The skilful employment of this substitutive process enables us to make measurements beyond the powers of our senses. No one can count the vibrations, for instance, of an organ-pipe. But we can construct an instrument called the siren, so that, while producing a sound of any pitch, it shall register the number of vibrations constituting the sound. Adjusting the sound of the siren in unison with an organ-pipe, we measure indirectly the number of vibrations belonging to a sound of that pitch. To measure a sound of the same pitch is as good as to measure the sound itself.
Sir David Brewster, in a somewhat similar manner, succeeded in measuring the refractive indices of irregular fragments of transparent minerals. It was a troublesome, and sometimes impracticable work to grind the minerals into prisms, so that the power of refracting light could be directly observed; but he fell upon the ingenious device of compounding a liquid possessing the same refractive power as the transparent fragment under examination. The moment when this equality was attained could be known by the fragments ceasing to reflect or refract light when immersed in the liquid, so that they became almost invisible in it. The refractive power of the liquid being then measured gave that of the solid. A more beautiful instance of representative measurement, depending immediately upon the principle of inference, could not be found.[29]
Throughout the various logical processes which we are about to consider—Deduction, Induction, Generalisation, Analogy, Classification, Quantitative Reasoning—we shall find the one same principle operating in a more or less disguised form.
Deduction and Induction.
The processes of inference always depend on the one same principle of substitution; but they may nevertheless be distinguished according as the results are inductive or deductive. As generally stated, deduction consists in passing from more general to less general truths; induction is the contrary process from less to more general truths. We may however describe the difference in another manner. In deduction we are engaged in developing the consequences of a law. We learn the meaning, contents, results or inferences, which attach to any given proposition. Induction is the exactly inverse process. Given certain results or consequences, we are required to discover the general law from which they flow.
In a certain sense all knowledge is inductive. We can only learn the laws and relations of things in nature by observing those things. But the knowledge gained from the senses is knowledge only of particular facts, and we require some process of reasoning by which we may collect out of the facts the laws obeyed by them. Experience gives us the materials of knowledge: induction digests those materials, and yields us general knowledge. When we possess such knowledge, in the form of general propositions and natural laws, we can usefully apply the reverse process of deduction to ascertain the exact information required at any moment. In its ultimate foundation, then, all knowledge is inductive—in the sense that it is derived by a certain inductive reasoning from the facts of experience.
It is nevertheless true,—and this is a point to which insufficient attention has been paid, that all reasoning is founded on the principles of deduction. I call in question the existence of any method of reasoning which can be carried on without a knowledge of deductive processes. I shall endeavour to show that induction is really the inverse process of deduction. There is no mode of ascertaining the laws which are obeyed in certain phenomena, unless we have the power of determining what results would follow from a given law. Just as the process of division necessitates a prior knowledge of multiplication, or the integral calculus rests upon the observation and remembrance of the results of the differential calculus, so induction requires a prior knowledge of deduction. An inverse process is the undoing of the direct process. A person who enters a maze must either trust to chance to lead him out again, or he must carefully notice the road by which he entered. The facts furnished to us by experience are a maze of particular results; we might by chance observe in them the fulfilment of a law, but this is scarcely possible, unless we thoroughly learn the effects which would attach to any particular law.
Accordingly, the importance of deductive reasoning is doubly supreme. Even when we gain the results of induction they would be of no use unless we could deductively apply them. But before we can gain them at all we must understand deduction, since it is the inversion of deduction which constitutes induction. Our first task in this work, then, must be to trace out fully the nature of identity in all its forms of occurrence. Having given any series of propositions we must be prepared to develop deductively the whole meaning embodied in them, and the whole of the consequences which flow from them.
Symbolic Expression of Logical Inference.
In developing the results of the Principle of Inference we require to use an appropriate language of signs. It would indeed be quite possible to explain the processes of reasoning by the use of words found in the dictionary. Special examples of reasoning, too, may seem to be more readily apprehended than general symbolic forms. But it has been shown in the mathematical sciences that the attainment of truth depends greatly upon the invention of a clear, brief, and appropriate system of symbols. Not only is such a language convenient, but it is almost essential to the expression of those general truths which are the very soul of science. To apprehend the truth of special cases of inference does not constitute logic; we must apprehend them as cases of more general truths. The object of all science is the separation of what is common and general from what is accidental and different. In a system of logic, if anywhere, we should esteem this generality, and strive to exhibit clearly what is similar in very diverse cases. Hence the great value of general symbols by which we can represent the form of a reasoning process, disentangled from any consideration of the special subject to which it is applied.