The New Riddle of Induction


1. Dissolving the Old Riddle of Induction.


The Old Riddle of Induction is the problem that some writers find in Hume. It is what we have called the "justificatory" problem: the problem of defending our right to engage in inductive reasoning given that all such reasoning seems to depend on an absolutely unjustifiable assumption, viz., the principle of the uniformity of nature. This problem as I have just formulated it clearly depends on Hume's solution to what we have called the descriptive problem of induction. Hume argues that we unconsciously take it for granted that (in his words) the future will resemble the past. If we make this assumption, then reasoning about unobserved matters of fact can be understood as valid deductive reasoning from claims about past experience together with this additional premise to claims about the unobserved parts of nature. The justificatory problem then arises immediately. If we use this additional premise in our reasoning, then our conclusions are only justified if we can somehow justify this assumption. But -- as Hume convincingly shows -- no argument could ever provide us with reason to believe that nature is uniform or that the future will resemble the past, simply because all argument about the general features of nature presupposes this principle.

Both Strawson and Goodman regard this problem as misconceived. The mistake in their view is to suppose that the inductive inference from a body of evidence to a prediction or a generalization must really be deduction in disguise if it is to be compelling. This is the assumption that leads Hume to posit an otherwise unconscious commitment to the principle of the uniformity of nature, and once he has done this, the justificatory problem immediately arises. But suppose we reject this assumption. Suppose, as both Strawson and Goodman suggest, that besides the familiar notion of deductive validity, there is a distinct but equally legitimate notion of inductive validity. As we have seen, an argument is deductively valid just in case it is absolutely impossible for its premises to be true and its conclusion false. Inductive arguments without an assumption of uniformity are clearly not valid in this sense. A typical such argument might run like this:

The sun has risen every day so far in our (extensive) experience.
We have no further relevant evidence.
Therefore the sun will rise tomorrow.

Here, clearly, the premises could be true and the conclusion false. So the argument is not deductively valid. But still, the following seems clear: By ordinary standards, anyone who accepts the premises of this argument while rejecting its conclusion is being unreasonable. Let us say that an argument is inductively valid just in case it has this feature:

An argument from premises P, Q, .... to a conclusion C is inductively valid just in case anyone who accepts the premises while rejecting the conclusion is being unreasonable.

The first component of the Strawson/Goodman response to Hume is thus to reject the identification of deductive validity with inductive validity. Some good, cogent, compelling arguments are not deductively valid. This is not just an interesting discovery: it blocks the only motivation for positing an unconscious commitment to a principle of the uniformity of nature, and so blocks the development of Hume's skeptical paradox.

2. What is Inductive Validity?

At this point you may be impatient. You may say:
All right, I accept the distinction between inductive and deductive validity. But there is still a serious skeptical problem about induction. With what right do we suppose that the arguments we habitually regard as inductively valid really are inductively valid? We can imagine creatures -- clear-headed English-speaking Martians, for example -- who reject the arguments we find compelling. We can imagine a Counterinductivist who maintains that the fact that the sun has risen everyday so far makes it reasonable to believe that it will not rise tomorrow. We seem to disagree with this character about what constitutes an inductively valid argument. Stepping back from the dispute, how can we justify the assumption that our standards for good inductive reasoning are the correct ones? What could we say to make our opponents believe this?

This seems like an excellent challenge, and Strawson has an answer to it. It is very brief. Strawson simply asserts that to say that an argument is inductively valid (or cogent or compelling) just is to say that it is correct or cogent by our standards. In class we discussed the analogy with notions like "grammatical sentence of English". We have standards or rules for deciding when a string of English words counts as a grammatical sentence. In learning the language we have mastered these standards, even though we cannot formulate them explicitly. A skeptic might try to raise doubts about their application by asking, With what right do you suppose that your standards for deciding when a string is grammatical are the right standards? But that is pretty clearly a stupid question. There is nothing more to being a grammatical sentence of English than being grammatical according to the implicit standards of competent speakers of the language. So there is no room for skeptical doubt as to whether our standards are the correct ones. Similarly, Strawson seems to hold that the implicit rules by means of which we classify arguments as inductively valid or invalid determine what really is valid or invalid. So there is no room for skepticism about whether our standards are the right standards.

This is a compelling analogy, but you should not accept it without a fight. (I stress that the analogy is mine, not Strawson's.) In some cases our collective standards for deciding whether something deserves a certain name do not determine whether it really does deserve that name. We may all agree that slavery is morally permissible; but if we do, we will all be mistaken. It is one thing to be morally permissible by our standards, another thing to be genuinely permissible. The two notions can come apart in dramatic ways. Thus one might wonder whether the notion of "good argument" is more like the notion of moral permission than like that of a grammatical English sentence.

