from being evident, that the logical student could not be trusted with their use. A cumbrous system of six, eight, or more rules of the syllogism was therefore made out, in order that the validity of an argument might thereby be tested; but, as even then the task was no easy or self-evident one, logicians formed a complete list of the limited number of forms obeying these artificial rules, and composed a curious set of mnemonic lines by which they might be committed to memory. These lines, the venerable Barbara celarent, &c., were no doubt creditable to the ingenuity of men who lived in the darkest ages of science, but they are altogether an anachronism in the present age. What should we think now of a writer of mathematical textbooks, who should select about a score of the commonest forms of mathematical equations, and invent a mnemonic by which both the forms of the equations and the steps of their solution might be carried in the memory? Instead of such an absurdity, we now find, even in purely elementary books, that the general principles and processes are impressed upon the pupil's mind, and he is taught by practice to apply these principles to indefinitely numerous and varied examples. So it should be in logic; the logical student need only acquire a thorough comprehension of the principle of substitution and the very primary laws of thought, in order to be able to analyse any argument and develop any form of reasoning which is possible. No subsidiary rules are needful, and no mnemonics would be otherwise than a hindrance. 40. I have yet a striking proof to offer of the truth of the views I am putting forward; for when once we lay down the primary laws of thought, and employ them by means of the principle of substitution, we find that an unlimited system of forms of indirect reasoning develops itself spontaneously. Of this indirect system there is hardly a vestige in the old logic, nor does any writer previous to Dr. Boole appear to have conceived its existence, though it must no doubt have been often unconsciously employed in particular cases. This indirect or negative method is closely analogous to the indirect proof, or reductio ad absurdum, so frequently used by Euclid and other mathematicians, and a similar method is employed by the old logicians in the treatment of the syllogisms called Baroko and Bokardo, by the reductio ad impossibile. But the incidental examples of the indirect logical method which can be found in any book previous to the "Mathematical Analysis of Logic" of Dr. Boole give no idea whatever of its all-commanding power; for it is not only capable of proving all the results obtained already by a direct method of inference, but it gives an unlimited number of other inferences which could not be arrived at in any other than a negative or indirect manner. In a previous little work1 I have given a complete, but somewhat tedious, demonstration of the nature and results of this method, freed from the difficulties and occasional errors in which Dr. Boole left it involved. I will now give a brief outline of its principles. 41. The indirect method is founded upon the law of the substitution of similars as applied with the aid of the fundamental laws of thought. These laws are not to be found in most textbooks of logic, but yet they are necessarily the basis of all reasoning, since they enounce the very nature of similarity or identity. Their existence is assumed or implied, therefore, in the complicated rules of the syllogism, whereas my system is founded upon an immediate application of the laws themselves. The first of these laws, which I have already referred to in an earlier part of this tract (p. 35), is 1 “Pure Logic, or the Logic of Quality apart from Quantity : with Remarks on Boole's System, and on the Relation of Logic and Mathematics." By W. Stanley Jevons, M. A. London: Edward Stanford, 1864. the LAW OF IDENTITY, that whatever is, is, or a thing is identical with itself; or, in symbols, A = A. The second law, THE LAW OF NON-CONTRADICTION, is that a thing cannot both be and not be, or that nothing can combine contradictory attributes; or, in symbols, —that is to say, what is both A and not A does not exist, and cannot be conceived. The third law, that of excluded middle, or, as I prefer to call it, the LAW OF DUALITY, asserts the self-evident truth that a thing either exists or docs not exist, or that everything either possesses a given attribute or does not possess it. Symbolically the law of duality is shown by in which the sign indicates alternation, and is equivalent to the true meaning of the disjunctive conjunction or. Hence the symbols may be interpreted as, A is either В or not B. These laws may seem truisms, and they were ridiculed as such by Locke; but, since they describe the very nature of identity in its three aspects, they must be assumed as true, consciously or unconsciously, and if we can build a system of inference upon them, their self-evidence is surely in our favour. 42. The nature of the system will be best learnt. from examples, and I will first apply it to several moods of the old syllogism. Camestres may thus. be proved and illustrated : (1) A sun is self-luminous, (2) A planet is not self-luminous, Now it is apparent that a planet is either a sun or it is not a sun, by the law of duality. But if it be a sun, it is self-luminous by (1), whereas by (2) it is not self-luminous; it would, if a sun, combine contradictory attributes. By the law of non-contradiction it could not exist, therefore, as a sun, and it consequently is not a sun. To represent this reasoning in symbols take A = sun. B = planet. C = self-luminous. Then the premises are (1) A = AC (2) B = Bc |