The text of this book has been preserved as closely as practicable to its original form. However, the author used some unusual symbols, and I have taken the liberty of using Unicode characters with similar appearance (ꖌ ᔕ) as substitutes, disregarding their official meaning and aware that they might not display on all devices. An archaic symbol used by the author to indicate the mathematical ‘factorial’ function has been replaced by the modern equivalent, viz. ! Unusual placements of some sub- and superscripted symbols remain as in the original text.
Inconsistencies of punctuation have been corrected silently, but inconsistent spellings such as Roemer, Römer, Rœmer have not been altered. A list of [corrected spellings] is appended at the end of the book.
>Footnotes have been renumbered consecutively and relocated to the end of the book. A missing footnote marker has been inserted on p.751 after tracking down the original document. A missing negative symbol has been added to an exponent in a formula on p.327.
There is a misleading calculation on p.194 and the table that follows, regarding progressive powers of two: ((22)2)2 is equivalent to (16)2 which equals 256 not 65,356 as stated, but 216 does equal 65,356.
[sic] has been inserted on p.179 alongside a statement that the alphabet contains 24 letters; however, the statement may well be correct given that it was written in 1704 by a Flemish author and the language is not specified.
New original cover art included with this eBook is granted to the public domain.
THE PRINCIPLES OF SCIENCE.
THE LOGICAL MACHINE.
THE PRINCIPLES OF SCIENCE: A TREATISE ON LOGIC AND SCIENTIFIC METHOD.
BY
W. STANLEY JEVONS,
LL.D. (EDINB.), M.A. (LOND.), F.R.S.
London:
MACMILLAN AND CO.
1883.
The Right of Translation and Reproduction is Reserved.
LONDON:
R. Clay, Sons, & Taylor, Printers,
BREAD STREET HILL.
Stereotyped Edition.
PREFACE
TO THE FIRST EDITION.
It may be truly asserted that the rapid progress of the physical sciences during the last three centuries has not been accompanied by a corresponding advance in the theory of reasoning. Physicists speak familiarly of Scientific Method, but they could not readily describe what they mean by that expression. Profoundly engaged in the study of particular classes of natural phenomena, they are usually too much engrossed in the immense and ever-accumulating details of their special sciences to generalise upon the methods of reasoning which they unconsciously employ. Yet few will deny that these methods of reasoning ought to be studied, especially by those who endeavour to introduce scientific order into less successful and methodical branches of knowledge.
The application of Scientific Method cannot be restricted to the sphere of lifeless objects. We must sooner or later have strict sciences of those mental and social phenomena, which, if comparison be possible, are of more interest to us than purely material phenomena. But it is the proper course of reasoning to proceed from the known to the unknown—from the evident to the obscure—from the material and palpable to the subtle and refined. The physical sciences may therefore be properly made the practice-ground of the reasoning powers, because they furnish us with a great body-of precise and successful investigations. In these sciences we meet with happy instances of unquestionable deductive reasoning, of extensive generalisation, of happy prediction, of satisfactory verification, of nice calculation of probabilities. We can note how the slightest analogical clue has been followed up to a glorious discovery, how a rash generalisation has at length been exposed, or a conclusive experimentum crucis has decided the long-continued strife between two rival theories.
In following out my design of detecting the general methods of inductive investigation, I have found that the more elaborate and interesting processes of quantitative induction have their necessary foundation in the simpler science of Formal Logic. The earlier, and probably by far the least attractive part of this work, consists, therefore, in a statement of the so-called Fundamental Laws of Thought, and of the all-important Principle of Substitution, of which, as I think, all reasoning is a development. The whole procedure of inductive inquiry, in its most complex cases, is foreshadowed in the combinational view of Logic, which arises directly from these fundamental principles. Incidentally I have described the mechanical arrangements by which the use of the important form called the Logical Alphabet, and the whole working of the combinational system of Formal Logic, may be rendered evident to the eye, and easy to the mind and hand.
The study both of Formal Logic and of the Theory of Probabilities has led me to adopt the opinion that there is no such thing as a distinct method of induction as contrasted with deduction, but that induction is simply an inverse employment of deduction. Within the last century a reaction has been setting in against the purely empirical procedure of Francis Bacon, and physicists have learnt to advocate the use of hypotheses. I take the extreme view of holding that Francis Bacon, although he correctly insisted upon constant reference to experience, had no correct notions as to the logical method by which from particular facts we educe laws of nature. I endeavour to show that hypothetical anticipation of nature is an essential part of inductive inquiry, and that it is the Newtonian method of deductive reasoning combined with elaborate experimental verification, which has led to all the great triumphs of scientific research.