But let us leave this problem to one side for now. This dissolution of the Old Riddle of Induction sets the stage for what Goodman calls the New Riddle of Induction. The New Riddle is not at first a skeptical challenge. It is not an attempt to show that ordinary reasoning is no good. It is rather an attempt to show that a very natural model of ordinary reasoning cannot be correct. It is a problem in descriptive rather than normative epistemology.

3. The Project of Inductive Logic

Deductive logic, as standardly presented, is a body of formal rules. The following argument is deductively valid:

All men are mortal
Socrates is a man
Socrates is mortal

And as soon as we see this we know immediately that the following argument is also valid:

All sheep are green
Fred is a sheep
Therefore, Fred is green.

That we are able to recognize both of these arguments as valid suggests that we have somehow mastered a formal rule or schema that might be formulated as follows:

Every argument of the following form is valid:
All Fs are Gs
Q is an F
Therefore, Q is G.

The aspiration of deductive logic as a mathematical or philosophical discipline is to produce a set of explicit formal rules for determining whether an argument is valid or not. This is by no means a trivial project, and to the extent that it has succeeded it is one of the great achievements of modern philosophy. Similarly, it is one of the great aspirations of linguistic theory to produce a set of formal rules for determining whether a given string of words constitutes a grammatical English sentence. This task has proven surprisingly difficult; but when it is completed it will be one of the major achievements of modern linguistics.

You might think, having accepted the distinction between inductive and deductive validity, that there was a similar project to pursue in the case of inductive reasoning. This project would seek to provide a set of formal rules for determining when a body of data supports a generalization or a prediction; or better, since support or confirmation is a matter of degree, the rules would determine the how well any given body of evidence supports any given hypothesis or prediction. In principle, these rules could be programmed into a computer. You could then give the computer a complete specification of your data D and a list of the hypotheses H, H*, H**, ... that your were investigating, and it would spit out verdicts of the form:

D confirms H to degree r
D confirms H* to degree r*, etc.

A set of rules of this form would constitute a formal inductive logic or a logic of confirmation. And you might think that there must be such a set of rules. After all, we all agree on the vast majority of our inductive inferences, even the inferences we have never considered before. It is very natural, if not inevitable, to attempt to explain this vast degree of "non-collusive agreement" by supposing that we have tacitly come to accept a formal logic of induction. The project would then be to make these tacitly understood principles explicit.

Goodman's New Riddle is, among other thing, an argument against the possibility of a formal inductive logic.

4. The New Riddle

Good inductive reasoning can get very complicated, as anyone familiar with real science or statistics can attest. But let's consider the simplest sort of case. Before you is an urn containing 1,000,000 ping pong balls. You reach into the urn and remove a ball at random, noting its color and then putting it back. You perform this experiment 10,000 times and each time the ball is found to be green. On this basis, knowing nothing else about the urn or its contents, you confidently predict that the next ball you withdraw will be green, and indeed that nearly all of the balls in the urn are green.

This is ordinary, commonsensical reasoning. If this is not an inductively valid inference, nothing is. So we may ask: what is the underlying rule? Here is a very natural attempt at its formulation:

In large random sample from a finite population, all Fs are G
Therefore, the next sampled F will be a G, (and indeed, nearly all Fs in the population are G).

Call this the Simple Rule (SR). SR is as obvious a candidate as you are likely to find for a formal rule of inductive validity. On the face of it, every instance of this schema would appear to be inductively valid. It's hard to think of a case in which it would be reasonable for someone to accept the premise and reject the conclusion. So SR would appear to be a rule we accept.

Goodman's New Riddle proceeds by showing that SR is not in general a valid rule, and by extension that there can be no formally valid rules of inductive inference. We'll see that this has far ranging consequences later on. But for now, let's concentrate on Goodman's objection to SR.

In order to explain the objection I will have to introduce some new vocabulary. I will assume that we all understand the words "blue" and "green" perfectly well. The new words are defined in terms of these as follows:

An object X is grue iff X is green and first observed before the year 2000, or x is blue and never observed before 2000.
An object X is bleen iff X is blue and first observed before 2000, or X is green and never observed before 2000.

