Chance, Love, and Logic · Charles S. Peirce
IV
We have examined the problem proposed by the conceptualists, which, translated into clear language, is this: Given a synthetic conclusion; required to know out of all possible states of things how many will accord, to any assigned extent, with this conclusion; and we have found that it is only an absurd attempt to reduce synthetic to analytic reason, and that no definite solution is possible. But there is another problem in connection with this subject. It is this: Given a certain state of things, required to know what proportion of all synthetic inferences relating to it will be true within a given degree of approximation. Now, there is no difficulty about this problem (except for its mathematical complication); it has been much studied, and the answer is perfectly well known. And is not this, after all, what we want to know much rather than the other? Why should we want to know the probability that the fact will accord with our conclusion? That implies that we are interested in all possible worlds, and not merely the one in which we find ourselves placed. Why is it not much more to the purpose to know the probability that our conclusion will accord with the fact? One of these questions is the first above stated and the other the second, and I ask the reader whether, if people, instead of using the word probability without any clear apprehension of their own meaning, had always spoken of relative frequency, they could have failed to see that what they wanted was not to follow along the synthetic procedure with an analytic one, in order to find the probability of the conclusion; but, on the contrary, to begin with the fact at which the synthetic inference aims, and follow back to the facts it uses for premises in order to see the probability of their being such as will yield the truth.
The use of this may be illustrated by an example. By the census of 1870, it appears that the proportion of males among native white children under one year old was 0.5082, while among colored children of the same age the proportion was only 0.4977. The difference between these is 0.0105, or about one in a 100. Can this be attributed to chance, or would the difference always exist among a great number of white and colored children under like circumstances? Here _p_ may be taken at 1/2; hence 2_p_(1-_p_) is also 1/2. The number of white children counted was near 1,000,000; hence the fraction whose square-root is to be taken is about 1/2000000. The root is about 1/1400, and this multiplied by 0.477 gives about 0.0003 as the probable error in the ratio of males among the whites as obtained from the induction. The number of black children was about 150,000, which gives 0.0008 for the probable error. We see that the actual discrepancy is ten times the sum of these, and such a result would happen, according to our table, only once out of 10,000,000,000 censuses, in the long run. It may be remarked that when the real value of the probability sought inductively is either very large or very small, the reasoning is more secure. Thus, suppose there were in reality one white ball in 100 in a certain urn, and we were to judge of the number by 100 drawings. The probability of drawing no white ball would be 366/1000; that of drawing one white ball would be 370/1000; that of drawing two would be 185/1000; that of drawing three would be 61/1000; that of drawing four would be 15/1000; that of drawing five would be only 3/1000, etc. Thus we should be tolerably certain of not being in error by more than one ball in 100.
It appears, then, that in one sense we can, and in another we cannot, determine the probability of synthetic inference. When I reason in this way: Ninety-nine Cretans in a hundred are liars; But Epimenides is a Cretan; Therefore, Epimenides is a liar:— I know that reasoning similar to that would carry truth 99 times in 100. But when I reason in the opposite direction: Minos, Sarpedon, Rhadamanthus, Deucalion, and Epimenides, are all the Cretans I can think of; But these were all atrocious liars, Therefore, pretty much all Cretans must have been liars; I do not in the least know how often such reasoning would carry me right. On the other hand, what I do know is that some definite proportion of Cretans must have been liars, and that this proportion can be probably approximated to by an induction from five or six instances. Even in the worst case for the probability of such an inference, that in which about half the Cretans are liars, the ratio so obtained would probably not be in error by more than 1/6. So much I know; but, then, in the present case the inference is that pretty much all Cretans are liars, and whether there may not be a special improbability in that I do not know.
Text from Project Gutenberg, public domain in the United States.