Pure logic and other minor works

Preface

In the present volume are contained some of the earliest and some of the latest contributions of the author to the science of logic. The several works and papers that have been included fall into two distinctly marked groups, corresponding on the whole to the difference in time of their composition, and presenting at first sight no very clear indication of common aim or principle. The first group consists of independently published works, the Pure Logic and the Substitution of Similars, with certain papers in which the general principles of the author's logical theory, stated in their most mature form in his main logical works, the Principles of Science and the Studies in Deductive Logic, are carried out in certain special directions. In the second group appear such portions of a general examination of J. S. Mill’s philosophy as had been completed at the date of the author’s death.

The essays in the first group seem to claim a place in a permanent collection of the author’s works, partly from the historic importance that must always attach to them, partly by reason of the relative fulness of discussion which they extend to certain fundamental and characteristic ideas in the general view of logic contained in them. They are reprinted as they originally appeared, with only the correction of one or two obvious misprints. In the author’s copy al PREFACE of the Pure Logic an alteration was introduced in his own hand on § 146. For the expressions there given of De Morgan’s two types of propositional form, viz.

Everything is either A or B A=b Some things are neither A nor B C0:

There are substituted—_ UA=b Ua =0U.

It has not been thought desirable to introduce into the text these alterations.

The arrangement of the four essays is chronological. Pure Logic or the Logic of Quality Apart from Quantity, with Remarks on Boole’s System, and on the Relation of Logie to Mathematics was published in 1864. The Substitution of Similars, the True Principle of Reasoning, Derwed from a Modification of Aristotles Dictum, was published in 1869. The memoir On the Mechanical Performance of Logical Inference was received by the Royal Society 16th October 1869, and read 20th January 1870. The paper On a General System of Numerically Definite Reasoning was communicated in 1870 to the Literary and Philosophical Society of Manchester.’

There have not been included in this collection one or two contributions of less extent to logical science, the contents of which have been reproduced in more complete fashion in other works of the author. A communication to the Manchester Literary and Philosophical Society on 3d April 1866 deals with the Logical Abacus, which is elaborately explained in the later work, Substetutcon of Similars. A paper On the Inverse, or Inductive, Logical Problem (Memoirs of the Literary and Philosophical Society of

1 As regards some points in this paper, see Principles of Science, second edition, p. 172.

PREFACE _ vil

Manchester, 1871-72), is incorporated with no substantive — alteration in the Principles of Science, 1877, pp. 137-143. A short paper entitled Who Discovered the Quantification of the Predwate ? appeared in the Contemporary Review, May 1873, pp. 821-824. It emphasises strongly the claims of Mr. George Bentham to recognition as the first English writer on Logic who had stated and applied the principle of Quantification. The paper is a comment on an article by Professor T. 8. Baynes in the same journal, and as it formed part of a somewhat extended controversy, it has been judged needless to reprint it here.

The author himself attached so much weight to his critical examination of J. S. Mill’s doctrines, and the labour bestowed on it played so large a part in the last ten or twelve years of his life, that the editors greatly regret their inability to add much to the four papers published during the author’s lifetime in the Contenvporary Review. From the large mass of MS. material for the further portions of the work, a general idea may be gathered as to the special lines of criticism that would have been followed; but with one or two exceptions no portion seemed to be in such a state that the editors without hesitation could print it as expressing the authors views. The four papers published by the author himself appeared in the Contemporary Review, Dec. 1877, Jan. 1878, April 1878, and Nov. 1879, under the title John Stuart Mill's Philosophy Tested. The fragment from the author’s MSS. on The Method of Difference deals with the point referred to by him in the third of these articles. It is proposed here to indicate the relation in which the criticism of Mill stands to the author’s logical theory, and to give a brief connected account of the lines it was to have followed.

