It is of course quite as practicable to arrive at new conclusions from facts assumed, as from facts observed; from fictitious, as from real, inductions. Deduction, as we have seen, consists of a series of inferences in this form--_a_ is a mark of _b_, _b_ of _c_, _c_ of _d_, therefore _a_ is a mark of _d_, which last may be a truth inaccessible to direct observation. In like manner it is allowable to say, _Suppose_ that _a_ were a mark of _b_, _b_ of _c_, and _c_ of _d_, _a_ would be a mark of _d_, which last conclusion was not thought of by those who laid down the premisses. A system of propositions as complicated as geometry might be deduced from assumptions which are false; as was done by Ptolemy, Descartes, and others, in their attempts to explain synthetically the phenomena of the solar system on the supposition that the apparent motions of the heavenly bodies were the real motions, or were produced in some way more or less different from the true one. Sometimes the same thing is knowingly done, for the purpose of showing the falsity of the assumption; which is called a _reductio ad absurdum_. In such cases, the reasoning is as follows: _a_ is a mark of _b_, and _b_ of _c_; now if _c_ were also a mark of _d_, _a_ would be a mark of _d_; but _d_ is known to be a mark of the absence of _a_; consequently _a_ would be a mark of its own absence, which is a contradiction; therefore _c_ is not a mark of _d_.
§ 5. It has even been held by some writers, that all ratiocination rests in the last resort on a _reductio ad absurdum_; since the way to enforce assent to it, in case of obscurity, would be to show that if the conclusion be denied we must deny some one at least of the premisses, which, as they are all supposed true, would be a contradiction. And in accordance with this, many have thought that the peculiar nature of the evidence of ratiocination consisted in the impossibility of admitting the premisses and rejecting the conclusion without a contradiction in terms. This theory, however is inadmissible as an explanation of the grounds on which ratiocination itself rests. If any one denies the conclusion notwithstanding his admission of the premisses, he is not involved in any direct and express contradiction until he is compelled to deny some premiss; and he can only be forced to do this by a _reductio ad absurdum_, that is, by another ratiocination: now, if he denies the validity of the reasoning process itself, he can no more be forced to assent to the second syllogism than to the first. In truth, therefore, no one is ever forced to a contradiction in terms: he can only be forced to a contradiction (or rather an infringement) of the fundamental maxim of ratiocination, namely, that whatever has a mark, has what it is a mark of; or, (in the case of universal propositions,) that whatever is a mark of anything, is a mark of whatever else that thing is a mark of. For in the case of every correct argument, as soon as thrown into the syllogistic form, it is evident without the aid of any other syllogism, that he who, admitting the premisses, fails to draw the conclusion, does not conform to the above axiom.