eminent beneficial uses of proper mathematical science in intellectual training and discipline, besides giving the promise of all those incomparably higher and richer benefits which a science of thought itself should yield as compared with a science of mere special forms.

Philosophy owes Hamilton a debt of lasting gratitude for having aided so effectually in establishing the proper limits of logic. The contracted views of its sphere presented by the leading British logicians, as by Dr. Whately, limiting it essentially to a fractional part of but one, and that one by far the least important, of the three grand departments of the science, to the deductive forms of the syllogism, — Hamilton utterly discards. The doctrine of concepts and that of judgments are departments of altogether higher importance and rank than that of syllogisms, in every view that can be taken of the matter. The doctrine of syllogisms concerns altogether a less important part of our thinking than that of judgments or that of concepts, and is founded on those doctrines, and without them must be baseless and futile; and deductive reasoning is by far the least important, in every view, whether of intrinsic rank or of beneficial promise as a study and discipline, of the various modes of reasoning. Hamilton has greatly enlarged the domain of logical science, as marked out by such logicians as Dr. Whately; he has greatly circumscribed its boundaries as defined by such logicians as Watts, Kirwan, et id omne genus. He has drawn its circumference in a clear, well-defined line, and marked out thus a science second to none in the entire circle of sciences, both in intrinsic worth and in utility as a study; a science outranking all others as lying at the foundation of all, and determining the validity and the methods of all, strictly and literally ars artium et scientia scientiarum. If any still think that logic should embrace in its sphere the laws of intelligence generally, or of the cognitive intelligence, all that they need do, will be to limit the term to its recognized sphere, and denominate what Hamilton would call logic without limitation, discursive logic, or the logic of mediate cognition.

Logic, then, or, if any so prefer, discursive logic, — the science of mediate cognition, is exclusively conversant with the acts of the discursive faculty, and its acts all come within its domain. Their spheres are commensurate. It will be serviceable to indicate more exactly, and from other points of view, the field of mental activity thus denominated.

The discursive faculty has otherwise been known as the understanding properly so called (German, verstand), as the comparative faculty, or the faculty of comparison, the faculty of relations, the faculty of thought in its narrower import. It is denominated by Hamilton the elaborative faculty.

Of the nature of the operations of this faculty, the profound and accurate discrimination of Hamilton has given us the most true and exact notions. It is a faculty of cognition, not of retention, not of reproduction, or as Hamilton (as we think) unhappily denominates it, of imagination, but of acquisitive cognition. But of acquisitive cognition there are two easily distinguishable species. There is the immediate, the direct, as in perception and intuition, recognized on the European continent as the intuitional, and, with some indefiniteness, in English science as that of simple apprehension; and there is the mediate, the indirect. In the one case, the object is given; in the other, it is thought. In the one, we know the object immediately and irrespectively as an individual,-"as a complement of certain qualities or characters considered simply as belonging to itself"; in the other, we know the object mediately and relatively, "as comprising qualities or characters common to it with other objects." The distinction is clear and unquestionable. It is, we will add, radical, and of as vital importance in the representation in discourse of an object of thought as in the apprehension of it in the mind itself. A "ship," as an object of immediate cognition, is known as an individual ship—the Pacific - with a certain size, color, rigging, etc. It is known mediately only as having

