theorem. Let us examine it closely and attentively. What is a proof? According to logicians, it is a syllogism. And what is a syllogism? A group of three propositions of this kind: "All men are mortal; Prince Albert is a man; therefore Prince Albert is mortal." Here we have the type of a proof, and every complete proof is conformable to this type. Now what is there according to logicians, in this proof? A general proposition concerning all men, which gives rise to a particular proposition concerning a certain man. From the first we pass to the second, because the second is contained in the first; from the general to the particular, because the particular is comprised in the general. The second is but an instance of the first; its truth is contained beforehand in that of the first, and this is why it is a truth. In fact, as soon as the conclusion is no longer contained in the premisses, the reasoning is false, and all the complicated rules of the Middle Ages have been reduced by the Port-Royalists to this single rule, "The conclusion must be contained in the premisses." Thus the entire process of the human mind in its reasonings consists in recognising in individuals what is known of a whole class; in affirming in detail what has been established for the aggregate; in laying down a second time, and piecemeal, what has been laid down once for all at first.

By no means, replies Mill; for if it were so, our reasoning would be good for nothing. It would not be a progress, but a repetition. When I have affirmed that all men are mortal, I have affirmed implicitly that Prince Albert is mortal. In speaking of the whole class, that is to say, of all the individuals of the class, I have spoken of each individual, and therefore of Prince

Albert, who is one of them. I say nothing new, then, when I now mention him expressly. My conclusion teaches me nothing; it adds nothing to my positive knowledge; it only puts in another shape a knowledge which I already possessed. It is not fruitful, but purely verbal. If, then, reasoning be what logicians represent it, it is not instructive. I know as much of the subject at the beginning of my reasoning as at the end. I have transformed words into other words; I have been moving without gaining ground. Now this cannot be the case; for, in fact, reasoning does teach us new truths. I learn a new truth when I discover that Prince Albert is mortal, and I discover it by dint of reasoning; for, since he is still alive, I cannot have learnt it by direct observation. Thus logicians are mistaken; and beyond the scholastic theory of syllogism, which reduces reasoning to substitutions of words, we must look for a positive theory of proof, which shall explain how it is that, by the process of reasoning, we discover facts.

For this purpose, it is sufficient to observe, that general propositions are not the true proof of particular propositions. They seem so, but are not. It is not from the mortality of all men that I conclude Prince Albert to be mortal; the premisses are elsewhere, and in the background. The general proposition is but a memento, a sort of abbreviative register, to which I have consigned the fruit of my experience. This memento may be regarded as a notebook to which we refer to refresh our memory; but it is not from the book that we draw our knowledge, but from the objects which we have seen. My memento is valuable only for the facts which it recalls. My general proposition

has no value except for the particular facts which it sums up.

"The mortality of John, Thomas, and company is, after all, the whole evidence we have for the mortality of the Duke of Wellington. Not one iota is added to the proof by interpolating a general proposition. Since the individual cases are all the evidence we can possess, evidence which no logical form into which we choose to throw it can make greater than it is; and since that evidence is either sufficient in itself, or, if insufficient for the one purpose, cannot be sufficient for the other; I am unable to see why we should be forbidden to take the shortest cut from these sufficient premisses to the conclusion, and constrained to travel the high priori road' by the arbitrary fiat of logicians." 1

"The true reason. which makes us believe that Prince Albert will die is, that his ancestors, and our ancestors, and all the other persons who were their contemporaries, are dead. These facts are the true premisses of our reasoning." It is from them that we have drawn the general proposition; they have taught us its scope and truth; it confines itself to mentioning them in a shorter form; it receives its whole substance from them; they act by it and through it, to lead us to the conclusion to which it seems to give rise. It is only their representative, and on occasion they do without it. Children, ignorant people, animals know that the sun will rise, that water will drown them, that fire will burn them, without employing this general proposition. They reason, and we reason, too, not from the general to the particular, but from particular to particular :

"All inference is from particulars to particulars; General propositions are merely registers of such inferences already made,

1 Mill's Logic, i. 211.

and short formulæ for making more: The major premiss of a syllogism, consequently, is a formula of this description: and the conclusion is not an inference drawn from the formula, but an inference drawn according to the formula: the real logical antecedent, or premisses, being the particular facts from which the general proposition was collected by induction. Those facts, and the individual instances which supplied them, may have been forgotten; but a record remains, not indeed descriptive of the facts themselves, but showing how those cases may be distinguished respecting which the facts, when known, were considered to warrant a given inference. According to the indications of this record we draw our conclusion; which is to all intents and purposes, a conclusion from the forgotten facts. For this it is essential that we should read the record correctly: and the rules of the syllogism are a set of precautions to ensure our doing so."1

"If we had sufficiently capacious memories, and a sufficient power of maintaining order among a huge mass of details, the reasoning could go on without any general propositions; they are mere formula for inferring particulars from particulars."2

Here, as before, logicians are mistaken: they gave the highest place to verbal operations, and left the really fruitful operations in the background. They gave the preference to words over facts. They perpetuated the nominalism of the Middle Ages. They mistook the explanation of names for the nature of things, and the transformation of ideas for the progress of the mind. It is for us to overturn this order in logic, as we have overturned it in science, to exalt particular and instructive facts, and to give them in our theories that superiority and importance which our practice has conferred upon them for three centuries past.

1 Mill's Logic, i. 218.

Ibid. i. 240.

VI.

There remains a kind of philosophical fortress in which the Idealists have taken refuge. At the origin of all proof are Axioms, from which all proofs are derived. Two straight lines cannot enclose a space; two things, equal to a third, are equal to one another; if equals be added to equals, the wholes are equal. These are instructive propositions, for they express, not the meanings of words, but the relations of things. And, moreover, they are fertile propositions; for arithmetic, algebra, and geometry are all the result of their truth. On the other hand, they are not the work of experience, for we need not actually see with our eyes two straight lines in order to know that they cannot enclose a space; it is enough for us to refer to the inner mental conception which we have of them: the evidence of our senses is not needed for this purpose; our belief arises wholly, with its full force, from the simple comparison of our ideas. Moreover, experience follows these two lines only to a limited distance, ten, a hundred, a thousand feet; and the axiom is true for a thousand, a hundred thousand, a million miles, and for an unlimited distance. Thus, beyond the point at which experience ceases, it is no longer experience which establishes the axiom. Finally, the axiom is a necessary truth; that is to say, the contrary is inconceivable. We cannot imagine a space enclosed by two straight lines: as soon as we imagine the space enclosed, the two lines cease to be straight; and as soon as we imagine the two lines to be straight, the space ceases to be enclosed In the assertion of axioms, the constituent ideas are irresistibly drawn together. In