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8. Conclusion
Jevons's method of inverse probability essentially agrees with Sherlock Holmes's method. (1) It can be regarded as eliminative in the sense that it eliminates all but the most probable hypothesis (conditional on the given event). (2) It is reasoning backward, because we infer the probabilities of hypotheses given the event as data. Or, to put it Jevons's way, it is reasoning the probability of a cause, given the effect and the probabilistic relations between the effect and its possible causes. (3) And finally, it tells us what the balance of probability is. What Holmes means by the 'scientific use of the imagination' is to invent many probable hypotheses (given the initial information), and to select among them according to the balance of probabilities (after testing them by all data obtained by investigation).
Maybe some words of caution are in order here. Although I assert that probabilistic element is essential in Holmes's method, I do not mean Herschel-Mill's method or Whewell's are incompatible with it. Thus, it is perfectly all right, for instance, if Holmes applies Mill's eliminative induction, or Whewell's colligation, first, and then uses his reasoning backward, in terms of probabilities.
As a further confirmation of the previous conclusion, I can quote Holmes's conversation with Watson in chapter 10 of The Sign of Four ; there he speaks of probabilistic or statistical method, and even of an a priori probability of a hypothesis. This is crucial, because without good knowledge of statistical method, he cannot use such a word.
(Q13) "We have no right to take anything for granted," Holmes answered. "It is certainly ten to one that they go downstream, but we cannot be certain. From this point we can see the entrance of the yard, and they can hardly see us. ..... We must stay where we are. See how the folk swarm over yonder in the gaslight."
"They are coming from work in the yard."
"Dirty-looking rascals, but I suppose every one has some little immortal spark concealed about him. You would not think it, to look at them. There is no a priori probability about it. A strange enigma is man!"
(Q14) "Winwood Reade is good upon the subject," said Holmes. "He remarks that, while the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to. Individuals vary, but percentages remain constant. So says the statistician."
The first of these two quotations is one of the hardest passages from Holmes, but I think the point is this: If we assign an extreme value (either zero or one) to any hypothesis as its a priori probability, we cannot learn from our experience. So Holmes is saying in (Q13) that, even to such a hypothesis as supposing an immortal spark in a dirty-looking rascal, we should not assign a priori probability zero. This interpretation is indeed in the spirit of his first remark that "we have no right to take anything for granted."
The second, (Q14), states his agreement with statisticians' observation; and since this is now familiar to us, I don't have to explain it any further.
From all this, we can conclude that Sherlock Holmes knew well the new probabilistic theory of scientific reasoning, and applied it, as his own method, to his criminal investigations. Moreover, since the major advocates of this new theory were also an expert of symbolic logic, we may conclude, in all probability, that Holmes knew Boolean symbolic logic as well. On the total evidence, the balance of probability is that he was a very good logician.
Bibliography
De Morgan, Augustus. (1847) Formal Logic.
Doyle, Arthur Conan. The Penguin Complete Sherlock Holmes, Penguin Books, 1981.
Eco, Umberto and Thomas A Sebeok, eds. (1983) The Sign of Three: Dupin, Holmes, Peirce. Indiana University Press.
Herschel, J. F. W. (1830) A Preliminary Discourse on the Study of Natural Philosophy. Johnson Reprint, New York, 1966.
Herschel, J. F. W. (1850) "Quetlet on Probabilities." Edinburgh Review, 92.
Jevons, William Stanley. (1874) The Principles of Science, Macmillan.
Mill, J. S. (1843) A System of Logic.
Uchii, S. (1988) Sherlock Holmes's Theory of Reasoning (in Japanese), Kodansha.
Whewell, W. (1837) History of the Inductive Sciences, 3 vols.
Whewell, W. (1847) Philosophy of the Inductive Sciences, 2 vols., 2nd ed.
For the web resources, begin with 221B Baker Street and Sherlockian Homepage.
Postscript
The materials presented in this paper are a part of my book on Holmes written in Japanese (Uchii, 1988); my treatment of Holmes's reasoning in this book is more comprehensive.
In reproducing this paper on the web page, I wish to thank Prof. Jerry Massey (then the Director of the Center for Philosophy of Science) for his warm hospitality.
June 21, 1998; last modified, April 16, 2006. (c) Soshichi Uchii
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