L Inductive Reasoning

If it is impossible to make scientific progress by deductive reasoning alone, how can we discover scientific truths? Traditionally, the most popular answer, which has innumerable variants, is induction; the idea that that we go from observed regularities in nature to general rules that we then take to be true.

1. L.1 Enumerative Induction

Learning about the natural world scientifically involves careful observation, data collection, and analysis. When this began to be done systematically, exper­imental science came into being. An early advocate of the empirical approach was Francis Bacon, whose major contribution to the philosophy of science was to emphasize the importance of experiment and inductive, rather than de­ductive, reasoning as a way of making scientific advances. Every day the earth turns and the sun rises in the sky. Supposedly, we reason by induction that the sun will reappear tomorrow. This sort of argument is known as enumerative induction because the general rule follows from the enumeration of individual cases. Larry Laudan classifies enumerative induction as a form of low-class, or­dinary induction—“plebian induction”⁴¹—and distinguishes it from high-class “aristocratic induction” that tries to assess the validity of a theory or hypothesis from a number of confirmatory tests of predictions that it makes. First, the low-class form.

1. L.2 The Problem of Induction

From Bacon onward until the nineteenth century, many thinkers considered that there was “no significant element of uncertainty or doubt attached to the conclusions of so-called inductive inference.”⁴² This opinion was anchored by the widespread belief that Isaac Newton had arrived at his famous laws of motion via induction.

If you take the claims of enumerative inductive reasoning too literally, you get the impression that experimental observations themselves lead to general rules. This is not true, of course. Generalizations do not arise from data; they are cre­ated by human minds thinking about data. Moreover, when the minds invent the generalizations, predictions are often the limit of what you get with induc­tion; patterns, not deep understanding. Richard Feynman notes that the ancient Mayans made repeated astronomical observations of lunar eclipses that allowed them to predict future lunar eclipses accurately but not to grasp the celestial me­chanics that would have explained eclipses.⁴⁶ Their reproducible observations could only take their astronomical knowledge so far.

Any serious analysis of induction has to start with the philosopher David Hume⁴⁷ who exposed its shortcomings by showing that its validity rests on the huge unproven assumption that Nature—the external, objective universe—has always been the way it is now and will always be as it is. The validity of induc­tion depends on Nature's being uniform in this sense, and the idea is called the Uniformity of Nature (UN) assumption. If the UN assumption were true, then you could legitimately argue by induction that the rule you generalized from past observations would hold true in the future. Therefore, to determine if induction is a valid way to argue, we need to know if the UN assumption is true.

We could try to prove the truth of the UN assumption using either deductive or inductive reasoning. We begin by asking, “Is the UN assumption a matter of reason; does it logically follow from premises that we know to be absolutely true?” No, because the only statements like that are parts of closed, self-contained, arti­ficial systems, such as mathematics or symbolic logic, or are tautologies. In con­trast, the UN assumption is explicitly about the world outside of these man- made systems; hence, deductive logic cannot prove it. Thousands of years of failed ef­fort to discover logically certain statements about nature was why Bacon turned to inductive reasoning in the first place.

If the UN assumption is not true as a matter of reason, could it be true as a matter of fact? Can we demonstrate its validity via empirical information gotten through experience, observations, experiments, etc.? One way to find out would be to look to the past and ask if Nature was always the same as it is now. If so, then, reasoning inductively, we could infer that Nature will remain the same in the future. Voila! We would have proved that the UN assumption is true. There is, of course, one snag. We said that the validity of inductive reasoning depends on the validity of the UN assumption, and now we're saying that the validity of the UN assumption depends on the validity of inductive reasoning. This is a com­pletely circular argument, and there is no way out. We can't prove the UN as­sumption is true as a matter of fact, either.

1. L.3 Proposed Solutions to the Problem of Induction

The Problem of Induction is a tough one. Philosophers tried to develop a work­around in the form of a Principle of Induction to shore up inductive reasoning. By adhering to the Principle, the truths of inductive reasoning could, it was hoped, become as secure as the truths of deductive reasoning. Every scheme like this was utterly dependent on the particular Principle it rested on; if the Principle wasn't true, the scheme couldn't work.

If we can't prove that a Principle of Induction is true, then why not simply assert that such a principle must be true? Just declare that it has to be true a priori—that is, true without proof—and go from there. Indeed, this strategy neatly escapes the problem of proving Principles of Induction, but if your goal is to justify the truths of scientific facts about the rough and changeable world, it isn't fair to call in reinforcements from an otherworldly realm of perfect, though unprovable, ideas. Only testable ones are allowed in science, so a priorism was out, too.

