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5. Numerical Evaluation of Case Probability
Case probability is not open to any kind of numerical evaluation. What is commonly considered as such exhibits, when more closely scrutinized, a different character.
On the eve of the 1944 presidential election people could have said:
(a) I am ready to bet three dollars against one that Roosevelt will be elected.
(b) I guess that out of the total amount of electors 45 millions will exercise their franchise, 25 millions of whom will vote for Roosevelt.
(c) I estimate Roosevelt's chances as 9 to 1.
Statement (d) is obviously inexact. If asked under oath on the witness stand whether he is as certain about Roosevelt's future victory as about the fact that a block of ice will melt when exposed to a temperature of 150 degrees, our man would have answered no. He would have rectified his statement and would have declared: I am personally fully convinced that Roosevelt will carry on. That is my opinion. But, of course, this is not certainty, only the way I understand the conditions involved.
The case of statement (a) is similar. This man believed that he risked very little when laying such a wager. The relation 3:1 is the outcome of the interplay of two factors: the opinion that Roosevelt will be elected and the man's propensity for betting.
Statement (b) is an evaluation of the outcome of the impending event. Its figures refer not to a greater or smaller degree of probability, but to the expected result of the voting. Such a statement may be based on a systematic investigation like the Gallup poll or simply on estimates.
It is different with statement (c). This is a proposition about the expected outcome couched in arithmetical terms. It certainly does not mean that out of ten cases of the same type nine are favorable for Roosevelt and one unfavorable. It cannot have any reference to class probability. But what else can it mean?
It is a metaphorical expression. Most of the metaphors used in daily speech imaginatively identify an abstract object with another object that can be apprehended directly by the senses. Yet this is not a necessary feature of metaphorical language, but merely a consequence of the fact that the concrete is as a rule more familiar to us than the abstract. As metaphors aim at an explanation of something which is less well known by comparing it with something better known, they consist for the most part in identifying something abstract with a better-known concrete. The specific mark of our case is that it is an attempt to elucidate a complicated state of affairs by resorting to an analogy borrowed from a branch of higher mathematics, the calculus of probability. As it happens, this mathematical discipline is more popular than the analysis of the epistemological nature of understanding.
There is no use in applying the yardstick of logic to a critique of metaphorical language. Analogies and metaphors are always defective and logically unsatisfactory. It is usual to search for the underlying tertium comparationis. But even this is not permissible with regard to the metaphor we are dealing with. For the comparison is based on a conception which is in itself faulty in the very frame of [p. 115] the calculus of probability, namely the gambler's fallacy. In asserting that Roosevelt's chances are 9:1, the idea is that Roosevelt is in regard to the impending election in the position of a man who owns 90 per cent of all tickets of a lottery in regard to the first prize. It is implied that this ratio 9:1 tells us something substantial about the outcome of the unique case in which we are interested. There is no need to repeat that this is a mistaken idea.
No less impermissible is the recourse to the calculus of probability in dealing with hypotheses in the field of the natural sciences. Hypotheses are tentative explanations consciously based on logically insufficient arguments. With regard to them all that can be asserted is: The hypothesis does or does not contradict either logical principles or the facts as experimentally established and considered as true. In the first case it is untenable, in the second case it is--under the present state of our experimental knowledge--not untenable. (The intensity of personal conviction is purely subjective.) Neither frequency probability nor historical understanding enters into the matter.
The term hypothesis, applied to definite modes of understanding historical events, is a misnomer. If a historian asserts that in the fall of the Romanoff dynasty the fact that this house was of German background played a relevant role, he does not advance a hypothesis. The facts on which his understanding is founded are beyond question. There was a widespread animosity against Germans in Russia, and the ruling line of the Romanoffs, having for 200 years intermarried exclusively with scions of families of German descent, was viewed by many Russians as a germanized family, even by those who assumed that Tsar Paul was not the son of Peter III. But the question remains what the relevance of these facts was in the chain of events which brought about the dethronement of this dynasty. Such problems are not open to any elucidation other than that provided by understanding.