Front Page Titles (by Subject) CHAPTER III.: IMPERFECT AND SIMPLE QUANTITATIVE REASONING. - The Principles of Psychology
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CHAPTER III.: IMPERFECT AND SIMPLE QUANTITATIVE REASONING. - Herbert Spencer, The Principles of Psychology 
The Principles of Psychology (London: Longman, Brown, Green and Longmans, 1855).
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IMPERFECT AND SIMPLE QUANTITATIVE REASONING.
§ 24. Ability to perceive equality implies a correlative ability to perceive inequality: neither can exist without the other. But though inseparable in origin, the cognitions of equality and inequality, whether between things or relations, altogether differ in this; that whilst the one is essentially definite, the other is essentially indefinite. There is but one equality; but there may be numberless degrees of inequality. To assert an inequality, involves the affirmation of no fact, but merely the denial of a fact; and hence, as positing nothing specific, the cognition of inequality can never be a premiss to any specific conclusion.
Thus it happens that reasoning which is perfectly quantitative in its results, proceeds wholly by the establishment of equality between relations, the members of which are either equal, or one a known multiple of the other: and that, conversely, if any of the magnitudes standing in immediate relation are neither directly equal, nor the one equal to so many times the other; or if any of the successive relations which the reasoning establishes are unequal; the results are imperfectly quantitative. This truth is illustrated in that class of geometrical theorems in which it is asserted of some thing that it is greater or less than some other; that it falls within or without some other; and the like. Let us take as an example the proposition—“Any two sides of a triangle are together greater than the third side.”
“Let A B C be a triangle; any two sides of it are, together, greater than the third side; namely, B A, A C, greater than B C; and A B, B C, greater than A C; and B C, C A, greater than A B.”
“Produce B A to D, and make A D equal to A C; and join D C.”
It will be observed, that throughout this demonstration, though the magnitudes dealt with are unequal, yet the relations successively established are always equal to certain other relations: though the primary relations (between things) are those of inequality, yet the secondary relations (between relations) are those of equality. And this holds in the majority of imperfectly quantitative arguments. Though, as we shall by and by see, there are cases in which both the magnitudes and the relations are unequal, yet they are comparatively rare; and are incapable of any but the simplest forms.
§ 25. Another species of imperfectly quantitative reasoning occupies a position in mathematical analysis, like that which the foregoing species does in mathematical synthesis. The ordinary algebraic inequation supplies us with a sample of it.
Thus, if it is known that is less than the argument instituted is as follows:—
Now, in this case, as in the case of equations, the reasoning proceeds by steps, of which each asserts the equality of the new relation to the relation previously established: with this difference, that instead of the successive relations being relations of equality, they are relations of inferiority. That the general process of thought, however, is alike in both, will be obvious on considering that as the inferiority of x to y can be known only by deduction from the inferiority of to and as it can be so known only by the intermediation of other relations of inferiority; the possibility of the argument depends upon the successive relations being recognized as severally equal. It is true that these successive relations need not be specifically equal; but they must be equal in so far as they are defined. In the above case, for example, the original form of the inequation expresses a relation in which the first quantity bears a greater ratio to the second, than it does in the subsequent transformations; seeing that when equals are taken from unequals, the remainders are more unequal than before. But though in the degree of inferiority which they severally express, the successive relations need not be equal; yet they must be equal in so far as being all relations of inferiority goes: and this indefinite inferiority is all that is predicated either in premiss or conclusion.
Here, too, should be specifically remarked the fact hinted in a previous chapter; namely, that the reasoning by which one of these inequations is worked out, palpably proceeds upon the intuition that relations which are equal to the same relation are equal to each other. The relations being those of inequality, the filiation of the last upon the first can only thus be explained: and the parallelism that subsists between inequations and equations, in respect of the mental acts effecting their solutions, confirms the conclusion before reached that in equations that intuition is involved, though less manifestly.
It remains to be pointed out that, of imperfect quantitative reasoning, the lowest type is that in which the inequality of the successive relations is expressed in its most general form—a form which does not define the relations as either those of superiority or inferiority. For instance:—
In this case the deductive process is the same as before: the successive relations are perceived to be alike in respect to their inequality; though it is not known whether the antecedents or the consequents are the greater. There is a definite co-ordination of the successive relations; though each relation is itself defined to the smallest possible extent. And, starting from this as the least developed type, we may see that the type previously exemplified, in which the antecedents are known to be greater or less than the consequents, is an advance towards those highest forms in which the antecedents and consequents are either directly equal, or the one equal to some specified multiple of the other.
