CHAPTER VIII.

PRINCIPLES OF NUMBER.

Not without reason did Pythagoras represent the world as ruled by number. Into almost all our acts of thought number enters, and in proportion as we can define numerically we enjoy exact and useful knowledge of the Universe. The science of numbers, too, has hitherto presented the widest and most practicable training in logic. So free and energetic has been the study of mathematical forms, compared with the forms of logic, that mathematicians have passed far in advance of pure logicians. Occasionally, in recent times, they have condescended to apply their algebraic instrument to a reflex treatment of the primary logical science. It is thus that we owe to profound mathematicians, such as John Herschel, Whewell, De Morgan, or Boole, the regeneration of logic in the present century. I entertain no doubt that it is in maintaining a close alliance with quantitative reasoning that we must look for further progress in our comprehension of qualitative inference.

I cannot assent, indeed, to the common notion that certainty begins and ends with numerical determination. Nothing is more certain than logical truth. The laws of identity and difference are the tests of all that is certain throughout the range of thought, and mathematical reasoning is cogent only when it conforms to these conditions, of which logic is the first development. And if it be erroneous to suppose that all certainty is mathematical, it is equally an error to imagine that all which is mathematical is certain. Many processes of mathematical reasoning are of most doubtful validity. There are points of mathematical doctrine which must long remain matter of opinion; for instance, the best form of the definition and axiom concerning parallel lines, or the true nature of a limit. In the use of symbolic reasoning questions occur on which the best mathematicians may differ, as Bernoulli and Leibnitz differed irreconcileably concerning the existence of the logarithms of negative quantities.[87] In fact we no sooner leave the simple logical conditions of number, than we find ourselves involved in a mazy and mysterious science of symbols.

[87] Montucla. *Histoire des Mathématiques*, vol. iii. p. 373.

Mathematical science enjoys no monopoly, and not even a supremacy, in certainty of results. It is the boundless extent and variety of quantitative questions that delights the mathematical student. When simple logic can give but a bare answer Yes or No, the algebraist raises a score of subtle questions, and brings out a crowd of curious results. The flower and the fruit, all that is attractive and delightful, fall to the share of the mathematician, who too often despises the plain but necessary stem from which all has arisen. In no region of thought can a reasoner cast himself free from the prior conditions of logical correctness. The mathematician is only strong and true as long as he is logical, and if number rules the world, it is logic which rules number.

Nearly all writers have hitherto been strangely content to look upon numerical reasoning as something apart from logical inference. A long divorce has existed between quality and quantity, and it has not been uncommon to treat them as contrasted in nature and restricted to independent branches of thought. For my own part, I believe that all the sciences meet somewhere. No part of knowledge can stand wholly disconnected from other parts of the universe of thought; it is incredible, above all, that the two great branches of abstract science, interlacing and co-operating in every discourse, should rest upon totally distinct foundations. I assume that a connection exists, and care only to inquire, What is its nature? Does the science of quantity rest upon that of quality; or, *vice versâ*, does the science of quality rest upon that of quantity? There might conceivably be a third view, that they both rest upon some still deeper set of principles.

It is generally supposed that Boole adopted the second view, and treated logic as an application of algebra, a special case of analytical reasoning which admits only two quantities, unity and zero. It is not easy to ascertain clearly which of these views really was accepted by Boole. In his interesting biographical sketch of Boole,[88] the Rev. R. Harley protests against the statement that Boole’s logical calculus imported the conditions of number and quantity into logic. He says: “Logic is never identified or confounded with mathematics; the two systems of thought are kept perfectly distinct, each being subject to its own laws and conditions. The symbols are the same for both systems, but they have not the same interpretation.” The Rev. J. Venn, again, in his review of Boole’s logical system,[89] holds that Boole’s processes are at bottom logical, not mathematical, though stated in a highly generalized form and with a mathematical dress. But it is quite likely that readers of Boole should be misled. Not only have his logical works an entirely mathematical appearance, but I find on p. 12 of his *Laws of Thought* the following unequivocal statement: “That logic, as a science, is susceptible of very wide applications is admitted; but it is equally certain that its ultimate forms and processes are mathematical.” A few lines below he adds, “It is not of the essence of mathematics to be conversant with the ideas of number and quantity.”

[88] *British Quarterly Review*, No. lxxxvii, July 1866.

[89] *Mind*, October 1876, vol. i. p. 484.