In the end, all of this became pretty frustrating. At one point the philosopher Bertrand Russell threw up his hands and declared that “induction is an inde­pendent logical principle, incapable of being inferred either from experience or from other logical principles, and that without this principle science is impos- sible.”⁵⁰ In other words, for Russell, induction was simultaneously indefensible and indispensable. An astonishing and revealing conclusion.

The nub of the problem of induction for the philosophy of science is the con­viction that scientific experimentation as a rational endeavor must rely on the predictability of prior experience to proceed.⁵¹ Is this true? And, whether it is or not, why do so many people seem so convinced that it must be true?

To start with, we must admit that the world does appear to be pretty predict­able, and if the world were totally chaotic, with fundamental physical constants changing daily, science could not proceed.

Nonetheless, for the sake of the argument, let us grant enough regularity that scientists and philosophers of science can evolve. How would science progress without trusting in the validity of inductive reasoning from past experience to develop solid generalizations about the future? Defenders of induction invite us to “Imagine a universe in which ‘computers might explode for no reason!' ”⁵² It is ridiculous, a reductio ad absurdum, to conceive of successful science in such a place, they say.

We can only hope that none of these critics was handling a laptop or a cell phone when it suddenly burst into flames.⁵³ Extensive experience with laptops and cell phones, coupled with the exercise of inductive reasoning, should have guaranteed that nothing of the kind could possibly happen. Still, it has happened many times. (Indeed, you might be wondering if it's time to draw the inductive inference that these devices are dangerous and, hence, that users should, at least wear oven-mittens when using them.)

What went wrong in the case of the exploding computers is what can always go wrong with inductive reasoning: unexpected things happen.

1. L.4 Other Forms of Induction

Given that philosophers gradually embraced the view that the problem of enu- merative induction “is insoluble,” ⁵⁴ that the attempt to solve it is a “dead end,” you might think that they would simply give up on it altogether—but that would underestimate the power of the conviction that we must, simply must, rely on something like induction to carry out science and the resourcefulness of philosophers who devised an entirely new approach to the problem.

Before going on, I want to digress to bring out a point that, although it becomes central to our story, does not get the emphasis it deserves. It is a matter of definition: What does “confirm” (or “verify”) mean? There are two related senses of these words: one is “to establish the indubitable truth of a statement,” and the other is “to strengthen or support a statement that might or might not be true.” Here's an example. You might be asked, say before testifying at a trial, to confirm your name. You know your name beyond doubt—you've got your birth certificate with your tiny footprints on it, assurances from your parents, testimonials from others who've known you forever, official papers and records documenting who you are, etc.—it's your name. Period. You attest to the cer­tainty of that fact. But “confirm” is sometimes used as an indicator of support for a statement having questionable truth value. In this case, evidence is said to make the statement more likely to be true without guaranteeing that it is true. For a philosopher, “a statement can be highly confirmed and still false.”⁵⁵ The am­biguity is why its best for scientists to avoid “confirm.” You might mean it in the soft sense of support for your hypothesis, but if a reviewer thinks you mean it in the hard sense—as proving your hypothesis—then he or she may wonder if you know what you're doing.

Keeping the two senses of “confirm” in mind makes it easier to understand the reinterpretations of induction that we'll turn to next. Until now, the discussion has centered on the first meaning: induction as a way of establishing scientific Truths beyond doubt. When that project failed, the goalposts shifted. Instead of trying to guarantee the logical validity of statements, proponents of induction argued that induction could increase the probability that statements would be true. From there, they moved on to trying to justify our confidence in the findings of science.

1.L.5 Induction and Probability as Criteria for Success

The strategy of the new attack was to supplement pure induction with proba­bilistic reasoning. By incorporating the idea of probability, inductive reasoning from true premises could lead to conclusions that would probably be true rather than certainly be true. Inductive logic would from this perspective let you assess how close competing scientific theories and hypotheses were to the truth. ⁵⁶ But without knowing the final Truth in advance, how can you say exactly how close you are to it? And you can't compute the probability of the Truth of competing theories the way you can compute the chance of rolling a seven with two dice (we'll review probability concepts in Chapters 5 and 6). When advocates of in­ductive reasoning say that the truth of scientific conclusions can be made more probable with further experiments, they are talking about subjective probability (also known as Bayesian probability, see Chapter 6). An outcome is probable to the extent that they believe it is.

