Tuesday, October 18, 2011

Further Reflections on Induction

Since a few confusions about the problem of induction persist — and the topic isn’t quite ready to go away — I thought I’d attempt some clarification. I’ll proceed in the time-honored form of an FAQ sheet. This isn’t meant to be exhaustive: I suggest circling back to the relevant Foster, Lipton, and Popper readings for further details. However, I’d be very happy to answer other questions you think should be included here (feel free to leave a comment or shoot me an email). And as usual, you’re welcome to join me in my office hours (or another time) to clarify any lingering puzzlement.

What is induction? 
Broadly speaking, induction is a non-deductive form of inference. People often have specific ideas about what sorts of inferences count as induction. For example, they may say that inductive inference proceeds from specific to general — while deductive arguments proceed from general to specific. Thus “All men are mortal; Socrates is a man; therefore, Socrates is mortal” is a classic deductive argument, while “This man is mortal; this other man is moral; that guy’s mortal, . . . ; therefore, all men are moral” is an exemplary inductive argument. 

However, a little reflection shows that this isn’t that great of a characterization — for either inductive or deductive arguments. Deductive arguments come in all shapes and sizes. Here’s one: “Professor Bermudez is either in his office or meeting with the dean; he’s not in his office, so he must be meeting with the dean.” Shoehorning this argument into the “general-to-specific” motto doesn’t seem right. We can find similar mismatches among inductive arguments. For example, consider one of the pieces of evidence for the Big Bang: “everywhere in the universe we look, there’s this hiss of background radiation; such a background would be nicely explained by the universe’s coming to be in a ‘Big Bang’; so therefore (probably) the big bang model of the universe’s origin is true.” We’ll look at arguments like this — sometimes called ‘abductive’ arguments or ‘inference to the best explanation’ — in more detail in a few weeks. If anything, we’re going from general to particular there, yet the argument is clearly supposed to be inductive in our broad sense. This example also demonstrates the inadequacy of another popular characterization of induction: that it is about prediction of the future events (or future observations). The arguments for the occurrence of a Big Bang are obviously not forward-looking. Science is about more than prediction: it is about finding out what happened and explaining it.   

You said that the inference to the best explanation above was clearly inductive? Why? Because the argument is not deductively valid?
Good question! Though it’s true that the argument is not deductively valid (the conclusion doesn’t follow from the premises as a matter of logic alone: we can imagine the premises about the background radiation being true but the Big Bang theory being false), this fact alone isn’t enough to make the argument inductive. We wouldn’t want to say that deductive arguments are necessarily valid: that is the standard to which deductive arguments “strive”, but there are plenty of invalid deductive arguments. For example: “If Professor Bermudez in his office, then he’s working; he’s not in his office; therefore, he’s not working.” The conclusion doesn’t follow from the premises in this case: it’s possible that Bermudez is working elsewhere. Thinking that this argument form is valid is a common enough mistake that it has its own name: ‘the fallacy of denying the antecedent’. What makes this argument deductive rather than inductive. The best answer has to do with the standard of evaluation that is likely intended by the arguer. 

So induction is a weaker, less demanding standard than deduction?
That’s a somewhat misleading way of putting it. Induction and deduction are simply different standards. In the case of deductive arguments, we can tell whether they are valid by more or less algorithmic means. That’s because validity has to do with logical form (roughly, the grammatical structure) of an argument rather than its content. (This is why it’s often called ‘formal logic’ — not because it dresses up nicely.) But the level of certainty that this standard gives us comes at a price: triviality. There’s a sense that we don’t really learn much when we derive the conclusion of a deductively valid argument from its premises. We might not have worked it out, but the information was (in a sense) already there, buried, as it were, in the form of the premises. That shows us that deduction alone won’t be a good way of expanding our knowledge. Inductive arguments, on the other hand, purport to do this. That is why some call inductive inference “ampliative”. 