In attempting to give an explanation of this view of Scientific Method, I have first to show that the sciences of number and quantity repose upon and spring from the simpler and more general science of Logic. The Theory of Probability, which enables us to estimate and calculate quantities of knowledge, is then described, and especial attention is drawn to the Inverse Method of Probabilities, which involves, as I conceive, the true principle of inductive procedure. No inductive conclusions are more than probable, and I adopt the opinion that the theory of probability is an essential part of logical method, so that the logical value of every inductive result must be determined consciously or unconsciously, according to the principles of the inverse method of probability.
The phenomena of nature are commonly manifested in quantities of time, space, force, energy, &c., and the observation, measurement, and analysis of the various quantitative conditions or results involved, even in a simple experiment, demand much employment of systematic procedure. I devote a book, therefore, to a simple and general description of the devices by which exact measurement is effected, errors eliminated, a probable mean result attained, and the probable error of that mean ascertained. I then proceed to the principal, and probably the most interesting, subject of the book, illustrating successively the conditions and precautions requisite for accurate observation, for successful experiment, and for the sure detection of the quantitative laws of nature. As it is impossible to comprehend aright the value of quantitative laws without constantly bearing in mind the degree of quantitative approximation to the truth probably attained, I have devoted a special chapter to the Theory of Approximation, and however imperfectly I may have treated this subject, I must look upon it as a very essential part of a work on Scientific Method.
It then remains to illustrate the sound use of hypothesis, to distinguish between the portions of knowledge which we owe to empirical observation, to accidental discovery, or to scientific prediction. Interesting questions arise concerning the accordance of quantitative theories and experiments, and I point out how the successive verification of an hypothesis by distinct methods of experiment yields conclusions approximating to but never attaining certainty. Additional illustrations of the general procedure of inductive investigations are given in a chapter on the Character of the Experimentalist, in which I endeavour to show, moreover, that the inverse use of deduction was really the logical method of such great masters of experimental inquiry as Newton, Huyghens, and Faraday.
In treating Generalisation and Analogy, I consider the precautions requisite in inferring from one case to another, or from one part of the universe to another part; the validity of all such inferences resting ultimately upon the inverse method of probabilities. The treatment of Exceptional Phenomena appeared to afford an interesting subject for a further chapter illustrating the various modes in which an outstanding fact may eventually be explained. The formal part of the book closes with the subject of Classification, which is, however, very inadequately treated. I have, in fact, almost restricted myself to showing that all classification is fundamentally carried out upon the principles of Formal Logic and the Logical Alphabet described at the outset.
In certain concluding remarks I have expressed the conviction which the study of Logic has by degrees forced upon my mind, that serious misconceptions are entertained by some scientific men as to the logical value of our knowledge of nature. We have heard much of what has been aptly called the Reign of Law, and the necessity and uniformity of natural forces has been not uncommonly interpreted as involving the non-existence of an intelligent and benevolent Power, capable of interfering with the course of natural events. Fears have been expressed that the progress of Scientific Method must therefore result in dissipating the fondest beliefs of the human heart. Even the ‘Utility of Religion’ is seriously proposed as a subject of discussion. It seemed to be not out of place in a work on Scientific Method to allude to the ultimate results and limits of that method. I fear that I have very imperfectly succeeded in expressing my strong conviction that before a rigorous logical scrutiny the Reign of Law will prove to be an unverified hypothesis, the Uniformity of Nature an ambiguous expression, the certainty of our scientific inferences to a great extent a delusion. The value of science is of course very high, while the conclusions are kept well within the limits of the data on which they are founded, but it is pointed out that our experience is of the most limited character compared with what there is to learn, while our mental powers seem to fall infinitely short of the task of comprehending and explaining fully the nature of any one object. I draw the conclusion that we must interpret the results of Scientific Method in an affirmative sense only. Ours must be a truly positive philosophy, not that false negative philosophy which, building on a few material facts, presumes to assert that it has compassed the bounds of existence, while it nevertheless ignores the most unquestionable phenomena of the human mind and feelings.