The solution of the difficulty is that Boole used the term mathematics in a wider sense than that usually attributed to it. He probably adopted the third view, so that his mathematical *Laws of Thought* are the common basis both of logic and of quantitative mathematics. But I do not care to pursue the subject because I think that, in either case Boole was wrong. In my opinion logic is the superior science, the general basis of mathematics as well as of all other sciences. Number is but logical discrimination, and algebra a highly developed logic. Thus it is easy to understand the deep analogy which Boole pointed out between the forms of algebraic and logical deduction. Logic resembles algebra as the mould resembles that which is cast in it. Boole mistook the cast for the mould. Considering that logic imposes its own laws upon every branch of mathematical science, it is no wonder that we constantly meet with the traces of logical laws in mathematical processes.

*The Nature of Number.*

Number is but another name for *diversity*. Exact identity is unity, and with difference arises plurality. An abstract notion, as was pointed out (p. 28), possesses a certain *oneness*. The quality of *justice*, for instance, is one and the same in whatever just acts it is manifested. In justice itself there are no marks of difference by which to discriminate justice from justice. But one just act can be discriminated from another just act by circumstances of time and place, and we can count many acts thus discriminated each from each. In like manner pure gold is simply pure gold, and is so far one and the same throughout. But besides its intrinsic qualities, gold occupies space and must have shape and size. Portions of gold are always mutually exclusive and capable of discrimination, in respect that they must be each without the other. Hence they may be numbered.

Plurality arises when and only when we detect difference. For instance, in counting a number of gold coins I must count each coin once, and not more than once. Let C denote a coin, and the mark above it the order of counting. Then I must count the coins

C′ + C″ + C‴ + C″″ + ....

If I were to count them as follows

C′ + C″ + C‴ + C‴ + C″″ + ...,

I should make the third coin into two, and should imply the existence of difference where there is no difference.[90] C‴ and C‴ are but the names of one coin named twice over. But according to one of the conditions of logical symbols, which I have called the Law of Unity (p. 72), the same name repeated has no effect, and

A ꖌ A = A.

[90] *Pure Logic*, Appendix, p. 82, § 192.

We must apply the Law of Unity, and must reduce all identical alternatives before we can count with certainty and use the processes of numerical calculation. Identical alternatives are harmless in logic, but are wholly inadmissible in number. Thus logical science ascertains the nature of the mathematical unit, and the definition may be given in these terms--*A unit is any object of thought which can be discriminated from every other object treated as a unit in the same problem.*

It has often been said that units are units in respect of being perfectly similar to each other; but though they may be perfectly similar in some respects, they must be different in at least one point, otherwise they would be incapable of plurality. If three coins were so similar that they occupied the same space at the same time, they would not be three coins, but one coin. It is a property of space that every point is discriminable from every other point, and in time every moment is necessarily distinct from any other moment before or after. Hence we frequently count in space or time, and Locke, with some other philosophers, has held that number arises from repetition in time. Beats of a pendulum may be so perfectly similar that we can discover no difference except that one beat is before and another after. Time alone is here the ground of difference and is a sufficient foundation for the discrimination of plurality; but it is by no means the only foundation. Three coins are three coins, whether we count them successively or regard them all simultaneously. In many cases neither time nor space is the ground of difference, but pure quality alone enters. We can discriminate the weight, inertia, and hardness of gold as three qualities, though none of these is before nor after the other, neither in space nor time. Every means of discrimination may be a source of plurality.

Our logical notation may be used to express the rise of number. The symbol A stands for one thing or one class, and in itself must be regarded as a unit, because no difference is specified. But the combinations AB and A*b* are necessarily *two*, because they cannot logically coalesce, and there is a mark B which distinguishes one from the other. A logical definition of the number *four* is given in the combinations ABC, AB*c*, A*b*C, A*bc*, where there is a double difference. As Puck says--

“Yet but three? Come one more; Two of both kinds makes up four.”

I conceive that all numbers might be represented as arising out of the combinations of the Logical Alphabet, more or less of each series being struck out by various logical conditions. The number three, for instance, arises from the condition that A must be either B or C, so that the combinations are ABC, AB*c*, A*b*C.

*Of Numerical Abstraction.*

There will now be little difficulty in forming a clear notion of the nature of numerical abstraction. It consists in abstracting the character of the difference from which plurality arises, retaining merely the fact. When I speak of *three men* I need not at once specify the marks by which each may be known from each. Those marks must exist if they are really three men and not one and the same, and in speaking of them as many I imply the existence of the requisite differences. Abstract number, then, is *the empty form of difference*; the abstract number *three* asserts the existence of marks without specifying their kind.

