This concept of sampling and accompanying application of the laws of probability finds extensive use in polls, public opinion polls. Polls to determine what radio or television programs are being watched and listened to, polls to determine house wives’ reaction to a new product, political polls, and the like. In most cases the sampling is carefully planned and often a margin of error is stated. Polls cannot, however, altogether eliminate the fact that certain people dislike being questioned and may deliberately conceal or give false information. In spite of this and other objections, the method of sampling often makes results available in situations where the cost of complete enumeration would be prohibitive both from the standpoint of time and of money.
Thus we can see that probability and statistics are used in insurance, physics, genetics, biology, business, as well as in games of chance, and we are inclined to agree with P.S. LaPlace who said: We see . . . that the theory of probabilities is at bottom only common sense reduced to calculation, it makes us appreciate with exactitude what reasonable minds feel by a sort of instinct, often being able to account for it . . . it is remarkable that [this] science, which originated in the consideration of games of chance, should have become the most important object of human knowledge.
It seems, that the most taken of are the paradoxes in the foundations of set theory as discovered by Russell in 1901. Some classes have themselves as members: The class of all abstract objects, for example, is an abstract object, whereby, others do not: The class of donkeys is not itself a donkey. Now consider the class of all classes that are, not members for themselves, are that, to concede that this class a member of itself, that, if it is, then it is not, and if it is not, then it is.
The paradox is structurally similar to easier examples, such as the paradox of the barber, such that of a village having a barber in it, who shaves all and only the people who do not have in themselves. Who shaves the barber? If he shaves himself, then he does not, but if he does not shave himself, then he does not. The paradox is actually just a proof that there is no such barber or in other words, that the condition is inconsistent. All the same, it is no too easy to say why there is no such class as the one Russell defines. It seems that there must be some restriction on the kind of definition that is allowed to define classes and the difficulty that of finding a well-motivated principle behind any such restriction.