late years in Germany. They take the following forms: I. The Law of Identity Proper, which requires us to recognize the same to be the same, presented it may be at different times, or in different circumstances, or in different forms. II. The Law of Contradiction, according to which it is impossible for the same thing to be and not to be at the same time; thus whatever the thing be, an independently existing object, or an attribute. III. The Law of Excluded Middle, which requires that when two propositions are in the relation of contradictories, one or other must be true, and yet both cannot be true. These laws have a great importance in Formal Logic. Being carried out and applied in special forms, they show what may be drawn from any proposition or set of propositions given, and they keep thought consistent with itself.

Synthetic (as distinguished from Analytic) Judgments are those in which the predicate affirms or denies something more than is embraced in the subject: as when we say "gold is yellow,” “body gravitates,” “sin will be punished." Some of these judgments are a posteriori; that is, we reach them by experience. Others of them are said to be a priori; that is, the mind, on the bare contemplation of the notions, at once pronounces the agreement or disagreement. As examples, there are the mathematical axioms, such as that two straight lines cannot enclose a space; and metaphysical principles, such as that every effect must have a cause. In this section, I have given Sir W. Hamilton's analysis of Identical or Analytic Judg-`

1 We may give the account by Sir W. Hamilton: “1. The Law, Principle, or Axiom, of Identity, which, in regard to the same thing, immediately or directly enjoins the affirmation of it with itself, and mediately or indirectly prohibits its negation: (A is A). 2. The Law, etc., of Contradiction (properly Non-Contradiction), which in regard to contradictories explicitly enjoining their reciprocal negation, implicitly prohibits their reciprocal affirmation: (A is not Not-A). In other words, contradictories are thought as existences incompatible at the same time, as at once mutually exclusive. 3. The Law, etc., of Excluded Middle or Third, which declares that whilst contradictories are only two, everything, if explicitly thought, must be thought as of these, either the one or the other: (A is either B or not Not-B). In different terms: Affirmation and Negation of the same thing, in the same respect, have no conceivable medium: whilst anything actually may, and virtually must, be either affirmed or denied of anything. In other words: Every predicate is true or false of every subject; or contradictories are thought as impossible, but at the same time one or other as necessary." Metaph. Vol. п. App. п.

I have applied these Laws to Logic at the close of The Laws of Discursive Thought.

ments. In the remaining sections, I am to endeavour to unfold the Synthetic Judgments a priori.

SECT. II.—RELATIONS OF WHOLE AND PARTS.

It is a fundamental principle of this treatise that the mind begins with the concrete,—a truth which should always go along with the other, which has, however, been more frequently noticed, that it begins with the individual. Being thus furnished with the concrete in its primary knowledge and beliefs,--and we may add, imaginations, the mind can consider a part of the concrete whole separate from the other parts. In doing so, it is much aided by the circumstance that the concrete whole seldom comes round in all its entireness. The child sees a man with a hat to-day and without his hat to-morrow, and is thus the better enabled to form a notion of the hat apart from the man that wore it.

In all abstraction there is judgment or comparison; that is, we discover a relation between two objects contemplated. We contemplate a concrete whole, and we contemplate a part, and observe a relation of the part as a part to the whole. It should be admitted that, without any exercise of comparison, we are capable of imaging a part of a whole, in cases where the part can be separated; thus, having seen a man on horseback, I can easily picture to myself the man separately, or the horse separately, without thinking of any relation between them; but in such processes there is no exercise of abstraction. Abstraction is eminently an intellectual operation. In it we contemplate a part as part of a whole, say a quality as a quality of a substance; for example, transparency as a quality of ice, or of some other substance. In all such exercises there is involved a Correlative Power. This power may be called Comprehension, inasmuch as it contemplates the whole in its relation to the parts; or Abstraction, inasmuch as it contemplates the part as part of a whole; and the Faculty of Analysis and Synthesis, inasmuch as it resolves the whole into its parts, and shows that the parts make up the whole. There is, if I do not mistake, intuition involved in every exercise of this power. The operations of the intuition are always singular, but they may be generalized, and being so, they will give us the following as involved in Abstraction.

1. The Abstract implies the Concrete. This arises from the very nature of abstraction. When an object is before it in the concrete, the mind can separate a quality from the object, and one quality from another. It can distinguish, for example, between a man taken as a whole, and any one quality of his, such as bodily strength; and distinguish between any one quality and another, as between his bodily strength and intellectual power, between his intellectual faculties and his feelings, and between any one feeling, such as joy, and any other feeling, such as sorrow. But we are not to suppose that, while we can thus distinguish between a whole and its parts, between an object and its qualities, between one quality and another, therefore the part can exist independent of the whole, or the quality of its object. Every abstracted quality implies some concrete object from which it has been separated in thought.

