Neither does it mend matters to subjoin that we abstract altogether from the reality of the thing which the concept represents. It is, therefore, the same whether we say that the concept is equal to all its characters, or that the thing is equal to itself For pure Logic not only abstracts from the reality of the thing; but from the thing itself. Nay, more; it only abstracts from the reality of the thing, because it abstracts altogether from the thing, (i.e. the object of the thought and from all that is representative in the thought). Hence, it is not the same whether we say that the concept is equal to all its characters, or that the thing is equal to itself." For Logic contemplates the logical characters only of thought, considered purely according to its form or law; not the characters, or constituent qualities, of the object represented. Considered under this second aspect, thought becomes the formal property of Ideology. When, however, it is question of the equality of a thing (or reality) with itself, we have entered within the proper sphere of Metaphysics. (6) It will be found, after careful investigation, that the whole and parts announced in the quotations, and which form so important an element in the Hamiltonian theory, are a metaphysical whole and metaphysical parts,-not a logical whole and logical parts. (c) The Principle of contradiction has been generally accepted by the ancients and by the School as the ultimate in scientific and metaphysical demonstration; and with reason. Since, then, Sir William Hamilton has claimed for the Principle of identity the same place in affirmative Judgments which he claims for the Principle of contradiction in negative Judgments, it follows that the sphere of the former must be the same as the sphere of the latter, viz. metaphysical. (d) The new system of syllogistic modes is strictly metaphysical; yet no one can fail to see its dependence on the Principle of identity which it is intended to subserve. Nor can it avail to urge, as in the objection, that Sir William Hamilton declares the Principle of contradiction to be logical, not real,— formal, not metaphysical; because, throughout his Logic, he has systematically confounded the two orders. This may, perhaps, account in some measure for the contradictory statements that he makes touching this Principle. Thus, in his Logic he tells us, as we have seen, that the law of Identity and the law of Contradiction are co-ordinate and reciprocally relative, and neither can be educed as second from the other as first.' Yet, in his Metaphysics,

1 Logic, Lect. V, Vol. I, p. 80.

he assures us that the Principle of contradiction is the highest of all logical laws, in other words, the supreme law of thought1;' and then, again, in a fragment already referred to, pronounces that it is partial, not thorough-going . . . . and is, therefore, all too narrow in its application as a universal criterion or instrument of judgment 2.'

PROPOSITION CXXII.

The Principle of equality cannot be the ultimate Principle in order of reduction.

PROLEGOMENON I.

The Law of equality is usually expressed in this wise: Things which are equal to the same are equal to one another. Here, at length, a Principle is set before us, which seems to carry on its front a capacity for becoming the basis of all demonstration; for it exhibits not two only but three terms. In this respect it stands on a par with the famous Dictum de Omni et Nullo. Moreover, from the nature of the axiom, its terms must be respectively distinct, one from another. For equality essentially connotes distinction. No one would ever dream of univocally predicating the equality of a thing with itself. Those things, therefore, which are conceived as equal, must be likewise conceived as distinct. Hence, equal things are simply and entitatively distinct; and the same, only in a certain respect. In what respect are they the same? In Quantity. Yet the sameness is not in the entitative Quantity of each; for that is as distinct in each from the rest of the quantities, as is the Being which it informs from the quantified others. It is identity of measure in quantity, that constitutes the sameness of things equal. Nevertheless, and this is the principal point to be now considered, -equality is limited to the Category of Quantity. We speak of equal height, of equal weight, of equal number, of equal age. When the term is otherwise employed, as it not unfrequently is, it is applied either metaphorically or analogously. That which, in the Category of Quality, answers to equality, is called likeness or similarity. That, again, in the Category of Substance, which comes nearest to the concept of equality, is specific or generic identity. For these latter include a real distinction between the entities that 2 Ibid. Appendix II, p. 534.

1 Metaph. Lect. XXXVIII, Vol. II, p. 368.

are the terms of identity. Since the Principle of equality is limited to the Category of Quantity, it is not surprising to find that it is the principal basis of mathematical demonstration.

PROLEGOMENON II.