This subjective way of assessing probability also makes it possible to distin­guish among degrees of certainty, which creates an aura of precision around in­ductive conclusions. For example, reasoning from the statements that “a recent tally shows that 50.82% of Americans are female” and “Pat is an American,” we could conclude that “Pat is probably a female.” Since most Americans are female, this must be true (assuming that the categories of male, female, and American are unambiguous). Still, the difference is very slight—49.18% of Americans would be male—and Pat could well be a male. You would not want to base any vitally im­portant decision, such as whether to assign Pat to the “Boys” or the “Girls” gym locker room on the first day ofjunior high school, on such reasoning. Arguments like these are sometimes classed as “weak inferences” because the conclusions are not very secure.

Reasoning from the information that “97% of American workers earn less than $250,000 per year” and “Pat is an American worker” generates the inductive inference: Pat probably earns less than $250,000 per year. This would be an ex­ample of “strong inference” because the probability of correctly guessing Pat's in­come range is high. Unfortunately, if we're thinking like basic scientists we don't care about that; we want the Truth. There is a 3% chance that our guess is wrong and that Pat makes a pile of money.

1.L.6 Does Science Seek Probable Truth?

People who put a lot of stock in probabilistic inferences argue that, say, a 97% certainty is an appropriate goal for science; it is realistic, and we should be satis­fied with some such standard. This may be true to for an applied scientist, where 97% certainty might be enough to justify taking action. On the contrary, basic scientists are not satisfied with probabilistic standards. Nor would you be if the stakes were high.

Imagine a scenario in which a healthcare official offers you a choice between two vaccines against a horrible, often fatal disease such as Ebola. Both vaccines have been carefully tested in large, well-designed and conducted clinical trials. The vaccines are identical in cost, availability, and ease of administration, but one is 90% effective and the other is 97% effective. Obviously, you are eager to get the 97% effective one, when you learn that a 99% effective vaccine has just become available. Now which vaccine do you choose? It is a no-brainer: When compared to 99% protection, 97% is not acceptable, and you would switch with equal eagerness at the chance for 99% safety. Effectively, you make no distinc­tion between 90% and 97% protection. And if a 99.9% effective vaccine came along, you'd have the same reaction—after all, in a country of over 325,000,000 people, an increase of 0.9% would represent approximately ~ 3,000,000 people who didn't get Ebola, and you would want to be among them. And this is the way it is in basic science, where scientists do not seek probable truth any more than they seek partial protection against disease.

1.L.7 Inference not Induction

Conclusions based on reasoning with probabilities do not represent enumera- tive induction. They are not reached by extrapolating a general rule from partic­ular instances. Still, some philosophers would classify probabilistic conclusions as coming from inductive reasoning because they are not certainly (i.e., deduc­tively) True. However, this stretches the definition of “induction” so far that it becomes meaningless.

Many philosophers have dispensed with the term “induction,” preferring var­ious forms of “inference” instead. In addition to encompassing a wider variety of logical constructions, “inference” is also less burdened by the disappointing his­tory of “induction.” Hence, we now have inductive inference, explanatory infer­ence, and inference to the best explanation⁵⁷; for example, explanatory inference is “inference from a set of data to a hypothesis about a structure or process.” From the variety of names, you might suppose that each kind of inference represents a distinct cognitive process. On the contrary, it appears that the kinds of inference are defined more by the behavioral contexts used to identify them than by the mental operations at work. To illustrate: In one situation, you’re shown a series of particular events and, after a pause, you say “Aha!” and state the generalization that predicts the next event. This would be inductive inference. In another situa­tion, you’re given some clues to a puzzle, and, after a pause, you say “Aha!” and solve the puzzle. This could be classified as explanatory inference. And so forth. The philosophical project here is to identify the conditions that call for using one term or the other, while the origins of the “Aha! ” moments remain equally myste­rious products of the mind. It is hard to see how the distinctions among kinds of inference, whatever they are, will be much help in trying to understand scientific thinking, especially hypothesis- based thinking, which is what we’re interested in.

1.L.8 Affirming the Consequent

At beginning of this chapter, I noted that Larry Laudan distinguished his own two kinds of induction: plebian and aristocratic. So far, we have been discussing his ordinary, plebian kinds and have not directly considered the high-class, aris­tocratic type of induction. While we won’t go into the topic in detail, there is a related philosophical principle we should know about.

