Groups of units are what we really treat in arithmetic. The number five is really 1 + 1 + 1 + 1 + 1, but for the sake of conciseness we substitute the more compact sign 5, or the name five. These names being arbitrarily imposed in any one manner, an infinite variety of relations spring up between them which are not in the least arbitrary. If we define four as 1 + 1 + 1 + 1, and five as 1 + 1 + 1 + 1 + 1, then of course it follows that five = four + 1; but it would be equally possible to take this latter equality as a definition, in which case one of the former equalities would become an inference. It is hardly requisite to decide how we define the names of numbers, provided we remember that out of the infinitely numerous relations of one number to others, some one relation expressed in an equality must be a definition of the number in question and the other relations immediately become necessary inferences.
In the science of number the variety of classes which can be formed is altogether infinite, and statements of perfect generality may be made subject only to difficulty or exception at the lower end of the scale. Every existing number for instance belongs to the class m + 7; that is, every number must be the sum of another number and seven, except of course the first six or seven numbers, negative quantities not being here taken into account. Every number is the half of some other, and so on. The subject of generalization, as exhibited in mathematical truths, is an infinitely wide one. In number we are only at the first step of an extensive series of generalizations. As number is general compared with the particular things numbered, so we have general symbols for numbers, and general symbols for relations between undetermined numbers. There is an unlimited hierarchy of successive generalizations.
Numerically Definite Reasoning.
It was first discovered by De Morgan that many arguments are valid which combine logical and numerical reasoning, although they cannot be included in the ancient logical formulas. He developed the doctrine of the “Numerically Definite Syllogism,” fully explained in his Formal Logic (pp. 141–170). Boole also devoted considerable attention to the determination of what he called “Statistical Conditions,” meaning the numerical conditions of logical classes. In a paper published among the Memoirs of the Manchester Literary and Philosophical Society, Third Series, vol. IV. p. 330 (Session 1869–70), I have pointed out that we can apply arithmetical calculation to the Logical Alphabet. Having given certain logical conditions and the numbers of objects in certain classes, we can either determine the numbers of objects in other classes governed by those conditions, or can show what further data are required to determine them. As an example of the kind of questions treated in numerical logic, and the mode of treatment, I give the following problem suggested by De Morgan, with my mode of representing its solution.
“For every man in the house there is a person who is aged; some of the men are not aged. It follows that some persons in the house are not men.”[92]
| Now let | A = person in house, |
| B = male, | |
| C = aged. |
By enclosing a logical symbol in brackets, let us denote the number of objects belonging to the class indicated by the symbol. Thus let
| (A) = | number of persons in house, |
| (AB) = | number of male persons in house, |
| (ABC) = | number of aged male persons in house, |
and so on. Now if we use w and w′ to denote unknown numbers, the conditions of the problem may be thus stated according to my interpretation of the words—
(AB) = (AC) - w, (1)