or pure logical judgments, and if it be truly possible to distinguish between them, otherwise than for the sake of convenience. The distinction is in any case convenient and does no harm at this point; and therefore for didactic reasons we allow it to stand; indeed we make use of it. [Sidenote: _The syllogism as fixed verbal form. Its use and abuse._] Just as in the case of definitions, so also in the case of the syllogism, it is to be noted that the verbal expression does not consist of an obligatory formula, but assumes the most varied forms, apparently very remote from syllogizing as commonly understood. The abuse of the syllogism as a formula continued for centuries, notably in mediæval Scholasticism, and notwithstanding the rebellion of the Renaissance, it has persisted among many philosophical schools, its last conspicuous manifestation being the didactic elaboration of the Leibnitzian philosophy, or Wolffianism. Certain of Wolff's demonstrations have remained famous, such as that concerning the construction of windows, contained in his _Manual of Architecture._ "A window must be large enough for two persons to lean against it, side by side," he developed it in this way: "_Demonstration._ It is customary to lean against a window with another person in order to look out. But the architect must serve the interests of his employer in everything. Therefore he must make the window large enough for two persons to be able to be there side by side.[1] _Q.E.D._" No more such syllogistic pedantries have been seen in our times, but (as has been already remarked in reference to pedantry of definition) contempt for the formula has too often resulted in contempt even for the correctness of the reasoning. So that it has sometimes been necessary to advise a bracing bath of scholasticism, and it has been observed and lamented of certain new civilizations (for example, of Russian culture, or of the Japanese people, who are so little addicted to mathematics), that they have not had a scholastic period, like that of the West, so general with them is the habit of incorrect, loose, and passionately impulsive and fantastic reasoning. Certainly the formula, the exercise of disputation in _forma,_ the _logica scholastica utens_ has its merits; and we must know how to have recourse to it when it is advantageous to do so, and to express thought in the brief and perspicuous formulæ of the syllogism, of the sorites, or of the dilemma. From this point of view the new methods of mathematical Logic or Logistic, upon which some are now working, and even the logical machines which have been constructed, would help; they would help--if they helped. For the point is just this: when formulæ, methods of demonstration, machines and the like, are recommended, expedients and instruments of practical or economic use are thereby proposed; and these cannot make good their existence otherwise than by getting themselves accepted for the utility--the saving of time and space, and