Problem of induction
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The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge understood in the classic philosophical sense, highlighting the apparent lack of justification for:
- Generalizing about the properties of a class of objects based on some number of observations of particular instances of that class (e.g., the inference that "all swans we have seen are white, and, therefore, all swans are white", before the discovery of black swans) or
- Presupposing that a sequence of events in the future will occur as it always has in the past (e.g., that the laws of physics will hold as they have always been observed to hold). Hume called this the principle of uniformity of nature.
- From causes which appear similar we expect similar effects. This is the sum of all our experimental conclusions. Now it seems evident that, if this conclusion were formed by reason, it would be as perfect at first, and upon one instance, as after ever so long a course of experience. But the case is far otherwise. Nothing so like as eggs; yet no one, on account of this appearing similarity, expects the same taste and relish in all of them. It is only after a long course of uniform experiments in any kind, that we attain a firm reliance and security with regard to a particular event. Now where is that process of reasoning which, from one instance, draws a conclusion, so different from that which it infers from a hundred instances that are nowise different from that single one? This question I propose as much for the sake of information, as with an intention of raising difficulties. I cannot find, I cannot imagine any such reasoning.
- David Hume, An Enquiry Concerning Human Understanding, 4. Sceptical doubts concerning the operations of the understanding
- Domestic animals expect food when they see the person who usually feeds them. We know that all these rather crude expectations of uniformity are liable to be misleading. The man who has fed the chicken every day throughout its life at last wrings its neck instead, showing that more refined views as to the uniformity of nature would have been useful to the chicken.
- The riddle of induction can be put thus: What rational basis is there for any of our beliefs about the unobserved?
The theory of personal probability touches on the domain of the riddle and can even be construed as giving a partial answer. The theory prescribes, presumably compellingly, exactly how a set of beliefs should change in the light of what is observed. It can help you say, "My opinions today are the rational consequence of what they were yesterday and of what I have seen since yesterday." In principle, yesterday's opinions can be traced to the day before, but even given a coherent demigod able to trace his present opinions back to those with which he was born and to what he has experienced since, the theory of personal probability does not pretend to say with what system of opinions he ought to have been born. It leaves him, just as Hume would say, without rational foundation for his beliefs of today.
Can there be any such foundation? The theory as such is silent, but I am led by study of it to doubt that there is a rational basis for what we believe about the unobserved. In fact, Hume's arguments, and modern variants of them such as Goodman's discussion of 'bleen' and 'grue', appeal to me as correct and realistic. That all my beliefs are but my personal opinions, no matter how well some of them may coincide with opinions of others, seems to me not a paradox but a truism. The grandiose image of a demigod tracing his beliefs back to the cradle only to find an impasse there seems a valid metaphor. If there is rational basis for beliefs going beyond mere coherency, then there are some specific opinions that a rational baby demigod must have. Put that way, the notion of any such basis seems to me quite counterintuitive.
- L. J. Savage, "Implications of Personal Probability for Induction", The Journal of Philosophy, Vol. 64, No. 19 (Oct. 5, 1967)