§ II.-RELATION OF JUDGMENT AND REASONING

Judgment Synthetic, Reasoning Analytic.-The rela tion of judgment and reasoning to each other becomes evident from what has been said of the nature of the reasoning process. Judgment is essentially synthetic. Reasoning, essentially analytic. The former combines, affirms one thing to be true of another; the latter divides, declares one truth to be contained in another. All reasoning involves judgment, but all judgment is not reasoning. The several propositions that constitute a chain of reasoning, are so many distinct judgments. Reasoning is the evolution or derivation of one of these judgments, viz., the conclusion, from another, viz., the premiss. It is the process by which we arrive at some of our judgments.

Mr. Stewart's View. - Reasoning is frequently defined as a combination of judgments, in order to reach a result not otherwise obvious. Mr. Stewart compares our several judg ments to the separate blocks of stone which the builder has prepared, and which lie upon the ground, upon any one of which a person may elevate himself a slight distance from the ground; while these same judgments, combined in a process of reasoning, he likens to those same blocks converted now, by the builder's art, into a grand staircase leading to the summit of some lofty tower. It is a simple combination of separate judgments, nor is there any thing in the last step of the series differing at all in its nature, says Mr. Stewart, from the first step. Each step is precisely like every other, and the process of reaching the top is simply a repetition of the act by which the first step is reached. This View called in Question. It is evident that this position is not in accordance with the general view which we have maintained of the nature of the reasoning process. According to this view, reasoning is not so much a combina tion as an analysis of judgments; nor is the last of the several propositions in a chain of argument of the same nature pre

cisely as the first. It is, like the first, a judgment, but unlike the first, it is a particular sort of judgment, viz., an inference or conclusion, a judgment involved in and derived from the former.

In the series of propositions, A is B, B is C, therefore A is C, the act of mind by which I perceive that A is B, or that B is C, is not of the same nature with that by which I perceive the consequent truth that A is C; no mere repeti tion of the former act would amount to the latter. There is a new sort of judgment in the latter case, a deduction from the former. In order to reach it, I must not merely perceive that A is B, and that B is C, but must also perceive the connection of the two propositions, and what is involved in them. It is only by bringing together in the mind these two propositions, that I perceive the new truth, not,otherwise obvious, that A is C, and the state or act of mind involved in this latter step seems to me a different one from that by which I reach the former judgments.

§ III. DIFFERENT KINDS OF REASONING.

Two Kinds of Truth.—The most natural division is that according to the subject-matter, or the materials of the work. The truths which constitute the material of our reasoning process are of two kinds, necessary, and contingent. That two straight lines cannot enclose a space, that the whole is greater than any one of its parts, are examples of the former. That the earth is an oblate spheroid, moves in an elliptical orbit, and is attended by one satellite, are examples of the latter.

The Difference lies in what. The difference is not that one is any less certain than the other, but of the one you cannot conceive the opposite, of the other you can. That three times three are nine, is no more true and certain, than that Cæsar invaded Britain, or that the sun will rise to-mor row a few minutes earlier or later than to-day. But the one

admits of the contrary supposition without absurdity, the other does not; the one is contingent, the other necessary Now these two classes of truths, differing as they do, in this important particular, admit of, and require, very different methods of reasoning. The one class is susceptible of demonstration, the other admits only that species of reasoning called probable or moral. It must be remembered, however, hat when we thus speak we do not mean that this latter class of truths is deficient in proof; the word probable is not, as thus used, opposed to certainty, but only to demonstra tion. That there is such a city as Rome, or London, is just as certain as that the several angles of a triangle are equal to two right-angles; but the evidence which substantiates the one is of a very different nature from that of the other. The one can be demonstrated, the other cannot. The one is an eternal and necessary truth, subject to no contingence, no possibility of the opposite. The other is of the nature of an event taking place in time, and dependent on the will of man, and might, without any absurdity, be supposed not to be as it is.

I. DEMONSTRATIVE REASONING.

Field of Demonstrative Reasoning. Its field, as we have seen, is necessary truth. It is limited, therefore, in its range, takes in only things abstract, conceptions rather than realities, the relations of things rather than things themselves, as existences. It is confined principally, if not entirely, to mathematical truths.

No degrees of Evidence. -There are no degrees of evi dence or certainty in truths of this nature. Every step follows irresistibly from the preceding. Every conclusion is mevitable. One demonstration is as good as another, so far as regards the certainty of the conclusion, and one is as good as a thousand. It is quite otherwise in probable rea soning.

Two Modes of Procedure.-In demonstration, we may

proceed directly, or indirectly; as, e. g., in case of two trian gles to be proved equal. I may, by super-position, prove this directly; or I may suppose them unequal, and proceed to show the absurdity of such a supposition; or I may make a number of suppositions, one or the other of which must be true, and then show that all but the one which I wish to establish are false.

Force of Mathematical reasoning. The question arises whence the peculiar force of mathematical, in distinction from other reasoning?-a fact observed by every one, but not easily explained: how happens this, and on what does it depend, this irresistible cogency which compels our assent? Is it owing to the pains taken to define the terms employed, and the strict adherence to those definitions? I think not; for other sciences approximate to mathematics in this, but not to the cogency of its reasoning. The explanation given by Stewart is certainly plausible. He ascribes the peculiar force of demonstrative reasoning to the fact, that the first principles from which it sets out, i. e., its definitions, are purely hypothetical, involving no basis or admixture of facts, and that by simply reasoning strictly upon these assumed hypotheses the conclusions follow irresistibly. The same thing would happen in any other science, could we (as we cannot) construct our definitions to suit ourselves, instead of proceeding upon facts as our data. The same view is ably maintained by other writers.

If this be so, the superior certainty of mathematical, over all other modes of reasoning, if it does not quite vanish, becomes of much less consequence than is generally supposed. Its truths are necessary in no other sense than that certain definitions being assumed, certain suppositions made, then the certain other things follow, which is no more than may be said of any science.

Confirmation of this View.-It may be argued, as a confirmation of this view, that whenever mathematical reasoning comes to be applied to sciences involving facts either

as the data, or as objects of investigation, where it is no longer possible to proceed entirely upon hypothesis, as, e. g., when you apply it to mechanics, physics, astronomy, practical geometry, etc., then it ceases to be demonstrative, and be comes merely probable reasoning.

Mathematical reasoning supposed by some to be iden tical. - It has been much discussed whether all mathematical reasoning is merely identical, asserting, in fact, nothing more than that a a; that a given thing is equivalent to itself, capable of being resolved at last into merely this. This view has been maintained by Leibnitz, himself one of the greatest mathematicians, and by many others. It was for a long time the prevalent doctrine on the Continent. Condillac applies the same to all reasoning, and Hobbes seems to have had a similar view, i. e., that all reasoning is only so much addition or subtraction. Against this view Stewart con tends that even if the propositions themselves might be represented by the formula a=a, it does not follow that the various steps of reasoning leading to the conclusion amount merely to that. A paper written in cipher may be said to be identical with the same paper as interpreted; but the evidence on which the act of deciphering proceeds, amounts to something more than the perception of identity. And further, he denies that the propositions are identical, e. g., even the simple proposition 2 x2=4. 2 x 2 express one set of quantities, and 4 expresses another, and the proposition that asserts their equivalence is not identical; it is not saying that the same quantity is equal to itself, but that two different quantities are equivalent.

II. PROBABLE REASONING.

Not opposed to Certainty. It must be borne in mind, as already stated, that the probability now intended is not opposed to certainty. That Cæsar invaded Britain is certain, but the reasoning which goes to establish it, is only probable reasoning, because the thing to be proved is