§ 26. Incidentally, simple quantitative reasoning has been to a considerable extent treated of in the course of the foregoing analyses. The successive steps into which every compound quantitative argument is resolvable are all simple quantitative arguments; and we have already found that they severally involve the establishment of equality or inequality between two relations. It will be convenient, however, to consider by themselves, a class of simple quantitative arguments which are of habitual occurrence in the compound ones: some of them axioms; some nearly allied to axioms.
Let us commence with the familiar one—“Things which are equal to the same thing, are equal to each other.” It may be shown by reasoning like that already used in a parallel case, that this truth is reached by an intuition of the equality of two relations. Thus, putting A, B and C, as the three magnitudes, it is clear that for the equality of A and C to be discerned, they must be presented to consciousness in two states, of which the one immediately succeeds the other. But if A and C are contemplated alone, in immediate succession, their equality cannot be recognized; seeing that it is only in virtue of their mutual equality to B, that they can be known as equal. And if, on the other hand, B is interpolated in consciousness, and the three are contemplated serially—A, B, C, or C, B, A,—then A and C do not occur in the required juxtaposition. There remains no alternative, therefore, but that of contemplating them in pairs, thus:—
When A and B are united together in the single concept—a relation of equality; and when B and C are united into another such concept; it becomes impossible to recognize the equality of these two relations of equality which possess a common term, without the equality of the other terms being involved in the intuition.
But, perhaps, the most conclusive mode of showing that the mental act is of the kind described, will be to take a case in which some of the magnitudes dealt with have ceased to exist. Suppose A to represent a standard unit of measure preserved by the State; and let a surveyor be in possession of a measure B, which is an exact copy of the original one A; suppose, further, that in the course of his survey the measure B is broken; and that in the meantime the building containing the standard measure A, has been destroyed by fire: nevertheless, by purchasing another measure C, which had also been made to match the standard A, the surveyor is enabled to complete his work; and is perfectly satisfied that his later measurements will agree with his earlier ones. What is the process of thought by which he perceives this? It cannot be by comparing B and C: for one of these was broken before he got the other. Nor can it be by comparing them serially—B, A, C, and C, A, B: for two of them have ceased to exist. Evidently, then, he thinks of B and C, as both copies of A: he contemplates the relations in which they respectively stood to A: and in recognizing the sameness or equality of these relations, he unavoidably recognizes the equality of B and C. And here it will be instructive to notice a fact having an important bearing, not only on this, but on endless other cases: the fact, namely, that the mind may retain a perfectly accurate remembrance of a relation, when it is unable to retain an accurate remembrance of the things between which it subsisted. Supposing that in the above case the surveyor has had opportunities, at the respective times when he bought them, of comparing B and C with A. It becomes possible for him, at any time afterwards, to remember with perfect precision the relation of equality in which B stood to A: he can see in thought that exact agreement which they displayed when placed side by side, with as much completeness as though he were again observing it. But it is impossible for him to remember the magnitudes themselves, with anything like this precision. He finds that by figuring in imagination two objects which he has seen at different times, but has never compared, he can form an approximate idea of their relative magnitudes, if they are markedly different; but, if they are nearly of a size, he is as likely to be wrong as right in saying which is the greater. If, then, two magnitudes separately observed, cannot afterwards be so distinctly represented in consciousness as that their equality or inequality can be determined; and if, on the other hand, a relation of equality that was once remarked between two magnitudes can be represented in consciousness with perfect distinctness, and recognized as equal to some other relation of equality; then it becomes manifest that, in cases like the above, the truth perceived cannot be reached by remembering the magnitudes, but can be reached by remembering the relations. And thus we have demonstrative proof that the process of thought is as was stated.
Diverging from this original type are certain intuitions in which the thing cognized is the equality, not of two relations of equality having a common term, but of two relations of inequality having a common term. Thus, if A is greater than B, and B greater than C, then A is greater than C: and the like holds if they are severally less instead of greater. The act of thought may be symbolized thus:—
The relation A to B being given as a relation of superiority, while that of C to B is given as a relation of inferiority, it is known that the relation A to B is greater than the relation C to B; and as the term B, is common to the two relations, the intuition that the relation A to B is greater than the relation C to B, cannot be formed without involving the intuition that A is greater than C.