characters in common with all ships, and is of course never realized in objective reality, either as having those characters only, or as wanting any one of them. "Fortitude," known immediately, is known as an individual action characterized by its relations to space, time, person, etc.; known mediately, is known as one of a class of virtues, with specific characteristics distinguishing it from other virtues, never realized just with these characters alone, and never realized without them all. Mediate cognition thus fastens only on what is common, what belongs to a plurality of objects. In other words, in mediate cognition, we know a character or complement of characters that a plurality of objects possess in common. A mediate cognition of "ship" knows it as an instrument of transportation by water, as the complement of these characters. But these several characters are common to many individuals. That is, many individual objects possess the same characters. The discursive faculty as the faculty of mediate cognition, applied to several objects, apprehends such a common character the same character- as belonging to the several objects. This character, as the same, identical in the many individuals, is the object with which it has to do. It is, in fact, in its essential nature an identifying faculty, apprehending the same in the many; and with all deference, we think that this name better indicates its proper character than the other denominations, discursive, comparative, elaborative, or the faculty of relations. The proper, the peculiar, the individual, it has nothing to do with, as such; it is the common, - what is the same, identical, in the plurality of individuals alone that it apprehends. All the modifications of its action, in comparison, in analysis and synthesis, in abstraction and generalization, are modifications of this one essential activity of identifying, of seizing the identical in the many. We compare, thus, by identifying the common character in the objects compared. We analyze and synthesize only that we may separate in a complement of characters some one character common to some other notion, or that is identical with

some one character in another notion, or that we may gather about a single character that is identical in a plurality of objects those objects that possess it in common. We generalize by identifying a property or character in all the individuals of the class. Identifying is the essential, characteristic operation of this so-called discursive faculty. Everything else that is ever associated with it is accidental, -constitutes no essential property.

It would be easy to substantiate this view of the essential nature of this mental activity from the expositions of psychologists. If we find it nowhere formally enunciated, it is necessarily involved in their best teachings. No satisfactory explication, for instance, of the process of generalization has ever been given which did not involve this as the essential element in the process. A single quotation from Hamilton's Metaphysics must answer our purpose. "A general notion is nothing but the abstract notion of a circumstance in which a number of individual objects are found to agree, that is, to resemble each other. In so far as two objects resemble each other, the notion we have of them is identical, and therefore, to us, the objects may be considered as the same. Accordingly, having discovered the circumstance in which objects agree, we arrange them by this common circumstance into classes, to which is also usually given a Generalization is thus but that modificacation of the identifying process in which we view the plurality of objects possessing the same character as one. And what is to be particularly remarked is the convertible use of the terms "resemblance" and "identity." "Similarity," "resemblance," is in fact but partial identity. Two objects are similar, resemble each other, when they are recognized as having any one property the same; and we say that in respect to that property they are the same. All individuals of a given class are the same in respect to that property which constitutes the principle of classification. James and John, and all individual men, are the same in respect of their 'Lecture XXXV. (Am. ed.), p. 475. 87

common name."

VOL. XXI. No. 4.

rationality and immortality. If any two of our notions embrace similar properties and relations in all respects, we of necessity think they alike respect the same object. Now, while it is a contradiction in terms to say that any two real objects are absolutely identical, and we say only that they are the same in respect to one or more properties, that they are partially identical, similar, we may in thought separate one or more properties and think them as the same, although pertaining to different real objects. We say "snow and paper are the same in whiteness"; "the white in the snow and the white in the paper are the same"; "snow and paper are white." In thus thinking, the discursive faculty has identified the common property "white," in the two objects; and by virtue of that identification thinks the objects as so far the same. All generalization is thus but an identifying

process.

But we may go further than this. In truth, all judgment is but an identifying act. When it is judged "snow is white," the judgment is true and actual only as the subject "snow" is, in a part of its meaning, identified with the quality "white"; and the real, necessary import of the proposition is that one of those characters that make up our notion of "snow" is the character "white." To predicate a quality of an object is nothing more or less than to judge that qual ity to be one of the qualities that together make up our notion of the object. We are thus prepared to accept the oracular enunciation of Hamilton, that "a proposition is always an equation of its subject and its predicate," in the exact, literal meaning of its terms, regretting ever that he has not carried out his principle in the development of his logical system, and has nowhere indicated whether he designed the language to be taken as rhetorical or strictly scientific. In the exactest mathematical import of the language, we maintain that every proposition, so far as true, is an identification, an equation, of the subject and the predicate. This is the foundation principle of all logic, of all discourse.