Problem of Induction

In this essay, I discuss the question what does it take for an argument to be a good inductive argument.

I. Introduction to Induction and the Problem of Induction

Induction, by its definition in the Oxford English Dictionary, is “a method of discovering general rules and principles from particular facts and examples”. In ordinary cases, those “particular facts and examples” are what we have experienced in the past, and we use them to make predictions about the future and the unobserved instances.

In an important sense, we cannot live without such a method of induction. We have bread fill our stomachs because we believe that it will nourish us. For it did so yesterday, last week, last year, and it did so to numerous ancestors of ours. In fact, a counter-inductive way of living can be ridiculous. D. H. Miller once depicted a hilarious picture of living a counter-inductive life in his article The Warrant of Induction:

“Counter-inductivists aren’t just odd: they’re mad. Imagine one. He won’t eat bread: he thinks it would poison him, because it never poisoned anyone before. He would eat cyanide, which he also expects to freeze in the oven and bake in the fridge; but not by swallowing it. He won’t use any language people so far understood, or breathe air, or drink water. And throughout his (brief) life he inconsistently defends his wholesale counter-inductivism that as it’s almost never worked yet, it will now.”

But ultimately, are there really any solid grounds for us to rely on inductions and trust the conclusions derived from such a method? In deductive arguments, the conclusions are already entailed in the premises, yet such is not the case with arguments which are inductive, making the validity of induction dubious. When we reason deductively, with the premises that “All Greek philosophers had beards” and “Socrates is a Greek philosopher” in mind, we can draw the conclusion that “Socrates had a beard” with certainty. Yet if we are to argue inductively, the argument shall be constructed as follows: “Pythagoras had a beard; Zeno had a beard; Thales had a beard; Aristotle had a beard… All Greek philosophers that we know of had a beard. Socrates is a Greek philosopher. Therefore, Socrates had a beard.” Though put in a similar form, the conclusion drawn from the inductive inference appears to be less certain than that drawn from the deductive inference. But why?

The problem of induction was first noticed and discussed by David Hume in his book An Enquiry Concerning Human Understanding. Putting it briefly, Hume attacked induction from two angles. Firstly, it is totally possible for unobserved cases to deviate from those observed regularities. Secondly, the reliability of projecting the past to the future can never be guaranteed. Following Hume, other vivid and potent examples where induction fails have been raised. Bertrand Russell once constructed an example where a peasant fed his chicken every day throughout its life, until one morning, he came and wrung its neck. From the chicken’s perspective, induction worked very well until that catastrophic morning. Additionally, Nelson Goodman composed an even more illustrative example: suppose there is a special colour called “grue”, which applies to things that are green if observed before time t and blue otherwise. Before time t arrives, all observed green emeralds are also grue, so induction appears to support equally well the claims that “all emeralds are green” and “all emeralds are grue”. But after time t, these two hypotheses diverge, for the latter predicts that emeralds first observed after t will be blue.

Personally, I hold the opinion that the problem of induction is inescapable. Despite various attempts to account for the problem of induction made by many genius philosophers, there’s no overwhelming argument which solves the problem convincingly. But the inescapability of the problem of induction brings about another serious question: there can be no reason for us to believe in scientific speculations and theories. Given that all inductive inferences are unreliable, are scientific inferences such as the law of gravitation any different from the chicken’s inference of being fed tomorrow? Our intuition refuses such an equation, but telling the difference between the two, in other words, telling a good inductive argument from a bad one, is no easy task. The main purpose of this essay is to try to work out the possible criteria to distinguish good and sound inductive arguments from poor and unreliable ones. Before looking into that, I will first list out some inductive arguments for the convenience of further consideration.

