ENTER NORTON
John Norton (2005) argues that there are no valid universal rules of inductive inference at all. He claims that particular inductive inferences are warranted by and only by empirical facts, and not at all by being shown to satisfy some formal schema or template in the manner of formal deductive inferences.
I think it is safe to say that he has in his sights Newton's methodological rules and Mill's ideas about induction, among many others. Indeed, I think that Norton would equally reject Whewell's rules pertaining to consilience and coherence, though he would probably agree with a number of Whewell's charges against Newton and Mill. So while Norton is a friend of particular inductions, he is also a foe of any general inductive principles.Again I want to take up the challenge to defend Newton and Mill. To begin with, what is the difference between formal rules of the type Norton has in mind in deductive logic and rules of the sort Newton and Mill propose? Are the latter formal in any sense? They can certainly be given what I would regard as a reasonably formal expression. For example, one of Mill's definitions of an inductive inference might be written like this: an inference from a sentence of the form “all observed As are Bs” to one of the form “all As are Bs.” However, as I have already emphasized, what is not the case, and what Mill explicitly denies, is that any argument representing an inference of this form that has true premises will have a conclusion that is true or probably true. Mill makes it clear that there are numerous arguments of this form that are really bad.
Similarly, one might express Newton's causal rule 2 as an inference from “e1 and e2 are effects of the same kind” to “e1 and e2 have the same cause.” However, what is not the case, as Newton recognizes, is that any argument of this form with a true premise will have a conclusion that is true or probably true.
Just think of simple cases, such as uniform motion in a straight line, which can be caused by a pair of forces or by the absence of a force.Are there any analogies in deductive logic? Consider a type of argument form in deduction called the syllogism. One standard way to define a syllogism is as a deductive argument with two premises and a conclusion, each of which has one of the four categorical forms, A, E, I, and O; the argument contains three terms, a different pair of which occurs in each of the three propositions. Put this way, a syllogism may be valid or invalid, as can readily be determined by a Venn diagram. In other words, there are good and bad syllogisms, all having a formal structure satisfying the formal rule defining a syllogism. Of course, you could define a syllogism as a valid argument satisfying the formal definition, in which case there wouldn't be any invalid syllogisms. But then you could do the same for “inductive generalization,” defining it as a valid inference of the form “all observed As are Bs, so all As are Bs.” Then there would be no invalid inductive generalizations. No one, certainly not Mill, wants to do that.
So I'll just keep the original definition I gave for the syllogism, and point out that for our purposes the important difference between this case and the inductive and causal ones is that whether a syllogism is valid can be determined entirely by a priori formal means. Whether an inductive generalization, or a causal inference, is valid or reasonable cannot be. In view of this, how should we understand inductive rules of the sort proposed by Newton and Mill? One suggestion is to say that while rules of deduction are exceptionless, rules of induction are just rules of thumb. They have numerous exceptions, but at the outset at least, when we can't think of any exceptions at the moment, they are generally to be followed. (This is Paul Feyerabend's one concession to rationality.) If so, we might write Mill's idea of inductive generalization as: if all the As you have observed have been Bs, then, pending exceptions, it is reasonable at the outset at least to assume that all As are Bs.
Similarly, we might think of Newton's causal rule 2 as something like this: if effects are of the same kind, then, pending exceptions, it is reasonable at the outset at least to assume they have the same cause.[35] This would make the rules in question pretty weak and rather vague—features of rules that I think both Newton and Mill would prefer to avoid.I want to suggest a very different way to regard the rules of Newton and Mill. Let me begin with Mill. When Mill characterizes induction as “the process by which we conclude that what is true of certain individuals of a class is true of the whole class,” what is he doing? For one thing, as I have said, he is giving a definition of “induction” or “inductive generalization.” And as he himself emphasizes, the definition permits both good and bad inductions. For another, as I have also said, he is claiming that whether the induction is good or bad depends on empirical considerations involving the size and variation of the sample, as well as on other information available. But he is doing something else as well. For Mill, one of the main aims of science is to “discover and prove” causal laws that enable one to explain and predict phenomena. Such discovery and proof requires making inductions to causal generalizations, which is what laws are for him. Accordingly, in giving what Norton calls a “formal inductive template” what Mill is doing, among other things, is identifying and characterizing an important type of scientific reasoning. He is not justifying any particular instance of reasoning of that form by saying that it has that form—any more than in deductive logic one justifies a particular syllogism by saying that it has the form of a syllogism.
Similarly, when Newton introduces his four methodological rules, he is at least implicitly claiming that causal reasoning of the sort expressed in rules 1 and 2 and inductive reasoning of the sort expressed in rules 3 and 4 are crucial in establishing propositions in empirical science.
To be sure, it would have been possible for Newton the scientist to argue for his law of gravity without explicitly invoking any of his four methodological rules. By invoking them, however, one of the things I take Newton to be doing is attempting to give us a better understanding of, as we might put it, the “logic” of his argument. And that, indeed, might help us to understand the argument and convince us that it is a good one, that its conclusion does in fact follow from the Newtonian “phenomena” cited, and how it does, which indeed was one of Newton's primary concerns.Look, e.g., at Newton's initial discussion of proposition 5 of book 3.[36] In this discussion he claims that the revolutions of the moons of Jupiter about Jupiter, of the moons of Saturn about Saturn, and of Mercury and Venus and the other known planets are phenomena of the same kind as the revolution of our moon about the earth and therefore (by rule 2) depend on causes of the same kind, especially since it has been proved that the forces on which those revolutions depend are directed toward the centers of Jupiter, Saturn, and the sun, and decrease according to the same ratio and law (in receding from Jupiter, Saturn, and the sun) as the force of gravity (in receding from the earth). Rule 2, to which Newton alludes, is his second rule of reasoning, according to which “the causes assigned to natural effects of the same kind must be, so far as possible, the same.” Newton is here characterizing what he is doing as arguing from the same kind of effects to the same unique cause. John Norton is right in saying that the empirical force of the argument comes when Newton argues that these are the same type of effects, namely, that these are all Keplerian motions. Newton is saying that in view of this he can infer that they have causes of the same kind, especially since he has already proved in book 1 that such motions must be governed by central inverse square forces. He is also saying that he is simply arguing from effects of the same kind to causes of the same kind, in accordance with his second rule, which he thinks will help to explain what he is doing.
If you want to attack Newton the physicist, argue against his empirical idea that these motions are all Keplerian or that such motions are produced by central inverse-square forces. If you want to attack Newton the methodologist, you need to show that he wasn't making an inference from effects to causes, or that pointing this out in the way that he does is of little if any value.If you think in the latter way, possibly a loose analogy with musicology might help you. The classical sonata has a rather definite formal structure that music theorists teach. It usually has three movements (an allegro, slow movement, and closing presto). In the first movement there is the introductory section, the exposition, the development, the recapitulation, and the coda. And so forth. Simply following this form will not make the resulting sonata good or bad. And, I suppose, it is possible to play a Mozart sonata well without knowing its structure. However, the musicologist helps us to understand what is going on in the sonata by explicitly invoking the formal structure and applying it to the particular sonata. So, perhaps, Newton is helping us to see the formal structure of his argument by appeal to his rules the way the musicologist is helping us to understand the Mozart sonata.
4.