Diverging again from this type and its converse are others, having in common with it the characteristic that the two compared relations are perceived to be not equal, but unequal. For example, if A is greater than B, and B is equal to C; we know that A is greater than C. Similarly, if A is less than B, we know it is less than C. And if the first relation is one of equality and the second is one of inequality, there is a parallel intuition. In these cases, or rather in the first of them, we may express the mental act thus:—
Here, as before, the magnitude B being common to both, the relation A to B cannot become known as greater than the relation C to B without the superiority of A to C being known. Two relations having a common term cannot be conceived unequal, unless the remaining terms are unequal. And just as two magnitudes placed side by side, cannot be perceived unequal without its being at the same time perceived which is the greater; so, of two conjoined relations, one cannot be perceived greater than the other, without its being at the same time perceived which includes the greater magnitude. Should any one hesitate as to the correctness of these analyses, he has but to revert to the method of inquiry before followed, and consider by what process the conclusion is reached when some of the magnitudes have ceased to exist, to at once see that no other acts of thought can suffice.
The species of intuition serving to establish the equality of the successive forms of an equation—a species of intuition by which are recognized the general truths that the sums of equals are equal; that the differences of equals are equal; that if equals be multiplied by equals the products are equal, and if divided by equals the quotients are equal—is also accompanied by a converse species of intuition, in which the fact recognized is the inequality of two relations. Perhaps the simplest cases are the antitheses of the foregoing ones. They are seen in such axioms as—If to equals, unequals be added, the sums are unequal; and—If equals be divided by unequals, the quotients are unequal. But some of the intuitions of this order exhibit a higher degree of complexity: instance those by which it is known that if from unequals, equals be taken, the remainders are more unequal; and conversely, that if to unequals, equals be added, the sums are less unequal. To which general cases may be added the specific ones in which the first pair of unequals being known to stand in a relation of superiority, the second pair are known to stand in a still greater relation of superiority, or a less relation, according to the operation performed; and similarly, when the relation is one of inferiority. Thus if A + c is greater than B + c, then in a still higher degree is A greater than B—an intuition which may be expressed in symbols as follows:—
For present purposes it is needless to detail the varieties of intuition belonging to this class. It will suffice to remark, alike of these cases in which the thing perceived is the inequality of two relations, and of the antithetical cases in which the equality of two relations is perceived, that they differ from the previous class in this; that the relations are not conjoined ones, but disjoined ones. There are never three magnitudes only: there are always four. Throughout the first series, of which the simplest type is the axiom—“Things which are equal to the same thing are equal to each other,” there is invariably one term common to the two relations; whilst throughout the second series, of which as a typical sample we may take the axiom—“If equals be added to equals, the sums are equal,” the compared relations have no term in common. Hence it happens that in this second series, the relations being perfectly independent and distinct, the mental processes into which they enter are more readily analyzable. It is at once manifest that the groups of axioms above given, severally involve an intuition of the equality or inequality of two relations; and indeed the fact is more or less specifically stated throughout: seeing that in each case there is a certain relation, the terms of which are modified after a specified manner, and there is then an assertion that the new relation is or is not equal to the old one—an assertion which, being based on no argument, expresses an intuition.
One further fact respecting these two groups of intuitions remains to be noticed; namely, that they have a common root with those which proportions express. The one group is related in origin to that species of proportion in which the second of three magnitudes is a mean between the first and third; and the other group to that species in which the proportion subsists between four separate magnitudes. Thus the axiom—“Things which are equal to the same thing are equal to each other,” may, if we call the things A, B and C, be written thus:—
And again, the axiom—“The sums of equals are equal,” may, if we put A and B for the first pair of equals, and C, D for the second pair, be expressed thus:—
This fundamental community of nature being recognized, it will at once be perceived that the intuitions by which proportions are established, differ from the majority of the foregoing ones, simply in their greater definiteness—in their completer quantitativeness. The two compared relations are always exactly equal, whatever the magnitudes may be—are not joined by the indefinite signs meaning greater than or less than: and when the proportion is expressed numerically, it not only implies the intuition that the two relations are equal; but the figures indicate what multiple, or submultiple, each magnitude is of the others.