CHAP. III.

DEMONSTRATIVE REASONING.

LET us turn, in the next place, to the second species of reasoning, in which certain things or facts lead us to discern other things or facts not immediately manifest by themselves. An example of it having been already given in the first chapter, and a still more detailed one being intended for the Appendix*, in order to avoid interrupting the exposition of the subject by too great particularity, the simplest instance will here suffice: the lines A and B are respectively equal to c, and therefore they are equal to each other.

Here the mind observing successively the equality of a to c and that of в to C, is thence led to discern the mutual equality of A and B, which is not self-evident or immediately discernible from the inspection of A and в alone.

It is plain that in reasoning of this second species, which is with great propriety termed demonstrative, we intuitively discern, at each step, that one fact implies another, and discern too that a denial of the implied fact involves a contradiction.

But demonstrative reasoning is not confined to

* See Appendix, Article 1.

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the science of quantity. It is to be found in all departments of human knowledge.

Whenever the mind discerns one fact to be implied in another, or the exclusion of a fact to be implied in another fact, it reasons demonstratively, whether they are facts of quantity or otherwise.

Examples of this truth might be multiplied without end, but the few which follow will be sufficient for illustration.

That portrait is a striking likeness of two different persons; therefore they must resemble each

other.

The two litigants cannot both be the exclusive owners of the property in dispute; therefore one of them must be urging a wrong claim.

The traveller who was attacked had no money with him; therefore he could not be robbed of a large sum as reported.

The planets are opaque bodies; therefore they must shine by light derived from an external source.

Under this species of reasoning must be ranked that which is usually denominated syllogistic, but which I shall venture to call class-reasoning, because perfect specimens of it, as I shall hereafter show, are found in the form of enthymemes.

Of class-reasoning, or at least of so much of it as exemplifies the maxim of Aristotle, termed the dictum de omni et nullo, the characteristic is, inferring some attribute to belong or not to belong to a given individual or to given individuals of a class, because it belongs to all or does not belong

to any of the individuals of the class. It would be clearly a self-contradiction to adınit the latter and to deny the former. All such reasoning is obviously demonstrative: it is, indeed, largely interfused in geometrical demonstration, in which general propositions not self-evident, but which have been shown to be implied in other propositions, are subsequently employed as major premises. Such, for example, are the propositions that the angles at the base of an isosceles triangle are equal; that the three angles of every triangle are together equal to two right angles; and that all equilateral triangles are equiangular.

That all demonstrative reasoning consists in discerning, and, when expressed in words, in asserting, one fact or one proposition to be implied in another, is plain. If we call one the implying fact, the other will be of course the implied fact, as in the following examples.

IMPLYING FACTS.

IMPLIED FACTS.

All horned animals are rumi- This horned animal is ruminant.

nant.

The lines A and B are severally The lines A and B are equal to

By the terms implying and being implied nothing is assumed they are merely expressive of the truth, that when the two facts so denominated are presented together to the mind, the proposition enunciating the second fact is at once seen to be true if the proposition enunciating the first is true, and the denial of it to involve a contradiction: nor is it pretended that this mode of stating an argument is superior for common purposes to the usual forms.

If we examine the general principles on which demonstrative reasoning proceeds, or which it exemplifies, we shall find less uniformity than in the case of contingent reasoning.

The general principle exemplified in the argument, that A and B are equal to each other because they are respectively equal to c, is, that things equal to the same thing are equal to each other. In the demonstration cited in the first chapter, that the opposite angles made by the intersection of two right lines are equal, the reasoning consists of two steps, the first of which proceeds on the same axiom, while the second exemplifies the axiom, that if equals are taken from equals the remainders are equal.

In the argument that because the culprit at the bar was in Edinburgh at a given time, he could not be guilty of a crime committed at that precise moment in London, the general principle exem

plified is, that a man cannot be in two places at the same time.

Axioms might easily be educed in the same way from the other examples of demonstrative inference furnished in the preceding pages; but as I purpose to resume the consideration of such maxims in a subsequent chapter, it would be superfluous to dwell upon them here. In that chapter, I shall enter into an express examination of the general principles exemplified in class-reasoning, one of which has become so noted under the name of the dictum de omni et nullo.

The remark before made regarding the cogency of the process in the first species of reasoning, may be repeated with regard to the second. Its cogency is not susceptible of proof. If the argument that "because A and B are respectively equal to c they are equal to each other," is not intuitively discerned to be true, nothing can make it appear so. It would be idle, too, in this case, as in the case of probable reasoning, to cite the general principle with a view to strengthen the force of the particular instance. The maxim that all things which are equal to the same thing are equal to each other springs up in the mind after the mutual equality of two particular things which are equal to a third thing has been discerned, and is merely a generalisation of what the particular fact implies; a truth which will be more fully elucidated in the two following chapters.