Induction Reconfigured

Induction Reconfigured

Some critics of induction have charged that it is simply a logical fallacy to reason inductively. If induction is inferring a general conclusion from particular cases, then the conclusion patently doesn’t follow from the premises. How can we validly infer that all swans are white from the premise that some swans are white? Clearly, the fact that some swans are white is compatible with some swans being black, so how can we infer that all swans are white merely from observing some white ones? Statistically speaking, we are moving from propositions about a sample to propositions about the population that contains the sample—and the sample may be small and biased relative to the population. In fact, in all cases in which we are particularly confident in our inductions the sample is tinycompared to the population: for example, the number of times we have seen the sun rise is minute compared to the number of times that it will (we think) rise. Any law of nature will apply to extremely large populations of objects or events (in the billions of billions) and yet we have observed only a small fraction of these cases; but we don’t seem to be deterred by the statistical boldness of our inference. What if our limited sample is highly unrepresentative of the totality of cases? The critic insists that it is simply irrational to infer such a vast generalization from a small number of positive instances. You can’t derive all from some! Nor does it help to limit inductive inference to judgments of probability: the critic will insist that not even probability follows from premises about particular cases, especially when the sample size is so small. Even if we can infer probabilities from our samples, we vastly overestimate those probabilities: we think it is close to certain that the sun will rise tomorrow and thereafter, and yet we have only observed a tiny proportion of the sun’s behavior, past and future. Our inductive reasoning is clearly defective, the critic maintains, given the nature of the premises we use and the conclusions we draw.

            It is hard to deny that there is something amiss about moving from some to all in the way we are represented as doing. We recognize that in many cases such a move is wildly inappropriate: just because some coins are copper it doesn’t remotely follow that all coins are copper. We are well aware that samples can be unrepresentative, so why do we seem to forget this in the cases where we confidently reason inductively? The answer I want to suggest is that the reasoning that is involved is not properly represented by the standard formulations: we don’t in fact try to derive all from some. The reasoning works differently: there is an intermediate premise from which the conclusion does logically follow. We don’t think, “x is F and y is F and z is F; therefore, everything is F”, which is certainly vulnerable to the critic’s complaint. We think something subtler, namely: “The things we have observed have a certain nature that ensures that they are F; therefore, everything of this type is F”. First let us focus on the relationship between the nature and the prediction: given that it is in the nature of the things we have observed to be F, anything of this type will be F. For instance, given that it is in the nature of material bodies to exert gravitational force, all material bodies will exert gravitational force. This is not an inference from some to all but an inference from nature to consequence: if it is in the nature of certain things to have a particular property, then all things of that type will exhibit that property. So if the proposition about nature is one of our premises, we are not drawing a conclusion that fails to follow from the premises. If it is in the nature of kind K to be F, then necessarily any instance of K will be F. Being F is a consequence of being K.

The question then is whether the premise itself follows from observations of particular instances. And to this I reply that we do not suppose that it does: we don’t think that truths about instances imply truths about natures and properties. Rather, this is a hypothesis we bring to bear on the instances: we think that it explains the regularities we have observed. We generally suppose that nature works by means of natures and we hypothesize that this is what is going on in a particular case—but we don’t think the hypothesis follows from the observations. So we are not making any fallacious inference. We can accept that other conceptions of nature may conceivably be true—there are no natures in nature and everything we observe is just a giant coincidence—but we believe that our general hypothesis is more likely. Why we believe that is another question (it might just be a matter of metaphysical faith); the point is that it is not arrived at by inductive inference in the classic style. It is not an inference from some to all: it is a high-level hypothesis about how the world works. No doubt the hypothesis is vulnerable to skeptical doubt, because it is difficult to rule out the alternatives, but it is not just a flagrant non sequitur, like the move from some to all. We have skepticism about the external world and other minds too, but it is not charged that our habitual beliefs in these areas are based on a logical fallacy. That is the charge I am anxious to rebut. Skepticism we can live with, but a non sequitur of numbing grossness is not something we can tolerate. The point is that our so-called inductive reasoning does not operate according to the model classically proposed: it is not an unmediated leap from some to all; rather, it ventures a hypothesis and then makes a logically valid inference from that hypothesis.