It is approximately certain that in freely employing illustrations drawn from many different sciences, I have frequently fallen into errors of detail. In this respect I must throw myself upon the indulgence of the reader, who will bear in mind, as I hope, that the scientific facts are generally mentioned purely for the purpose of illustration, so that inaccuracies of detail will not in the majority of cases affect the truth of the general principles illustrated.
December 15, 1873.
PREFACE
TO THE SECOND EDITION.
Few alterations of importance have been made in preparing this second edition. Nevertheless, advantage has been taken of the opportunity to revise very carefully both the language and the matter of the book. Correspondents and critics having pointed out inaccuracies of more or less importance in the first edition, suitable corrections and emendations have been made. I am under obligations to Mr. C. J. Monro, M.A., of Barnet, and to Mr. W. H. Brewer, M.A., one of Her Majesty’s Inspectors of Schools, for numerous corrections.
Among several additions which have been made to the text, I may mention the abstract (p. [143]) of Professor Clifford’s remarkable investigation into the number of types of compound statement involving four classes of objects. This inquiry carries forward the inverse logical problem described in the preceding sections. Again, the need of some better logical method than the old Barbara Celarent, &c., is strikingly shown by Mr. Venn’s logical problem, described at p. [90]. A great number of candidates in logic and philosophy were tested by Mr. Venn with this problem, which, though simple in reality, was solved by very few of those who were ignorant of Boole’s Logic. Other evidence could be adduced by Mr. Venn of the need for some better means of logical training. To enable the logical student to test his skill in the solution of inductive logical problems, I have given (p. [127]) a series of ten problems graduated in difficulty.
To prevent misapprehension, it should be mentioned that, throughout this edition, I have substituted the name Logical Alphabet for Logical Abecedarium, the name applied in the first edition to the exhaustive series of logical combinations represented in terms of A, B, C, D (p. [94]). It was objected by some readers that Abecedarium is a long and unfamiliar name.
To the chapter on Units and Standards of Measurement, I have added two sections, one (p. [325]) containing a brief statement of the Theory of Dimensions, and the other (p. [319]) discussing Professor Clerk Maxwell’s very original suggestion of a Natural System of Standards for the measurement of space and time, depending upon the length and rapidity of waves of light.
In my description of the Logical Machine in the Philosophical Transactions (vol. 160, p. 498), I said—“It is rarely indeed that any invention is made without some anticipation being sooner or later discovered; but up to the present time I am totally unaware of even a single previous attempt to devise or construct a machine which should perform the operations of logical inference; and it is only, I believe, in the satirical writings of Swift that an allusion to an actual reasoning machine is to be found.” Before the paper was printed, however, I was able to refer (p. 518) to the ingenious designs of the late Mr. Alfred Smee as attempts to represent thought mechanically. Mr. Smee’s machines indeed were never constructed, and, if constructed, would not have performed actual logical inference. It has now just come to light, however, that the celebrated Lord Stanhope actually did construct a mechanical device, capable of representing syllogistic inferences in a concrete form. It appears that logic was one of the favourite studies of this truly original and ingenious nobleman. There remain fragments of a logical work, printed by the Earl at his own press, which show that he had arrived, before the year 1800, at the principle of the quantified predicate. He puts forward this principle in the most explicit manner, and proposes to employ it throughout his syllogistic system. Moreover, he converts negative propositions into affirmative ones, and represents these by means of the copula “is identic with.” Thus he anticipated, probably by the force of his own unaided insight, the main points of the logical method originated in the works of George Bentham and George Boole, and developed in this work. Stanhope, indeed, has no claim to priority of discovery, because he seems never to have published his logical writings, although they were put into print. There is no trace of them in the British Museum Library, nor in any other library or logical work, so far as I am aware. Both the papers and the logical contrivance have been placed by the present Earl Stanhope in the hands of the Rev. Robert Harley, F.R.S., who will, I hope, soon publish a description of them.[1]
By the kindness of Mr. Harley, I have been able to examine Stanhope’s logical contrivance, called by him the Demonstrator. It consists of a square piece of bay-wood with a square depression in the centre, across which two slides can be pushed, one being a piece of red glass, and the other consisting of wood coloured gray. The extent to which each of these slides is pushed in is indicated by scales and figures along the edges of the aperture, and the simple rule of inference adopted by Stanhope is: “To the gray add the red and subtract the holon,” meaning by holon (ὅλον) the whole width of the aperture. This rule of inference is a curious anticipation of De Morgan’s numerically definite syllogism (see below, p. [168]), and of inferences founded on what Hamilton called “Ultra-total distribution.” Another curious point about Stanhope’s device is, that one slide can be drawn out and pushed in again at right angles to the other, and the overlapping part of the slides then represents the probability of a conclusion, derived from two premises of which the probabilities are respectively represented by the projecting parts of the slides. Thus it appears that Stanhope had studied the logic of probability as well as that of certainty, here again anticipating, however obscurely, the recent progress of logical science. It will be seen, however, that between Stanhope’s Demonstrator and my Logical Machine there is no resemblance beyond the fact that they both perform logical inference.
