have asserted that probability is concerned with degree or quantity of belief. De Morgan says,1 “By degree of probability we really mean or ought to mean degree of belief.” The late Professor Donkin expressed the meaning of probability as “quantity of belief;” but I have never felt satisfied with such definitions of probability. The nature of belief is not more clear to my mind than the notion which it is used to define. But an all-sufficient objection is, that the theory does not measure what the belief is, but what it ought to be. Few minds think in close accordance with the theory, and there are many cases of evidence in which the belief existing is habitually different from what it ought to be. Even if the state of belief in any mind could be measured and expressed in figures, the results would be worthless. The value of the theory consists in correcting and guiding our belief, and rendering our states of mind and consequent actions harmonious with our knowledge of exterior conditions.
This objection has been clearly perceived by some of those who still used quantity of belief as a definition of probability. Thus De Morgan adds—“Belief is but another name for imperfect knowledge.” Donkin has well said that the quantity of belief is “always relative to a particular state of knowledge or ignorance; but it must be observed that it is absolute in the sense of not being relative to any individual mind; since, the same information being presupposed, all minds ought to distribute their belief in the same way.”2 Boole seemed to entertain a like view, when he described the theory as engaged with “the equal distribution of ignorance;”3 but we may just as well say that it is engaged with the equal distribution of knowledge.
I prefer to dispense altogether with this obscure word belief, and to say that the theory of probability deals with quantity of knowledge, an expression of which a precise explanation and measure can presently be given. An event is only probable when our knowledge of it is diluted with ignorance, and exact calculation is needed to discriminate how much we do and do not know. The theory has been described by some writers as professing to evolve knowledge out of ignorance; but as Donkin admirably remarked, it is really “a method of avoiding the erection of belief upon ignorance.” It defines rational expectation by measuring the comparative amounts of knowledge and ignorance, and teaches us to regulate our actions with regard to future events in a way which will, in the long run, lead to the least disappointment. It is, as Laplace happily said, good sense reduced to calculation. This theory appears to me the noblest creation of intellect, and it passes my conception how two such men as Auguste Comte and J. S. Mill could be found depreciating it and vainly questioning its validity. To eulogise the theory ought to be as needless as to eulogise reason itself.
Fundamental Principles of the Theory.
The calculation of probabilities is really founded, as I conceive, upon the principle of reasoning set forth in preceding chapters. We must treat equals equally, and what we know of one case may be affirmed of every case resembling it in the necessary circumstances. The theory consists in putting similar cases on a par, and distributing equally among them whatever knowledge we possess. Throw a penny into the air, and consider what we know with regard to its way of falling. We know that it will certainly fall upon a side, so that either head or tail will be uppermost; but as to whether it will be head or tail, our knowledge is equally divided. Whatever we know concerning head, we know also concerning tail, so that we have no reason for expecting one more than the other. The least predominance of belief to either side would be irrational; it would consist in treating unequally things of which our knowledge is equal.
The theory does not require, as some writers have erroneously supposed, that we should first ascertain by experiment the equal facility of the events we are considering. So far as we can examine and measure the