§ 21. So far then we do not seem to have made the slightest approximation to a solution of the particular
question now under examination. The more closely we have analysed special examples, the more unmistakeably are we brought to the conclusion that in the individual instance no justification of anything like quantitative belief is to be found; at least none is to be found in the same sense in which we expect it in ordinary scientific conclusions, whether Inductive or Deductive. And yet we have to face and account for the fact that common impressions, as attested by a whole vocabulary of common phrases, are in favour of the existence of this quantitative belief. How are we to account for this? If we appeal to an example again, and analyse it somewhat more closely, we may yet find our way to some satisfactory explanation.
In our previous analysis (§ 18) we found it sufficient to stop at an early stage, and to give as the justification of our belief the fact of the proposition being true. Stopping however at that stage, we have found this explanation fail altogether to give a justification of partial belief; fail, that is, when applied to the individual instance. The two states of belief and disbelief correspond admirably to the two results of the event happening and not happening respectively, and unless for psychological purposes we saw no reason to analyse further; but to partial belief there is nothing corresponding in the result, for the event cannot partially happen in such cases as we are concerned with. Suppose then we advance a step further in the analysis, and ask again what is meant by the proposition being true? This introduces us, of course, to a very long and intricate path; but in the short distance along it which we shall advance, we shall not, it is to be hoped, find any very serious difficulty. As before, we will illustrate the analysis by first applying it to the case of ordinary full belief.
§ 22. Whatever opinion then may be held about the essential nature of belief, it will probably be admitted that a
readiness to act upon the proposition believed is an inseparable accompaniment of that state of mind. There can be no alteration in our belief (at any rate in the case of sane persons) without a possible alteration in our conduct, nor anything in our conduct which is not connected with something in our belief. We will first take an example in connection with the penny, in which there is full belief; we will analyse it a step further than we did before, and then attempt to apply the same analysis to an example of a similar kind, but one in which the belief is partial instead of full.
Suppose that I am about to throw a penny up, and contemplate the prospect of its falling upon one of its sides and not upon its edge. We feel perfectly confident that it will do so. Now whatever else may be implied in our belief, we certainly mean this; that we are ready to stake our conduct upon its falling thus. All our betting, and everything else that we do, is carried on upon this supposition. Any risk whatever that might ensue upon its falling otherwise will be incurred without fear. This, it must be observed, is equally the case whether we are speaking of a single throw or of a long succession of throws.
But now let us take the case of a penny falling, not upon one side or the other, but upon a given side, head. To a certain extent this example resembles the last. We are perfectly ready to stake our conduct upon what comes to pass in the long run. When we are considering the result of a large number of throws, we are ready to act upon the supposition that head comes every other time. If e.g.[ TN: space] we are betting upon it, we shall not object to paying £1 every time that head comes, on condition of receiving £1 every time that head does not come. This is nothing else than the translation, as we may call it, into practice, of our belief that head and tail occur equally often.