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only legitimate illation in formal logic is that regulated by the law of reason and consequent, which connects thought into a reciprocally dependent series, each necessarily inferring the other, it is at once manifest that the distinction of matter into possible, actual, and necessary, is a doctrine wholly extralogical. Logical illation never differs in degree-never falls below that of absolute necessity. The necessary laws of thought, constraining an inevitable illation, are the only principles known to the logician.

We have just seen that Sir William Hamilton is the first to signalise the fact, that reasoning from the parts to the whole is just as necessary, and exclusive of material considerations, as reasoning from the whole to the parts. And he has evolved the laws of the inductive syllogism, and correlated them with those of the deductive syllogism.

We now proceed to another important addition which he has made to logic. He has shown that there are two logical wholes, instead of one, as the logicians had supposed. These two wholes are, the whole of comprehension, called by Sir William, depth; and the whole of extension, called by him breadth. These two wholes are in an inverse ratio of each other. The maximum of depth and the minimum of breadth are found in the concept of an individual (which in reality is not a concept, but only a single representation), while the minimum of breadth and the maximum of depth are found in a simple concept-the concept of being or existence. Now, the depth of notions affords one of two branches of reasoning, which, though overlooked by logicians, is at least equally important as that afforded by their breadth, which alone has been developed by the logicians. The character of the former is, that the predicate is contained in the subject; of the latter, that the subject is contained under the predicate. All reasoning, therefore, is either from the whole to the parts, or from the parts to the whole, in breadth; or from the whole to the parts, or from the parts to the whole, in depth. The quantity of breadth is the creation of the mind, the quantity of depth is at once given in the very nature of things. The former, therefore, is factitious, the latter is natural. The same proposition forms a different premise in these different quantities, they being inverse ratios; the sumption in breadth being the subsumption in depth.

Another fundamental development of logic made by Sir William, is, that the categorical syllogism, though mentally one (for all mediate inference is one, and that categorical), is either analytic or synthetic, from the necessity of adopting the one order or the other, in compliance with that condition of language which requires that a reasoning be distinguished

into parts and detailed in order of sequence; because explication is sometimes better attained by an analytic and sometimes by a synthetic enouncement, as is shown in common language. The Aristotelic syllogism is exclusively synthetic. Sir William Hamilton thus relieves the syllogism from a onesided view; and also rescues it from the objection of petitio principii, or of an idle tautology, which has been so often urged against it. Such objection does not hold against the analytic syllogism, in which the conclusion is expressed first, and the premises are then stated as its reasons. And this form of reasoning being shown to be valid, the objection of petitio principii is at once turned off, as applicable only to the accident of the external expression, and not to the essence of the internal thought. The analytic syllogism is not only the more natural, but is pre-supposed by the synthetic. It is more natural to express a reasoning in this direct and simple way, than in the round-about synthetic way.

We will next consider the most important doctrine, perhaps, which Sir William Hamilton has discovered in the domain of logic. Logicians had admitted that the subject of a proposition has a determinate quantity in thought, and this was, accordingly, expressed in language. But logicians had denied that the predicate in propositions has a determinate quantity. Sir William Hamilton has therefore the honour to have first disclosed the principle of the thorough-going quantification of the predicate in its full significance, in both affirmative and negative propositions. By keeping constantly in view that logic is conversant about the internal thought, and not the external expression, he has detected more of what it is common to omit in expression, of that which is efficient in thought, than any other philosopher. Inferences, judgments, problems, are often occult in the thought which are omitted in the expression. The purpose of common language is merely to exhibit with clearness the matter of thought. This is often accomplished best by omitting the expression of steps in the mental process of thinking; as the minds of others will intuitively supply the omitted steps, as they follow the meaning of the elliptical expression. This elliptical character of common language has made logicians overlook the quantification of the predicate. The purpose of common language does not require the quantity to be expressed. Therefore it was supposed that there is no quantification in the internal thought. When we reflect that all thought is a comparison of less and more, of part and whole, it is marvellous that it should not have been sooner discovered that all thought must be under some determinate quantity. And as all predication is but the expression of the internal thought, predication must have a determinate quantity-the

quantity of the internal thought. But such has been the iron rule of Aristotle, that, in two thousand years, Sir William Hamilton has been the first logician who, while appreciating the labours of the Stagyrite in this paramount branch of philosophy, has been in no degree enslaved by his authority, and has made improvements in, and additions to logic, which almost rival those of the great founder of the science himself.

The office of logic is to exhibit with exactness the form of thought, and therefore to supply in expression the omissions of common language; whose purpose is merely to exhibit with clearness the matter of thought. Logic claims, therefore, as its fundamental postulate, That we be allowed to state in language what is contained in thought. This is exemplified in the syllogism, which is a logical statement of the form of thought in reasoning, supplying in expression what has been omitted in. common language. Apply this rule to propositions, and it is at once discovered that the predicate is always of a given quantity in relation to the subject.

Upon the principle of the quantification of the predicate, Sir William Hamilton has founded an entirely new analytic of logical forms. The whole system of logic has been remodelled and simplified. The quantification of the predicate reveals that the relation between the terms of a proposition is one not only of similarity, but of identity; and there being consequently an equation of subject and predicate, these terms are always necessarily convertible. So that simple conversion takes the place of the complex and erroneous doctrine, with its load of rules, heretofore taught by logicians.

