When Absence of Evidence is Evidence of Absence

There is a popular and catchy saying which I myself have been caught repeating in the past, but which, for all its intuitive appeal, is false; namely, that the absence of evidence isn’t evidence of absence. Many a new-atheist has repeated the mantra that there is no evidence for God’s existence, insinuating thereby that this absence of evidence is good evidence for atheism. William Lane Craig, a noted philosopher, theologian and tireless Christian apologist has responded as follows:

[Atheists] insist that it is precisely the absence of evidence for theism that justifies their claim that God does not exist. The problem with such a position is captured neatly by the aphorism, beloved of forensic scientists, that “absence of evidence is not evidence of absence.” The absence of evidence is evidence of absence only in cases in which, were the postulated entity to exist, we should expect to have more evidence of its existence than we do.1

He has reiterated as much more informally (but more elaborately) on his podcast, ReasonableFaith, where he says:

The absence of evidence will count as evidence of absence when if the thing existed, then having surveyed the grounds, so to speak, we would expect to see evidence of their existence, and we don’t see it. And so, for example, in the case of fairies, if they existed then we ought to be able to find traces of their existence – their dead bodies when they die, their carcasses, other sorts of remains, little clothing factories where they build their clothes, and we ought to detect them flying about just as we detect dragon flies and bumblebees – but we don’t. So this would be a case where I think the absence of evidence would count as evidence of absence.”2

On this view, the absence of evidence only counts as evidence of absence when we have some reason to expect to see the evidence ex hypothesi. This has enormous intuitive appeal; consider the hypothesis that there is at least one tiger in India. Can the fact that I, sitting in Canada, currently see no tiger really count as evidence that there is not at least one tiger in India? Surely not; presumably because that evidence isn’t expected on the assumption of the relevant hypothesis’ truth. Elliott Sober, reflecting on absence of evidence, notes that in the case of arguments from absence “it is easy to see how each can be turned into a valid argument by adding a premise. The arguments have the form:

I do not have any evidence that p is true.
p is false.

Just add the premise

(P1) If p were true, then I’d have evidence that p is true.”3

This further highlights the fact that it is natural for us to think that absence of evidence is evidence of absence only when we expect the evidence ex hypothesi.

For years I found this response intellectually satisfying, but in recent years I have come to think that it is woefully mistaken. It is true that my failure to observe a tiger in Canada provides no evidence against there being at least one tiger in India, but it is not because I wouldn’t have anticipated seeing a tiger in Canada given that there is at least one tiger in India. All my affection and respect for Craig notwithstanding, if Craig means that absence of evidence E for hypothesis H is only evidence of absence (i.e., not-H) when the probability of E on H is greater than 0.5, then he is, I think, incorrect. In what follows I will try to explain why, as well as explore what to me seem interesting corollaries of Bayesianism.4

John Hawthorne, speaking about probability theory and the fine-tuning argument at a conference back in 2015, warned:

“Human beings, even intelligent human beings, are terrible at reasoning about probabilities. There’s enormous empirical evidence that human beings are terrible at reasoning about probabilities, and so we have to proceed with care.”5

Playfully picking on (presumably) a student in the audience, Hawthorne says: “Justin gave us the kind of awesome sounding principle… [that] if you don’t see something then that can be evidence of its absence only if you expect that you would get evidence were the thing there.”6 Not the cleanest off the cuff articulation, but clearly Hawthorne had in mind the principle for which W.L. Craig advocates. He continues; “that’s wrong… and I can prove to you that it’s wrong.”7 He proceeds to give an illustration using a hypothetical creature he calls a Dynx, where he stipulates that 75% of Dynx are invisible to the naked eye, and the probability that there is a Dynx in a box placed before us is 50%. We open the box, and we see no Dynx. The probability that there is no Dynx given our background knowledge and this new piece of information (namely that we do not see any Dynx) is approximately 57%. You can satisfy this for yourself by simply dividing up the space of possibilities (i.e., ‘seeing a Dynx in the box,’ ‘not seeing the Dynx in the box,’ and ‘there being no Dynx in the box’), eliminating the possibility of ‘seeing a Dynx in the box,’ and then expressing your updated probability assessment accordingly. So, even though we ought not to expect to see a Dynx in the box if there is one in the box, our failure to observe one is still evidence for their being no Dynx. This simple illustration (and others like it) seems to be entirely compelling. What, then, is the genuinely Bayesian determination of evidence?

