Is an Increase in Probability Always an Increase in Evidential Support?

Abstract

Peter Achinstein has argued at length and on many occasions that the view according to which evidential support is defined in terms of probability-raising faces serious counterexamples and, hence, should be abandoned. Proponents of the positive probabilistic relevance view have remained unconvinced. The debate seems to be in a deadlock. This paper is an attempt to move the debate forward and revisit some of the central claims within this debate. My conclusion here will be that while Achinstein may be right that his counterexamples undermine probabilistic relevance views of what it is for e to be evidence that h, there is still room for a defence of a related probabilistic view about an increase in being supported, according to which, if p(h|e) > p(h), then h is more supported given e than it is without e. My argument relies crucially on an insight from recent work on the linguistics of gradable adjectives.

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Notes

  1. 1.

    Achinstein also thinks that while a high probability of h given e is necessary, it is not sufficient for e being evidence that h; see, for instance, Achinstein (2001: 71). The focus of the present discussion is on the probabilistic relevance views of evidential support, not on the high probability view of e being evidence that h, according to which e is evidence that h if and only if the probability of h given e is higher than some specific threshold of high probability, e.g. .5, .75 or .9.

  2. 2.

    The subtleties of Achinstein’s view on ‘potential’ evidence are, I suggest, not central for our purposes here. Similarly, for what matters here, we can focus on the simple relevance view without always making explicit the reference to possible background information.

  3. 3.

    Note that in later writings (starting from 1983) Achinstein’s formulation of the view in (1b) includes explicitly a reference to background information. See, for instance: “Probabilists may then relativize evidence statements to some set b of background assumptions and say that (7) e is evidence that h, given b, if and only if p(h/e&b) > p(h/b).” (Achinstein 2001: 46). Thanks to an anonymous reviewer for drawing my attention to this aspect.

  4. 4.

    More precisely, a central focus of Achinstein’s work over the years has been on what it is for e to be evidence that p. However, he has also written on topics related to comparative aspects of evidence. In a paper on stronger evidence (Achinstein 1994), he does consider the question of what makes a piece of evidence el stronger evidence (for h) than another piece of evidence e2 (for h). Achinstein’s main thesis there is that a probabilistic account (like Carnap’s) of stronger evidence cannot be maintained. The focus is on the view according to which “Where el and e2 are both evidence for h, given b, el is stronger evidence than is e2 iff p(h/el&b) > p(h/e2&b)” (Achinstein 1994: 331). However, the discussion there is still closely related to Achinstein’s objections to probabilistic views of evidence that h (for instance, in his appeal to evidence being a reason to believe, see: “These combined results provide a reason for believing h that is better, more convincing, than that provided by the first study alone” (Achinstein 1994: 335). We will come back to Achinstein’s appeal to good reasons and reasonable belief in characterizing evidence in what follows.

  5. 5.

    See, for instance, Patrick Maher, a prominent critic of Achinstein’s proposal: “As we see from this example, Achinstein talks of evidence that a hypothesis is true, whereas I talk of evidence for a hypothesis. For Achinstein’s objections to be relevant to the account of confirmation defended here, this difference needs to be regarded as merely stylistic and I will so regard it” (Maher 1996, footnote 11).

  6. 6.

    See, for instance: “econfirms or supportsh just in case P(h|e) > P(h)” (Howson and Urbach 2006: 91–92, original emphasis).

  7. 7.

    In various places Achinstein provides a number of further similar examples against positive relevance, e.g. a swimming champion entering a pool and thereby raising the probability of his death by drowning, without it being the case that his entering the pool is evidence that he will drown and so on. For the sake of the argument, I will assume that all these examples are sufficiently similar for the purpose of the present discussion.

  8. 8.

    See also Achinstein (1981, 1992, 1996, 2001, 2004) for his responses to these replies.

  9. 9.

    It is often presented as a distinction between incremental confirmation and absolute confirmation.

  10. 10.

    See, for instance: “Let us say that E raises the probability of H if the probability of H given E is higher than the probability of H not given E. According to many confirmation theorists, ‘E confirms H’ means that E raises the probability of H. This conception of confirmation will be called incremental confirmation. Let us say that H is probable given E if the probability of H given E is above some threshold. (This threshold remains to be specified but is assumed to be at least one half.) According to some confirmation theorists, ‘E confirms H’ means that H is probable given E. This conception of confirmation will be called absolute confirmation” (Maher 2005: 433). See also Hájek and Joyce (2008) and Kelly (2014), among others.

  11. 11.

