2. All certain reasoning, commonly called demonstration, must begin with a comparison of two Ideas expressed by an intuitive proposition; and every proposition, expressive of the agreement of any two intermediate Ideas, or of every successive step of the demonstration, must be intuitive.

3. These are the chief cases of intuitive truth. But before we leave this topic, we must observe, that some axioms which philosophers seem to be so fond of holding forth as the foundations of all science, appear so far from being such, that no reasoning is ever founded on them, and that they are of no essential use in the course of reasoning. This leads us to ask, What is an axiom? It is evidently a general proposition, including a number of particular cases, and declarative of an intuitive truth. This truth must be as obvious, when surveyed in any of the particular cases, as it is in the general proposition. If this therefore be true, the axiom can be of little use, for its application to the particular case affords no light which the mind did not possess before that application.

Example 1. If you say that two and three are equal to four and one, I am perfectly satisfied of the equality of these two quantities, before the application of the axiom, that "Things equal to the same thing are equal to one another," and before I add, that they are both equal to five. The axiom adds no light to my conceptions. It merely repeats, in general terms, what was expressed more simply, if not more intelligibly, in particular terms.

2. If from two lines, each a mile long, you take away respectively two half miles, I cannot hesitate a moment, that the remaining half miles are equal to one another, although I had never heard of the axiom, "If equals are taken from equals, the remainders will be equal."

3. If from a field, of an acre in extent, you take away half an acre, and throw it into an adjacent field, I have the most entire conviction that the extent of the first field will be much less than it was before the division, without having recourse to the axiom, that "The whole is greater than a part."

4. If you infer that something must have existed from eternity, because something now exists, your conviction is complete, before you reflect on, or perhaps know, the scholastic maxim, "Ex nihilo nihil sit," Nothing can produce nothing.

5. If you are certain that the sun is above the horizon, you conclude, with entire confidence, that he is not also below it, although you may be unacquainted with the axiom, "Bodies cannot be in different places at the same time."

6. If, having two lines, one half a mile, and the other a quarter of a mile long, you add to each a whole mile, you are perfectly satisfied that the new line, composed of the mile and the half mile, is longer than that composed of the mile and the quarter. Nor do you procure any additional conviction whatever from the application of the axiom, "If equals be added to unequals, the wholes will be unequal."

Corol. From all these examples it is apparent, that axioms are general expressions of truths, obvious in particular cases included under general expressions. In a word, an axiom is applicable when we have found, by other means than by its aid, that under it is comprehended the particular case about which we are reasoning.

473. REASONING supports an exceedingly numerous class of propositions, more numerous than all the other kinds of evidence put together. But we do not now discuss its nature, nor explain the different degrees of evidence it supplies.

Obs. Almost all the propositions of science, most of those of the arts, and of business; in a word, those propositions of all cases in which the mind receives certain or probable conviction by the exercise of its rational faculties, belong to this class: but we have already explained the nature of these propositions under the different kinds of evidence by which they are supported, when we treated of the different kinds of evidence in Chapter XI. Book II.

474. TESTIMONY was the last source of knowledge, and the last species of evidence, which we purposed to explain. Testimony, founded in the trust which we repose in the veracity of our fellow creatures, and in their intercourse with one another, is of very extensive use.

Illus. All the credit of history, all the intelligence of places, men, and things, we cannot in person examine; all the security society can confer on life and property in courts of justice; all the information of business and social life; depend entirely on the opinion we have, that men will tell truth in their communications to one another. (See Art. 315 and 440.) In many cases, the evidence of testimony affords a high degree of satisfaction; but the degrees of satisfaction decrease, till they degenerate into that equivocal state, in which probability for and against truth are so equally poised, as to leave the mind in a state of suspense. (See Art. 315 and 439.)

475. Two causes chiefly induce us to distrust the credibility of testimony, 1st. suspicion that the relater was not fully informed; or, 2dly. that his interest might influence him to utter falsehood. The presence of either, or of both these causes, is a sufficient reason for hesitation. But where neither takes place, we seem to have no reason to distrust the information of testimony. Truth is congenial to the mind of man. It is more easy to tell truth than to utter falsehood. It is not easy to utter falsehood with success. Some time must elapse before the mind can acquire those habits, and that composure, which are necessary to secure falsehood from the inconsistency and embarrassment which instantly proclaim its baseness and its insincerity. (Art. 442.)

Illus. Though the evidence of testimony cannot be deemed equivalent to that of demonstration, or to that of the senses, yet in most cases it would be ridiculous to indulge the least suspicion.

.

Example. That there are such cities as Paris, Rome, or Pekin,

that Alexander conquered a great part of the western quarter of Asia, and that Julius Cæsar was killed in the senate-house, are all facts of which we cannot entertain the smallest doubt. (Art. 309. Illus. 2. and Example.)

Corol. The conviction which we have of the truth of such facts is called certainty, and the impression made on the mind by the evidence of testimony in general, is termed belief. The impression which results from divine testimony, or the evidence of revelation, has obtained the name of faith.

IV. Of Mathematical, Moral, Political, and Prudential Reasoning.

476. All knowledge is either intuitive, demonstrative, or probable. Intuitive knowledge is extremely circumscribed, and reasoning therefore begins where intuition ends, and consists in finding out the truth of a proposition, or the agreement or disagreement of its subject and predicate, by the help of intermediate ideas. The intermediate ideas form the steps, or links, by which the mind passes from the first of the primary ideas to the last, or from the subject of any proposition to its predicate; and finally perceives their relation.

