§ 37. The above, which may be considered tolerably complete as a definition, might equally well have been given in the last chapter. It has been deferred however to the present place, in order to connect with it at once a proposition involving the conceptions introduced in this chapter; viz.
the state of our own minds, in reference to the amount of belief we entertain in contemplating any one of the events whose probability has just been described. Reasons were given against the opinion that our belief admitted of any exact apportionment like the numerical one just mentioned. Still, it was shown that a reasonable explanation could be given of such an expression as, ‘my belief is 1/10th of certainty’, though it was an explanation which pointed unmistakeably to a series of events, and ceased to be intelligible, or at any rate justifiable, when it was not viewed in such a relation to a series. In so far, then, as this explanation is adopted, we may say that our belief is in proportion to the above fraction. This referred to the purely intellectual part of belief which cannot be conceived to be separable, even in thought, from the things upon which it is exercised. With this intellectual part there are commonly associated various emotions. These we can to a certain extent separate, and, when separated, can measure with that degree of accuracy which is possible in the case of other emotions. They are moreover intelligible in reference to the individual events. They will be found to increase and diminish in accordance, to some extent, with the fraction which represents the scarcity of the event. The emotion of surprise does so with some degree of accuracy.
The above investigation describes, though in a very brief form, the amount of truth which appears to me to be contained in the assertion frequently made, that the fraction expressive of the probability represents also the fractional part of full certainty to which our belief of the individual event amounts. Any further analysis of the matter would seem to belong to Psychology rather than to Probability.
[1] In the ordinary signification of this term. As De Morgan uses it he makes Formal Logic include Probability, as one of its branches, as indicated in his title “Formal Logic, or the Calculus of Inference, necessary and probable.”
[2] Formal Logic. Preface, page v.
[3] An illustration of the points here insisted on has recently [1876] been given in a quarter where few would have expected it; I allude, as many readers will readily infer, to J. S. Mill's exceedingly interesting Essays on Theism. It is not within our province here to criticise any of their conclusions, but they have expressed in a very significant way the conviction entertained by him that beliefs which are not justified by evidence, and possibly may not be capable of justification (those for instance of immortality and the existence of the Deity), may nevertheless not only continue to exist in cultivated minds, but may also be profitably encouraged there, at any rate in the shape of hopes, for certain supposed advantages attendant on their retention, irrespective even of their truth.
[4] It is necessary to take an example in which the man is forced to act, or we should not be able to shew that he has any belief on the subject at all. He may declare that he neither knows nor cares anything about the matter, and that therefore there is nothing of the nature of belief to be extracted out of his mental condition. He very likely would take this ground if we asked him, as De Morgan does, with a slightly different reference (Formal Logic, p. 183), whether he considers that there are volcanoes on the unseen side of the moon larger than those on the side turned towards us; or, with Jevons (Principles of Science, Ed. II. p. 212) whether he considers that a Platythliptic Coefficient is positive. These do not therefore seem good instances to illustrate the position that we always entertain a certain degree of belief on every question which can be stated, and that utter inability to give a reason in favour of either alternative corresponds to half belief.
[5] Except indeed on the principles indicated further on in §§ 24, 25.
[6] For a fuller discussion of this, see the Chapter on Causation.