Numerical abstraction is thus seen to be a different process from logical abstraction (p. 27), for in the latter process we drop out of notice the very existence of difference and plurality. In forming the abstract notion *hardness*, we ignore entirely the diverse circumstances in which the quality may appear. It is the concrete notion *three hard objects*, which asserts the existence of hardness along with sufficient other undefined qualities, to mark out *three* such objects. Numerical thought is indeed closely interwoven with logical thought. We cannot use a concrete term in the plural, as *men*, without implying that there are marks of difference. But when we use an abstract term, we deal with unity.

The origin of the great generality of number is now apparent. Three sounds differ from three colours, or three riders from three horses; but they agree in respect of the variety of marks by which they can be discriminated. The symbols 1 + 1 + 1 are thus the empty marks asserting the existence of discrimination. But in dropping out of sight the character of the differences we give rise to new agreements on which mathematical reasoning is founded. Numerical abstraction is so far from being incompatible with logical abstraction that it is the origin of our widest acts of generalization.

*Concrete and Abstract Number.*

The common distinction between concrete and abstract number can now be easily stated. In proportion as we specify the logical characters of the things numbered, we render them concrete. In the abstract number three there is no statement of the points in which the *three* objects agree; but in *three coins*, *three men*, or *three horses*, not only are the objects numbered but their nature is restricted. Concrete number thus implies the same consciousness of difference as abstract number, but it is mingled with a groundwork of similarity expressed in the logical terms. There is identity so far as logical terms enter; difference so far as the terms are merely numerical.

The reason of the important Law of Homogeneity will now be apparent. This law asserts that in every arithmetical calculation the logical nature of the things numbered must remain unaltered. The specified logical agreement of the things must not be affected by the unspecified numerical differences. A calculation would be palpably absurd which, after commencing with length, gave a result in hours. It is equally absurd, in a purely arithmetical point of view, to deduce areas from the calculation of lengths, masses from the combination of volume and density, or momenta from mass and velocity. It must remain for subsequent consideration to decide in what sense we may truly say that two linear feet multiplied by two linear feet give four superficial feet; arithmetically it is absurd, because there is a change of unit.

As a general rule we treat in each calculation only objects of one nature. We do not, and cannot properly add, in the same sum yards of cloth and pounds of sugar. We cannot even conceive the result of adding area to velocity, or length to density, or weight to value. The units added must have a basis of homogeneity, or must be reducible to some common denominator. Nevertheless it is possible, and in fact common, to treat in one complex calculation the most heterogeneous quantities, on the condition that each kind of object is kept distinct, and treated numerically only in conjunction with its own kind. Different units, so far as their logical differences are specified, must never be substituted one for the other. Chemists continually use equations which assert the equivalence of groups of atoms. Ordinary fermentation is represented by the formula

C^{6} H^{12} O^{6} = 2C^{2} H^{6} O + 2CO^{2}.

Three kinds of units, the atoms respectively of carbon, hydrogen, and oxygen, are here intermingled, but there is really a separate equation in regard to each kind. Mathematicians also employ compound equations of the same kind; for in, *a* + *b* √ - 1 = *c* + *d* √ - 1, it is impossible by ordinary addition to add *a* to *b* √ - 1. Hence we really have the separate equations *a* = *b*, and *c* √ - 1 = *d* √ - 1. Similarly an equation between two quaternions is equivalent to four equations between ordinary quantities, whence indeed the name *quaternion*.

*Analogy of Logical and Numerical Terms.*

If my assertion is correct that number arises out of logical conditions, we ought to find number obeying all the laws of logic. It is almost superfluous to point out that this is the case with the fundamental laws of identity and difference, and it only remains to show that mathematical symbols do really obey the special conditions of logical symbols which were formerly pointed out (p. 32). Thus the Law of Commutativeness, is equally true of quality and quantity. As in logic we have

AB = BA,

so in mathematics it is familiarly known that

2 × 3 = 3 × 2, or *x* × *y* = *y* × *x*.

The properties of space are as indifferent in multiplication as we found them in pure logical thought.

Similarly, as in logic

triangle or square = square or triangle,

or generally A ꖌ B = B ꖌ A, so in quantity 2  +  3  =  3  +  2, or generally *x* + *y* = *y* + *x*.

The symbol ꖌ is not identical with +, but it is thus far analogous.

