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ance with the current conception, which insists that in Reasoning something unknown must be reached; nor in accordance with the conception which insists on Inference as the essence of Reasoning. Take for example the demonstration of Euclid (XI. 18) of the proposition, “If a straight line be at right angles to a plane, every plane which passes through it shall be at right angles to that plane." This is not Reasoning at all, according to any accepted definition; no sooner are the terms clearly presented to the mind than the conclusion is intuited. We cannot mentally see a straight line at right angles to a plane without seeing that any plane passing through that line will be a plane of such lines, and that what is true of the one is necessarily true of the other. Doubt is excluded here, because by the terms of the proposition no variation is possible: there is no inference. But now contrast this with a case of Reasoning, which to many minds would have equal cogency, because not only is it founded on an induction from millions of observations, with no contradictory cases, but because the terms are presented so clearly to the mind, that the conclusion would be irresistible could we be quite certain of the induction, which we never can be so long as it remains an induction. The case is this: All observations of animals having separate sexes record the fact that these animals reproduced their kind only by the sperm cells of the male fecundating the germ cells of the female; hence the induction that offspring are the products of fecundated germs furnishes the deductive. conclusion that any animal belonging to this group of bisexual animals must have been so produced. Here are two acts of Reasoning, inductive and deductive; and till a few years ago every naturalist would have held these conclusions to be irresistible; although no one profoundly versed in Logic would have overlooked the fact that both

induction and deduction were inferences, and possibly inexact. The discovery of Parthenogenesis, wherein the female dispenses with the co-operation of the male, and the virgin aphis, or moth, not only produces aphides and moths, but these products of virgins themselves produce others, without the aid of the males; this, which is now recognized as a mode of reproduction, destroys the unconditional generalization of the induction. We need scarcely add, that while Euclid's proposition is absolutely true, because it is reducible to an identical proposition, and is not a truth of Reasoning, since there is no Inference; in like manner the naturalist's proposition will be absolutely true, if we exclude Inference by limiting the terms to those of the identical proposition, "All products of fecundated germs are products of fecundation."

63. Here we return once more to the unsatisfactory notion of Reasoning being characterized by the passage from the known to the unknown, and evolving from its premises a new and distinct conclusion. If it be said that when I infer that an alkaloid will have poisonous properties, the fact being certainly not known to me before trial, and being only concluded by me because of the resemblance of the new substances to substances known as poisonous, I have reached the unknown by Inference; the answer simply is, that the unknown fact is not reached at all, but remains unknown until it be known, which is to be effected by a very different process. If it be said that the conclusion is something new and distinct from the premises, and therefore must be what was unknown before, the answer has already been given in treating of the Syllogism, namely, that the conclusion simply restates what has been stated, explicitly or implicitly, in the premises; and if it bring anything in which was not already there, the conclusion is illogical. 64. Having rejected the distinction between Perfect

and Imperfect Induction and Deduction, we must also reject that between Perfect and Imperfect Reasoning, unless we are speaking of the products, not the process. In this latter sense we may say that such or such Reasoning is not valid, or is not sufficiently buttressed by fact; but the process is none the less perfectly performed. Reasoning from Analogy, for example, is the same process as that by which the most valid induction is formed; it differs only in the symbols operated on.

Finally, we may note that reasonings pass into reasons, from which all contingency is excluded, and which are therefore intuitions, truths seen by the Intellect as, to speak metaphorically, objects are seen by Sense, very inuch as intelligent actions pass into instincts when the discursive element of choice is lapsed. (Compare what is said on Instinct, Vol. I. p. 208 et seq.) A conclusion is an inference until it is established as a truth; once verified, it takes its place among the data of positive knowledge. Observe the parallelism here between the Logic of Feeling and the Logic of Signs. From sensations we pass to inferences, which are representations of what will be, or would be, presentations; and the proof of the correctness of such inferences is the conversion of re-presentation into presentation. Thus Sensation, Inference, and Sensation again are the three terms in the progression of Knowledge; and in the ideal sphere this progression is Datum, Hypothesis, and Verification: a starting-point, a search, and a finding.

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CHAPTER VI.

ON THE EXTENSION OF KNOWLEDGE THROUGH REASONING.

65. THE discussion just concluded has not been undertaken for the somewhat trivial purpose of rectifying the ambiguities of logical theories, but for the important purpose of exhibiting the psychological foundations of Speculation. We have there seen, in the nature of Reasoning, how inexorably Knowledge is limited to Experience; and how all suprasensible conceptions are metempirical and vain. Hence the attempt to penetrate the secrets of Nature by Reasoning alone has always been, and must forever be, a failure.

And we are now in a position to answer the question, proposed some time since, How is it possible to extend Knowledge by means of a process which is only valid when it is a restatement of what is already known? Our exposition of Reasoning may seem to lead to Plato's conclusion that all Knowledge is nothing but Reminiscence; Discovery seems taken out of its hands. Yet on reconsideration it will appear that we have only specified the kind of instrument which Reasoning is, and that we have only taken Discovery out of its hands when Reasoning pretends to be all-sufficient. Discovery is reasoned Experience. It must be verified by the reduction of Inference to Sensation or Intuition, otherwise it remains mere guesswork, not Knowledge.

66. This may seem a truism. Yet the constant practice of metempiricists, and the teaching of most mathe

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maticians, show that the truism is disregarded. The belief that physical or metaphysical discovery can be made à priori, and by Reasoning alone,* is sustained by the belief that Mathematics is a science of pure Reasoning, and is independent of Experience. The two beliefs fall together. I have already (Vol. I. p. 396) pointed out that Mathematics employs the Method of all Science, and has equally to find its data in Experience, being unable to stir a step without the aid of Observation, Induction, Hypothesis, and Experiment. There is no doubt a certain sense in which we may say, with De Morgan, that "all mathematical theorems are concealed truisms, the mere repetition and echo of our definitions of the quantities. about which we are busied, and of the laws of the operations we perform on them"; † and in this sense Bailly's description of Mathematics, " cette immense postérité d'un même père," may be allowed. But these phrases must be interpreted. To suppose that new mathematical truths are evolved deductively from axioms or definitions, irrespective of the intuition of the new relations given in the new figures or terms, is equivalent to supposing that the human race issued from Adam and the sons of Adam, without the co-operation of Eve and the daughters of Eve. Let those who hold that mathematical truths are simple deductions from axioms, unaided by intuition of the relations of the figures, try this in some case unknown to them. Let them, for example, take the definition of a cycloid, and, aided by all the axioms, let them discover the ratio of its area to the generating circle. It will be as futile as attempting from the axiom of causation and the definition of alcohol to deduce what the effect of a dose of alcohol would be on an organism, before experiments had revealed the kind of effect.

* On this common error compare the remarks of TAIT, Thermodynamics, 1868, § 4, also § 82.

+ DE MORGAN, Theory of Algebraical Expression, p. 26.

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