Appendix B. Jaynes and 'common sense'

Jaynes and 'common sense'
In his 1955 paper, How Does the Brain Do Plausible Reasoning?, the physicist E.T. Jaynes justified Laplace's principle of insufficient reason by the use of a symbolic representation of the logic behind conditional probabilities.

(AB|C) = (A|BC)(B|C) = (B|AC)(A|C)
[(A+B) |C)] = (A|C) + (B|C) - (AB|C)

In current notation, we have

[C --> (A & B)]  <--> {[(B & C)-->A] & (C --> B)}

<--> {[(A & C)-->B] & (C-->A)}

[C v (A v B)] <--> [(C-->A) v (C -->B)] and ~(C --> (A and B)]

This logic, Jaynes thought, establishes the "common sense" behind Laplacian probability theory, which he found to be correct though incomplete.

Yet, Jaynes insisted, there is a proposition X that stands for all one's past experience. X, which cannot have a rationally based numerical value, undergirds Laplacian probability and science in general, he argued. X means that there is no possibility of an absolute or correct probability for any particular case. "All probabilities are conditional on X," which varies from person to person and for one person over time.Those who object to Laplace's principle tend to leave out X. Take an example, writes Jaynes, by George Polya:

A boy is ten years old today. According to Laplace's law of succession, he has a probability of 11/12 of living one more year. His grandfather is 70. According to the same law, he has the probability 71/72 of living one more year. Obviously the result contradicts common sense. Laplace's law of succession, however, applies only to the case where we have absolutely no prior information about the problem. In this example, it is obvious that we do have a great deal of prior information relevant to this question, which our common sense used, but we do not allow Laplace's theory to use. Laplace's theory gives the result of consistent plausible reasoning on the basis of the information which was put into it.

We note that X has changed considerably since Jaynes's mid-1950s paper and since Polya's  mid-1940s book How to Prove It (Princeton Vol. II). With advances in medicine, life American life expectancies have changed. According to the Social Security Administration calculator, statistically speaking, the boy has 72.3 more years ahead of him and the grandfather has 15.1 more years. But when Polya was writing "common sense" said that Granddad would be lucky to live through the current year.  Had these men been living in 19th Century America, "common sense" would have said that a child of ten had a fairly high chance of not surviving another year.