## Sunday, June 24, 2007

### Jottings on How We Teach Induction

It is a crazy peculiarity of the way we talk about inductive reasoning these days that we virtually never talk about induction. For instance, this is the sort of thing given as an example of induction:

Duck #1 has webbed feet.
Duck #2 has webbed feet.
Duck #3 has webbed feet.
Duck #4 has webbed feet.
&c.
Therefore all ducks have webbed feet.

But this is not, in fact, an inductive argument; it is an analogical argument, in which we extrapolate from all ducks we've considered to all ducks we haven't by taking the latter on the analogy of the former.

The reason for this confusion of inductive and analogical arguments, I think, comes in part from thinking entirely in terms of simple enumeration, and in part from thinking incorrectly that any sort of argument from particulars to universals is inductive. Both of these come, I think, from missing the fact that the key moves in genuinely inductive arguments are the circumscription and division of possibilities. For instance, if you wanted to do an induction by simple enumeration on the above model, you'd need to circumscribe the possible ducks. For instance, I could circumscribe it to the sixteen ducks in the pond. Now, the key distinguishing feature of simple enumeration is that it is a division to individuals, so in induction by simple enumeration, I would simply go through all the ducks in the pond and verify that, in fact, they all have webbed feet. And I would conclude, rightly, that all the ducks in the pond have webbed feet.

It is easy to see why induction by simple enumeration is so much less interesting than analogical inference from particular cases. It is much more important, of course, since we could hardly get along without it, but it doesn't get you anything surprising. Of course, other forms of induction can be a bit more robust, and this will be because they circumscribe and divide things differently. Things get more interesting if our division is into kinds rather than into particular individual instances. For instance, Aquinas has an inductive argument that everything that is moved is moved by another: circumscribing the field of possibilities to cases where something is moved, he divides that field into what is supposed to be an exhaustive division of these possibilities according to kinds of motion, and argues that for each kind of motion there is operative a version of the principle that everything that is moved is moved by another. The conclusion then follows. It does not follow 'deductively'; this is not a deductive argument:

Type A of Domain D has property p
Type B of Domain D has property p
Type C of Domain D has property p
Therefore all of Domain D has property p.

It's common enough to say that any argument is deductive if the conclusion adds no information to the premises; this is even at the most optimistic assessment confused, since in every non-trivial deductive argument the conclusion adds information to the premises, if only in the form of the link between subject and predicate, which was not in the premises. This is why we do not consider conjunctions of premises to say exactly the same thing as conclusions that can be derived from them. They are different constructions. What is meant, I suppose, is that the result(s) of the construction on the left-hand side (the premises) includes the result of the construction on the right-hand side (the conclusion); which is simply a truism, and one that tells us nothing about deduction.

Another way to put the matter is to say that deductions deliver their conclusions with absolute certainty whereas inductions do not. This is not, in fact, equivalent to the former claim, although it often is treated as if it were. If taken rigorously, this criterion would get us absurd results, since what would count as deductive would depend entirely on the formal system you are using. For instance, in Heyting's system, which lacks any rule for elimination of double negation, the conclusion of an inference making use of double-negation elimination does not follow with any certainty in the system. In classical systems, however, it does, since they allow us to move from not-not-p to p. The same inference would be deductive in one case and not deductive in the other; and of two people making the same inference in the same circumstances, one could be deducing and the other not simply on the basis of what background rules they were assuming.

Another common criterion is that induction reasons from particulars to universals and deduction from universals to particulars. If stated with the proper qualifications, this is no doubt true, but in unqualified form it is manifestly false. For instance, I can infer from the particular, "The pope said yesterday that all true Catholics dance a merry jig" to "All true Catholics dance a merry jig"; this is not an inductive inference, but an argument from authority. It is, however, an argument from a particular claim to a universal claim. In any case, this criterion would put inductive inferences of the sort I describe above clearly on the inductive side, not the deductive side.

An interesting side question arises, as to whether mathematical induction is a genuine form of induction. The common consensus, based as far as I can see on no good reason, is that it is not (although people who hold that induction is from particulars to universals and deduction the reverse can't, as far as I can see, hold this consistently). But, of course, if the view that I give above is true, or, indeed, anything remotely like it is true, it looks very much like it is a form of inductive inference. In a typical instance of mathematical induction, you have a class of cases related by a successor relation; and you prove something of the minimal case, and then prove that all the cases related to it by that successor relation would have to have the same property. Looks inductive to me.

My point here is not that there is a fundamental issue about the terms, deduction and induction. In fact, it seems that there are several senses of 'deduction' in which you can make any inference you please deductive, simply by making the right premises explicit; everything becomes a proper deduction or an enthymeme, i.e., an abbreviated deduction. Likewise, there are several senses of 'induction' in which you can make any inference you please inductive, simply by suppressing the right premises; inductions turn out to be nothing but enthymemes. Similarly, it is clear that many people mean by 'induction' nothing more than an analogical argument. But this, I would suggest, is not a useful way to consider the matter. There are perfectly good reasons for calling 'mathematical inductions' inductive; the fact that they don't look like analogical arguments arbitrarily labeled as 'inductive' is not one of them. The real difference between induction and deduction, which is to say, the most useful way to distinguish the two labels, is found, I would suggest, in the obvious differences between their abbreviations, i.e., examples and enthymemes. Regardless, you don't have to hold this to see that the way we talk about 'deduction' and 'induction' to undergraduates is simply muddled. (I haven't even scratched the surface. For instance, there's another way of talking about induction, in terms of probability, that, if used consistently, would imply that any result in probability theory is obtained by induction. Again, this makes the label virtually useless for classifying reasoning. Many, many other cases could be educed.)