By the new analytic Sir William Hamilton has also amplified logic. The narrower views of logicians, in accordance with which an unnatural art had been built up, have been superseded by a wider view, commensurate with nature. Logic should exhibit all the forms of thought, and not merely an arbitrary selection; and especially where they are proclaimed as all. The rules of the logicians ignore many forms of affirmation and negation which the exigencies of thinking require, and which are constantly used, but have not been noted in their abstract generality. Accordingly, Sir William Hamilton has shown that there are eight necessary relations of propositional terms; and, consequently, eight propositional forms performing peculiar functions in our reasonings, which are implicitly at work in our concrete thinking, and not four only, as has been generally taught. Logic has been rescued from the tedious minuteness of Aristotle and his one-sided view, and from the trammels of technicality, and restored to the amplitude and freedom of the laws of thought.

The analysis of Sir William Hamilton enables us also to dis

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criminate the class, and to note the differential quality, of each of those syllogisms whose forms are dependent on the internal essence of thought, and not on the contingent order of external expression, such as the disjunctive, hypothetical, and dilemmatic syllogism; and to show the special fundamental law of thought by which each distinctive reasoning is more particularly regulated. And those forms of syllogism which are dependent on the contingent order of the external expression, embraced in the three figures of Aristotle, are expounded anew; and while their legitimacy is vindicated, the fourth figure, which has been ingrafted on the system by some alien hand, is shown to be a mere logical caprice. But we cannot particularise further. In fact, the work-shop of the understanding has been laid open, and the materials, the moulds and the castings of thought, in all their variety of pattern, have been exhibited, and the great mystery of thinking revealed, by this great master, on whom the mantle of Aristotle has fallen in the nineteenth century.

Logic may be discriminated into two grand divisions-the doctrine of elements, and the doctrine of method. Thought can only be exerted under the general laws of identity, contradiction, and excluded middle, and reason and consequent ; and through the general forms of concepts, judgments, and reasonings. These, therefore, in their abstract generality, are the elements of thought; and that part of logic which treats of them is the doctrine of elements. To this part of logic we have thus far confined our remarks. And the writings of Sir William Hamilton treat only of this part of logic. But in order to show the historical position of Sir William, and to exhibit the relation which, we have said, his philosophy bears to the philosophy of Aristotle and the philosophy of Bacon, as an initial, or step of progress towards harmonising the logic of the one with the method of the other, it becomes necessary to remark briefly upon the second part of logic, the doctrine of method.

Method is a regular procedure, governed by rules which guide us to a definite end, and guard us against aberrations. The end of method is logical perfection; which consists in the perspicuity, the completeness, and the harmony of our knowledge. As we have shown, our knowledge supposes two conditions; one of which has relation to the thinking subject, and supposes that what is known is known clearly, distinctly, completely, and in connection; the second has relation to what is known, and supposes that what is known has a veritable or real existence. The former constitutes the logical or formal perfection of knowledge; the latter, the scientific or material perfection of knowledge. Logic, as we have shown, is

conversant about the form of thought only; it is therefore confined exclusively to the formal perfection of our knowledge, and has nothing to do with its scientific or material truth, or perfection. Method, therefore, consists of such rules as guide to logical perfection. These rules are, definition, division, and concatenation, or probation. The doctrine of these rules is method.

Logic, as a system of rules, is only valuable as a mean towards logic as a habit of the mind-a speculative knowledge of its doctrines, and a practical dexterity with which they may be applied. Logic, therefore, both in the doctrine of elements and the doctrine of method, is discriminated into abstract or pure, and into concrete or applied. We have thus far only had reference to abstract or pure logic; and Sir William Hamilton treats only of this. It becomes, however, necessary for our purpose to pass into concrete or applied logic. Now, as the end of abstract, or pure logical method, is merely the logical perfection of our knowledge, having reference only to the thinking subject, the end of concrete, or applied logical method, is real or material truth, having reference only to the real existence of what is thought about. Concrete logic is, therefore, conversant about the laws of thought as modified by the empirical circumstances, internal and external, in which man thinks; and also about the laws under which the objects of existence are to be known. We beg our readers to remember these distinctions, and that all that now follows is about concrete or applied logic.

In order to show how the improvements and developments in formal logic, which we have exhibited, that have been made by Sir William Hamilton, conciliate the deductive or explicative logic of Aristotle with the inductive or ampliative logic of Bacon, it becomes necessary to state the difference of the philosophical methods of the two philosophers.

The great difficulty, with the ancient philosophers of the Socratic school, was to correlate logically the a priori and the a posteriori elements of our knowledge. The difficulty seems to have been suggested by the question, How can we know a thing for the first time? This question raised the doubt, that it is vain to search after a thing which we know not, since, not knowing the object of our search, we should be ignorant of it when found, for we cannot recognise what we do not know. Plato and Socrates, perhaps, solved the difficulty by the doctrine, that to discover, or to learn, is but to remember what had been known by us in a prior state of existence. Investigation was thus vindicated as a valid process; and also a useful one, as it is important to recall to memory what has been forgotten. Upon this theory of knowledge, Plato made

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