Inductive Arguments

(i) The sun will rise tomorrow.
(ii) The peasant has fed the chicken every day throughout its life, so he will feed it tomorrow.
(iii) All crows are black.
(iv) All emeralds are green.
(v) All metals are conductive.
(vi) All objects on my desk are conductive.
(vii) The pompous buildings found in the desert were constructed by human beings.
(viii) Stonehenge was built by human beings.
(ix) Having been travelling for 24 hours, he must be very tired now.
(x) The mechanic at the bike store told me that my brake was about to fail.

II. Distinguishing in the Falsifiability Approach

Since the problem of induction is insoluble, and hence induction itself is not something that can differentiate between scientific arguments and random arguments like “the peasant is going to feed the chicken tomorrow” and ascertain the dependability of science, we need to look elsewhere for the affirmation of valid inductive arguments. In fact, Hume tried to justify our belief in propositions like “the sun will rise tomorrow” and “the stone will fall if you let go of it” by saying that nature has its uniformity. Though our preference for induction is no more than animal instinct, following inductive conclusions cannot do us any wrong as it conforms to the uniformity of nature. Yet resorting to the uniformity of nature can be ambiguous and indecisive. For example, are propositions like (iii) and (iv) conforming to the uniformity of nature? Before we actually find a crow that is not black, we can well agree that (iii) obeys laws of nature, which is totally another story once we find a crow that’s of another colour.

Hume’s problem of induction has inspired many of his successors to develop various systems in response to the difficulties the validity of science faces, among whom Karl Popper is quite an eminent one. Popper advocated that we should abandon the inductive explanation of knowledge altogether and invented the concept of falsifiability. According to the theory of falsifiability, good scientific arguments are ones which are falsifiable, yet have not been falsified. Being falsifiable means that a theory can be contradicted by evidence. For example, “the sun rises every day” is falsifiable, for it can be contradicted if the sun does not rise on a certain day. Yet “the soul exists” is not falsifiable, for we can neither prove nor disprove it by evidence. Falsifiability draws a line between science and pseudo-science, and in a sense replaces induction in justifying the validity of scientific theories. If we examine the ten inductive arguments listed above, we can see that all of them are falsifiable. As a matter of fact, all inductive inferences, which are drawn from daily experiences, are falsifiable.

But being falsifiable is not the only condition that restricts a theory or an argument to be a scientific one. To be identified as scientific, we also require an argument to be unfalsified. For instance, if we have not found a crow which is not black up to now, we can grant (iii) as an inductively justified statement. Goodman’s grue emerald example, however, shows a difficulty for this approach. Before time t, as the proposition “all emeralds are green” is falsifiable and not falsified, we can grant (iv) as an inductively justified statement. But the very same observations also support the proposition that “all emeralds are grue”. The problem, then, is that falsifiability by itself does not tell us why “all emeralds are green” should count as the better inductive inference. It may appear a bit weird to allow such a conflict in scientific statements to happen. But in reality, science itself is conditional, and old theories do need to be replaced by more illustrative and powerful new ones when the context of discussion changes.

But the falsification theory still encounters serious problems, for it cannot successfully tell a bad inductive argument from a good one until it has been falsified. The argument “all objects on my desk are conductive” is no weaker an argument than “all metals are conductive” until we find, say, an eraser on my desk. But it is odd to say that the validity of an argument depends on the discovery of counterexamples. Rather, whether an argument is a valid one should be determined by its internal structure.

III. Multiple Verification

The greatest problem of falsifiability analysis lies in the fact that it checks every argument, or hypothesis, separately without fitting them into the whole picture of the system of a certain science. It is impossible to justify the validity of a single argument without looking at it from a global perspective.

