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Updated: May 26, 2025
The inductions of arithmetic are of two sorts: first, those which we have just expounded, such as One and one are two, Two and one are three, &c., which may be called the definitions of the various numbers, in the improper or geometrical sense of the word Definition; and secondly, the two following axioms: The sums of equals are equal, The differences of equals are equal.
New conclusions may be got as well from fictitious as from real inductions; and this is even consciously done, viz. in the reductio ad absurdum, in order to show the falsity of an assumption. It has even been argued that all ratiocination rests, in the last resort, on this process.
That is exactly the reason why it has made the theory of evolution the basis of its inductions and why it thus marks the truly living and final phase and, therefore, the only phase recognized by the intelligence of the collectivist democracy of socialism which had theretofore remained floating in the nebulosities of sentiment and why it has taken as its guide the unerring compass of scientific thought, rejuvenated by the works of Darwin and Spencer.
The theologians have frequently been very highly incensed against those who believed they had discovered the most forcible proof of the existence of their gods, because they were obliged to discover that their adversaries could make very contrary inductions from their propositions; they did not perceive that it was next to impossible not to lay themselves open to attack, in establishing principles visibly founded upon that which each man sees variously.
Cicero became a mere magazine of instances, and the main use of the river was to feed the canal. I am unable to say whether these elaborate inductions would profit any one else, but I have a vivid recollection of the great utility they were at that time to my own mind.
If we find any means of deducing one of the less strong inductions from stronger ones, it acquires, at once, all the strength of those from which it is deduced; and even adds to that strength; since the independent experience on which the weaker induction previously rested, becomes additional evidence of the truth of the better established law in which it is now found to be included.
Still I rejoice with humility at feeling myself, in that order of understandings which, although utterly incapable of following the chain of your reasonings, calculations, and inductions utterly deprived of the powers necessary sic itur ad astra am yet informed, enlightened, and entertained with the series of sublime truths to which you conduct me.
As little did his scheme partake of the enthusiasm of some natural philosophers, who hoped, by physiological and chemical inductions, to arrive at a knowledge of the source of life, and so qualify themselves to manufacture and improve upon it.
Right regulation of the actions of so complex a being as man, living under conditions so complex as those presented by a society, evidently forms a subject-matter unlikely to admit of specific statements throughout its entire range. The principal inductions drawn from the data collected in the first of these volumes may be set forth in a few sentences.
Now he cannot add to these inductions, for he has parted with his instrument for inductive reasoning, and therefore his deductive reasoning in the purely subjective state which he has now entered is necessarily limited to the consequences which may be deducted from the premises which he has brought along with him.
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