Immediate Inference. Probably the simplest of all forms of inference is that which has been called Immediate Inference, because it can be performed upon a single proposition. It consists in joining an adjective, or other qualifying clause of the same nature, to both sides of an identity, and asserting the equivalence of the terms thus produced. For instance, since Conductors of electricity Non-electrics, it follows that = Liquid conductors of electricity = Liquid non-electrics. If we suppose that Plants Bodies decomposing carbonic acid, = it follows that Microscopic plants = Microscopic bodies decomposing carbonic acid. In general terms, from the identity This is but a case of plain substitution; for by the first Law of Thought it must be admitted that AC = AC, and if, in the second side of this identity, we substitute for A its equivalent B, we obtain AC = BC. In like manner from the partial identity we may obtain A = AB AC = ABC by an exactly similar act of substitution; and in every other case the rule will be found capable of verification by the principle of inference. The process when performed as here described will be quite free from the liability to error which I have shown 1 to exist in "Immediate Inference by added Determinants," as described by Dr. Thomson.2 1 1 Elementary Lessons in Logic, p. 86. Inference with Two Simple Identities. One of the most common forms of inference, and one to which I shall especially direct attention, is practised with two simple identities. From the two statements that "London is the capital of England" and "London is the most populous city in the world," we instantaneously draw the conclusion that "The capital of England is the most populous city in the world." Similarly, from the identities Hydrogen Substance of least density, = = Hydrogen Substance of least atomic weight, we infer Substance of least density = Substance of least atomic weight. The general form of the argument is exhibited in the symbols We may describe the result by saying that terms identical with the same term are identical with each other; and it is impossible to overlook the analogy to the first axiom of Euclid that "things equal to the same thing are equal to each other." It has been very commonly supposed that this is a fundamental principle of thought, incapable of reduction to anything simpler. But I entertain no doubt that this form of reasoning is only one case of the general rule of inference. We have two propositions, A =B and B= C, and we may for a moment consider the second one as affirming a truth concerning B, while the former one informs us that B is identical with A; hence by substitution we may affirm the same truth of A. It happens in this particular case that the truth affirmed is identity to C, and we might, if we preferred it, have considered the substitution as made by means of the second identity in the first. Having two identities we have a choice of the mode in which we will make the substitution, though the result is exactly the same in either case. Now compare the three following formula, In the second formula we have an identity and a difference, and we are able to infer a difference; in the third we have two differences and are unable to make any inference at all. Because A and C both differ from B, we cannot tell whether they will or will not differ from each other. The flowers and leaves of a plant may both differ in colour from the earth in which the plant grows, and yet they may differ from each other; in other cases the leaves and stem may both differ from the soil and yet agree with each other. Where we have difference only we can make no inference; where we have identity we can infer. This fact gives great countenance to my assertion that inference proceeds always through identity, but may be equally well effected in propositions asserting difference or identity. Deferring a more complete discussion of this point, I will only mention now that arguments from double identity occur very frequently, and are usually taken for granted, owing to their extreme simplicity. In regard to the equivalence of words this form of inference must be constantly employed. If the ancient Greek yaλrós is our copper, then it must be the French cuivre, the German kupfer, the Latin cuprum, because these are words, in one sense at least, equivalent to copper. Whenever we can give two definitions or expressions for the same term, the formula applies; thus Senior defined wealth as "All those things, and those things only, which are transferable, are limited in supply, and are directly or indirectly productive of pleasure or preventive of pain." Wealth is also equivalent to "things which have value in exchange;" hence obviously, "things which have value in exchange all those things, and those things only, which are transferable, &c." Two expressions for the same term are often given in the same sentence, and their equivalence implied. Thus Thomson and Tait say,1 The naturalist may be content to know matter as that which can be perceived by the senses, or as that which can be acted upon by or can exert force." I take this to = what can be perceived by the senses; 1 Treatise on Natural Philosophy, vol. i. p. 161. For the term "matter" in either of these identities we may substitute its equivalent given in the other definition. Elsewhere they often employ sentences of the form exemplified in the following: "The integral curvature, or whole change of direction of an arc of a plane curve, is the angle through which the tangent has turned as we pass from one extremity to the other." This sentence is certainly of the form = The integral curvature the whole change of direction, &c. the angle through which the tangent has turned, &c. = Disguised cases of the same kind of inference occur throughout all sciences, and a remarkable instance is found in algebraic geometry. Mathematicians readily show that every equation of the form y = mx + c corresponds to or represents a straight line; it is also easily proved that the same equation is equivalent to one of the general form Ac + By + C = o, and vice verså. Hence it follows that every equation of the form in question, that is to say, every equation of the first degree, corresponds to or represents a straight line." Inference with a Simple and a Partial Identity. A form of reasoning somewhat different from that last considered consists in inference between a simple and a partial identity. If we have two propositions of the forms A = B, B = BC, we may then substitute for B in either proposition its equivalent in the other, getting in both cases A = BC; in this we may if we like make a second substitution for B, getting A = AC. Thus, since "The Mont Blanc is the highest mountain in Europe, and the Mont Blanc is deeply covered with snow," we infer by an obvious substitution that "The highest mountain in Europe is deeply covered with snow." These propositions when rigorously stated fall into the forms above exhibited. This mode of inference is constantly employed when for 1 Treatise on Natural Philosophy, vol. i. p. 6. * Todhunter's Plane Co-ordinate Geometry, chap ii. pp. 11—14. The very Thus, pro a term we substitute its definition, or vice versa. purpose of a definition is to allow a single noun to be employed in place of a long descriptive phrase. when we say "A circle is a curve of the second degree," we may substitute a definition of the circle, getting “A curve, all points of which are at equal distances from one point, is a curve of the second degree." The real forms of the positions here given are exactly those shown in the symbolic statement, but in this and many other cases it will be sufficient to state them in ordinary elliptical language for sake of brevity. In scientific treatises a term and its definition are often both given in the same sentence, as in "The weight of a body in any given locality, or the force with which the earth attracts it, is proportional to its mass." The conjunction or in this statement gives the force of equivalence to the parenthetic phrase, so that the propositions really are Weight of a body force with which the earth attracts it. Weight of a body mass. A slightly different case term B. Thus from A = = weight, &c. proportional to its of inference consists in substitut = AC. ing in a proposition of the form A = AB, a definition of the AB and B = C we get A For instance, we may say that " Metals are elements" and "Elements are incapable of decomposition." Metal Hence metal element. Metal metal incapable of decomposition. = It is almost needless to point out that the form of these arguments does not suffer any real modification if some of the terms happen to be negative; indeed in the last example "incapable of decomposition" may be treated as a negative term. Taking C capable of decomposition c incapable of decomposition; the propositions are of the forms C= |