On the Bayesian theory of confirmation,8 some evidence E will count as evidence for some hypothesis H (given background knowledge B) just in case E (conjoined with B) raises the (prior) conditional probability of H. To put it more formally, E will count as evidence for H just in case: P(H|E&B)>P(H|B). However, [P(H|E&B)>P(H|B)]⊃[P(~H|~E&B)>P(~H|B)]. In other words, if E provides any evidence for H, then ~E provides some evidence against H. It needn’t, of course, be the case that E provides as much evidence for H as ~E does for ~H, but it strictly follows from Bayesianism itself that ~E would be evidence against H just in case E would be evidence for H.

To illustrate with an example, let us take a hypothesis H1: “that aliens exist,” and evidence E1: “I am being abducted by aliens.” Obviously P(H1|E1&B)>>P(H1|B). What is not so obvious is that P(H1|~E1&B)<P(H1|B). The reason it isn’t so obvious is that ~E1 provides negligible evidence for ~H1 (even though E1 would provide compelling evidence of H1). If aliens abduct me, that’s really good evidence that they exist. If aliens do not abduct me that’s really poor evidence that they don’t exist. It may be some evidence, but it isn’t very much evidence.

Not only can the absence of evidence be negligible evidence of absence while the presence of that evidence would be altogether compelling, but the absence of evidence can even be inscrutable evidence of absence while the presence of evidence is scrutable and enormously supportive of the hypothesis in question. Take the example of a miracle, and for simplicity let us use the miracle of the bodily resurrection of Jesus of Nazareth. The bodily resurrection of Jesus, if it did occur, would be relatively good evidence for God’s existence; P(G|R&B)>>P(G|B). However, if Jesus had not been raised from the dead, would that provide any evidence against God’s existence? According to Bayesianism it would, but it seems like it would be not only negligible evidence, but even inscrutable evidence. There is no way one could put a figure (with any justification) on how much more confident it should make us in atheism that some miracle, like Jesus’ resurrection, did not occur. If we could give any estimate of what the probability is that God would perform a miracle when called upon to do so, for instance, then we could make some predictions about how many hospitalized people with terminal diseases (according to medical diagnosis) under observation get better when prayed for. We can’t make these predictions not because there is no actual probability of God doing a miracle, but because we aren’t at an epistemic vantage point from which we can assess that probability with any level of confidence at all.

Further, the evidence may not be merely negligible, but can in special instances be literally infinitesimal (an infinitesimal is a non-zero infinitely small quantity). Consider Hempel’s paradox9 for a moment; any observation of a pink shoe provides some evidence for the hypothesis that all ravens are black. The hypothesis that all ravens are black is logically equivalent to the statement that all non-black things are non-ravens. It follows, therefore, that any observation of a black raven is evidence that all non-black things are non-ravens, and any observation of a non-black non-raven is evidence that all ravens are black. An observation can’t be evidence for one without being evidence for the other precisely because they are logically equivalent statements, at least interpreted at face value; this is just what Hempel called “the equivalence condition.”10 However, it seems as though there are potentially infinitely many things which are non-black non-ravens which, at any moment, we will fail to observe. If this is so, then each of these instances of absence of evidence will count as instances of infinitesimal evidence of absence (or, at least, infinitely many of these instances will count as instances of infinitesimal evidence of absence). One thinks of the infinitely many miracles God could have performed at any given moment (e.g., growing a lost limb, bringing a dead child back to life, parting the Atlantic ocean); is it really the case that every instance of a miracle not happening provides some evidence against God’s existence? If so, and if there are infinitely many opportunities for God to perform a miracle of some kind (in infinitely many of which God decides to perform no miracle), does that not entail that the probability of theism is literally infinitesimal, or else that each instance (or, at least, infinitely many instances) of a non-miracle provides at most infinitesimal evidence against theism? This gets a little tricky, of course, because Bayesian theory isn’t really equipped to deal with cases of what we might call ‘transfinite probabilities,’11 but if we take its implications seriously even in such cases we will plausibly think that at least some things provide literally infinitesimal evidence for a conclusion or hypothesis.