In the first edition I inserted a section (vol. i. p. 25), on “Anticipations of the Principle of Substitution,” and I have reprinted that section unchanged in this edition (p. [21]). I remark therein that, “In such a subject as logic it is hardly possible to put forth any opinions which have not been in some degree previously entertained. The germ at least of every doctrine will be found in earlier writings, and novelty must arise chiefly in the mode of harmonising and developing ideas.” I point out, as Professor T. M. Lindsay had previously done, that Beneke had employed the name and principle of substitution, and that doctrines closely approximating to substitution were stated by the Port Royal Logicians more than 200 years ago.
I have not been at all surprised to learn, however, that other logicians have more or less distinctly stated this principle of substitution during the last two centuries. As my friend and successor at Owens College, Professor Adamson, has discovered, this principle can be traced back to no less a philosopher than Leibnitz.
The remarkable tract of Leibnitz,[2] entitled “Non inelegans Specimen Demonstrandi in Abstractis,” commences at once with a definition corresponding to the principle:—
“Eadem sunt quorum unum potest substitui alteri salva veritate. Si sint A et B, et A ingrediatur aliquam propositionem veram, et ibi in aliquo loco ipsius A pro ipso substituendo B fiat nova propositio æque itidem vera, idque semper succedat in quacunque tali propositione, A et B dicuntur esse eadem; et contra, si eadem sint A et B, procedet substitutio quam dixi.”
Leibnitz, then, explicitly adopts the principle of substitution, but he puts it in the form of a definition, saying that those things are the same which can be substituted one for the other, without affecting the truth of the proposition. It is only after having thus tested the sameness of things that we can turn round and say that A and B, being the same, may be substituted one for the other. It would seem as if we were here in a vicious circle; for we are not allowed to substitute A for B, unless we have ascertained by trial that the result is a true proposition. The difficulty does not seem to be removed by Leibnitz’ proviso, “idque semper succedat in quacunque tali propositione.” How can we learn that because A and B may be mutually substituted in some propositions, they may therefore be substituted in others; and what is the criterion of likeness of propositions expressed in the word “tali”? Whether the principle of substitution is to be regarded as a postulate, an axiom, or a definition, is just one of those fundamental questions which it seems impossible to settle in the present position of philosophy, but this uncertainty will not prevent our making a considerable step in logical science.
Leibnitz proceeds to establish in the form of a theorem what is usually taken as an axiom, thus (Opera, p. 95): “Theorema I. Quæ sunt eadem uni tertio, eadem sunt inter se. Si A ∝ B et B ∝ C, erit A ∝ C. Nam si in propositione A ∝ B (vera ea hypothesi) substituitur C in locum B (quod facere licet per Def. I. quia B ∝ C ex hypothesi) fiet A ∝ C. Q. E. Dem.” Thus Leibnitz precisely anticipates the mode of treating inference with two simple identities described at p. 51 of this work.
Even the mathematical axiom that ‘equals added to equals make equals,’ is deduced from the principle of substitution. At p. 95 of Erdmann’s edition, we find: “Si eidem addantur coincidentia fiunt coincidentia. Si A ∝ B, erit A + C ∝ B + C. Nam si in propositione A + C ∝ A + C (quæ est vera per se) pro A semel substituas B (quod facere licet per Def. I. quia A ∝ B) fiet A + C ∝ B + C Q. E. Dem.” This is unquestionably the mode of deducing the several axioms of mathematical reasoning from the higher axiom of substitution, which is explained in the section on mathematical inference (p. [162]) in this work, and which had been previously stated in my Substitution of Similars, p. 16.