Sir William Hamilton has apparently assumed that the Principle of equality, in its simplest expression, and that of identity are one and the same; for he proposes indifferently the two formulas, as legitimate expressions of his favourite law, viz. A is A, and A = A. He further declares, in the quotation already given, that the law of identity may be also thus enounced,-Everything is equal to itself. But, surely, there is here some considerable confusion of ideas and terms. For equality, as has been already pointed out, is limited to the Category of Quantity; identity, on the other hand, is co-extensive with Being. Again: equality postulates distinct terms, real or individual; identity (which Sir William Hamilton contemplates) essentially supposes one term only. Hence, the concept of identity may be reasonably represented by the formula, A is A; but that of equality can only be represented by A=B, as indicative of the necessary distinction between the terms. It might possibly be objected by a disciple of the Hamiltonian theory, that in analytical Judgments, identity, like equality, supposes a real distinction in the terms identified. For, in such Judgments, the identity is necessarily either generic or specific; it cannot be individual. But, according to the admission made in the preceding Prolegomenon, species supposes individuals who are really distinct but conceptually identical in their nature; and genus equally supposes species which are mutually distinct, but conceptually identical, in the material part of their essence. This plea, however, is not solid. For the specific identity of individuals, (and the same may be said of the generic identity of several species), qua identity, is conceptual, not real. That which is the real foundation of the concept is, a similarity in the essential notes of each respectively. But similarity, like equality, connotes two distinct terms. In applying, however, the Principle of identity or of equality as measured by extension, the idea of similarity and of the thereby connoted distinction of subordinates disappears. Again: In the theory at present under examination, the specific or generic identity is not taken distributively, but collectively. Hence there arises a conceptual singularity, --not numerical, but specific or generic. Thus, the Judgment, Man

is the same as animality in man, or, as some animality (which is the way in which we are lessoned to read the Judgment, All men are animals, in order to reduce it to the new theory), singularizes the genus, as it were, by the act of identifying it with one of its own species. It should be added that if the genus were not thus singularized, it is inconceivable how the formula, A is A, could from any point of view symbolize the Judgment. To make the matter clearer, let us take an example from a concrete, not an abstract, Judgment; and let the predicate be quantified in the recently approved fashion. The old instance will serve our turn; All men are some animals. Under this form, the subject is not distributed. For, though it is true to say that all men, taken collectively, are some animals; it is not true that this man is some animals. Yet, if the particularizing prefix be omitted, there is no equality or identity.

THE PRESENT PROPOSITION IS PROVED BY A TWOFOLD ARGUMENT.

I. That Principle, whose motive is limited to a particular Category, cannot be the ultimate in order of reduction; since the ultimate must exhibit a motive common to all analytical Principles and Judgments of whatever Category. But the Principle of equality is limited to the Category of Quantity. If, however, it should be said that the term, equality, is used analogously according to the tenor of Sir William Hamilton's explanation, the answer is apparent. In such case, the equality is the equality of sameness, which is no equality at all; and we are referred back to the already discarded Principle of identity.

II. The law, or canon, of equality, viz. Those things which are equal to the same, are equal to one another, is not the basis of scientific demonstration; as it certainly is not the basis of the syllogism. It is sufficiently obvious, and otherwise stands confessed, that it could not be applied to indirect demonstration, or Reduction to the absurd. But can it be legitimately applied to ostensive demonstration? Thus much may be at once admitted that, if the said equality is measured by the logical whole, (i. e. by the whole of extension), this canon is verified in the instance of most powerful demonstration, (as it is called), i. e. of that primary demonstrative syllogism from which all the other successive syllogisms in one and the same series proceed. For, in this mother-syllogism, all the propositions-the conclusion as well as the two premisses,-are simply

convertible. In the Major Premiss, the attribute is predicated of the definition; in the Minor, the definition is predicated of the subject defined; while, in the Conclusion, the attribute or passion is predicated of the subject. Evidently, therefore, there is an equality of extension between the two terms of each Judgment and, in consequence, between the three terms of the syllogism. If, then, we represent the subject by S, the attribute by A, the definition by D, according to the whole of extension, we shall have D = A, SD,.. SA; that is to say, Things that are equal to the same are equal to one another. But, first of all, it should be remembered that, although the equality is logical, it is quite accidental to the laws, or forms, of thought; and owes its origin to the matter, i. e. to that in the thought which is representative. Hence it is that demonstration finds no place in pure Logic. For it is the application. of the universal syllogistic forms to a definite subject-matter; and the subject-matter is extra-logical. Then, again, Metaphysic has nothing to do with the logical whole. Yet our present search is for the ultimate metaphysical Principle, as exhibiting the motive common to all scientific, or analytical, Judgments. The measure of equality, therefore, ought to be the whole of comprehension; not the whole of extension. But, thus measured, not even will the most powerful demonstration satisfy the canon of equality. For, although it may be allowed that the Minor exhibits a certain sort of equality; nevertheless, it is impossible to affirm the same either of the Major or of the Conclusion. In the Minor, the definition, (as we have said), is predicated of the subject defined; therefore, the reality, represented by each term, is equal. Not without reason, however, has it been said that this premiss only exhibits a certain sort of equality; for, though the reality represented is equal, the respective representation of the reality by each term is not equal. In the Major, on the other hand, and in the Conclusion, there is no pretension to such equality. For, in the former, the attribute is predicated of the definition; in the latter, of the subject. But no one can fail to see that an attribute or passion, which is outside the essence, does not exhaust the reality of the subject and its definition. Thus, for instance, in the following demonstration,All rational animals are capable of laughter: Man is a rational animal : . Man is capable of laughter,-who would seriously maintain that capacity for laughter exhausts all the reality represented in the concept of man or in that of rational animal? If the Principle of