contraposition, but I shall suppose for the sake of argument that these could be resolved, e.g. by adding more existential content. Of course, any such additional content will not stop them being contraries.

25 One might argue that restricted statistical generalisations like ‘All the ravens in Australia in the twentieth century were black’ do not have comparable generality with ‘All ravens are black’, but that does not affect my arguments with respect to unrestricted statistical generalisations.

26 To have evidence that is intuitively disconfirms the statistical generalisation, I am switching here from unit-sample reports like (Ra ^ Ba) to large sample reports. Arguably, the same effect could be achieved by making the interval in the statistical generalisation very close to zero, but I do not want to go too deep into the logic of statistical inference here.

27 An anonymous referee helpfully points out that, unlike my earlier arguments regarding Selective Confirmation (1), the problems I raise here work even if we do not suppose that (Ra → Ba) should selectively confirm ∀x(Rx → Bx). This occurs because the problem is with Specificity rather than Avoid Overkill.

28 Something similar can also be achieved with ‘All ¬S are C and all S are ¬C’ instead of H_c but statistical generalisations have the feature that, unlike universal generalisations, they are always deductively consistent with our sample reports, so it is easier to think of cases where Hc is consistent with our background knowledge.

29 It is somewhat similar to the analysis of selective confirmation by Goodman and Scheffler, though they have a non-probabilistic analysis of confirmation simpliciter.

30 For instance, ‘Everything is Y if it is X’ would satisfy this clause.

31 They write ‘If we were to maintain that genuine evidence must be able to discriminate between contrary hypotheses, that is to say, cannot be evidence for both, then we could reject white shoes as evidence for the hypothesis that all ravens are black, and with them the paradox.’ (2006, 276). The remark ‘cannot be evidence for both’ might be interpreted as only requiring that E does not support both ∀x(Rx → Bx) and ∀x(Rx → ¬Bx). Nevertheless, they never explicitly restrict the scope of their analysis to universal generalisations or subtypes of universal generalisations, and therefore Selective Confirmation (4) cannot be attributed to them.

32 For example, the restriction to universal generalisations is unmotivated except to make the Selective Confirmation Answer work and so is the modification of Clause (2) to include cases where E is probabilistically irrelevant to the putative rivals.

Additional information

Funding

Philosophy of Pharmacology: Safety, Statistical standards and Evidence Amalgamation; Funding programme: H2020-EU.1.1.; Grant number: 639276