reading notes of《Probability Theory - The Logic of Science》

1.Plausible reasoning

• Suppose some dark night a policeman walks down a street, apparently deserted. Suddenly he hears a burglar alarm, looks across the street, and sees a jewelry store with a broken window. Then a gentleman wearing a mask comes crawling out through the broken window, carrying a bag which turns out to be full of expensive jewelry. The policeman doesn’t hesitate at all in deciding that this gentleman is dishonest. But by what reasoning process does he arrive at this conclusion? Let us first take a leisurely look at the general nature of such problems.

1.1.Deductive and plausible reasoning

• The evidence did not make the gentleman’s dishonesty certain, but it did make it extremely plausible.

• As is generally credited to the Organon of Aristotle (fourth century bc) deductive reasoning (apodeixis) can be analyzed ultimately into the repeated application of two strong syllogisms:

  • if A is true, then B is true - $\frac{Aistrue}{therefore,Bistrue}$
  • if A is true, then B is true - $\frac{Bisfalse}{therefore,Aisfalse}$

• In almost all situations confronting us we do not have the right kind of information to allow this kind of reasoning. We fall back on weaker syllogisms (epagoge):

  • if A is true, then B is true - $\frac{Bistrue}{therefore,Abecomesmoreplausible}$
  • if A is true, then B is true - $\frac{Aisfalse}{therefore,Bbecomeslessplausible}$

• The reasoning of our policeman was not even of the above types. It is best described by a still weaker syllogism: if A is true, then B becomes more plausible - $\frac{Bistrue}{therefore,Abecomesmoreplausible}$. But in spite of the apparent weakness of this argument, the policeman’s conclusion has a very strong convincing power.

• This show that the brain, in doing plausible reasoning, not only decides whether something becomes more plausible or less plausible, but that it evaluates the degree of plausibility in some way. And the brain also makes use of old information (prior information) as well as the specific new data of the problem.

• The same idea is expressed in a remark of S. Banach (quoted by S. Ulam, 1957): Good mathematicians see analogies between theorems; great mathematicians see analogies between analogies.

1.2.Analogies with physical theories

• In physics, we learn quickly that the world is too complicated for us to analyze it all at once. We can make progress only if we dissect it into little pieces and study them separately. Sometimes, we can invent a mathematical model which reproduces several features of one of these pieces (prior information), and whenever this happens we feel that progress has been made. These models are called physical theories. As knowledge advances (specific new data of the problem), we are able to invent better and better models (more accurate conclusions), which reproduce more and more features of the real world, more and more accurately.
• The analogy with physical theories is deeper than a mere analogy of method. Often, the things which are most familiar to us turn out to be the hardest to understand.