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 judgments 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 combination as an analysis of judgments; nor is the last of the several propositions in a chain of argument of the same nature precisely 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 repetition 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-morrow 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 that 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 demonstration. 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 evidence or certainty in truths of this nature. Every step follows irresistibly from the preceding. Every conclusion is inevitable. 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 reasoning.
Two Modes of Procedure.—In demonstration, we may proceed directly, or indirectly; as, e. g., in case of two triangles 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.