There are one or two other brief tracts in which Leibnitz anticipates the modern views of logic. Thus in the eighteenth tract in Erdmann’s edition (p. 92), called “Fundamenta Calculi Ratiocinatoris”, he says: “Inter ea quorum unum alteri substitui potest, salvis calculi legibus, dicetur esse æquipollentiam.” There is evidence, also, that he had arrived at the quantification of the predicate, and that he fully understood the reduction of the universal affirmative proposition to the form of an equation, which is the key to an improved view of logic. Thus, in the tract entitled “Difficultates Quædam Logicæ,”[3] he says: “Omne A est B; id est æquivalent AB et A, seu A non B est non-ens.”
It is curious to find, too, that Leibnitz was fully acquainted with the Laws of Commutativeness and “Simplicity” (as I have called the second law) attaching to logical symbols. In the “Addenda ad Specimen Calculi Universalis” we read as follows.[4] “Transpositio literarum in eodem termino nihil mutat, ut ab coincidet cum ba, seu animal rationale et rationale animal.”
“Repetitio ejusdem literæ in eodem termino est inutilis, ut b est aa; vel bb est a; homo est animal animal, vel homo homo est animal. Sufficit enim dici a est b, seu homo est animal.”
Comparing this with what is stated in Boole’s Mathematical Analysis of Logic, pp. 17–18, in his Laws of Thought, p. 29, or in this work, pp. [32]–35, we find that Leibnitz had arrived two centuries ago at a clear perception of the bases of logical notation. When Boole pointed out that, in logic, xx = x, this seemed to mathematicians to be a paradox, or in any case a wholly new discovery; but here we have it plainly stated by Leibnitz.
The reader must not assume, however, that because Leibnitz correctly apprehended the fundamental principles of logic, he left nothing for modern logicians to do. On the contrary, Leibnitz obtained no useful results from his definition of substitution. When he proceeds to explain the syllogism, as in the paper on “Definitiones Logicæ,”[5] he gives up substitution altogether, and falls back upon the notion of inclusion of class in class, saying, “Includens includentis est includens inclusi, seu si A includit B et B includit C, etiam A includet C.” He proceeds to make out certain rules of the syllogism involving the distinction of subject and predicate, and in no important respect better than the old rules of the syllogism. Leibnitz’ logical tracts are, in fact, little more than brief memoranda of investigations which seem never to have been followed out. They remain as evidence of his wonderful sagacity, but it would be difficult to show that they have had any influence on the progress of logical science in recent times.
I should like to explain how it happened that these logical writings of Leibnitz were unknown to me, until within the last twelve months. I am so slow a reader of Latin books, indeed, that my overlooking a few pages of Leibnitz’ works would not have been in any case surprising. But the fact is that the copy of Leibnitz’ works of which I made occasional use, was one of the edition of Dutens, contained in Owens College Library. The logical tracts in question were not printed in that edition, and with one exception, they remained in manuscript in the Royal Library at Hanover, until edited by Erdmann, in 1839–40. The tract “Difficultates Quædam Logicæ,” though not known to Dutens, was published by Raspe in 1765, in his collection called Œuvres Philosophiques de feu Mr. Leibnitz; but this work had not come to my notice, nor does the tract in question seem to contain any explicit statement of the principle of substitution.
It is, I presume, the comparatively recent publication of Leibnitz’ most remarkable logical tracts which explains the apparent ignorance of logicians as regards their contents and importance. The most learned logicians, such as Hamilton and Ueberweg, ignore Leibnitz’ principle of substitution. In the Appendix to the fourth volume of Hamilton’s Lectures on Metaphysics and Logic, is given an elaborate compendium of the views of logical writers concerning the ultimate basis of deductive reasoning. Leibnitz is briefly noticed on p. 319, but without any hint of substitution. He is here quoted as saying, “What are the same with the same third, are the same with each other; that is, if A be the same with B, and C be the same with B, it is necessary that A and C should also be the same with one another. For this principle flows immediately from the principle of contradiction, and is the ground and basis of all logic; if that fail, there is no longer any way of reasoning with certainty.” This view of the matter seems to be inconsistent with that which he adopted in his posthumous tract.