But how can induction “amplify” our knowledge if the conclusions of inductive arguments are underdetermined by our evidence?
It is important to realize that the fact of underdetermination alone should not scupper our confidence in induction. For example, when some people are first exposed to the problem of induction they seem a little too eager to relinquish claims about knowledge of the future. It seems to me that they are confusing knowledge and certainty. Fair enough: I cannot be certain that, stay, the sun will rise tomorrow. My evidence thus far underdetermines whether it will (perhaps a rogue star will sweep through our solar system and disrupt everything tonight!). But on the other hand, our evidence suggests pretty strongly that no such freakish occurrence will take place. Underdetermination alone should not get us to relinquish inductive inference. We use it all the time. It has been successful.

What is the justificatory problem of induction? Doesn’t it stem from underdetermination? 
Underdetermination is only part of the story. Hume’s skeptical argument is roughly this: the fact that inductive inferences are underdetermined by evidence shows us that no deductive justification of the reliability of inductive inference will be forthcoming. But what’s the alternative? Induction!? If we say something like “when we’ve previously used induction, it’s shown itself to be more or less reliable”, we’re using induction. So if the reliability of induction is already an open question, we can’t use induction to defend its reliability. Otherwise, we reason in a circle — or “beg the question” —, taking for granted what is in question. Ditto for attempts to justify particular inductive arguments by adding premises about the “Uniformity of Nature”. This gambit is in even worse trouble than using induction to justify induction. For one, presumably, we’d need induction in order to show that nature is uniform, running into the same problem circularity problem. And for two, it looks pretty doubtful that we can put specific enough sense to the claim that nature is uniform in order to make it come out both true and useful. It’s either going to be true but too weak (e.g., “Nature is uniform for the most part, in certain respects”) or strong enough to be of some use, but false (e.g., “Nature is uniform in all respects relevant to induction”). 

So does Hume’s skeptical argument show that induction is unreliable?
No. For one, just because he gives us an argument whose conclusion is that inductive inference cannot be justified, doesn’t mean that we ought to believe that conclusion. We might try to show where the argument goes wrong. Or we might try to finesse the issue in a less direct way. But anyway, even if we did accept the conclusion that we cannot justify our use of inductive inference — that, as Lipton puts it, there is a “deep symmetry” between induction and other “counterinductive” principles — we shouldn’t confuse this with the claim that induction is unreliable. Lipton’s analogy to lying is revealing here: if you are wondering whether I am honest, there’s not a lot that I can say that should help you decide. But importantly, this fact — that I cannot effectively testify to my own honesty — does not show that I am not honest.

How does the descriptive problem fit into this picture?
Here’s one way of thinking about this story: we have general reasons for being skeptical about the possibility of justifying inductive inference’s reliability. We seem forced by underdetermination to use induction to justify itself, but this has us committing the fallacy of arguing in a circle. Subtle minds begin to ask whether we haven’t been asking the wrong questions. Does inductive inference need to be justified? What exactly is it that we’re looking for here? Compare deduction: what justifies a particular rule of deduction? It seems unlikely that we’ll be able to say anything here without using deductive inference. [Hey, maybe we could argue inductively that deduction is reliable — homework (replace a question of your choosing in an SWA): can this idea go anywhere?] So perhaps we should do with induction what we’ve done with deduction: simply articulate the rules very carefully and follow them.

However, it turns out that this is easier to say than to do. Not too surprising: it’s often harder to describe what we have a knack for doing (playing a musical instrument, shooting a basket, cooking the perfect omelet, whatever). Problems like the Ravens Paradox, the Tacking Problem, and Goodman’s New Riddle offer further challenges to particular descriptions of how we in fact reason inductively. Notice that these problems do not call into question the ways in which we reason (suggesting that we shouldn’t reason in those ways, say). Instead, they cast doubt on the accuracy of those descriptions. The justification of our practices needn’t enter into it at this point.

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