ABSTRACT
The above quotation was found in a daily bridge column by Phillip Alder (Calgary Sun, 25 February, 2001); the column was entitled “Don’t jump to conclusions”. (Alder says that Lichtenberg was an 18th century German physicist and philosopher.)
Every cow that I have seen has four legs, and so it would be easy for me to conclude that all cows have four legs. Such reasoning is called empirical inductionempirical evidence suggesting a pattern that holds in all cases. (Come to think of it, I have seen a variety of cow with no legs-it’s called “ground beef”.) Okay, perhaps a better example is that since the sun has risen every day this century, it will rise again tomorrow, and hence the expression “is as certain as the sun rising tomorrow.” Quoting Martin Gardner [214, p. 137], Charles Sanders Peirce once wrote “I like that phrase, for its great moderation because it is infinitely far from certain that the sun will rise tomorrow.” Gardner continues: “There is not a single truth of science, Peirce said, on which he would ‘bet more than about a million of millions to one.’ ”
Exercise 19. Give an example of a statement S(n) that is true only for n = 1 to n = 1, 000, 000, but fails at n = 1, 000, 001.