An interesting objection to this suggests that there is not, even potentially, an infinite number of unobserved observables. Given the limited bandwidth of the human body as a kind of measuring apparatus,12 there may be infinitely many different but observationally indistinguishable events. Imagine, for instance, two pairs of pink shoes whose colours or sizes differ by so little as to make it impossible for any human being to tell the difference between them. For any of the attributes assessed by the five senses, there will be limited empirical bandwidth given the human body as a tool of observation. What this seems to entail is that there is not a potentially infinite number of different possible observations, in which case we needn’t concede the absurdity of infinitesimal probabilities. This objection is appreciably practical, but I’m not entirely confident that it settles the matter. After all, I can imagine a human being with “electron-microscope eyes”13 or with any number of other physical alterations which would allow them to observe an apparently potentially infinite number of different events. For any such alteration, I can imagine God miraculously bringing it about that observer S has precisely the alterations necessary to observe some miracle M1 which would have previously been indistinguishable from miracle M2, but is not now indistinguishable from M2 for S. Moreover, I’m not convinced that observational indistinguishability is terribly relevant; there are infinitely many possible pink shoes which I could now be observing, but am not, and even if infinitely many of them would be indistinguishable to me, failing to observe any one provides some evidence against the hypothesis that all ravens are black. So it seems to me that we’re stuck with conceding that at least some things provide literally infinitesimal evidence.

In summary, I think we have seen why the absence of evidence is evidence of absence in all cases except those in which the presence of so-called evidence would do nothing to raise the conditional probability of the hypothesis in question. Thus, my failing to observe a tiger in Canada provides no evidence against the hypothesis that there is at least one tiger in India not because I wouldn’t expect that evidence if there were at least one tiger in India, but because even if I were observing a tiger in Canada it would provide no evidence that there is at least one tiger in India.14 We have also seen that even when absence of evidence is negligible evidence of absence, or inscrutable evidence of absence, or infinitesimal evidence of absence (or any combination of those three), it will still provide some evidence of absence; if E would have been evidence for H, then the absence of E provides evidence against H.

Post Scriptum: I want to thank Tim Blais, Cale Nearing and Sean Boivin who provided me, in discussions subsequent to the original article, with food for thought without which I would never have made the improvements I have lately introduced above.

1 William Lane Craig, “Theistic Critiques of Atheism” The Cambridge Companion to Atheism. Edited by Michael Martin (Cambridge University Press, 2006): 70.

3 Elliott Sober, “Absence of Evidence and Evidence of Absence: Evidential Transitivity in Connection with Fossils, Fishing, Fine-Tuning, and Firing Squads,” in Philosophical Studies 143, no. 1 (2009): 64.

4 As a cautionary caveat lector; though I’m pretty confident that what I’m about to say is correct, I have not taken any class on probability theory (yet); if anyone thinks there’s some subtle mistake somewhere, they are encouraged to share it. I am more than open to updating my views.

8 Elliott Sober, “Absence of Evidence and Evidence of Absence: Evidential Transitivity in Connection with Fossils, Fishing, Fine-Tuning, and Firing Squads,” in Philosophical Studies 143, no. 1 (2009): 66.

9 James Fetzer, “Carl Hempel,” in The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta, (Spring 2017 Edition), accessed April 2, 2017. https://plato.stanford.edu/archives/spr2017/entries/hempel/

10 James Fetzer, “Carl Hempel,” in The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta, (Spring 2017 Edition), accessed April 2, 2017. https://plato.stanford.edu/archives/spr2017/entries/hempel/

11 If one dislikes this term because they think that probabilities can be no higher than 1, which makes them finite, I would suggest they think about how the conditions I just stipulated could imply that some hypothesis H is infinitely likely without having probability 1. However, if that doesn’t mollify the critic, I could agree to change the term to ‘non-finite’ probabilities.

12 I borrow here from Bas C. van Fraassen, who notes insightfully that “the human organism is, from the point of view of physics, a certain kind of measuring apparatus.” See: Bas C. van Fraassen, The Scientific Image, (Oxford: Clarendon Press, 1980), 17. 

13 Bas C. van Fraassen, The Scientific Image, (Oxford: Clarendon Press, 1980), 17.

14 If one thinks that observing a tiger somewhere raises the conditional probability that one may be observed anywhere then one will reject this conclusion, but they needn’t, in so doing, reject the principle this example is being employed to illustrate.