    For instance, Carnap proposes to distinguish two triples of concepts of confirmation/evidence. According to Carnap, on one hand there are concepts that are “concerned with the question of how probable the hypothesis h is on the basis of the evidence e” (Carnap 1962: xv–xvi) and on the other hand there are concepts that “relate to the question as to whether and how much the probability of h is increased when new evidence i is acquired” (Carnap 1962: xvi). Carnap calls the former ‘concepts of firmness’ and the latter ‘concepts of increase in firmness’ (cf. Carnap 1962: xvi). And for each of these Carnap defines further distinctions. Namely, both concepts of firmness and concepts of increase in firmness admit, according to Carnap, three further conceptual distinctions: for both of these there are classificatory concepts, comparative concepts, and qualitative concepts. A final complication in Carnap is that among the comparative concepts we can make even further distinctions with respect to what are we comparing. But for our purposes here it suffices to note only the general distinction between concepts of firmness and those of increase in firmness.

  12. 12.

    See Achinstein: “A necessary condition that must be satisfied for hypothesis h to have any acceptability, foundation, or firmness, and before I have any confidence in it, is that h’s probability exceed some threshold” (Achinstein 2001: 73–74).

  13. 13.

    See also: “There is not enough probability here to exceed a threshold necessary for me to have any confidence (for the hypothesis to have any firmness, support, etc.)” (Achinstein 2001: 74, emphasis added).

  14. 14.

    As I am reading this passage I interpret “etc.” and “the other concepts invoked” above as including a reference to the concept of being supported.

  15. 15.

    Which h and e count as relevant is left out of discussion here. This qualification is intended to deal with the fact that according to many, including Achinstein and Maher, a priori or necessary truth will not count as evidence. What exactly this amounts to is not the main object of our discussion. However, we might want to respect the idea that only empirical propositions can count as evidence.

  16. 16.

    Achinstein is right that supported is different from adjectives like tall, rich, good, etc. It is evident that nothing similar to (Bridge Support) would hold for typical gradable adjectives like tall, rich, good, and so on. From the mere fact that this five-year-old child is taller than that three-year-old child, it doesn’t follow at all that the five-year-old child is tall (she may well be short for her age). And from the fact that to kill a victim painlessly is better than to kill the victim painfully, it doesn’t follow at all that to kill a victim can be good.

  17. 17.

    At this point, one might object that even if there is a natural use of supported in our ordinary language that doesn’t validate the Bridge Support principle, we are still not entitled to give up the Bridge Support principle. The thought here would be that the conciliatory position (see below for details) according to which, roughly, one could provide a probabilistic account of comparative support (e.g., ‘x is more supported than y’) but not of categorical support (e.g., ‘x is supported’) is ultimately unwarranted, because it is misleading. In some cases where a hypothesis (e.g., ‘I will win the lottery’) is not supported (i.e. ‘that I will win is supported’ is false), the conciliatory view still implies that the hypothesis is comparatively supported (e.g., it is true that ‘the hypothesis that I will win is more supported, given that I’ve bought a ticket in a 1 million lottery, than it is without my having bought a ticket’). But providing a claim about there being comparative support in cases where there is no categorical support might be misleading and counterintuitive in many scientific and nonscientific contexts. There are two things that I would like to propose in reply to this worry. First, I acknowledge that it might indeed be uncooperative and misleading to reply that the notion that one will win the lottery is more supported now that one has a ticket compared to when one didn’t even have a ticket, when one is asked whether the claim that one will win the lottery is supported. Of course, the same would go for any other case of a similar structure (although I would also like to point out that there might be situations where providing the comparative claim is not misleading, for instance, when historians of science are investigating which one of two competing outdated theories was more supported). But this aspect can be perfectly explained by appeal to Gricean maxims of communication without accepting the Bridge Support principle (by appeal to the maxim of relation/relevance, for instance, cf. Grice 1975). Strictly speaking, comparative claims about being supported can be true without corresponding categorial claims being true. Second, I would like to note that this line of defense of the Bridge Support principle appeals to something beyond our common-sense judgements and ordinary language use. Namely, it questions the theoretical utility of claims about comparative support. The part of Achinstein’s argument that we are focusing on here is his appeal to our ordinary understanding of support, firmness and so on. Hence, unfortunately, a full reply to this challenge lies beyond the scope of the present article. I would like to thank an anonymous reviewer for this journal for making me aware of this potential worry.

  18. 18.

    The sign ‘?’ indicates that the expression is odd.

  19. 19.

    There is a sense, I think, in which what Achinstein calls ‘threshold’ concepts might actually correspond to the broad category of what linguists call ‘absolute gradable adjectives’. However, we lack textual grounds to claim this with any certainty.

  20. 20.