2. When the Concrete is Real, the Abstract is also Real. In this respect there is a truth in the now exploded doctrine of realism. Abstraction, if it proceeds on a reality and is properly conducted, ever conducts to realities. It is thus a most important intellectual exercise for the discovery of truth, enabling us to discover the permanent amidst the fleeting, the real amidst the phenomenal. As I look on a piece of magnetized iron, I know it to be a real existence, and I think of it as having a certain form, and of its attracting certain objects, and I must believe that this figure is a reality quite as much as the iron which has the form, and that the attractive power is not a mere fiction, any more than the iron of which it is a property. But it is to be carefully observed that this abstract thing, while it has an existence, has not necessarily an independent existence. We have already seen that when it is a quality it must always be the quality of a substance. Beauty is certainly reality, but it has no existence apart from a beautiful person or scene, of whom or of which it has an attribute.

A philosopher, says Kant, was asked, What is the weight of smoke? and he answered,—Subtract the weight of the ashes from the weight of the fuel burned, and we have the weight of smoke. At the basis of his judgment is the intuitive maxim that the whole is equal to the sum of its parts. The individual intuitive

judgments which the mind pronounces on looking at whole and parts may perhaps be all generalized into two principles. (1.) The parts make up the whole. (2.) The whole is equal to the sum of its parts. From the first of these we may derive the rules, that the abstract part is involved in the concrete whole, and that the abstract, as part of a real concrete thing, is also a real. From the first we have the rule that the parts are less than the whole, and from the second the maxim that the whole is greater than the parts. It is of importance to have such maxims as these accurately enunciated in mathematical demonstration and logical and metaphysical science. Spontaneously, however, the mind does not form any such general axioms, which are merely the generalized expression of its individual judgments.

Still, the maxim is underlying many of our thoughts in all departments of investigation. Thus in Natural History it urges us to seek for a classification in which all the members of any subdivision will make up the whole. It impels the chemist to look out for all the elements which go to constitute the compound substance. In psychology and metaphysics it prompts us to analyze a concrete mental state into parts, and insists that in the synthesis the parts be equal to the whole. In logic it demands, as a rule of division, that the members make up the class, and is involved in all those processes in which we infer (in subalternation) that what is true of all must be true of some; or (in disjunctive division) that what is true of one of two alternatives (A and B), and is not true of one (A), must be true of the other (B). In most of such cases the more prominent elements are got from experience; in some of them, other intuitions act the more important part; but in all of them there are intuitions of whole and parts underlying the mental processes,—unconsciously and covertly, no doubt, but still capable of being brought out to view for scientific purposes.

SECT. III.—RELATIONS OF SPACE.

I have endeavoured to show that the mind in sense-perception has a knowledge of objects as occupying space, and that round these original cognitions there gather certain native beliefs. Upon

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the contemplation of the objects thus apprehended, the mind is led at once and necessarily to pronounce certain judgments. They may be arranged as follows:—

1. There are all the mathematical axioms which relate to limited extension, such as, "The shortest distance between any two points is a straight line;” “Two straight lines cannot enclose a space;” “Two straight lines which when produced the shortest possible distance are not nearer each other, will not, if produced ever so far, approach nearer each other;” “All right angles are equal to one another." Under the same head are to be placed the postulates involved in the definitions and in the propositions founded on them, such as the following, put in the form of maxims: "A straight line may be drawn from any one point to any other point;” “A straight line may be produced to any length in a straight line;""There may be such a figure as a circle, that is, a plane figure such that all straight lines drawn from a certain point within the figure are equal to one another;” and that “A circle may be described from any centre at any distance from that centre." I shall have occasion, in speaking of the application of the principles laid down in this treatise to mathematics, to return to axioms, and shall then show that the intuitive judgments pronounced by the mind in regard to the relations of space are all individual, and that the form assumed by them in the axioms of geometry is the result of the generalization, not indeed of an outward experience, but of the individual decisions of the mind.

2. There are certain axioms in regard to motion, such as that "All motion is in space;” “All motion is from one part of space to another;” “All motion is by an object in space;” “A body in passing from one part of space to another must pass through the whole intermediate space."

3. There are the primitive truths which arise from the relation of objects to space, such as "Body occupies space;” “Body is contained in space;” “Body occupies a certain portion of space;” and thus "Body has a defined figure.” But what, it may be asked, do our intuitive convictions say as to the relation of mind and space? I am inclined to think that our intuition declares of spirit, that it must be in space. It is clear, too, that so far as mind acts