# Goodman and the Grue Emeralds – an Induction Discussion

In this post I will talk about Goodmanized predicates and raise some concerns over their legitimacy. Subsequently, I will dismiss those concerns and argue that Goodmanized predicates are just as legitimate as normal predicates. I will conclude by discussing what Goodmanized predicates reveal about how we acquire beliefs about the unobserved.

Consider the following. Every emerald observed so far has been green. Each time we observe a new emerald, it is found to be green. Each instance of finding a green emerald (and none of any other color), is supposed [^1] to confirm the hypothesis that all emeralds are green. “Green” is a predicate that applies to every observed emerald. Now let’s define a new predicate, “grue”. The definition of “grue” is: x is grue iff it is first observed before t and is green, or else first observed after t and is blue [^2] (74). So something is grue if it is first observed to be green before some arbitrary time t, but if it is first observed after t it must be blue (and not green) to be grue. Suppose t is 7/2/17. Every emerald, then, is green. But every emerald has also been grue. So every observed emerald has (so it seems) supported the hypothesis that : all emeralds are green, and the hypothesis that all emeralds are grue. Of course, it seems that after 7/2/17 each newly observed emerald will not be grue – certainly no emerald could be both grue and green (provided that it is observed after t), for the definitions would conflict. If all we do is infer from particular instances to general claims, then we have equal evidence for two competing hypotheses [^3]. The mystery is why we are wont to believe that emeralds after t will be green and not grue.

But if “grue” is not a legitimate predicate, then the mystery disappears. Goodmanized predicates have some odd features that warrant closer inspection like (1) they are disjunctive definitions and include a time reference and (2) Goodmanized predicates are not natural terms for things actually in the world. When a predicate contains a time-dependent disjunction, it applies to one set of objects before t and a different set of objects after t. In this way, it appears poorly defined.

Another worry is that “grue” is not a term for a natural phenomenon, whereas “green” is. Because of this, we cannot legitimately predicate “grue” of emeralds. Here’s the thought. Green seems to be a natural property. Naturally existing in the world are green things, even if there is no one in the world to observe it. An emerald is naturally green – we just need to look and see. But we cannot just see that something is grue, because we must also know t and where we are in relation to it. We can’t see anything about t in the emerald. Green predicates a naturally occurring property, where grue predicates an artificial and contrived property that does not reflect our natural ontology.

But these worries can be dismissed; Goodmanized predicates are not illegitimate. It is a mistake to think that grue is poorly defined. The definition of grue is exactly the same before and after t. Which part of the disjunction is effective in the application of the predicate is determined by t, but not the definition. Objects that are grue before t are still grue after t – they do not become suddenly not-grue. Moreover, while grue is admittedly an artificial term, that does not mean that it is illegitimate to predicate it of natural objects. Consider the set of green things, e.g. some grass, the bushes in your mother’s yard, emeralds. This is easy and legitimate. Now consider the set of grue things, e.g. grass and emeralds first observed before t, and peacocks and blueberries first observed after t. Note that it is just as easy and legitimate to consider this set as it was to consider the green set. When we say X is grue, we say that X belongs to the set of grue things. If there was something wrong in the consideration of the set of grue things, then there would be something illegitimate about predicating grue of X. But there is not.

Now that we have preserved the legitimacy of Goodmanized predicates, we can ask what they reveal about how we come to get the beliefs we have about the unobserved. Let’s return to our consideration of the grue and green emeralds. Recall the state-of-play: every emerald thus far observed has been both green and grue, so each observation has (ostensibly) supported two conflicting hypotheses, viz. that all emeralds are green and that all emeralds are grue. After t, one of the two hypotheses must fail, for their predictions contradict each other [^4].

But which hypothesis fails? Presumably, the rules of induction are what enable us to project into the future – that is, to be able to make accurate predictions with regard to each subsequent, unobserved instance. The traditional view of induction works like this. ($\Phi\,\!$) Every A thus far observed has been found to be B. Each newly observed instance of an A as a B (and assuming that no A has been observed as not B) confirms a hypothesis, viz. all As are Bs. So, anyone who has observed a long positive correlation between things of any two kinds A and B in a wide variety of circumstances over a long interval of time, has reason (or will come) to believe that all As are B. The thought is that each new instance provides us with more reason to subscribe to the hypothesis – that is, it confirms the hypothesis, raising the probability of the hypothesis being true. But this way of thinking about induction cannot decide between all emeralds being grue or all emeralds being green, because an equally long positive correlation has been observed between emeralds and greenness as has been observed between emeralds and grueness.

What the competing hypotheses “all emeralds are grue” and “all emeralds are green” show us is that this way of thinking about projection into the future is wrong – that this is an inadequate way to explain how we form beliefs about unobserved cases. Why? Because intuitively we believe that after t, the emeralds will persist in being green and cease to be observed as grue.  We suppose that “green” can easily and legitimately figure in our inductive inferences.  And we suspect that “grue” cannot easily and legitimately figure in our inductive inferences.  Consequently, we believe in the green hypothesis over the grue hypothesis.  But what is it about “green”, rather than “grue”, that allows it to legitimately figure in our inductive inferences?