            Nor is that hypothesis inherently absurd or logically questionable. Nature is full of things with natures, and their properties flow from these natures. This is what natural laws depend on. We view nature in this way as a kind of antecedent commitment; we don’t infer it from a limited sample of regularities. If we did, we would be trying to derive powers from mere regularities. But that is not how our reasoning works: we bring the concept of power toobjects; we don’t derive it from objects. As Hume taught us, we can’t observe powers in objects. We commit no non sequitur; we just make an assumption. At no point are we inferring from the fact that some things are F that all things are F, even when reasoning “inductively”. We don’t induce the general from the particular. Our reasoning is subtler: we impute natures and powers and then we logically derive general conclusions. The conclusion does logically follow from the premises, since if it is in the nature of Ks to be F then all Ks will be F (near enough). If it is in the nature of swans to be white, they will all be (naturally) white; and if it is in the nature of material bodies to attract other material bodies, they will all do so. Of course, it may not be in the nature of these things to have those properties, in which case our projections will turn out to be mistaken: we are certainly not infallible in our inductive reasoning, and so skepticism can gain a foothold. But at no point have we made the numbingly gross inference from some to all. In a sense, then, we don’t use induction (“enumerative induction”) when reasoning “inductively”.  [1]

            This explains why the accumulation of positive instances does not increase our inductive confidence—because it was never based on the frequency of such instances. We are no more confident today that the sun will always rise than our ancestors were, though we have witnessed more confirming instances; and we don’t need to observe more effects of gravity in order to strengthen our conviction that gravity will continue to operate. The reason is that just a few instances can elicit from us the hypothesis of natures and powers, and from that we can derive a generalization. We are not adding up the instances and calculating that the conclusion is more probable the more instances we have; rather, we postulate powers and then derive the general conclusion they support. It is not that we believe that bread nourishes (will always nourish) simply because we have observed that it has nourished many times in the past; we believe it because we postulate a nourishing nature (the power to nourish) and conclude that bread will continue to nourish in virtue of that nature. We could make this postulation based on a single instance (and often do make such postulations); we don’t believe it based on observed frequencies. Indeed, we recognize that observed frequencies don’t entail the power in question, since bread may not nourish and yet be accidentally correlated with things that do (such as the air inside it). What we never do is reason that since bread has nourished on a number of past occasions it must nourish forever—any more than we infer that all coins must be copper because some are. We need the premise that it is in the nature of bread to nourish or for coins to be copper (which in this case it is not). Again, we can be wrong about such things, thus inviting skepticism, but we are not wrong because we have made an illogical leap from a meager sample to a large population. We have not supposed that facts about some things can establish facts about all things.

            Both the supporter of induction and the critic of induction make a common assumption, namely that our reasoning about the future and the unobserved is based on inferring general conclusions from premises concerning particulars. One side says that we can infer unlimited general propositions from propositions about limited samples; the other side denies that such inferences are legitimate. But the common assumption is mistaken: we just don’t reason in that simple way. Insofar as anyone does, they are indulging in questionable ratiocinative practices. There is no need to reason in that dubious way; we can provide an alternative reconstruction of the nature of the reasoning involved. Perhaps we should stop speaking of “induction” or “inductive reasoning”, because it suggests the all-from-some model of the reasoning at issue. Instead, we might speak of “projection” or “projective reasoning”, thus staying neutral about the precise nature of the inference. What is called “the problem of induction”—the problem of justifying moving from a limited set of specific facts to an unlimited general fact—does not apply to the kind of reasoning I have sketched, since no such move is made in that kind of reasoning.

The classic problem of induction arises from empiricist assumptions about where our knowledge must come from, namely from sensory observation and that alone.  [2] This is what leads to the idea that there is a logical problem about induction, since observation of a finite number of cases falls woefully short of the kind of conclusion we seek to derive from such observation. But we don’t reason in this empiricist way (unless we are convinced empiricists); we inject a much richer set of assumptions into our reasoning about the natural world (natures, powers). Empiricism leads to the idea that projective knowledge depends on enumerative induction from positive instances, but this kind of reasoning is a highly questionable procedure; better to abandon such reasoning and the empiricism that goes with it. There is no problem of induction in the classic sense because there is no induction in that sense.

  [1] I hope I have made it abundantly clear that my aim is not to answer the skeptical problem of justifying induction—there is plenty of room for skeptical doubt in the account of induction I favor. Induction really is vulnerable to skeptical challenge, like our beliefs in the external world and other minds. My point is just that induction is not based on the grotesque unmediated leap from some to all.

  [2] Suppose we grant that a certain rationalist assumption is known to be true, namely that the world consists of objects with natures that necessitate their properties and effects (maybe this assumption is implanted in us by God at birth): then we will not have a problem of induction of the classic type, since that assumption enables us to make inferences that avoid any dubious move from some to all. We get a serious problem of induction only when we seek to move from the observation of a small set of regularities to a conclusion concerning a more exensive population.