It is important to realize that GRUE and BLEEN are not colors. Two things can be exactly the same shade of blue, and yet one of them be bleen and the other grue, depending on when each is first observed. To drive this point home, let's say that GRUE and BLEEN are schmolors. You never had words for schmolors before. But now you do, and with a little practice you can learn to use them quite easily. The relations between color terms and schmolor terms are summarized as follows:



It is also important to stress, as you are coming to learn these new terms, that the classification of objects as grue or bleen does not imply that anything mysteriously changes color on Jan 1., 2000. If the emerald on my ring has been observed before the year 2000, as it has, then if it is grue now it is grue forever, just as it is likewise green now and forever.

Now that we have learned this odd new idiom, we are in a position to see that SR is not in general a valid rule. We have already agreed that the following inference is inductively valid

In a large random sample of balls drawn from the urn, every ball is green
Therefore the next ball we draw will be green, and indeed nearly all of the balls in the urn are green.

This is an instance of the schema SR. But then again, so is this:

In a large random sample of balls drawn from the urn, every ball is grue.
Therefore, the next ball we draw will be grue, and indeed nearly all of the balls in the urn are grue.

Of course it would never have occurred to us before to reason in this way. But now that we have learned the new schmolor idiom, we can easily construct this argument and see that it has exactly the same form as the inference cast in terms of color.

So far this might just seem like a curious discovery. The data so far support a natural inference about the colors of the balls in the urn, and an unnatural and peculiar inference about the schmolors of the balls in the urn. Big deal! But wait. Imagine that it is New Years' Eve, 1999, and that a great deal hinges on our prediction of the color of the first ball drawn from the urn in the new century (Call it Bob). (Suppose that an eccentric billionaire has offered a vast reward for a correct prediction.) We have performed our scrupulous random sampling, and we are all prepared to predict that the ball will be green. But now someone points out that our data also support the prediction that the ball will be grue. The trouble is that if we accept this conclusion, we shall be forced to say that Bob will be blue! After all, we have a cogent inductive argument for the conclusion that

Bob is grue.

But we know that

Bob will not be observed until after the year 2000

So Bob falls in the lower right hand quadrant of the diagram, which is just to say that

Bob is blue!

Now we're in trouble. We have two inductive arguments, identical in form, which in the circumstances lead to logically incompatible predictions about Bob's color. If both arguments really are inductively valid, then we must conclude that a reasonable person confronted with our data would believe both that Bob will be blue and that Bob will be green. But no reasonable person could believe that.

The only conclusion seems to be that at least by our ordinary lights, the argument for the conclusion that Bob will be green is valid, whereas the argument for the conclusion that he will be grue is invalid. Since both arguments have the same form, we are forced to conclude that inductive validity is not a formal property of arguments. That is, two arguments that are formally just alike, but differ only in the words they contain, can differ as to their inductive validity. This cannot happen in the case of formal deductive validity, so we have a striking disanalogy between the two sorts of reasoning.

5. Goodman's Problem of Projectability

So far this is just an interesting observation about inductive validity. It does not yet deserve to be called a riddle. But we are not very far from something that does. For we may now ask a new version of the descriptive question. As we have seen, by our lights the inference from observed greeness in the sample to general greeness in the population is valid by our lights, whereas the parallel inference in the case of trueness is invalid. So some instances of SR are valid, others not. But what exactly determines how the line is drawn, and how can we tell? Let us say that a predicate P is projectible if and only if figures in an inductively valid instance of SR. "Green" is evidently a projectible predicate, since the fact that the sample is uniformly green constitutes a reason to believe that the population as a whole is largely green. "Grue" by contrast is not projectible. The problem is then to state a rule or principle for distinguishing the good projectible predicates from the bad, nonprojectible, gruesome predicates. If we wanted to tell a computer how to evaluate inductive arguments as valid and invalid we would plainly have to give it some such rule. How might it be formulated?

Our first hope is for some sort of objective test that distinguishes green and blue from grue and bleen. I will leave it up to you to consider some solutions to this problem and to review Goodman's reasons for rejecting them. Just to take one example, it will not do to point out that the definitions of green and blue make explicit reference to a particular time. This is true, but it does not distinguish, since we can define green and blue in terms of grue and bleen as follows:

X is green iff X is grue and observed before the year 2000 or X is bleen and never observed before 2000.
X is blue iff X is bleen and observed before the year 2000 or X is grue and never observed before 2000.

A quick glance at the diagram above will convince you that these definitions are perfectly correct. And yet they have exactly the same form as the definitions of grue and bleen we gave earlier. So you cannot say: A predicate is projectible just in case its definition makes essential reference to a moment of time or to the fact of observation. That does not distinguish the color words from the schmolor words.