Dr. Thomson, indeed, was acquainted with Leibnitz’ tracts, and refers to them in his Outline of the Necessary Laws of Thought. He calls them valuable; nevertheless, he seems to have missed the really valuable point; for in making two brief quotations,[6] he omits all mention of the principle of substitution.
Ueberweg is probably considered the best authority concerning the history of logic, and in his well-known System of Logic and History of Logical Doctrines,[7] he gives some account of the principle of substitution, especially as it is implicitly stated in the Port Royal Logic. But he omits all reference to Leibnitz in this connection, nor does he elsewhere, so far as I can find, supply the omission. His English editor, Professor T. M. Lindsay, in referring to my Substitution of Similars, points out how I was anticipated by Beneke; but he also ignores Leibnitz. It is thus apparent that the most learned logicians, even when writing especially on the history of logic, displayed ignorance of Leibnitz’ most valuable logical writings.
It has been recently pointed out to me, however, that the Rev. Robert Harley did draw attention, at the Nottingham Meeting of the British Association, in 1866, to Leibnitz’ anticipations of Boole’s laws of logical notation,[8] and I am informed that Boole, about a year after the publication of his Laws of Thought, was made acquainted with these anticipations by R. Leslie Ellis.
There seems to have been at least one other German logician who discovered, or adopted, the principle of substitution. Reusch, in his Systema Logicum, published in 1734, laboured to give a broader basis to the Dictum de Omni et Nullo. He argues, that “the whole business of ordinary reasoning is accomplished by the substitution of ideas in place of the subject or predicate of the fundamental proposition. This some call the equation of thoughts.” But, in the hands of Reusch, substitution does not seem to lead to simplicity, since it has to be carried on according to the rules of Equipollence, Reciprocation, Subordination, and Co-ordination.[9] Reusch is elsewhere spoken of[10] as the “celebrated Reusch”; nevertheless, I have not been able to find a copy of his book in London, even in the British Museum Library; it is not mentioned in the printed catalogue of the Bodleian Library; Messrs. Asher have failed to obtain it for me by advertisement in Germany; and Professor Adamson has been equally unsuccessful. From the way in which the principle of substitution is mentioned by Reusch, it would seem likely that other logicians of the early part of the eighteenth century were acquainted with it; but, if so, it is still more curious that recent historians of logical science have overlooked the doctrine.
It is a strange and discouraging fact, that true views of logic should have been discovered and discussed from one to two centuries ago, and yet should have remained, like George Bentham’s work in this century, without influence on the subsequent progress of the science. It may be regarded as certain that none of the discoverers of the quantification of the predicate, Bentham, Hamilton, Thomson, De Morgan, and Boole, were in any way assisted by the hints of the principle contained in previous writers. As to my own views of logic, they were originally moulded by a careful study of Boole’s works, as fully stated in my first logical essay.[11] As to the process of substitution, it was not learnt from any work on logic, but is simply the process of substitution perfectly familiar to mathematicians, and with which I necessarily became familiar in the course of my long-continued study of mathematics under the late Professor De Morgan.
I find that the Theory of Number, which I explained in the eighth chapter of this work, is also partially anticipated in a single scholium of Leibnitz. He first gives as an axiom the now well-known law of Boole, as follows:—
“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.”
These remarks appear to me to be well founded, and I must state why it is that I have ventured to publish an extensive work on logic, without properly making up my mind as to the fundamental nature of the reasoning process. The fault after all is one of omission rather than of commission. It is open to me on a future occasion to supply the deficiency if I should ever feel able to undertake the task. But I do not conceive it to be an essential part of any treatise to enter into an ultimate analysis of its subject matter. Analyses must always end somewhere. There were good treatises on light which described the laws of the phenomenon correctly before it was known whether light consisted of undulations or of projected particles. Now we have treatises on the Undulatory Theory which are very valuable and satisfactory, although they leave us in almost complete doubt as to what the vibrating medium really is. So I think that, in the present day, we need a correct and scientific exhibition of the formal laws of thought, and of the forms of reasoning based on them, although we may not be able to enter into any complete analysis of the nature of those laws. What would the science of geometry be like now if the Greek geometers had decided that it was improper to publish any propositions before they had decided on the nature of an axiom? Where would the science of arithmetic be now if an analysis of the nature of number itself were a necessary preliminary to a development of the results of its laws? In recent times there have been enormous additions to the mathematical sciences, but very few attempts at psychological analysis. In the Alexandrian and early mediæval schools of philosophy, much attention was given to the nature of unity and plurality chiefly called forth by the question of the Trinity. In the last two centuries whole sciences have been created out of the notion of plurality, and yet speculation on the nature of plurality has dwindled away. This present treatise contains, in the eighth chapter, one of the few recent attempts to analyse the notion of number itself.