If we form beliefs about unobserved events based solely on the flat-footed understanding of induction (marked [$\Phi\,\!$]), then we couldn’t have the belief that emeralds observed after t are going to be green (but we do), because there is an equally well-supported competing hypothesis. We have no more reason to believe that they will be grue than that they will be green, when considering just the evidence from our experience on the traditional ($\Phi\,\!$) view of induction.  For all emeralds observed thus far have been both green and grue.

According to Goodman, legitimate inductive inferences are the ones that are performed on lawlike correlations rather than accidental correlations. A lawlike inductive hypothesis is confirmed by its positive instances [^5]; a coincidental inductive hypothesis is not confirmed by its positive instances.  If the green hypothesis or the grue hypothesis is lawlike, then that correlation is confirmed by its positive instances.  Intuitively, we think that the green hypothesis will be the lawlike one.

So in order to have beliefs about the unobserved, it seems that we must have some way of determining whether a given correlation is lawlike or not – or at least there must be some way we come to believe that one hypothesis is lawlike rather than another.  Positive instances are not enough.

I will try to show that we can have a perceptual experience of a causal relation.  If this is so, then we may have a way of determining between lawlike and coincidental correlations, and an explanation of why we believe in the green hypothesis over the grue hypothesis.

Consider the fact that we have a concept of causation. How did we acquire it [^6]? Consider how you acquired a concept of a chair. You first saw the chair, and then you abstracted from it. Now consider a pot of water on a flaming stovetop. You see the pot, the water starting to bubble, and you see the fire and the stovetop. But in addition to this, I submit that you also see (or have some perceptual awareness of the fire boiling the water. You don’t just see the individual elements (e.g. the pot, the fire) isolated from one another, you see the elements interacting. That is, you have perceptual awareness of the causal relation between the fire and the boiling water. If you saw the scene as composed just of isolated elements and didn’t see the fire boiling the water, then you couldn’t abstract the causal relation and come to have a concept of causation. But you do have a concept of causation, and this suggests that it comes from our perceptual experience.

If we have a concept of causation, then we can believe two things to be connected causally and apply the concept to the situation. I can believe that A caused B and form the inductive hypothesis that: all As are Bs. Without a concept of causation, I cannot experience a causal relation between A and B and will not come to believe or hypothesize that all As are Bs. When I see that every emerald I have observed has been green, I am inclined to form the belief that something about the emerald causes it to be green. However, when I see that every emerald I have observed has been grue, I am not inclined to form the belief that there is something about the emerald that causes it to be grue. Why the difference? Because I do not have a perceptual experience of something about the emerald causing it to be grue [^7] – my experience is of the green.  Why is my experience of the green and not the grue?  Because looking at a green emerald, I believe that there is something that causes it to be green – there is a reason or an explanation of it.  But when I look at a grue emerald, I do not believe that something caused it to be grue.  For part of what it is to be grue is to be first observed at a certain time.  But the time of first observation is something entirely coincidental.  I do not believe that there is something that causes the emerald to be grue, other than when I happened to first observe it – which could have been any other time t.  Consequently, the causal relation I perceive is between green and the emerald, not grue and the emerald.

But what happens when we think we have an experience of a causal relation, but the relation turns out to be coincidental? This worry is not devastating. We are often mistaken as to the contents of our perceptions. Consider a man who thinks he sees a bird in the distance when it is in fact a small plane. His perception is mistaken, but we do not dismiss him as a poor perceiver or suddenly begin to doubt his judgment of his perception. Likewise, should some coincidences mistakenly be seen as causal, this is no reason to doubt our general ability to perceive causal relations. It just means that we will sometimes be wrong. Some inductive hypotheses with positive instances will fail, even when we thought they were lawlike, but this is how we come to develop our system of scientific law. But if we can in fact perceive causal relations, then this will be how we come to differentiate between the lawlike and coincidental and thus come to have beliefs about the unobserved.

[^1]: Whether it actually confirms the hypothesis or not is less clear. This will be discussed later.

[^2]:  “Bleen” is another example of a Goodmanized predicate. An object is bleen iff it is first observed before t and blue, or else first observed after t and green.

[^3]: Note that no emerald changes color, even after t.

[^4]: Because an emerald cannot be both blue and green. After t, if an emerald is first observed to be grue, then it is not green, because by the definition of grue it must be blue. But after t, if an emerald is first observed to be green, then it cannot be grue, because if it is green then it is not blue.

[^5]: A fair coin flipped 10 times and always lands on heads is accidental. If this happened with a loaded coin, it would be lawlike.

[^6]: Assuming that we do not innately possess concepts like causation. (Indeed, it is hard to see how we could innately possess them.)

[^7]: Furthermore, it is difficult to see how I could ever acquire a concept of grue through perception. Because grue includes a time-dependent disjunction, when I see the green/grue thing I only see the green because the time of first observation is not included in my perceptual experience of the emerald and so I wouldn’t abstract the concept of grue from it. I see the greenness, but I don’t see the grueness.