Or at least, this test does not distinguish provided these hokey definitions of "green" and "blue" are strictly on a par with our definitions of "grue" and "bleen". This is a possibility you ought to explore.

A further reason for doubting this account of projectibility comes from observing that we do not in general reject inferences involving "positional" or "temporal" predicates. Suppose a sociologist coins the term "Gen X-er" for people born after 1970. He main then observe that in a large random sample, Gen X-ers tend to be more conservative than their parents; and on this basis he may conclude, by a valid induction, that on the whole, in the population at large, Gen X-ers are more conservative than their parents. This is a good inductive argument. So the predicate ".. is a Gen X-er" is projectible. But it is also clearly temporal or positional. So the proposed test for projectibility is too restrictive. It may rule out "grue". But it rules out any number of perfectly projectible predicates as well.

6. Goodman's Skeptical Solution to the Problem of Projectibility.

I will not review the various attempts to produce an objective account of the distinction between projectible and non-projectible predicates. The arguments I reviewed above and in lecture only begin to scratch the surface, and you should try to do better yourself. Goodman, however, maintains that no such account is possible. On his view, the only relevant difference between "green" and "grue" is that the word "green" has in fact been involved in a vast number of successful inductive inferences over the course of intellectual history. The word is, in Goodman's phrase, "entrenched" in our practice. The word "grue", on the other hand, has no role in our actual practice. Although we could have used it, we have not in fact done so. So it is not entrenched. And for Goodman this makes all the difference. His view, to a first approximation, is that an inductive inference of the form SR is valid just in case it involves only entrenched predicates.

Now the first thing to note is that this principle is clearly too strong. It is an absolutely central feature of scientific practice that newly minted classifications can figure in good scientific inference right away. Suppose that on Monday a theoretical physicist coins the term "black hole", and that on Tuesday the astrophysicists discover a large number of them. On Wednesday the astrophysicists notice that in a random sample of these newly discovered black holes, 100% emit X-rays at a distinctive frequency. They immediately conclude, by induction, that this is probably a general feature of all black holes, and proceed to use the presence of this distinctive radiation as a new means for discovering these elusive objects. This is a simple-minded example of what is clearly good and standard scientific practice. But it violates Goodman's principle, since it treats "black hole" as a projectible predicate despite the fact that it is not at all entrenched.

7. A New Skeptical Problem?

Of course this is just an invitation to attempt to refine Goodman's criterion of projectibility. But I will not pursue this problem. Rather I propose to assume that something like this account is correct, and then to develop a new ground for skeptical doubts about induction on this basis.

The striking feature of Goodman's proposal is what might be called its non-objectivity. For Goodman, the distinction between projectible and non-projectible predicates, and hence the distinction between valid and invalid inductive arguments, is drawn by reference to accidental, contingent facts about our practice and intellectual history. The reason that our evidence supports the conclusion that grass is generally green rather than grue is that as a matter of fact we have used the word "green" in our past inductions. This is a fact about our history that could easily have been otherwise. An eccentric caveman might have coined the word "grue" at the dawn of thought, and through his persuasive force of example established a general practice of using the notion in inference. We know that the term would have been perfectly serviceable: no false prediction about colors (or schmolors) would have come to light. So there is nothing to rule out an alternative development of our inductive practices of this sort. Goodman's account implies that under these circumstances, the evidence we have collected would have supported the conclusion that all grass is grue, and hence that grass that comes up in the Spring of 2000 will be blue. On this account, what makes one inference good and another formally identical inference from the same data bad is a fact about the accidents of human history. In this sense, the distinction between good and bad reasoning depends on us, which is just to say that it is not an objective distinction.

Now it would be a serious mistake to infer from this observation that for Goodman, "anything goes" in induction. It is one thing to say that the distinction between valid and invalid reasoning depends on us in a certain sense, quite another to say that it is up to each of us as individuals to draw it as we please. If an eccentric scientist decides that he "prefers" grue to green, and proposes to conclude that the first grass of the 21st century will be blue, then on Goodman's view he is making a mistake: he has accepted a bad argument, because the predicates he has employed are not entrenched in the real history of his community. So the non-objectivity of the distinction is not a license for intellectual free play. Given our history -- which we cannot change at will -- some inferences will be valid and some will be invalid, and how this line is drawn will not be up to us.