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8 thoughts on “When Absence of Evidence is Evidence of Absence

  1. Hi .
    I would just like to briefly ask you what kind of reply do you prefer to Evidential Argument From Evil? Because it seems,based on your considerations here that any instances of Prima Facie gratuitous evil should count as a good evidence against theism..Unless you think Existence of God and Gratuitous evil are compatible.

    • So for nearly any position you can think of, somebody somewhere at some time has defended it; so too here, there are defenders of the compatibility of gratuitous evil and theism. I, however, am not one of those – I accept that if there is a single gratuitous evil, God does not exist.

      I tend to think that prima facie gratuitous evils provide evidence against the existence of God, but that they provide negligible (that is, extraordinarily weak) evidence against the existence of God. I am also so skeptical of our ability to measure or detect gratuity that I think even when considering evils with prima facie no explanation (of which we can think or have thought) we are a long ways off from having any good reason to believe that they are gratuitous. The suspicion of gratuity may come naturally to us when we can think of no possible morally justifying reason for some evil, but I think this belief is easily defeated by noticing that if an evil is morally justified, we are extremely unlikely to have access in principle to its morally sufficient reason. Our confidence in how likely any evil is to be gratuitous, I think, is a byproduct of our confidence in (a)theism. The theist will argue that if theism is true, no evil is gratuitous, whereas the atheist will argue that if an evil is gratuitous theism is false; the theist is going to believe they have much better reasons for thinking that God exists than for thinking that any given evil is gratuitous, while the atheist may be inclined to think that they have better grounds for concluding that at least one evil is gratuitous than they do for concluding that God exists. The argument becomes an intuition pump, a sort of philosophical measuring instrument for determining how confident we are that theism is true relative to how confident we are in our ability to perceive gratuity.

      I’m also willing, for the sake of argument, to sacrifice one argument for another – I think that the evidential argument from evil for atheism is about as good as the fine-tuning argument for theism, and I’m willing to say that if those land us in a philosophical stalemate, then we have to move on to other arguments. Obviously the theist has a more impressive reservoir here.

      • So for nearly any position you can think of, somebody somewhere at some time has defended it; so too here, there are defenders of the compatibility of gratuitous evil and theism. I, however, am not one of those – I accept that if there is a single gratuitous evil, God does not exist.

        Yes,I am aware so some very innovative and subtle arguments for this conclusion given by some esteemed philosophers such as John Hick,Van Inwagen,William Hasker etc..though obviously they are very controversial and heavily criticised ..

        I was expecting you to give a similar response given your considerations in the post so I am surprised you instead opt for retreating into Sceptical theism and here the problem is that this move seems no less problematic for theist.. your claim that prima facie gratuitous evils provide evidence against the existence of God, but that they provide negligible (that is, extraordinarily weak) evidence against the existence of God needs some defending and It can be argues that it is indeed mistaken..

        Matt Banton,Hawthrone and Issacs in one their recent paper make similar considerations,its their claim that presence of evil is not only evidence against theism it is indeed very good evidence(because its absence would be good evidence forit,as you observe in above post too) they then consider various moves(which would include yours too) by which Sceptical theists try to circumvent this problem and find each problematic(though they do give sceptical theists some advise in the end,so I do recommend checking this paper out)..

        So what I want to say is that it seems to me that unless Theists reject this whole methodology of dealing with evidence,the problem of evil becomes very very hard to deal with,the question though is that can they do that?

        ohh and I do realize that the post isn’t exactly about problem of evil so I am sorry about thread-jacking I don’t really like doing it but I am genuinely curious about how theist could deal with these tensions ..

      • Hello again,

        Thanks for your reactions; I appreciate them.

        “your claim that prima facie gratuitous evils provide evidence against the existence of God, but that they provide negligible (that is, extraordinarily weak) evidence against the existence of God needs some defending and It can be argue[d] that it is indeed mistaken..”

        While anything can be argued against, I think it relatively easy to argue in defense of the position that evils for which we have no apparent morally sufficient explanation provide extremely weak evidence against the existence of God just in case we can establish that if such reasons did exist, we would be unlikely to be privy to them. I think if such reasons did exist, we actually would be unlikely to be privy to them, for the reasons can range from future consequences unforeseeable to us, to past consequences (if we admit the possibility of temporally backwards causation, or literally retro-active causation), to present consequences unknowable to us. It also seems pretty easy for me to imagine morally sufficient reasons for nearly any evil so long as (i) Utilitarianism is possibly correct, and (ii) God has knowledge of (relevant) counterfactuals about the future.