The concept of fallibilism assures us that we can’t prove the truth of a scientific hypothesis or theory. We’re going to take up the question of the hypothesis in ear­nest in Chapter 2, but an intuitive feel for it will do at the moment. The problem is that aristocratic induction wants to describe the relationship between empirical evidence and the Truth of theories. The philosopher Peter Godfrey-Smith frames the issue like this: “What connection between an observation and a theory makes that observation evidence for the theory?” and he considers it to be possibly “the fundamental problem in the last hundred years of the philosophy of science.”⁵⁸ Your theory makes a prediction that you test, and the results are consistent with it. Now what? How many tests would it take before you could reasonably conclude that a theory is correct, or at least better than another one? As with other problems with induction, the rigorous answer is that such conclusions are never logically justified, and, in fact, concluding otherwise would constitute a logicalfallacy.

There are two related problems for aristocratic induction. The first is that no one theory is uniquely capable of accounting for a real-world phenomenon be­cause it is always possible that other theories might do as well.

To appreciate the difficulty, we can consider a car that wouldn’t start as a stand-in for a real scientific problem. Your car mechanic takes a look and makes a diagnosis: “The car won’t start because the battery is dead.” This is his hypo­thesis, and it makes a prediction: if the battery is dead, the headlights won’t come on when you flip the light switch. So he flips the switch and lights don’t come on. Does this prove that the hypothesis was correct? Obviously not; many other possible explanations (broken or corroded wires, stolen battery, etc.) also predict that the lights will not turn on. The information he got by flipping the switch was not sufficient to restrict the conclusion to “the battery is dead.” To give it a name, this problem is called under- determinism, and it means that the data cannot be uniquely accounted for by one hypothesis; another one would do as well. It is a serious matter that is not confined to situations involving inert automobiles. When Einstein noted⁵⁹ that, for all of the successes of the extremely complex and precise theory of quantum mechanics, there might be an infinite number of alternative theories that could do as good a job, he was saying that the quantum theory was under-determined by the data.

Under-determinism is clearly a hindrance, but where is the logical fallacy in aristocratic induction? This is the second problem. To illustrate it, we can go back to the discussion of deduction, where true conclusions necessarily follow from true premises. It turns out that we cannot reverse the order and go backward. You can't start from a true conclusion and argue that the premises that led to it must be true—they might or might not be. Let's look at the car problem again and reformulate it like this: if the car battery were dead, then the headlights would not light (premise), and the headlights do not light (premise). Would the conclusion— the battery must be dead—follow logically (i.e., would it be indisputably true)? No. One of the other explanations could explain the unlit headlights even if the battery were fine. The conclusion in this case could be false even though the premises were true because the conclusion was not entailed by the premises: un­derstanding the terms in the premises did not automatically lead you to the con­clusion, and thinking it does is the logical fallacy, which is known (for reasons that don't matter here) as the fallacy of affirming the consequent. We commit this fallacy when we try to argue that getting a predicted result means that the hypo­thesis that made the prediction must be true.

The desire to avoid the fallacy of affirming the consequent is why, when scientists get a result that agrees with their hypothesis, they say that the results are “consistent” with the hypothesis, rather than that the results prove its truth, or “confirm” it.

1.L.9 Is Induction a Philosophical Problem?

If it is not possible to integrate induction into the philosophy of science, maybe it's time to take a different tack. Maybe induction is a psychological, neuroscien­tific, and cognitive phenomenon that can be best understood in the contexts of those fields. David Hume⁶⁰ was, as usual, ahead of his time with this insight. He reasoned that, if induction from experience does not take us to an “effect from an impression of its cause” by true understanding or reasoning, then “the imagina­tion”—the largely unconscious, nonrational part of the mind—must be respon­sible. We can stretch the insight a bit further with an example.

With his canine intellect, your puppy, Snowball, infers from his past experi­ence that the sounds of a can of dog food being opened mean that dinner will soon appear and comes bounding into the kitchen. If we call what Snowball does induction, then the erudite term loses its luster, and yet we'd need a good reason not to call Snowball’s cognitive feat a form of induction, especially since we don't understand our own thought processes any better than we understand his. Both dogs and people observe, learn about, and generalize from regularities in our environments and act accordingly. Naturally, we experience our minds differ­ently, but we cannot explain where our ideas come from any more than we can explain where his come from.

If induction is not an exclusive product of the human mind, why should it remain within the purview of philosophy? The study of planetary motions was initially carried out within philosophy but eventually left philosophy and transmogrified into physics, astronomy, and cosmology. One of philosophy's functions is to serve as an “incubator” of ideas,⁶¹ nurturing and fostering them until they can be expressed as testable hypotheses and segue into other fields. Surely the incubation period for induction must be up by now, and this problem should move into the neurocognitive sciences?

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