If further illustration is needed, I may refer to the differential calculus. Nobody calls in question the formal truth of the results of that calculus. All the more exact and successful parts of physical science depend upon its use, and yet the mathematicians who have created so great a body of exact truths have never decided upon the basis of the calculus. What is the nature of a limit or the nature of an infinitesimal? Start the question among a knot of mathematicians, and it will be found that hardly two agree, unless it is in regarding the question itself as a trifling one. Some hold that there are no such things as infinitesimals, and that it is all a question of limits. Others would argue that the infinitesimal is the necessary outcome of the limit, but various shades of intermediate opinion spring up.
Now it is just the same with logic. If the forms of deductive and inductive reasoning given in the earlier part of this treatise are correct, they constitute a definite addition to logical science, and it would have been absurd to decline to publish such results because I could not at the same time decide in my own mind about the psychology and philosophy of the subject. It comes in short to this, that my book is a book on Formal Logic and Scientific Method, and not a book on psychology and philosophy.
It may be objected, indeed, as the Spectator objects, that Mill’s System of Logic is particularly strong in the discussion of the psychological foundations of reasoning, so that Mill would appear to have successfully treated that which I feel myself to be incapable of attempting at present. If Mill’s analysis of knowledge is correct, then I have nothing to say in excuse for my own deficiencies. But it is well to do one thing at a time, and therefore I have not occupied any considerable part of this book with controversy and refutation. What I have to say of Mill’s logic will be said in a separate work, in which his analysis of knowledge will be somewhat minutely analysed. It will then be shown, I believe, that Mill’s psychological and philosophical treatment of logic has not yielded such satisfactory results as some writers seem to believe.[18]
Various minor but still important criticisms were made by Professor Robertson, a few of which have been noticed in the text (pp. [27], [101]). In other cases his objections hardly admit of any other answer than such as consists in asking the reader to judge between the work and the criticism. Thus Mr. Robertson asserts[19] that the most complex logical problems solved in this book (up to p. 102 of this edition) might be more easily and shortly dealt with upon the principles and with the recognised methods of the traditional logic. The burden of proof here lies upon Mr. Robertson, and his only proof consists in a single case, where he is able, as it seems to me accidentally, to get a special conclusion by the old form of dilemma. It would be a long labour to test the old logic upon every result obtained by my notation, and I must leave such readers as are well acquainted with the syllogistic logic to pronounce upon the comparative simplicity and power of the new and old systems. For other acute objections brought forward by Mr. Robertson, I must refer the reader to the article in question.
One point in my last chapter, that on the Results and Limits of Scientific Method, has been criticised by Professor W. K. Clifford in his lecture[20] on “The First and the Last Catastrophe.” In vol. ii. p. 438 of the first edition (p. [744] of this edition) I referred to certain inferences drawn by eminent physicists as to a limit to the antiquity of the present order of things. “According to Sir W. Thomson’s deductions from Fourier’s theory of heat, we can trace down the dissipation of heat by conduction and radiation to an infinitely distant time when all things will be uniformly cold. But we cannot similarly trace the Heat-history of the Universe to an infinite distance in the past. For a certain negative value of the time, the formulæ give impossible values, indicating that there was some initial distribution of heat which could not have resulted, according to known laws of nature, from any previous distribution.”
Now according to Professor Clifford I have here misstated Thomson’s results. “It is not according to the known laws of nature, it is according to the known laws of conduction of heat, that Sir William Thomson is speaking. . . . All these physical writers, knowing what they were writing about, simply drew such conclusions from the facts which were before them as could be reasonably drawn. They say, here is a state of things which could not have been produced by the circumstances we are at present investigating. Then your speculator comes, he reads a sentence and says, ‘Here is an opportunity for me to have my fling.’ And he has his fling, and makes a purely baseless theory about the necessary origin of the present order of nature at some definite point of time, which might be calculated.”