The distinguishing feature of the doctrine first expounded

Vill PREFACE

in the Pure Logic is the restriction of logical treatment to the qualitative aspect of thoughts. It was in this that the author found his difference from the system of Boole to consist; and from this follow the main features that characterise his logical method as a whole. There are certain consequences fairly involved in it, moreover, which though not formally stated by the author himself, are yet implied in his general theory of logic, and from which the precise ground of his emphatic dissent from Mill may be made tolerably clear. There is so much of agreement between the views of Mull and those expressed by the author in regard to the ultimate foundation of knowledge, both seeming to follow the lines of English empirical philosophy, that the strenuous dissent of the author from Mill’s whole doctrine of logic at first presents a problem. But if the conception of a logic of quality be followed out, it will be found to involve an opposition between pure or formal logic and the theory of empirical or concrete reasoning, which for Mill constitutes logic, so sharp as to be irreconcileable. One or the other must yield. Briefly, it is the first that has pre-eminence in Jevons’s view, the second is the all-important with Mill. Only within the limits of the first is there, according to Jevons, reasoning characterised by cogency and universality, but within its limits falls no determination of concrete existence; all its contents are, in reference to concrete existence, hypothetical. According to Mill there is no other field for reasoning than that of concrete existence; a pure logic is but a subordinate and relatively valueless abstraction from the actual processes of concrete reasoning.

To restrict reasoning, with its elements, the proposition and term, to quality is a position which seems to connect itself naturally with the familiar logical distinction between the comprehension and extension of notions. In his initial

PREFACE 1x statements (Pure Logic, §§ 1, 3, 4) the author refers to this distinction, and though his expressions are not entirely in accordance with the current logical doctrine, it is easy to understand his position as resting upon it, and to treat his system as an effort to develop logical laws from the side of comprehension. But it may fairly be argued, both from the nature of the case and from many expressions of the author himself, that his system is wholly independent of that distinction, and indeed proceeds from a point of view which renders the distinction meaningless or inapplicable. The distinction as ordinarily expressed makes reference more or less explicitly to classification, and whether the basis be nominalist or conceptualist, leads the logician to contrast terms or notions which possess both comprehension and extension with those which are wanting in one or other aspect. It would seem

_ Impossible to work out consistently from this point without assigning to extension a certain measure of concrete existence, which, when closely investigated, has always been found to involve difficulty. Now it is abundantly evident. that from all the implications of the view which finds expression in the distinction of comprehension from extension, the author’s fundamental positions are entirely free. In his development of logic it is wholly superfluous to recognise the obscure and baffling differences that appear in the exposition of the various kinds of terms. Whatever be their origin, whether grammatical or metaphysical, these differences rest on concrete details of matter, and not on anything in the pure form of thought. The familiar distinctions on which the logician dwells, between Proper or Singular Terms and Common Terms, are of necessity rejected on the ground that quantitative difference is a secondary and derived aspect, not affecting the fundamental laws of thought, and dependent in each case on the special data implied. Ina x PREFACE similar fashion, when the proposition is reached, it becomes evident that what it has to convey is altogether independent of the currently accepted distinctions of universal and particular, and that if these be retained at all, they must be interpreted in accordance with the fundamental laws of thinking that concerns itself with the identity or non-identity of the characterising qualities.

In adopting such a point of view, the logician is by no means prohibited from using forms of expression that seem to imply reference to concrete existences, to classes or single objects marked by the possession of qualities; and it is a further question, on which the author has hardly touched, whether it is possible to formulate the fundamental axioms of thinking in terms which do not imply such hypothetical objects. What is fundamental in his view is that the objects so referred to are hypothetical merely, are determined by the qualities thought as characterising them, and that they are absolutely to be distinguished from real concrete existences. On this account he emphasises so strenuously the position he shares with other logicians who have advanced to the same conclusion from widely different points of view, that in logic there is no question of existence,’ and insists that logically non-existence is equivalent to contradictoriness. Such existence as may be assigned in logic to an object or class of objects is invariably possible existence. Only on material data then, and with reference to a defined universe of objects, could it be concluded that the universe possessed one quality or combination of qualities to the exclusion of its contradictory.

That it is very difficult to retain the position thus taken may at once be admitted, and it is evident that the author at various points of his exposition felt the difficulties. The

1 Studies in Deductive Logic, pp. 141, 142.

al ot”

PREFACE xl particular proposition is an awkward adversary; there is some obscurity in the section of the Pure Logic (§ 146), where De Morgan’s types of propositions are expressed in qualitative fashion; and a little hesitancy is observable in

‘stating and defending the general criterion of consistency which is a legitimate and inevitable consequence of the view (Pure Logic, § 159, note; Studies, p. 181). But it does not seem that he ever wavered in regard to the fundamental position, that logically all judgments are nonexistential.

