Logic


THE PROBLEM OF INDUCTION

or: How we can learn from experience.

by Maarten Maartensz
 

    

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Introduction: The problem of induction I intend to solve in this essay is stated by Bertrand Russell in the following words (and I quote a few of many similar passages he penned):

"The great scandals in the philosophy of science ever since the time of Hume have been causality and induction. We all believe in both, but Hume made it appear that that our belief is a blind faith for which no rational ground can be assigned." (Bertrand Russell, Let the people think p. 50)
(Hume's) "almost unexampled combination of acuteness with intellectual honesty led him to certain devastating conclusions: that induction is a habit without logical justification, and that the belief in causation is little better than a superstition." (idem p.62-3)
"This state of affairs is profoundly unsatisfactory, and becomes more so as science becomes more entangled with philosophy. We must hope that an answer will be found; but I am quite unable to believe it has been found." (idem p.50)

These words were written between 1935 and 1941, and although much has been written on the problem of induction since then (Note 1), it remains unsolved, and in fact has received an even sharper form than before, since Nelson Goodman published his sharpened form of the problem in the 1940-ies, which we shall explain and consider in the body of the text.

There is also a certain widespread unclarity about what the problem of induction is, and it is useful to distinguish three different aspects of the problem of induction, namely:

a. The problem of how we learn from experience
b. The problem of the validity of generalisations
c. Goodman's sharpening of the problem of induction

The problem of induction is best construed as arising from the question "How do we learn from experience?", since this is clearly a very fundamental problem. The other two statements are more specific, and are in fact misleading.

Now we learn from experience - wholly apart from induction as under (b) or (c) - by setting up explanations for the facts in our experience we want to explain, and do this as follows:

  • We make assumptions to the effect that there is some kind of things beyond our experience - "the cause" - that logically entails the facts in our experience - "the effects" - we seek to explain, and

  • we test the assumptions we made by deriving further consequences from them, and verifying that these also hold in our experience.

  • If they do hold in experience, we have found an explanation for the facts we seek to explain, and may express this by saying we found the cause of these facts;

  • if they don't hold in experience we withdraw our assumptions, and try to find other assumptions that logically entail the facts we seek to explain, the logical consequences of which do hold in experience.

Something like this - called abduction, discussed in a little more detail at the end of this essay - does happen when we attempt to explain things in our experience, and the two problems Hume found with it (both causality and induction) stem from the fact that by introducing assumptions we go beyond the given evidence our experience affords: generally what we assume as cause is not fully or not at all given in experience, and neither are many of the predicted effects we deduce from these assumptions.

So let us see what in fact are these problems. Here is a preview of the 5 parts and 35 (or 37 counting the present file and the notes) sections:

0. Introduction to the problem of induction

1. The problem of induction

Section 1: Generalizing from the evidence
Section 2: Inductive generalizations are not deductively valid
Section 3: Inductive generalizations are question-begging
Section 4: The new problem of induction

Section 5: Logical form of the new problem of induction
Section 6: Logical relevance and logical necessity
Section 7: All inductive generalizations are suspect
Section 8: Summary of initial conclusions
Section 9: Summary of initial clues

2. The mathematics of probabilitiy

Section 10: Introduction to probability
Section 11: Basic theorems
Section 12: Basic conditional theorems
Section 13: Irrelevance

3. Probability, confirmation and induction

Section 14: Introduction to Learning from Experience
Section 15: Confirmation
Section 16: Undermining
Section 17: Competition
Section 18: Support
Section 19: Probability and induction

4. The interpretations of probability

Section 20: Introduction - Interpretations of probability
Section 21: Two mathematical facts
Section 22: Four interpretations of probability
Section 23. Remarks on logical probability
Section 24: Remarks on empirical probability
Section 25: Remarks on subjective probability
Section 26: Remarks on real probability
Section 27: Summary of the interpretations of probability

5. The solution of the problem of induction

Section 28: Introduction - the new problem of induction
Section 29: How to abstract from irrelevant conditions
Section 30: Algebraic interpretation of the Inductive Postulate
Section 31: How to attribute probabilities to theories
Section 32: Making further sense of the Inductive Postulate
Section 33: Some remarks on Newton and Hume
Section 34: Abduction and the problem of causality
Section 35: Theories and abduction, deduction and induction

Notes

)Note that the notes in the present version only moves one to the right file, but not anymore to the right section or note.)

 

 

    

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 Colofon: Written in 1980-83, lectured about in 1989, but not previously published. (Note 0)

Maartens@xs4all.nl