“Axioma I. Si idem secum ipso sumatur, nihil constituitur novum, seu A + A ∝ A.” Then follows this remarkable scholium: “Equidem in numeris 4 + 4 facit 8, seu bini nummi binis additi faciunt quatuor nummos, sed tunc bini additi sunt alii a prioribus; si iidem essent nihil novi prodiret et perinde esset ac si joco ex tribus ovis facere vellemus sex numerando, primum 3 ova, deinde uno sublato residua 2, ac denique uno rursus sublato residuum.”
Translated this would read as follows:—
“Axiom I. If the same thing is taken together with itself, nothing new arises, or A + A = A.
“Scholium. In numbers, indeed, 4 + 4 makes 8, or two coins added to two coins make four coins, but then the two added are different from the former ones; if they were the same nothing new would be produced, and it would be just as if we tried in joke to make six eggs out of three, by counting firstly the three eggs, then, one being removed, counting the remaining two, and lastly, one being again removed, counting the remaining egg.”
Compare the above with pp. [156] to 162 of the present work.
M. Littré has quite recently pointed out[12] what he thinks is an analogy between the system of formal logic, stated in the following pages, and the logical devices of the celebrated Raymond Lully. Lully’s method of invention was described in a great number of mediæval books, but is best stated in his Ars Compendiosa Inveniendi Veritatem, seu Ars Magna et Major. This method consisted in placing various names of things in the sectors of concentric circles, so that when the circles were turned, every possible combination of the things was easily produced by mechanical means. It might, perhaps, be possible to discover in this method a vague and rude anticipation of combinational logic; but it is well known that the results of Lully’s method were usually of a fanciful, if not absurd character.
A much closer analogue of the Logical Alphabet is probably to be found in the Logical Square, invented by John Christian Lange, and described in a rare and unnoticed work by him which I have recently found in the British Museum.[13] This square involved the principle of bifurcate classification, and was an improved form of the Ramean and Porphyrian tree (see below, p. [702]). Lange seems, indeed, to have worked out his Logical Square into a mechanical form, and he suggests that it might be employed somewhat in the manner of Napier’s Bones (p. 65). There is much analogy between his Square and my Abacus, but Lange had not arrived at a logical system enabling him to use his invention for logical inference in the manner of the Logical Abacus. Another work of Lange is said to contain the first publication of the well known Eulerian diagrams of proposition and syllogism.[14]
Since the first edition was published, an important work by Mr. George Lewes has appeared, namely, his Problems of Life and Mind, which to a great extent treats of scientific method, and formulates the rules of philosophising. I should have liked to discuss the bearing of Mr. Lewes’s views upon those here propounded, but I have felt it to be impossible in a book already filling nearly 800 pages, to enter upon the discussion of a yet more extensive book. For the same reason I have not been able to compare my own treatment of the subject of probability with the views expressed by Mr. Venn in his Logic of Chance. With Mr. J. J. Murphy’s profound and remarkable works on Habit and Intelligence, and on The Scientific Basis of Faith, I was unfortunately unacquainted when I wrote the following pages. They cannot safely be overlooked by any one who wishes to comprehend the tendency of philosophy and scientific method in the present day.
It seems desirable that I should endeavour to answer some of the critics who have pointed out what they consider defects in the doctrines of this book, especially in the first part, which treats of deduction. Some of the notices of the work were indeed rather statements of its contents than critiques. Thus, I am much indebted to M. Louis Liard, Professor of Philosophy at Bordeaux, for the very careful exposition[15] of the substitutional view of logic which he gave in the excellent Revue Philosophique, edited by M. Ribot. (Mars, 1877, tom. iii. p. 277.) An equally careful account of the system was given by M. Riehl, Professor of Philosophy at Graz, in his article on “Die Englische Logik der Gegenwart,” published in the Vierteljahrsschrift für wissenschaftliche Philosophie. (1 Heft, Leipzig, 1876.) I should like to acknowledge also the careful and able manner in which my book was reviewed by the New York Daily Tribune and the New York Times.
The most serious objections which have been brought against my treatment of logic have regard to my failure to enter into an analysis of the ultimate nature and origin of the Laws of Thought. The Spectator,[16] for instance, in the course of a careful review, says of the principle of substitution, “Surely it is a great omission not to discuss whence we get this great principle itself; whether it is a pure law of the mind, or only an approximate lesson of experience; and if a pure product of the mind, whether there are any other products of the same kind, furnished by our knowing faculty itself.” Professor Robertson, in his very acute review,[17] likewise objects to the want of psychological and philosophical analysis. “If the book really corresponded to its title, Mr. Jevons could hardly have passed so lightly over the question, which he does not omit to raise, concerning those undoubted principles of knowledge commonly called the Laws of Thought.... Everywhere, indeed, he appears least at ease when he touches on questions properly philosophical; nor is he satisfactory in his psychological references, as on pp. 4, 5, where he cannot commit himself to a statement without an accompaniment of ‘probably,’ ‘almost,’ or ‘hardly.’ Reservations are often very much in place, but there are fundamental questions on which it is proper to make up one’s mind.”