How far, now, is it true that mathematical symbols obey the Law of Simplicity expressed in the form

AA = A,

or the example

Round round = round?

Apparently there are but two numbers which obey this law; for it is certain that

*x* × *x* = *x*

is true only in the two cases when *x* = 1, or *x* = 0.

In reality all numbers obey the law, for 2 × 2 = 2 is not really analogous to AA = A. According to the definition of a unit already given, each unit is discriminated from each other in the same problem, so that in 2′ × 2″, the first *two* involves a different discrimination from the second *two*. I get four kinds of things, for instance, if I first discriminate “heavy and light” and then “cubical and spherical,” for we now have the following classes--

heavy, cubical. light, cubical. heavy, spherical. light, spherical.

But suppose that my two classes are in both cases discriminated by the same difference of light and heavy, then we have

heavy heavy = heavy, heavy light = 0, light heavy = 0, light light = light.

Thus, (heavy or light) × (heavy or light) = (heavy or light).

In short, *twice two is two* unless we take care that the second two has a different meaning from the first. But under similar circumstances logical terms give the like result, and it is not true that A′A″ = A′, when A″ is different in meaning from A′.

In a similar manner it may be shown that the Law of Unity

A ꖌ A = A.

holds true alike of logical and mathematical terms. It is absurd indeed to say that

*x* + *x* = *x*

except in the one case when *x* = absolute zero. But this contradiction *x* + *x* = *x* arises from the fact that we have already defined the units in one x as differing from those in the other. Under such circumstances the Law of Unity does not apply. For if in

A′ ꖌ A″ = A′

we mean that A″ is in any way different from A′ the assertion of identity is evidently false.

The contrast then which seems to exist between logical and mathematical symbols is only apparent. It is because the Laws of Simplicity and Unity must always be observed in the operation of counting that those laws seem no further to apply. This is the understood condition under which we use all numerical symbols. Whenever I write the symbol 5 I really mean

1 + 1 + 1 + 1 + 1,

and it is perfectly understood that each of these units is distinct from each other. If requisite I might mark them thus

1′+ 1″ + 1‴ + 1″″ + 1″‴.

Were this not the case and were the units really

1′ + 1″ + 1″ + 1‴ + 1″″,

the Law of Unity would, as before remarked, apply, and

1″ + 1″ = 1″.

Mathematical symbols then obey all the laws of logical symbols, but two of these laws seem to be inapplicable simply because they are presupposed in the definition of the mathematical unit. Logic thus lays down the conditions of number, and the science of arithmetic developed as it is into all the wondrous branches of mathematical calculus is but an outgrowth of logical discrimination.

*Principle of Mathematical Inference.*

The universal principle of all reasoning, as I have asserted, is that which allows us to substitute like for like. I have now to point out how in the mathematical sciences this principle is involved in each step of reasoning. It is in these sciences indeed that we meet with the clearest cases of substitution, and it is the simplicity with which the principle can be applied which probably led to the comparatively early perfection of the sciences of geometry and arithmetic. Euclid, and the Greek mathematicians from the first, recognised *equality* as the fundamental relation of quantitative thought, but Aristotle rejected the exactly analogous, but far more general relation of identity, and thus crippled the formal science of logic as it has descended to the present day.

Geometrical reasoning starts from the axiom that “things equal to the same thing are equal to each other.” Two equalities enable us to infer a third equality; and this is true not only of lines and angles, but of areas, volumes, numbers, intervals of time, forces, velocities, degrees of intensity, or, in short, anything which is capable of being equal or unequal. Two stars equally bright with the same star must be equally bright with each other, and two forces equally intense with a third force are equally intense with each other. It is remarkable that Euclid has not explicitly stated two other axioms, the truth of which is necessarily implied. The second axiom should be that “Two things of which one is equal and the other unequal to a third common thing, are unequal to each other.” An equality and inequality, in short, give an inequality, and this is equally true with the first axiom of all kinds of quantity. If Venus, for instance, agrees with Mars in density, but Mars differs from Jupiter, then Venus differs from Jupiter. A third axiom must exist to the effect that “Things unequal to the same thing may or may not be equal to each other.” *Two inequalities give no ground of inference whatever.* If we only know, for instance, that Mercury and Jupiter differ in density from Mars, we cannot say whether or not they agree between themselves. As a fact they do not agree; but Venus and Mars on the other hand both differ from Jupiter and yet closely agree with each other. The force of the axioms can be most clearly illustrated by drawing equal and unequal lines.[91]

[91] *Elementary Lessons in Logic* (Macmillan), p. 123. It is pointed out in the preface to this Second Edition, that the views here given were partially stated by Leibnitz.