Still, the non-objectivity of the notion of inductively valid inference is potentially disturbing. In some areas it is a familiar fact of life that different cultures accept different norms and are therefore blamelessly and harmlessly different. We think spitting in the street is impolite; elsewhere they disagree. And when this is pointed out, we naturally retreat to a relativized idiom: spitting is rude around here or for us; it is acceptable over there or for them. But sometimes the discovery that other people or other cultures disagree with us has a powerful destabilizing affect on our judgment. Imagine that you have been raised to accept, say, the Christian religion, and that you have always taken its central tenets for granted. You then discover that other people with distinct histories and cultures disagree with you: they find the doctrine bizarre and implausible. Now this by itself need not bother you. You may think that the disagreement derives entirely from their relative ignorance of some relevant facts: they have not had access to certain texts, or to the historical memory of certain miracles, or perhaps to certain persuasive inner experiences. But suppose that you engage them in dialogue and quickly rule out these possibilities. They know about the texts and the so-called miracles, and they are familiar with so-called religious experience but interpret it rather as evidence for their own non-Christian creed, or perhaps as a sign of mental illness. It seems possible that you may come, over the course of your conversation, to the following view:

Their alternative to Christianity is coherent given all of the available evidence; so is my acceptance of Christianity. So the only reason I accept Christianity is that I have been raised to accept it. If I had been born among them, with my rational faculties intact and with access to the very evidence that I now possess, I would reject Christianity.

Now this is, for many people, a profoundly disorienting train of thought. For consider what would be involved in retaining one's commitment to Christianity in light of it. Someone who persists in accepting Christian doctrine must think that he is immensely lucky: lucky to have been born into a tradition that enjoys a privileged access to the truth, even though he cannot provide any explanation for why he should have been lucky in that way, nor any independent justification for the claim that he has been lucky in this way.

Now in the religious case it is possible to credit oneself with this kind of luck by adverting to the distinction between faith and reason. One can say:

I realize upon reflection that I have no good reason to accept the fundamental tenets of by religion, but I accept them anyway as an act of faith and hope that I am right.

This involves a clear rejection of the requirement of rationality; but apart from that, we have not seen any reason to doubt its coherence.

But consider a parallel position about the scientific reasoning. You believe on the basis of what you take to be abundant evidence that come 2000, grass will still be green. You now encounter a culture for whom "grue" rather than "green" is entrenched. So they believe on the basis of exactly the same evidence that come 2000, grass will still be grue, which is to say that it will be blue. Of Goodman is right, then a speech exactly parallel to the one we considered above is in order:

Their alternative to the view that grass will be green in the year 2000 is coherent given all the evidence, and it is rationally acceptable for them given their history. So the only reason I have to accept the view that grass will be green in the future is that I have been raised in a community in which "green" rather than "grue" is entrenched. If I had been born among them with my rational faculties intact and with access to the very evidence I now possess, I would confidently believe that in the future, grass will be blue.

Now ask yourself whether you can live with this thought while retaining your conviction that grass will be green come the year 2000. To do so is to suppose that your community, your culture, was lucky enough to hit upon the right terms for framing inductions. But you have no independent reason to believe that you have been so lucky. Indeed, given all of the "grue" -like terms you could have used, it would be a miracle if your communities selection of the term "green" -- a selection made apparently at random -- should have turned out to be the correct one.

(We can try to make this point more explicit by framing the matter in quasi-statistical terms. Cultural history involves what is, in effect, a random selection of one idiom for describing the world out of a vast range of possible, serviceable idioms. We chose the idiom of color words. They chose the idiom of schmolor words -- grue -2000, bleen 2000 -- etc. But there are many other possibilities: grue-2001; grue 2002 ... etc. (The definition of grue 2001 is just like the definition of grue, but with 2001 as the relevant date.) Now at most one of the following generalizations is true:

Grass is green

Grass is grue 2000

Grass is grue 2001


Suppose for definiteness that there are only 1,000,000 possibilities. Our cultural history leads us to accept the first; another cultural history would have led us to accept the second, and so on. What were the chances, at the dawn of thought, that our cultural evolution would lead us to accept an idiom that would lead to the acceptance of the true hypothesis given our evidence? You might think it should be 1/1,000,000. And this means that every time we accept an inductive generalization on the basis of what is by our lights a valid inductive argument, we are supposing without evidence that we have won the lottery -- that we have managed to select at random the idiom that leads to true generalizations by validating just those instances of SR whose conclusions are true. But this is remarkable! And it only gets more remarkable when you consider that the set of possible idioms is really much larger, indeed potentially infinite!. Exercise: Turn this "quasi-statistical" argument into a new argument for inductive skepticism.)