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The Problem of Induction
by Merlin Jetton

Wikipedia says: "The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge. That is, what is the justification for either:

1. generalizing about the properties of a class of objects based on some number of observations of particular instances of that class (for example, the inference that "all swans we have seen are white, and therefore all swans are white," before the discovery of black swans) or
2. presupposing that a sequence of events in the future will occur as it always has in the past (for example, that the laws of physics will hold as they have always been observed to hold). Hume called this the Principle of Uniformity of Nature."

The Stanford Encyclopedia of Philosophy says it in a more symbolic way, but essentially similar. The Logical Leap asks: "When and why is the inference from "some" to "all" legitimate?"

Induction is an attempt to infer 'All S is P', given 'Some S is P.' How is that ever justified? It seems to me the justification must be at least one additional premise of the form 'Why each S is P'. As the Stanford Encyclopedia says -- substituting S and P for F and G -- enumerative induction is an inference from particular instances:  a1, a2, , an are all S that are also P 
to a general law or principle:   All S are P. 

Note that if all the particular known instances is a "closed" set, then an inference about one instance is not an inductive one. It is a deductive one. For example, if I say "if I draw a coin from my pocket, its date will be after 1963" after verifying that all coins in my pocket are dated after 1963, that is a deductive inference.

Enumerative induction is tantamount to a mockery of thoughtful induction. It often succeeds inferring to the next instance in mundane matters. However, in effect and by omission, enumerative induction says 'Why each S is P' is not important. Also, the formal statement from the Stanford Encylopedia says nothing about the variety of instances a1, a2, , an. The particular instances could all be precisely identical, but that by itself carries little weight to conclude a generalization therefrom will be true for all instances of that kind. I will not elaborate on the specifics of "identical" and "kind", but the gist of my point should be clear.

Consider the generalization 'All swans are white'. (Incidentally, the black-necked swan is both black and white (link).) It was merely an enumerative induction, based simply on all observed swans being white. There was no additional justification why they must be white, i.e. of the form 'Why each S is P'. Not only that, there was room for doubting that all swans are white. Many species of birds come in different colors. Even the closest relatives of swans, geese, come in different color patterns.

Let's move on to 'All swans have necks'. While somebody might believe it is a mere enumerative induction, I think not. The swan's neck connects its head and its body. It has within a channel between its mouth and lungs for breathing, a channel between its mouth and stomach for ingesting food, and part of its central nervous system. Fish don't have necks, but is that relevant? Swans and fish have quite different DNA and modes of existence. Incidentally, when John Stuart Mill commented on 'all swans are white' he contrasted it with 'all men's heads grow above their shoulders, and never below'.

I could proceed to many more scientific examples, but will give only one. In all instances water boils at 100C (212 F) at standard pressure, i.e. at sea level. I'm confident somebody else could give a better explanation than me. Regardless, there is a good reason why having to do with molecular motion and heat that justifies this induction.

Narratives of people arriving at various inductive generalizations, such as those in The Logical Leap and many in the writing of William Whewell, show a great variety of premises. There is variety in the instances used for a particular generalization. In many cases there are a great variety of facts which all point to the same generalization. There is also a great deal of variety between the grounds for different generalizations.

Some have tried to provide one more comprehensive premise -- answering to 'Why each S is P' -- to justify induction via a syllogism. That additional premise is the Principle of Uniformity of Nature. Especially notable was Richard Whately (1787-1863). There is a presentation about this here titled Whence the Uniformity Principle. (Note: 'ppsx' is a PowerPoint file extension introduced with Microsoft Office 2007. If you have an older version of PowerPoint, there is a downloadable viewer here.)  John Stuart Mill devoted several pages to this principle in A System of Logic. (They are mostly in Book III, Chapter III -- On The Ground of Induction. The book is fully viewable at Google Books.) "It would yet be a great error to offer this large generalization as any explanation of the inductive process. On the contrary, I hold it to be itself an instance of induction, and induction by no means of the most obvious kind. Far from being the first induction we make, it is one of the last, or at all events one of those which are latest in attaining strict philosophical accuracy."

The problems with the Uniformity of Nature premise are several. The principle itself is an induction, as Mill said. It is very vague, and nature shows numerous non-uniformities, i.e. differences. "The course of nature, in truth, is not only uniform, it is also infinitely various. Some phenomena are always seen to recur in the very same combinations in which we met with them at first; others seem altogether capricious; while some, which we had been accustomed to regard as bound down exclusively to a particular set of combinations, we unexpectedly find detached from some of the elements with which we had hitherto found them conjoined, and united to others of quite a contrary description" (Mill, III, III, 2). Very specific additional premises are needed.

Critics of induction often claim that all induction is merely enumerative, and that while all induction is invalid (or not logical), sound deductions (true premises and valid inference) are guaranteed true. Sometimes the claim that induction is invalid (or not logical) boils down to: it is invalid (or not logical) because it isn't deduction.  Anyway, deductive premises of the form "All S is P" and 'No S is P' -- S being "open-ended" -- don't fall like manna from heaven. So the critics use sleight-of-hand, using such premises and ignoring that they were established inductively. Also, because some inductive generalizations have been falsified does not imply all inductive generalizations are false. To claim that is self-refuting. Last, but far from least, the critic is obviously guilty of making many inductive generalizations while labeling the all inductive generalization others make as invalid. In other words, it is a kind of hypocrisy.

Francis Bacon wrote: "The induction which proceeds by simple enumeration is puerile, leads to uncertain conclusions, and is exposed to danger from one contradictory instance, deciding generally from too small a number of facts, and those only the most obvious" (Novum Organum, I 105). To summarize, the problem of induction is when is 'Why each S is P' sufficiently strong? Pretty clearly no uniform strength test can be given for every kind of endeavor. Mill's Methods are for arriving at generalizations and are not strength tests. Sometimes strength comes with small numbers and fails to come with large numbers. "Why is a single instance, in some cases, sufficient for a complete induction, while in others, myriads of concurring instances, without a single exception known or presumed, go such a very little way towards establishing an universal proposition? Whoever can answer this question knows more of the philosophy of logic than the wisest of the ancients, and has solved the great problem of induction" (Mill, III, III, 2).
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