The general conclusion then must be that where there is equality there may be inference, but where there is not equality there cannot be inference. A plain induction will lead us to believe that *equality is the condition of inference concerning quantity*. All the three axioms may in fact be summed up in one, to the effect, that “*in whatever relation one quantity stands to another, it stands in the same relation to the equal of that other*.”

The active power is always the substitution of equals, and it is an accident that in a pair of equalities we can make the substitution in two ways. From *a* = *b* = *c* we can infer *a* = *c*, either by substituting in *a* = *b* the value of *b* as given in *b* = *c*, or else by substituting in *b* = *c* the value of *b* as given in *a* = *b*. In *a* = *b* ~ *d* we can make but the one substitution of *a* for *b*. In *e* ~ *f* ~ *g* we can make no substitution and get no inference.

In mathematics the relations in which terms may stand to each other are far more varied than in pure logic, yet our principle of substitution always holds true. We may say in the most general manner that *In whatever relation one quantity stands to another, it stands in the same relation to the equal of that other.* In this axiom we sum up a number of axioms which have been stated in more or less detail by algebraists. Thus, “If equal quantities be added to equal quantities, the sums will be equal.” To explain this, let

*a* = *b*, *c* = *d*.

Now *a* + *c*, whatever it means, must be identical with itself, so that

*a* + *c* = *a* + *c*.

In one side of this equation substitute for the quantities their equivalents, and we have the axiom proved

*a* + *c* = *b* + *d*.

The similar axiom concerning subtraction is equally evident, for whatever *a* - *c* may mean it is equal to *a* - *c*, and therefore by substitution to *b* - *d*. Again, “if equal quantities be multiplied by the same or equal quantities, the products will be equal,” For evidently

*ac* = *ac*,

and if for *c* in one side we substitute its equal *d*, we have

*ac* = *ad*,

and a second similar substitution gives us

*ac* = *bd*.

We might prove a like axiom concerning division in an exactly similar manner. I might even extend the list of axioms and say that “Equal powers of equal numbers are equal.” For certainly, whatever *a* × *a* × *a* may mean, it is equal to *a* × *a* × *a*; hence by our usual substitution it is equal to *b* × *b* × *b*. The same will be true of roots of numbers and ^{c}√*a* = ^{d}√*b* provided that the roots are so taken that the root of *a* shall really be related to *a* as the root of *b* is to *b*. The ambiguity of meaning of an operation thus fails in any way to shake the universality of the principle. We may go further and assert that, not only the above common relations, but all other known or conceivable mathematical relations obey the same principle. Let Q*a* denote in the most general manner that we do something with the quantity *a*; then if *a* = *b* it follows that

Q*a* = Q*b*.

The reader will also remember that one of the most frequent operations in mathematical reasoning is to substitute for a quantity its equal, as known either by assumed, natural, or self-evident conditions. Whenever a quantity appears twice over in a problem, we may apply what we learn of its relations in one place to its relations in the other. All reasoning in mathematics, as in other branches of science, thus involves the principle of treating equals equally, or similars similarly. In whatever way we employ quantitative reasoning in the remaining parts of this work, we never can desert the simple principle on which we first set out.

*Reasoning by Inequalities.*

I have stated that all the processes of mathematical reasoning may be deduced from the principle of substitution. Exceptions to this assertion may seem to exist in the use of inequalities. The greater of a greater is undoubtedly a greater, and what is less than a less is certainly less. Snowdon is higher than the Wrekin, and Ben Nevis than Snowdon; therefore Ben Nevis is higher than the Wrekin. But a little consideration discloses sufficient 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. We need something more than 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 what other conditions are requisite, we find ourselves lapsing into the use of equalities or identities.

In the second place, every argument by inequalities may be represented in the form of equalities. We express that *a* is greater than *b* by the equation

*a* = *b* + *p*, (1)

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

*b* = *c* + *q*, (2)

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* - *r* - *s*.

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* > *f* and *g* > *f*, or *e* < *f* and *g* < *f*, we can infer no relation between *e* and *g*. And if the reader take the equations *x* = *y* = 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 may 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 the others, and similarly with the people, then there arises a resemblance between the chairs and the 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 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 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]

[92] *Syllabus of a Proposed System of Logic*, p. 29.