    These (and other) observations or tests in this and the next section for gradable adjectives are to be taken with a grain of salt. It is not claimed that all gradable adjectives in all contexts will validate this sort of behaviour. These are patterns of typical behaviour in normal contexts.

  21. 21.

    More specifically, the interpretation of ‘tall’, according to this account, is given as (cf. Kennedy and McNally 2005: 349): ⟦tall⟧ = λdλx.tall(x) = d, where tall is a measure function that takes an individual, x, as an input and gives x’s degree of tallness as the output. The degree of tallness here can be understood as a degree on the scale of vertical extension. Measure functions have the type <e, d>; they take individuals (in input) and provide degrees (in output).

  22. 22.

    Michael Glanzberg sums up the specificity of this approach insightfully, as follows: “[A] little bit of mathematics can be applied to get further explanations. In particular, Kennedy and McNally (2005) note that the basic topological properties of scales can offer some explanations of semantic properties of adjectives. For instance, scales can be open or closed. This provides a typology of adjective meanings, which helps explain some of their interesting semantic properties” (Glanzberg 2014: 274).

  23. 23.

    Note though that it provides some mitigated results in comparative questions. According to native English language speakers that the author has consulted on this point, ‘How supported is this view?’ sounds clearly odd. This is somewhat confirmed by poor results from Google searches for expressions like ‘How supported is the idea/thought/view’.

  24. 24.

    According to a referee for this journal and another native English speaker, the ‘more than’ constructions (e.g. 16) are a bit awkward. Other native English speakers that the author has consulted on this found (16) to be fine. Note also that (at least some) comparative constructions with ‘unsupported’ appear to be fine. ‘The 9/11 conspiracy theory is as unsupported as the Loch Ness Monster Myth’ is felicitous. One line of thought suggested by the referee here is that there might be a general worry about tests for the gradability of ‘supported’ given that ‘supported’ is a participle. This might explain some of the specific behaviour of ‘supported’ compared to other (non participle) gradable adjectives. See also footnote 25 for a related line of thought. Unfortunately, a proper treatment of this question goes beyond the scope of the present discussion. Nevertheless, I would like to stress here that typically non-gradable adjectives pass none of the relevant tests for gradability. Hence, as long as we have some data speaking in favour of the gradability of the adjective ‘supported’, we may reasonably take it to be gradable. Thanks to the referee for pointing out this aspect of tests with ‘supported’ and to Edgar Phillips for a very useful discussion on this.

  25. 25.

    An anonymous referee for this journal observes that while (19) is odd, a parallel claim with the scalar noun ‘support’ seems fine. See, for instance: (a) ‘This theory has (only) slight support’. It is reasonable to think that the scalar noun ‘support’ would share the same scale structure as ‘supported’. Thus, the difference in our intuitions about (19) and (a) here requires some extra explanation. One quick response is to observe that this oddity seems not to be reserved to ‘supported’ alone. Other paradigmatic absolute adjectives seem to have a similar problem. It does seem odd to say ‘This sample is slightly pure’. However, apparently, ‘Our uranium enrichment will not be limited to 3.67 percent of purity’ (see http://www.xinhuanet.com/english/2019-07/03/c_138195642.htm) is fine (which, I take it, implies that true ascriptions of ‘purity’ to an item doesn’t require the attribution of maximal purity to the item in question). Compare also ‘His mouth was minimally dry’ to ‘His mouth had some dryness’, where the former seems odd, but the latter appears to be fine. Thus, whatever is going on here, it might be a general worry about how to properly connect scalar nouns to (some) gradable adjectives within the scalar approaches to gradable adjectives. Another more constructive line of reply would be to point towards the specificities of gradable adjectives that are participles. Thus, the specificities of ‘supported’ (and the noun ‘support’) and the like might be explained by an appeal to specificities of the event structure of the counterpart verb. How exactly such an explanation would go is beyond our present discussion. The discussion in Kennedy (2007: 36–40) and in Kennedy and McNally (2005) on deverbal adjectives may contain some useful suggestions for further work on this issue. Unfortunately, a fully satisfactory treatment of this question cannot be achieved within the limits of the present article and has to be left for another occasion. Thanks to the referee for making me aware of this point.

  26. 26.

    Thanks to an anonymous referee and Fabrice Teroni for very useful remarks and suggestions on the last two paragraphs.

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Acknowledgements

Many thanks to Davide Fassio, Edgar Phillips, Fabrice Teroni and two anonymous referees for this journal for discussion and comments on earlier version of this paper. The research work that led to this article was supported by the Swiss National Science Foundation Grants Number 169293 and 186137.

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Logins, A. Is an Increase in Probability Always an Increase in Evidential Support?. Erkenn (2020). https://doi.org/10.1007/s10670-020-00241-4

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