        “Matt Banton,Hawthrone and Issacs in one their recent paper make similar considerations,its their claim that presence of evil is not only evidence against theism it is indeed very good evidence(because its absence would be good evidence for it,as you observe in above post too) ”

        You sound like you’ve really enjoyed reading on this particular topic. I’ll be sure to to ask you for reading materials when I have the time and inclination to focus on the problem of evil in the future. Although I haven’t read their paper, I think that their mistake is already obvious: the absence of evils would not be particularly good evidence for theism. At least I’d have to see a surprisingly powerful argument for that in order to change my mind about it, but it just seems to me that that’s mistaken.

  2. Hey Tyler. I favor this basic likelihoodist or bayesian view of evidence that Sober, Collins, and many others use. I also like Sober’s paper and how he explains that not finding of a transitional fossil is small evidence against common ancestry, and that’s because of the low probability of finding it in the first place.

    I have to confess that I skimmed your article and didn’t read it carefully. Can you tell me the difference between “negligible evidence of absence, inscrutable evidence of absence, and infinitesimal evidence?” I take inscrutable to be the way Plantinga uses it in his EAAN, which means we can’t even put a number on it, we’re completely in the dark as skeptical theists say.

    • Thanks for the comment. I was actually directed to Sober’s paper (which was a pleasure to revisit) due to another commentator who contacted me on facebook and made a comment which looks like a carbon copy of yours.

      So, by inscrutable evidence I mean, as you suspect, evidence on which we can put no figure (with any justified confidence). Skeptical theism comes in some varieties but it is certainly fair to draw the analogy from skeptical theism; the skeptical theist maintains that evil provides inscrutable evidence against God’s existence (even if she concedes that it provides *some* evidence). By negligible evidence, I mean that the evidence in question raises or lowers the probability of the considered hypothesis by so little as to be negligible; that is to say, it is practically irrelevant to the final analysis, and provides at best an extremely weak inferential basis for any conclusions. Finally, by infinitesimal evidence I mean, quite literally, evidence which raises or lowers the conditional probability of a considered hypothesis by an infinitely small non-zero quantity. These are special cases, but I am worried that they may arise from cases where there are infinitely many absences of what would have qualified as positive evidence.

  3. Tyler,

    I have a brief question (and so tackle it any way you like): The claim of divine hiddenness–at least one formulation of it–involves the claim that *if* God has as one of His superlative attributes maximal-love (or being all-loving), *then* there would be no nonresistant nonbelief. I am wondering if your paper here might shed light on the problem i.e., in what we *should* expect given an all-loving God. I’d love to know what you think. I see the previous questioners question and see a connection, but I am not sure if you’d like to add in some further thought.

    Thanks,
    Rashad.

    • Hey Rashad, it’s nice to hear from you again.

      So, as a knee jerk reaction, I’m not sure my ruminations about the absence of evidence are particularly relevant to the problem of divine hiddenness except insofar as they may suggest that we could consider the conditional probability of non-resistant non-belief (on theism) to be either inscrutable or negligible (which is what the skeptical theist is likely to insist). I’ll think it over in the meantime, but if you see some other way in which you think there’s a connection you’d be welcome to share it.

      There is, perhaps, one disadvantage here for the theist. It is more difficult for us to imagine morally sufficient reasons for non-resistant non-belief than to come up with relatively plausible theodicies, unless there really are subjunctive counterfactual conditionals of libertarian freedom. If there are truths of that variety, then the argument from divine hiddenness is mired in difficulty escape from which seems hopeless. If, however, there are not such facts, then many of the viable apologetic responses to the problem of divine hiddenness get severely undercut. That doesn’t leave us with no responses to the problem, but it may put us in a different position than we seem to be in when confronting the problem of evil. The techniques of skeptical theists can still be trotted out, of course, but I do think that the problem of divine hiddenness is actually a much stronger argument against theism than is any version of the problem of evil.

      In any case, as I said, those are just some of my immediate reactions.

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