Sometimes, however, the number of possible alternatives is overrated. Thus, visitors to London often remark that they never go there without meeting somebody from their own locality, and they are surprised at this as if they had the same chance of meeting their fellow-visitors and any other of the four millions of the metropolis. But really the possible alternatives of rencounter are far less numerous. The places frequented by visitors to London are filled by much more limited numbers: the possible rencounters are to be counted by thousands rather than by millions.
[Footnote 1:] See De Morgan's Essay on Probabilities, c. vi., "On Common Notions of Probability".
Chapter IX.
PROBABLE INFERENCE TO PARTICULARS—THE MEASUREMENT OF PROBABILITY.
Undoubtedly there are degrees of probability. Not only do we expect some events with more confidence than others: we may do so, and our confidence may be misplaced: but we have reason to expect some with more confidence than others. There are different degrees of rational expectation. Can those degrees be measured numerically?
The question has come into Logic from the mathematicians. The calculation of Probabilities is a branch of Mathematics. We have seen how it may be applied to guide investigation by eliminating what is due to chance, and it has been vaguely conceived by logicians that what is called the calculus of probabilities might be found useful also in determining by exact numerical measurement the probability of single events. Dr. Venn, who has written a separate treatise on the Logic of Chance, mentions "accurate quantitative apportionment of our belief" as one of the goals which Logic should strive to attain. The following passage will show his drift.[1]
A man in good health would doubtless like to know whether he will be alive this time next year. The fact will be settled one way or the other in due time, if he can afford to wait, but if he wants a present decision, Statistics and the Theory of Probability can alone give him any information. He learns that the odds are, say five to one that he will survive, and this is an answer to his question as far as any answer can be given. Statisticians are gradually accumulating a vast mass of data of this general character. What they may be said to aim at is to place us in the position of being able to say, in any given time or place, what are the odds for or against any at present indeterminable fact which belongs to a class admitting of statistical treatment.
Again, outside the regions of statistics proper—which deal, broadly speaking, with events which can be numbered or measured, and which occur with some frequency—there is still a large field as to which some better approach to a reasoned intensity of belief can be acquired. What will be the issue of a coming war? Which party will win in the next election? Will a patient in the crisis of a given disease recover or not? That statistics are lying here in the background, and are thus indirectly efficient in producing and graduating our belief, I fully hold; but there is such a large intermediate process of estimating, and such scope for the exercise of a practised judgment, that no direct appeal to statistics in the common sense can directly help us. In sketching out therefore the claims of an Ideal condition of knowledge, we ought clearly to include a due apportionment of belief to every event of such a class as this. It is an obvious defect that one man should regard as almost certain what another man regards as almost impossible. Short, therefore, of certain prevision of the future, we want complete agreement as to the degree of probability of every future event: and for that matter of every past event as well.
Technically speaking, if we extend the name Modality ([see p. 78]) to any qualification of the certainty of a statement of belief, what Dr. Venn here desiderates, as he has himself suggested, is a more exact measurement of the Modality of propositions. We speak of things as being certain, possible, impossible, probable, extremely probable, faintly probable, and so forth: taking certainty as the highest degree of probability[2] shading gradually down to the zero of the impossible, can we obtain an exact numerical measure for the gradations of assurance?
To examine the principles of all the cases in which chances for and against an occurrence have been calculated from real or hypothetical data, would be to trespass into the province of Mathematics, but a few simple cases will serve to show what it is that the calculus attempts to measure, and what is the practical value of the measurement as applied to the probability of a single event.