Snowdon; therefore Ben Nevis is higher than the Wrekin. But a little consideration discloses much reason for believing that even in such cases, where equality does not apparently enter, the force of the reasoning entirely depends upon underlying and implied equalities.

In the first place, two statements of mere difference do not give any ground of inference. We learn nothing concerning the comparative heights of St. Paul's and Westminster Abbey from the assertions that they both differ in height from St. Peter's at Rome. Thus we need something more than mere inequality; we require one identity in addition, namely the identity in direction of the two differences. Thus we cannot employ inequalities in the simple way in which we do equalities, and, when we try to express exactly what other conditions are requisite, we shall find ourselves lapsing into the use of equalities or identities.

In the second place, every argument by inequalities may be represented with at least equal clearness and force in the form of equalities. Thus we clearly express that a is greater than b by the equation

where Р is an intrinsically positive quantity, denoting the difference of a and b. Similarly we express that bis greater than c by the equation

and substituting for b in (1) its value in (2) we have

a = c + q + p.

(3)

Now as p and q are both positive, it follows that a is greater than c, and we have the exact amount of excess specified. It will be easily seen that the reasoning concerning that which is less than a less will result in an equation of the form

c = a -q - p.

Every argument by inequalities may then be thrown

into the form of an equality; but the converse is not true. We cannot possibly prove that two quantities are equal by merely asserting that they are both greater or both less than another quantity. From e>ƒ and g>f, or e<ƒ and g <f, we can infer no relation between e and g. And if the reader take the equations xy=3 and attempt to prove that therefore x = 3, by throwing them into inequalities, he will find it impossible to do so.

From these considerations I gather that reasoning in arithmetic or algebra by so-called inequalities is only an imperfectly expressed reasoning by equalities, and when we want to exhibit exactly and clearly the conditions of reasoning, we are obliged to use equalities explicitly. Just as in pure logic a negative proposition, as expressing mere difference, cannot be the means of inference, so inequality can never really be the true ground of inference. I do not deny that affirmation and negation, agreement and difference, equality and inequality, are pairs of equally fundamental relations, but I assert that inference is possible only where affirmation, agreement, or equality, some species of identity in fact, is present, explicitly or implicitly.

Arithmetical Reasoning.

It might seem somewhat inconsistent that I assert number to arise out of difference or discrimination, and yet hold that no reasoning can be grounded on difference. Number, of course, opens a most wide sphere for inference, and a little consideration shows that this is due to the unlimited series of identities which spring up out of numerical abstraction. If six people are sitting on six chairs, there is no resemblance between the chairs and the people in logical character. But if we overlook all the qualities both of a chair and a person, and merely remember that there are marks by which each of six chairs

may be discriminated from each other, and similarly with the people, then there arises a resemblance between the chairs and people, and this resemblance in number may be the ground of inference. If on another occasion the chairs are filled by people again, we may infer that these people must resemble the others in number, though they need not resemble them in any other points.

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 indefinite variety of relations spring up between them which are not in the least arbitrary. If we define four as I + 1 + 1 + 1, and five as I + I + I + I + I, 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 arithmetical or mathematical truths, is an indefinitely wide one. In

number we are only at the first step of an extensive series of generalizations. A number is general as compared with the particular things numbered, so we may have general symbols for numbers, or general symbols not for numbers, but for the relations between undetermined numbers. There is, in fact, an unlimited hierarchy of successive generalizations.

Numerically Definite Reasoning.

It was first discovered by Prof. de Morgan that many arguments are valid which combine logical and numerical reasoning, although they could in no way 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). Dr. 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 Abecedarium. Having given certain logical conditions and the numbers of objects in certain classes, we can either determine the number 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 f.

f It has been pointed out to me by Mr. A. J. Ellis, F.R.S., that my solution, as given in the Memoirs of the Manchester Philosophical Society, does not exactly answer to the conditions of the problem, and I therefore substitute above a more satisfactory 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 meng.'

By enclosing logical symbols in brackets, let us denote the number of objects belonging to the class indicated by the symbol. Thus let

(A)

(AB) (ABC)

=

=

=

number of persons in-house,
number of male persons in house,

number of aged male persons in house, and so on. Now if we use w and w' to denote unknown and indefinite numbers, the conditions of the problem may be thus stated according to my interpretation of the

words

(AB) = (AC) — w,

(1)

that is to say, the number of persons in the house who are aged is at least equal to, and may exceed, the number of male persons in the house;

that is to say, the number of male persons in the house who are not aged is some unknown positive quantity. If we develop the terms in (1) by the Law of Duality (pp. 87, 95, 97), we obtain

(ABC) + (ABC) = (ABC) + (AbC) — w.

Subtracting the common term (ABC) from each side and substituting for (ABc) its value as given in (2), we get at

once

(AbC) = w + w,

and adding (Abc) to each side, we have

(Ab) = Abc + w + w'.

The meaning of this result is that the number of persons in the house who are not men is at least equal to w+w',

g Syllabus of a proposed System of Logic,' p. 29.