Illus. 1. Reasoning assumes different names, according to the nature of the steps, or of the links, which display the relation between the primary ideas. Thus, if the mind attain complete satisfaction in every step of its progress, or in the successive comparison of every pair of ideas, it is said to acquire certainty of the agree ment or disagreement of the two primary ideas; and the reasoning is called demonstrative. (See Art. 303, 304, and 305.)

2. If the agreement of the intermediate ideas with one another, and with the extremes, is not perfectly satisfactory, that is, if the steps of the reasoning leave the mind under some degree of hesitation, the reasoning is denominated probable; and the reasoner attains probability only of the truth of the proposition he investigates. Where certainty terminates, probability commences; and the latter admits numerous degrees, from the highest degree, which stands next to certainty, to the lowest, which makes so little impression, as to permit the mind to remain in a state of suspense. (See Art. 306 and 307, with their Illustrations.)

477. If a proposition, supported by probable evidence, relate to speculation, the judgment formed concerning it is often called opinion; if it relate to facts, chiefly supported by testimony, the judgment is generally called belief. (See Art. 278, with all its Illustrations and Examples.)

Mus. 1. In explaining, therefore, the branch of logic now before us, all we have to do, is, to reduce to practice, first, the analysis we have given of demonstrative reasoning (Art. 303); secondly, that

of probable reasoning (Art. 306.); and point out the sciences and arts in which they are respectively employed.

2. All reasoning is either of the one kind or the other; and in every science or art, in which conviction does not come up to certainty, we must be content with probability (Art. 308.)

478. Mathematics and Arithmetic are the only sciences susceptible of demonstrative proof, which is so satisfactory and cogent as to exclude even the supposition of falsehood. (Art. 304.) Other sciences, in their principles, may perhaps furnish proofs nearly, if not completely demonstrative; but in the detail they exhibit nothing better than probability. The high evidence of the science of quantity, independent of the importance of the truths which it teaches, renders them good exemplifications of the rules of logic; and one of the best methods of becoming a good reasoner, is, to be familiar with the processes of investigation which they supply. (See the Illustrations and Examples to Art. 304.)

Illus. To reduce to practice demonstrative reasoning, we shall now analyze some propositions of the Elements of Euclid. Reasoning is a successive comparison of every pair of ideas, from the first to the last, or from the idea which forms the subject of the proposition, to the one which forms the predicate; and in demonstration every comparison is intuitively certain. When these ideas are found to agree, the demonstration is finished, and the reasoning is concluded. (Art. 298.)

Example 1. Suppose we begin with the first proposition of the first book of the Elements, which proposes "To describe an equilateral triangle on a given straight line." Let us pass over the operations by which the triangle in the figure is described, because we mean to analyze only the reasoning of the proposition.

Argument. After the figure has been constructed on the given line, the proposition to be proved is, that "The triangle so constructed is equilateral, or has all its sides equal." The subject of the proposition, or the first idea of it, is, that of the triangle described; the predicate of the proposition, or the second idea of it, is, that of the equality of the sides of the triangle. Now, it is not intuitively certain that the three sides are all equal to one another; therefore some intermediate ideas must be placed between the subject and the predicate of the proposition, to show their agreement. The process consists of two steps; that is to say, one intermediate idea is necessary to prove the proposition. The first step is the comparison of the base A B* with one of the sides A C; and of their equality we have intuitive certainty, because, by the description of the figure, they are radii of the same circle. The second step is the comparison of the same side A B, with the other side B C; and of their equality, also, we have intuitive certainty, as they are both semi-diameters of another circle of the same radius with the former. This step finish

* See the Figure in Simson's Euclid.

es the demonstration. The base is found to agree with both the sides; and the triangle must be equilateral, because all the sides are equal; the subject and predicate of the proposition are found exactly to agree.

Example 2. In the forty-seventh proposition of the first book of the Elements, the truth to be established is, "That in a right-angled triangle, the square of the side opposite to the right angle is equal in quantity to the sum of the squares of the other two sides." The square opposite to the right angle is the subject, the sum of the two other squares is the predicate, and the idea of the extent of the first square is to be compared with the idea of the sum of the other two squares.

Argument I. The first step is to prove, that G A C* is one straight line, and H A B another, in order to lay a foundation for demonstrating that the triangle F B C is equal to half the square F A, and the triangle A B D equal to half the parallelogram B L.

II. The next step is to prove the triangle A B D equal to the triangle F B C.

III. The third step is to prove the triangle A B D equal to half the parallelogram B L, and the triangle F B C equal to half the square F A; and hence to infer the equality of the square F A to the parallelogram B L.

IV. Three similar steps are necessary to find the square A K equal to the parallelogram C L; and hence to infer the equality of the whole square B E to the two squares FA and A K, which establishes the agreement of the subject and predicate of the proposition; or that the square of the side opposite to the right angle, is equal to the squares of the two other sides.

Corol. To complete this process, then, there are necessary these six capital steps, and each of these includes one or more subordinate steps, so that the sum of the subordinate steps amounts to no fewer than twelve; and if these are added to the six capital ones, it appears, that, to prove this proposition, there are requisite eighteen intermediate ideas. The mind has a clear and distinct perception of the agreement of every pair of ideas; and the effect is proportional to the cause, for the mind obtains the most complete certainty of the truth of the proposition.

479. All reasoning has this in common with demonstration, that the agreement or disagreement of the primary ideas must be proved by intermediate ideas; the difference is, that the agreement of the intermediate ideas with one another, and with their primary ideas, amounts not to certainty; it is no more than probable.

Corol. From this view it will appear, that the far greater part of knowledge, and even the most interesting and important part, that which concerns morality, politics, the useful arts, and business, is not supported by better evidence than probability. (See Art. 211.) The probability, however, in many cases is highly convincing,

* See the Figure.