For any inductive argument, we need to resort to deductive reasoning to assure its validity. Consider the first argument, that “the sun will rise tomorrow”. We may say so out of two considerations. Firstly, the sun rose today, yesterday, the day before yesterday, and so forth. Therefore, we form the belief that the sun will rise tomorrow. And secondly, we may hold the belief that it is so because of the self-rotation of the earth and the law of gravitation, etc. What guarantees the validity of the inductive argument is not the first consideration but the second one, which deductively links it to other established scientific arguments. The object of the first consideration can be any inductive argument listed above, yet not every one of them can be considered in the second way. For instance, the proposition that “the peasant will feed the chicken tomorrow” cannot be concluded deductively from other established arguments, and hence cannot be seen as a good inductive argument. Other arguments may be, or may not be, good inductive arguments, depending on whether a deductive explanation can be given about the fact stated by the argument. For instance, if the proposition “all emeralds are green” is merely inferred from the fact that all emeralds observed up to now are green, it is not enough to say that such an argument is a good and reliable one. Yet if such a proposition can be further demonstrated by arguments about the crystal structure of the emerald and the way it interacts with light, etc., then we can regard argument (iv) as a good inductive argument. I refer to this way of determining whether an argument is a good one as the method of multiple verification.

Such a method may receive objections, saying that we cannot guarantee the validity of “the sun will rise tomorrow” by resorting to the law of gravitation, as scientific laws as such are themselves purely inductive ones when they are first established. As a matter of fact, if we seriously ponder over inductive and deductive inferences, we would surprisingly find that all deductive arguments, once given any content, incorporate premises which are inductive conclusions. For example, when we construct the argument that “all human beings die”, “Socrates is a human being”, and “therefore, Socrates dies”, the first premise is one generated by induction. But admitting that most fundamental scientific laws are initially constructed from pure inductive inferences does not impair the effectiveness of the method of multiple verification, because after deductive conclusions are drawn from those initial propositions, initial arguments purely derived from induction come to be deductively linked to their conclusions as well. At the same time, when conclusions are getting strengthened in empirical experiments, the validity of premises gets enhanced as well. For example, if we consider F=ma as a theory drawn from purely inductive inference, and we build a gadget of mass m, with a force F applied to it, whose acceleration is measurable. We repeatedly apply the same force to the gadget and constantly find its acceleration to be a in numerous experiments. In these experiments, the validity of “the gadget of mass m moves with acceleration a when a force F is applied to it” is verified, because we can draw the conclusion inductively from our observations as well as deductively from “F=ma”. Meanwhile, the validity of “F=ma” gets reinforced when the conclusions deductively linked to it get repeatedly confirmed by empirical experiences.

Lastly, we are able to examine the validity of other arguments listed in Section I using the method of multiple verification. “All objects on my desk are conductive” is clearly not a good inductive argument, for it cannot be verified deductively. “Everyone who has travelled for 24 hours is very tired” may be a relatively good inductive argument, for it is possible for us to find some biological evidence supporting it. Arguments (vii) and (viii) can be seen as derived from the general inductive argument that “all organized constructions are fruits of human activity”, yet the validity of each case should be measured by whether sufficient evidence can be found in support of such inferences. The last argument, that “the mechanic at the bike store told me that my brake was about to fail”, is tricky. The mechanic in question may get the conclusion inductively from his previous experience of fixing bikes, and there might be some coherent theoretical explanations of the relationship between the mechanic’s observation and the failure of the brake. But if those theoretical explanations are not recognized by the mechanic, we still say that the conclusion that “the brake is about to fail” is no more valid than “the peasant is going to feed the chicken tomorrow”.

References

[1] Hume, David. An Enquiry Concerning Human Understanding. Charles River Editors, 2018. (Sec IV, Sec V)
[2] Russell, Bertrand. The Problems of Philosophy. OUP Oxford, 2001. (Chap 6)
[3] Strawson, P. F. Introduction to Logical Theory. Routledge, 2011. (Chap 9)
[4] Mellor, D. H. “Inaugural Lecture: The Warrant of Induction.” 1988.
[5] Goodman, N. The New Riddle of Induction. 1955.
[6] Curd, M., and Cover, J. A. Philosophy of Science: The Central Issues. New York: W. W. Norton, 1998. (pp. 409–411, pp. 426–432)
[7] Popper, K. R. “The Problem of Induction.” 1953.