and all of these emeralds are green, we infer that all emeralds are green. However, notice this. All of the emeralds we have seen are also grue. So, we’ve observed the exact same number of grue emeralds as we have green emeralds. So, we should also be willing to make a second inductive generalization: given that all of the emeralds that we have seen are grue, and we have seen a heck of a lot of emeralds, we should be willing to infer that all emeralds are grue too. So, as the result of inspecting the same sample of emeralds, we conclude both that all emeralds are green and that all emeralds are grue.
1.40 The problem is that the claim that all emeralds are green is incompatible with the claim that all emeralds are grue, since there are emeralds that we have never seen and that won’t be observed during our lifetime. Consider an unobserved emerald, which I will call “Eddy.” If all emeralds are green and all emeralds are grue, Eddy is both green and grue. But no one has or will observe Eddy in our lifetime. So, Eddy can’t be green and observed in our lifetime. Since Eddy is grue, and not green and observed in our lifetime, it follows that Eddy must be blue and not observed in our lifetime. So, Eddy is blue. But just a moment ago we said Eddy is green. Nothing can be both blue and green at the same time. So, the same sample set and the same method of forming more general beliefs from that sample set led to inconsistent results. Not good. That’s the puzzle of the new riddle of induction.
1.41 What seems plausible then is that we shouldn’t indiscriminately make inductive inferences. It is ok to infer from our sample size that all emeralds are green. It is not ok to infer from that same sample that all emeralds are grue, even though every emerald in that sample is both green and grue. Don’t believe that whenever we encounter a sufficiently large sample of things that each have a certain feature F, and everything that has F also has another feature G, we should generalize and so believe that all things that have F also have G. Instead, consider something more complex and careful. Only some of those Fs and Gs are ones that license inductive inferences. We are finally ready to define “projectible”: an expression is projectable if and only if it stands for a feature that we can justifiably make inductive inferences about. “Green” is plausibly a projectable expression, but if it is, then “grue” can’t also be projectable.
1.42 But what makes an expression a projectible expression? One answer to this question is that an expression is projectible to the extent that the expression corresponds to objects that objectively belong together. If this is the correct answer, we also have an argument that some objects objectively belong together. The argument is this. We are sometimes justified in making inductive inferences. But we are sometimes justified in making inductive inferences only if some of the words we use to make those inferences are projectible. And those words are projectible only if they classify objects that objectively belong together. So, some objects objectively belong together.
1.5 Two Questions about Classification
1.43 I hope that we now have a decent enough grip on the distinction between what I have called “subjectively belonging together” and “objectively belong together” that we can proceed to ask some interesting questions about this idea of things objectively belonging together. Let’s now turn to two of the most important general questions.
1.44 I want to distinguish the question of what it takes for some things to objectively belong together from the question of which things objectively belong together. Both questions are important questions, and knowing the answer to one could help us learn the answer to the other. But the answers to these questions are by no means guaranteed to be the same.
1.45 The first question, the one that asks “What does it take for some things to objectively belong together?,” is an example of a type of question that philosophers are prone to ask. The type of question of which this first question is an instance is “What does it take for a thing to be X?” where X is some feature that a thing could have. Here are some other examples of this type of question: “What does it take for a belief to count as knowledge?”; “What does it take for an action to be one that we morally ought to do?”; and “What does it take for an action to be done freely?” Let’s briefly think about one of these questions so that we can get a feel for this type of question in general. We’ll focus on the first one, the question of what it takes for a belief to count as knowledge.
1.46 There are some beliefs that don’t count as knowledge. For example, some people have false beliefs, and no false belief can be knowledge. The other day, I was trying to find my wallet. I had a very strong belief that I had left it on top of the fridge the previous evening. My wife insisted that I hadn’t, and I replied that I knew that I had. It turns out that I had left my wallet in my office. Given where my wallet in fact was, I didn’t know that I had left my wallet on top of the fridge. I only thought that I knew. Reflecting on this story makes it clear that a belief counts as knowledge only if that belief is true. But it’s also clear that merely having a true belief is not sufficient for that belief to count as knowledge. Suppose someone, who we will call “Fred,” reads the story I just told and as a consequence comes to believe that my wallet had been in my office. Fred’s belief is true, but does it count as knowledge? Well, you might think that Fred actually doesn’t have that great of a reason for believing that I left my wallet in my office. It is a well‐known fact that philosophy professors make up stories all the time just to provide vivid illustrations of some point that they want to make. Fred doesn’t have enough evidence to rule out the possibility that I am just making up a story. So, even though Fred’s belief is true, Fred’s belief doesn’t count as knowledge. It just doesn’t have what it takes to count as knowledge. Ok, so what does it take for a belief to count as knowledge? In other words, what are the necessary and sufficient conditions that something has to meet in order for it to be an instance of knowledge? This might be the most important question of epistemology, and if this were an epistemology book, we’d focus on it further.
1.47 The question “What does it take for a belief to count as knowledge?” is a different question from “Which beliefs count as knowledge?” The first question asks for insight into what knowledge is, while the second question asks for a list of beliefs that are known. In principle, someone could answer the second question simply by exhaustively listing all of the beliefs that count as knowledge, which include Jason’s knowledge that the sky is blue, Marcy’s knowledge that 2 + 5 = 7, Raul’s knowledge that World War II took place prior to the first Gulf war, Shamik’s knowledge that emeralds are green, and so on. But this sort of list doesn’t by itself answer the question “What does it take for a belief to count as knowledge?” If we had an answer to that question, we’d know why all the items on the list of beliefs that count as knowledge belong on this list.
1.48 The same idea applies to the two questions “What does it take for some things to objectively belong to each other?” and “Which groups of things objectively belong together?” An answer to the first question provides necessary and sufficient conditions that a group of things have to meet in order to objectively belong to each other, while an answer to the second question could simply take the form of a list: these things objectively belong together, and so do those other things, and so on.
1.49 Even though we’ve distinguished the two questions, as I said earlier, finding the answer to one of these questions might help us find an answer to the other. If we had an accurate list of which groups of things objectively belonged together, we could use that list to test alleged answers to the first question. Suppose, for example, that a proposed answer to the first question said that having X is what it takes for some group of things to objectively belong together, but one of the groups on our accurate list did not have X. We’d be able to deduce that this proposed answer to the first question is mistaken. On the other hand, if we had an answer to the first question, we’d hopefully have some guidance on how to go about making our list of things that do objectively belong together.
1.50 So, we should distinguish our two questions, but keep in mind that the strategies for answering them might not be completely independent of each other. In the following section, we are going to focus on the first question of what it takes for some things to objectively belong to each other.