assert that it is true. But the truth is not guaranteed by even the best evidence. Thus, there can’t be literally a proof of any empirical belief – not the Big Bang, quantum mechanics, plate tectonics, or evolution. Proof requires conclusive reasons which are limited to mathematics and logic. This seemingly technical point is borne out by the history of science. Most, if not all, empirical theories of the past have turned out to be strictly speaking false, even though many were in the right ballpark and much better than the alternatives. Had the evidence used to support them guaranteed their truth, they couldn’t have turned out false. But they did. Let’s look at the infamous “substance” caloric, the stuff heat is allegedly made of – a once entirely reasonable and well-supported hypothesis that was ultimately wrong.
There was once good evidence for the theory that heat is a sort of subtle fluid, exchanged between bodies. This fluid was called “caloric.” We still use the term “calories,” which derives from that old theory of heat. It was partially based on the observation that if two objects at different temperatures are in contact, they will eventually reach thermal equilibrium: the hotter object will cool off and the cooler object will heat up, until they both have the same temperature. This looks a lot like what happens when one opens the valve of a hose connecting two buckets that hold different amounts of water. The water will run from the bucket with more water to the other one, until both hold the same amount of water. If heat was also a fluid, we would have a good explanation of the fact that adjacent bodies eventually reach thermal equilibrium. So based on this evidence and reasoning from what seemed to be an analogous system, scientists inferred that heat is such a fluid and called it caloric.
Of course, now we know that heat is, roughly speaking, micromotion or the movement of unobservably small particles (atoms and molecules). This understanding arose from the production of canons in the eighteenth century.1 Using a cold drill to bore a hole into cold metal produced a lot of heat. But where was the heat coming from? Caloric was assumed to be a conserved quantity that could only be redistributed among objects, but neither created nor destroyed. The “new” heat from the canons was tantamount to running a hose between two half-full buckets and having them both overflow. You might think that this problem would’ve been apparent from other instances of friction, but the phenomenon was dramatically evident in the manufacturing of canons. What this episode shows is that even good evidence for a theory provides at best a defeasible reason for accepting that theory. New evidence can force us to retract our theory, exactly like knowledge of weird lighting conditions in a cave can force us to retract our belief about the color of rock formations.
1.2 Reasoning from Evidence
Scientists most often make use of induction, or drawing conclusions from evidence. Generally speaking, induction is an inference from the observed to gain information about the unobserved (or unexamined), and it takes three different forms: statistical inference, inductive generalization, and inference to the best explanation (IBE).
It is universally recognized that what these three forms of reasoning have in common is that they are defeasible. Beyond that, the terminology here is unfortunately not as widely agreed upon as it is with respect to deductive reasoning, but we can at least try to distinguish clearly between the three different forms of inductive reasoning just mentioned. Notice though that some textbooks in the sciences define inductive inferences as inferences from the particular to the general. This is misleading, because it covers only a tiny fraction of inductive inferences. We start with statistical inference.
1.2.1 Statistical Inference (SI)
The simplest example of an inductive inference is that of inferring something about an entire population from observing only some of its members. Recall the earlier example involving koalas: I inferred from having observed 20 of them munch exclusively on eucalyptus leaves that all members of the species Phascolarctos cinereus (that’s the koala’s scientific name) feed on eucalyptus leaves. Of course, such inferences are not restricted to biological populations. We might conclude that all igneous rocks are black, after we have seen many lava fields and observed that all of those were black. We can characterize the nature of statistical inferences in the following way:
A statistical inference is an inference from the observed frequency of a property in a sample to the claim that the same frequency holds for the population from which the sample was taken, within a certain margin of error.
Here is an example in explicit form:
| Premise 1: | The frequency of red marbles in a sample of 200 balls drawn from an urn was 49%. |
| Premise 2: | The urn contains exactly 1,000 marbles which are either red or black. |
| Conclusion: | The frequency of red balls in the urn is 50%, with a margin of error of ± 2%. |
Obviously, SI is an inference from the observed (the sample) to the unobserved (the population). Suppose you randomly picked up the first one hundred plants in a meadow and every one of them was a grass. You might well infer that every plant in the field was a grass. As we all know, beliefs (or hypotheses) based on SI can turn out false. Not all igneous rocks are black, and it’s unlikely that all plants in a meadow are grasses, although koalas seem to invariably eat eucalyptus. Often, this is due to sampling problems, which can never be fully eliminated (maybe all the tall plants that are easily accessed are grasses, but some small, ground-hugging plants are broad-leaved species). But even if the sampling doesn’t involve any bias, evidence from samples provides only defeasible reasons for beliefs about the relevant population, as the deviation of election results from predictions based on sampling (called polling) clearly demonstrates. There is much more to be said about SI, some of which you’ll find in later chapters.
1.2.2 Inductive Generalization (IG)
This form of inference is a bit more difficult to characterize to any great degree of precision. In fact, not even the name is widely agreed upon. Sometimes, IG is used to refer to what we call SI. Since nomenclature is a matter of convention, nothing really turns on it, as long as we are reasonably clear about the differences among the kinds of inferences. In order to begin developing a good understanding for what we decided to call IG, it’s best to start with an example.
Suppose you are interested in determining the functional relation between the period of a pendulum (how long it takes to pass through one cycle) and the length of its string. Dutifully, you plot changes in the dependent variable (the period) against variations in the independent variable (the length of the string). Unavoidably, you’ll get a general trend with a somewhat messy point distribution. If you were to precisely connect all the points, you’d end up with a jittery line. “Nature can’t be that crazy,” you mutter to yourself, as you begin accepting that some of the points might not fall exactly on the line describing the actual relationship. You know about air resistance, the variable elasticity of the string due to changes in ambient humidity levels, the imprecision of your starting and stopping the timing device, and other factors that really have nothing to do with the true relation between length and period (that’s why we call those factors “noise”). Thus, you decide to go for a nice, neat line – a section of a parabola, as it were. Then, you find the algebraic expression that generates that line. Finally, you make an inductive generalization and conclude that the period T of all pendula is related to the length l of their respective strings as follows: T=2π√l/g, where g is the gravitational acceleration. In fact, you are proud to have discovered the ideal pendulum law, which holds for all pendula with sufficiently small angular displacement. We will say more about the question of what a law of nature is in Chapter 12.
1.2.3 Inference to the Best Explanation (IBE)
This form of inference is exemplified by the story of caloric. The