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)

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;

(AB*c*) = *w*′, (2)

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. 74, 81, 89), we obtain

(ABC) + (AB*c*) = (ABC) + (A*b*C) - *w*.

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

(A*b*C) = *w* + *w*′,

and adding (A*bc*) to each side, we have

(A*b*) = (A*bc*) + *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*′, and exceeds it by the number of persons in the house who are neither men nor aged (A*bc*).

It should be understood that this solution applies only to the terms of the example quoted above, and not to the general problem for which De Morgan intended it to serve as an illustration.

As a second instance, let us take the following question:--The whole number of voters in a borough is *a*; the number against whom objections have been lodged by liberals is *b*; and the number against whom objections have been lodged by conservatives is *c*; required the number, if any, who have been objected to on both sides. Taking

A = voter, B = objected to by liberals, C = objected to by conservatives,

then we require the value of (ABC). Now the following equation is identically true--

(ABC) = (AB) + (AC) + (A*bc*) - (A). (1)

For if we develop all the terms on the second side we obtain

(ABC) = (ABC) + (AB*c*) + (ABC) + (A*b*C) + (A*bc*) - (ABC) - (AB*c*) - (A*b*C) - (A*bc*);

and striking out the corresponding positive and negative terms, we have left only (ABC) = (ABC). Since then (1) is necessarily true, we have only to insert the known values, and we have

(ABC) = *b* + *c* - *a* + (A*bc*).

Hence the number who have received objections from both sides is equal to the excess, if any, of the whole number of objections over the number of voters together with the number of voters who have received no objection (A*bc*).

The following problem illustrates the expression for the common part of any three classes:--The number of paupers who are blind males, is equal to the excess, if any, of the sum of the whole number of blind persons, added to the whole number of male persons, added to the number of those who being paupers are neither blind nor males, above the sum of the whole number of paupers added to the number of those who, not being paupers, are blind, and to the number of those who, not being paupers, are male.

The reader is requested to prove the truth of the above statement, (1) by his own unaided common sense; (2) by the Aristotelian Logic; (3) by the method of numerical logic just expounded; and then to decide which method is most satisfactory.

*Numerical meaning of Logical Conditions.*

In many cases classes of objects may exist under special logical conditions, and we must consider how these conditions can be interpreted numerically. Every logical proposition gives rise to a corresponding numerical equation. Sameness of qualities occasions sameness of numbers. Hence if

A = B

denotes the identity of the qualities of A and B, we may conclude that

(A) = (B).

It is evident that exactly those objects, and those objects only, which are comprehended under A must be comprehended under B. It follows that wherever we can draw an equation of qualities, we can draw a similar equation of numbers. Thus, from

A = B = C

we infer

A = C;

and similarly from

(A) = (B) = (C),

meaning that the numbers of A’s and C’s are equal to the number of B’s, we can infer

(A) = (C).

But, curiously enough, this does not apply to negative propositions and inequalities. For if

A = B ~ D

means that A is identical with B, which differs from D, it does not follow that

(A) = (B) ~ (D).

Two classes of objects may differ in qualities, and yet they may agree in number. This point strongly confirms me in the opinion which I have already expressed, that all inference really depends upon equations, not differences.

The Logical Alphabet thus enables us to make a complete analysis of any numerical problem, and though the symbolical statement may sometimes seem prolix, I conceive that it really represents the course which the mind must follow in solving the question. Although thought may outstrip the rapidity with which the symbols can be written down, yet the mind does not really follow a different course from that indicated by the symbols. For a fuller explanation of this natural system of Numerically Definite Reasoning, with more abundant illustrations and an analysis of De Morgan’s Numerically Definite Syllogism, I must refer the reader to the paper[93] in the Memoirs of the Manchester Literary and Philosophical Society, already mentioned, portions of which, however, have been embodied in the present section.

[93] It has been pointed out to me by Mr. C. J. Monroe, that section 14 (p. 339) of this paper is erroneous, and ought to be cancelled. The problem concerning the number of paupers illustrates the answer which should have been obtained. Mr. A. J. Ellis, F.R.S., had previously observed that my solution in the paper of De Morgan’s problem about “men in the house” did not answer the conditions intended by De Morgan, and I therefore give in the text a more satisfactory solution.

The reader may be referred, also, to Boole’s writings upon the subject in the *Laws of Thought*, chap. xix. p. 295, and in a paper on “Propositions Numerically Definite,” communicated by De Morgan, in 1868, to the Cambridge Philosophical Society, and printed in their *Transactions*, vol. xi. part ii.