If you see a person eating a horse, it seems you have evidence which justifies the belief that at least one person has eaten horse. But it wouldn’t be evidence to justify the belief that everybody has eaten horse – this belief goes beyond the evidence, in a way that seems unjustified. The problem of induction says: it’s not possible to justify any inference which takes you to a conclusion that goes beyond the evidence. Whether the problem of induction can be solved or not, it still seems possible to say that some ‘ampliative inferences’ are better than others. In fact, it seems that most people have an intuitive grasp of what makes one induction better than another. Consider the following:
1. So far I’ve taken out ten tangerines from this bag of 15 tangerines, and they’ve all had pips in them
Conclusion: All the tangerines in this bag of tangerines will have pips in them
2. So far, I’ve taken out ten tangerines from this bag of 15 tangerines, and they’ve all had pips in them
C. All tangerines have pips in them
1 seems to support its conclusion more strongly than 2. Similarly:
3. So far I’ve taken out ten tangerines from this bag of 15 tangerines, and they’ve all had pips in them
C. the next tangerine I take from this bag will have pips in it
seems to be a stronger induction than 1. We could call this feature the ‘ampliative size’ of the induction: the more an inference amplifies beyond its evidential base, the weaker the inductive support the base provides for that conclusion. That the ampliative size of an induction is relevant to the quality of the induction is also suggested by intuitions about contrast cases between 3 and 4:
4. So far I’ve taken out one tangerine from this bag of 15 tangerines, and it had pips in it
C. the next tangerine I take from this bag will have pips in it
Since the evidential base in 4 is smaller than the evidential base in 3, but they have the same conclusion, we feel that the case with the larger evidential base provides more support for the conclusion (since the ampliative size is smaller).
Another intuition we might have about what makes for a good inductive inference is called the ‘diversity’ of the induction. Ernest Nagel, in the Principle of the Theory of Probability Theory (1939), gives an example: if you’re inspecting a large shipment of coffee beans, rather than inspect a large number of coffee beans from one part of the shipment (one bag or sack or chest or whatever), you’d do better to inspect lots of smaller samples taken from lots of different parts of the shipment. The more diverse your evidential base, the better the support it generates for the conclusion.
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We could wonder whether these two intuitions could be in tension with each other, and try to construct cases in which raising the diversity of the evidential base necessarily increases the ampliative size of the inference, but I’m not sure that such cases (if possible) would show very much. All we’d learn is that there are some inductive inferences which are less good than others, and we already know that. I’m more interested in why we might think that these features make for better inductive inferences.
In particular, I’m interested in the intuition about diversity because I think it can be hard to determine whether an evidential base is diverse or not. Andrew Wayne puts the point in his article ‘Bayesianism and Diverse Evidence’ (Philosophy of Science, 62.1, 1995, 111-121)
Clearly, judgements of diversity are always made relative to a given theoretical context; what look like disparate phenomena in one context may appear closely akin in another. (115)
The problem here isn’t just to do with the possibility that we can be mistaken in our judgements of diversity. The diversity intuition (that more diverse evidential bases make for better induction inferences than narrower pools of evidence) seems to be predicated on the actual diversity of the base, rather than its apparent diversity. Think about Nagel’s example. Suppose someone decides that to test the quality of the shipment of coffee, instead of taking 100 coffee beans from one bag, they’ll take 20 coffee beans from 5 randomly selected different bags. According to the diversity intuition, this should provide better support for a conclusion about the quality of the shipment’s coffee. But suppose, unbeknowst to them, all 100 beans that they test have come from the very same, super-high quality coffee plant, and that none of the rest of the shipment includes any more of these beans. They’ve been unlucky to select the only high-quality beans (and selecting 100 beans from any one bag would have been more useful). This doesn’t show anything about the diversity intuition failing, since we can hang on to the notion that an actual diverse sample would have been better than this apparently-diverse-but-actually-uniform sample. We still think that diversity is important, it’s just that in this unfortunate example the test has failed to be diverse.
The worry I have goes slightly wider than this. Take the following induction:
5. This brown bear likes porridge
C. All brown bears like porridge
To strengthen the inductive support, we could increase the size of the inductive base:
6. These three brown bears like porridge
C. All brown bears like porridge
and we could increase its diversity:
7. This large brown bear likes porridge, this medium-sized brown bear likes porridge, and this small brown bear likes porridge
C. All brown bears like porridge
Does 7 provide more support than 6? Perhaps you think that 7 is no more diverse than 6; both might have exactly the same evidential base (the same three bears), but 7 just describes them as being more diverse. So perhaps this case is like with the coffee example I just gave; apparently diverse, but not really. Another reason you might hesitate about whether 7 gives more support than 6 is because you think that brown bear size doesn’t really represent anything significant about what brown bears are like (at least in so far as their eating habits are concerned). In order to get a more diverse sample, you might prefer something like:
8. This brown bear from Russia likes porridge, this brown bear from Alaska likes porridge, and this brown bear from the Hundred Acre Wood likes porridge
C. All brown bears like porridge
Is this a more diverse sample than 6? Does 8 provide more support for the conclusion than 6 does? Is brown-bear-geography a relevant feature of the ‘diversity’ of what brown bears are like? If you think not, is it because you think that 8 just tells us that there are 3 brown bears like porridge (just as 6 and 7 do)? How about this one:
9. This brown bear named Boris likes porridge, this brown bear named Belinda likes porridge, and this brown bear named Brose likes porridge
C. All brown bears like porridge
My concern is that when we start trying to arbitrate between whether 7, 8 or 9 provides better inductive support for the conclusion than 6 or not, it’s not simply because we’re particularly interested in the degree of the diversity of the samples. Rather, we’re interested in the appropriateness of the modality (bear-size, bear-geography, bear-name) to the nature of the conclusion. Some instances of more diverse evidential bases may add no further support to a conclusion beyond that generated by a less-diverse evidential base if the type of diversity is inappropriate.
But what do you think? Do you think that any of 7 or 8 or 9 are an example of an actual diverse evidence base (as opposed to a merely apparent diversity)? Do you think that any of 7 or 8 or 9 has a more diverse evidence base than 6 does? Do you think that any of them provide more support for the conclusion than 6 does?
Thanks for this post Joe. Is the worry simply that there is not going to be any principled way of specifying what would count as relevant diversity with respect to any given class of evidential base? If so, isn’t this just going to be an empirical matter? Granted, relevant diversity is going to vary widely depending on the evidential base, but there must be some fact of the matter, in most cases at any rate, as to what kind of diversity is relevant; and, further, it seems this fact would be discoverable by the appropriate subject matter expert. Perhaps this is bit hard to envision with respect to the porridge preference of bears, but not so in the coffee example. A subject matter expert who knew the packing and shipping methods of coffee producers would surely be able to determine what sampling method would provide the most diverse (in the relevant sense) evidential base practicable.
But now I wonder whether I am just begging the question against the problem you raise, since it appears that a subject matter expert must apply induction in order to determine relevant diversity, and if that inductive inference requires a diverse evidential base, then we appear to have embarked on a vicious regress …
Or perhaps your worry is altogether different? Maybe you think that the diversity intuition is just too vague and fuzzy to be accorded the justificatory status that our every day inductive practices seem to accord it. After all, most of us are not subject matter experts in the areas in which we employ induction, and yet we intuitively think that a diverse evidential base makes a difference to those inferences. If that’s the worry, then I’m not sure what to say in response…maybe we’re just deluded into thinking that diversity is important to the strength of our inductive inferences!
So is this supposed to be a problem for the scientist or just for the person in the street?