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number, and quantity, it is not difficult to gather the regulating principle out of the individual judgments. It is different with other of our original convictions, such as those which relate to cause and effect; the greater complexity of the objects renders it more difficult to seize 、 on the principle involved, and there is greater room for dispute as to any given formula whether it is an exact expression of the facts. We see the reason why we cannot have demonstration in such sciences as physics and ethics; it is because of the concreteness and complexity of the objects. The problem of "three bodies has been found a difficult one; how much more perplexing must be one in which there are a considerable number and variety of concrete things to be considered.

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(a) It has been shown by Kant that the axioms of geometry are synthetic and not analytic judgments. Thus, in the axiom, "Two straight lines cannot enclose a space," the predication that "they cannot enclose a space," is not contained in the bare notion of "two straight lines." Starting with axioms which involve more than analytic judgments, we are reaching throughout the demonstration more than identical truth. The propositions in the Books of Euclid are all evolved out of the definitions and axioms, but are not identical with them, or with one another (Kritik, p. 145). Dr. Mansel (Proleg. Log. 2d ed. p. 103) maintains that such axioms as that "Things which are equal to the same are equal to each other are analytic. But does not this confound equality with identity? D. Stewart remarks (Elem. Vol. 11. Chap. ii.) that most of the writers who have maintained that all mathematical evidence resolves ultimately into the perception of identity "have imposed on themselves by using the words identity and equality as literally synonymous and convertible terms. This does not seem to be at all consistent, either in point of expression or fact, with sound logic." Certain modern logicians have fallen into a still greater confusion, when they make the relation between subject and predicate merely one of identity or of equality. The proposition "Man is mortal" is not interpreted fully when it is said "Man is identical with some mortal," or that "All men some mortals." By all means let logicians use symbols,

but let them devise symbols of their own, and not turn to a new use the symbols of mathematics, which have a meaning, and a welldefined one, simply as applied to quantity, and should not be made to signify the relations of extension and comprehension in logical propositions.

(b) There is truth, then, in a statement of D. Stewart: “The doctrine which I have been attempting to establish, so far from degrading axioms from that rank which Dr. Reid would assign them, tends to identify them still more than he has done, with the exercise of our reasoning powers; inasmuch as, instead of comparing them with the data, on the accuracy of which that of our conclusion necessarily depends, it considers them as the vincula which give coherence to all the particular links of the chain; or (to vary the metaphor) as component elements, without which the faculty of reasoning is inconceivable and impossible" (Elem. Vol. 11. Chap. i.).

If this view be correct, we see how inadequate is the representation of those who, like D. Stewart and Mr. J. S. Mill, represent mathematical definitions as merely hypothetical, and represent the whole consistency and necessity as being between a supposition and the consequences drawn from it. This is to overlook the concrete cognitions or beliefs from which the definition is derived. It is likewise to overlook the fact that these refer to objects, and the further fact that the abstractions from the concretes also imply a reality. This theory also fails to account for the circumstance that the conclusions reached in mathematics admit of an application to the settlement of so many questions in astronomy, and in other departments of natural philosophy. Thus, what was demonstrated of the conic sections by Apollonius is found true in the orbits of the planets and comets, as revealed by modern discovery. All this can at once be explained if we suppose that the mind starts with cognitions and beliefs, that it abstracts from these, and discovers relations among the things thus abstracted: the reality that was in the original conviction goes on to the farthest conclusion.

(c) Mr. Mill maintains (Logic, II. v. 4, 5) that the proposition, "Two straight lines cannot enclose a space, ," is a generalization from observation, "an induction from the evidence of the senses.' That observation is needed I have shown in this treatise; but there is intuition in the observation. That there is generalization in the general maxim I have also shown; but it is not a gathering of outward instances. Observation can of itself tell us that these two lines before us do not enclose a space, and that any other couplets of

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lines examined by us, twenty, or a hundred, or a thousand, do not enclose a space; but experience can say no more without passing beyond its province. An intellectual generalization of such experience might allow us to affirm that very probably no two lines enclose a space on the earth, but could never entitle us to maintain that two lines could not enclose a space in the constellation Orion. Mr. Mill, in order to account for the necessity which attaches to such convictions, refers to the circumstance that geometrical forms admit of being distinctly painted in the imagination, so that we have "mental pictures of all possible combinations of lines and angles." We might ask him what he makes of algebraic and analytic demonstrations of every kind, where there is no such power of imagination and yet the same necessity. But without dwelling on this I would have it remarked, that in the very theory which he devises to show that the whole is a process of experience, he is appealing to what no experience can ever compass, "to all possible combinations of lines and angles." Intuitive thought, proceeding on intuitive perceptions of space, may announce laws of the "possible combinations" of geometrical figures; but this cannot be done by observation, by sense, or imagination. Supposing, he says, that two straight lines, after diverging, could again converge, we can transport ourselves thither in imagination, and can frame a mental image of the appearance which one or both of the lines must present at that point, which we may rely on as being precisely similar to the reality." Most freely do I admit all this. We may "rely on it, but surely it is not experience, nor imagination, but thought looking at things which tells us what must be at that point, and that it is a "reality." The very line of remark which he is pursuing might have shown him that the discovery of necessary spatial and quantitative relations is a judgment in which the mind looks upon objects intuitively known, and now presented, or more frequently represented to the mind.

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CHAPTER IV.

THE METAPHYSICS OF FORMAL LOGIC.

METAPHYSICS and Logic are to be carefully distinguished. The former deals with First Principles, of which it seeks to give an account. The latter treats of the laws of Discursive Thought, in which we proceed from something given or allowed to something derived from it by thinking. The two, though separate, have points of connection. There are primitive truths at the basis of secondary or discursive processes. It is part of the office of Metaphysics to unfold and express these.

Logic deals with the Notion, the Proposition, and Reasoning. Each of these involves principles which are perceived to be true on the bare contemplation of the notions. Thus the Abstract implies the Concrete, and the Universal implies Singulars. Logic should take up these principles, explain, and apply them.

Logic deals with the Proposition, which may be Affirmative or Negative, Universal or Particular. In the logical use of the proposition there are involved the laws of Identity, of Contradiction, and Excluded Middle, as explained under the primitive judgment of Identity.

Reasoning may be in Extension or Comprehension. Each of these has its fundamental laws. The regulative principle of reasoning in Extension is the Dictum of Aristotle, "Whatever is true of a class is true of each member of the class." The regulating principle of reasoning in Comprehension is attributive, "All that is in an attribute is in the thing that contains the attribute," or

as Leibnitz expresses it, "Nota notae est nota rei ipsius." All these are self-evident. The metaphysician should supply these to the logician, who takes up and applies them to the various forms of reasoning, Categorical, Hypothetical, and Disjunctive. In doing this a science has been constructed which I regard as the most perfect, next to geometry.

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