Professor Clifford proceeds to explain that Thomson’s formulæ only give a limit to the heat history of, say, the earth’s crust in the solid state. We are led back to the time when it became solidified from the fluid condition. There is discontinuity in the history of the solid matter, but still discontinuity which is within our comprehension. Still further back we should come to discontinuity again, when the liquid was formed by the condensation of heated gaseous matter. Beyond that event, however, there is no need to suppose further discontinuity of law, for the gaseous matter might consist of molecules which had been falling together from different parts of space through infinite past time. As Professor Clifford says (p. 481) of the bodies of the universe, “What they have actually done is to fall together and get solid. If we should reverse the process we should see them separating and getting cool, and as a limit to that, we should find that all these bodies would be resolved into molecules, and all these would be flying away from each other. There would be no limit to that process, and we could trace it as far back as ever we liked to trace it.”
Assuming that I have erred, I should like to point out that I have erred in the best company, or more strictly, being a speculator, I have been led into error by the best physical writers. Professor Tait, in his Sketch of Thermodynamics, speaking of the laws discovered by Fourier for the motion of heat in a solid, says, “Their mathematical expressions point also to the fact that a uniform distribution of heat, or a distribution tending to become uniform, must have arisen from some primitive distribution of heat of a kind not capable of being produced by known laws from any previous distribution.” In the latter words it will be seen that there is no limitation to the laws of conduction, and, although I had carefully referred to Sir W. Thomson’s original paper, it is not unnatural that I should take Professor Tait’s interpretation of its meaning.[21]
In his new work On some Recent Advances in Physical Science, Professor Tait has recurred to the subject as follows:[22] “A profound lesson may be learned from one of the earliest little papers of Sir W. Thomson, published while he was an undergraduate at Cambridge, where he shows that Fourier’s magnificent treatment of the conduction of heat [in a solid body] leads to formulæ for its distribution which are intelligible (and of course capable of being fully verified by experiment) for all time future, but which, except in particular cases, when extended to time past, remain intelligible for a finite period only, and then indicate a state of things which could not have resulted under known laws from any conceivable previous distribution [of heat in the body]. So far as heat is concerned, modern investigations have shown that a previous distribution of the matter involved may, by its potential energy, be capable of producing such a state of things at the moment of its aggregation; but the example is now adduced not for its bearing on heat alone, but as a simple illustration of the fact that all portions of our Science, especially that beautiful one, the Dissipation of Energy, point unanimously to a beginning, to a state of things incapable of being derived by present laws [of tangible matter and its energy] from any conceivable previous arrangement.” As this was published nearly a year after Professor Clifford’s lecture, it may be inferred that Professor Tait adheres to his original opinion that the theory of heat does give evidence of “a beginning.”
I may add that Professor Clerk Maxwell’s words seem to countenance the same view, for he says,[23] “This is only one of the cases in which a consideration of the dissipation of energy leads to the determination of a superior limit to the antiquity of the observed order of things.” The expression “observed order of things” is open to much ambiguity, but in the absence of qualification I should take it to include the aggregate of the laws of nature known to us. I should interpret Professor Maxwell as meaning that the theory of heat indicates the occurrence of some event of which our science cannot give any further explanation. The physical writers thus seem not to be so clear about the matter as Professor Clifford assumes.
So far as I may venture to form an independent opinion on the subject, it is to the effect that Professor Clifford is right, and that the known laws of nature do not enable us to assign a “beginning.” Science leads us backwards into infinite past duration. But that Professor Clifford is right on this point, is no reason why we should suppose him to be right in his other opinions, some of which I am sure are wrong. Nor is it a reason why other parts of my last chapter should be wrong. The question only affects the single paragraph on pp. [744]–5 of this book, which might, I believe, be struck out without necessitating any alteration in the rest of the text. It is always to be remembered that the failure of an argument in favour of a proposition does not, generally speaking, add much, if any, probability to the contradictory proposition. I cannot conclude without expressing my acknowledgments to Professor Clifford for his kind expressions regarding my work as a whole.
2, The Chestnuts,
West Heath,
Hampstead, N. W.
August 15, 1877.