According to this view, then, there is conceivable a pure logic, the complete statement of the ultimate conditions to which apprehension of qualities, as the same or different, is subject, and of the consequences that flow necessarily from such apprehension. Within the realm of pure logic we have absolute security and certainty. The laws to which thought necessarily conforms in dealing with identical or non-identical contents presented to it are at once the primary and the ultimate tests of truth. From these laws whatsoever absolute truth is attainable must be derived, and absolute certainty can only be assigned to deductions from them. Whenever we pass from the region of pure thought to regions of more concrete matter, we find that such measure of absolute certainty as may there be attainable depends on the completeness of conformity between the most general laws of such matter and the laws of pure thought. The elementary axioms of abstract arithmetic exhibit a perfect correspondence with the laws of pure thought, and are indeed derivatives from these laws. Number rests on logical discrimination or discrimination by pure thought, and its typical forms and laws are derivable from the conditions of pure thought when apphed to the abstract schema of difference. The author has nowhere discussed b xil PREFACE with such fulness as the nature of the question demands, the connection between geometrical axioms and the laws of thought, and the grounds on which are to be rested the certainty and universality of geometrical reasoning,

There follows from the view thus described an immediate consequence which, when expressed as a logical doctrine, has become familiarly associated with the author’s name. All reasoning, or, more exactly, all proof is deductive in character, and involves general propositions of absolute certainty. Proof is the more exact and appropriate term, for it remains within the limits of logical science or pure thought ; whereas the term reasoning is vaguely extended to cover the natural, psychological process of arriving from data at a conclusion,—a process, therefore, of natural fact, about which the psychologist, not the logician, has to concern himself. The generality of the principle involved in any proof is not to be construed after the concrete fashion, as an assertion found to hold good about a number of concrete, and possibly not exhausted, particulars, It is either the quasi-concrete expression of an established identity of qualities, or the summary expression of exhaustive enumeration of a strictly limited collection of instances, It is either a qualitative or a numerical identity. In strictness, indeed, no concrete or synthetical proposition which is more than a collective expression for a determinate number of observed cases, can ever possess certainty. Whenever we pass beyond the pure laws of thought, with their numerical derivatives, or beyond the statement of actually observed fact, we are in the region of assumption and probability, The utmost that can be achieved is to determine with what degree of probability the general assumptions we make can be held. ‘The rules for estimating such probability are numerical in character, and possess all the certainty of the laws of thought.

PREFACE xan

The range of assured knowledge is thus of narrow extent as compared with the indefinite expanse of concrete existence. But within that range it possesses absolute certainty. There and there only is proof possible. So soon as the consideration of concrete existence enters in, we are dependent on the hazardous data of intuitive experience, as to which we can never feel assured that they contain all that is required for complete insight. Within the region of concrete existence, reasoning, in the strict sense, is impossible. All apparent reasoning in which data and conclusions are assertions about concrete fact is hypothetical in character, . and the hypotheses made, however various in concrete expression, are essentially the same in all cases,—viz. assumptions that the concrete propositions correspond perfectly to the conditions of pure abstract thought. It is impossible for us to know accurately that they do correspond; but, knowing the types of valid inference, we can gauge by the rules of probability the extent to which they approximate to the desired correspondence. Induction, then, as ordinarily described, is not a special mode of reasoning. Psychologically, we may no doubt pass in thought without further query from isolated particulars to a generality ; but particulars can in.no way substantiate what is not contained in them. There is only one method of reasoninge—the deductive,— and it 1s used in concrete material as in abstract thought. But in the former case we have to note that our data involve assumptions that cannot be completely justified, and that are only more or less probable in a partially determinable degree.

Now it is the pointed opposition which the author makes between the narrow but secure region of pure, abstract thought and the wide sphere of more or less probable assumptions regarding concrete existence, an opposition in-

Xiv | PREFACE volved in and springing from his conception of pure logic, that constitutes the real difference between his views and those of Mill. In the long run both agree in respect to the character of what is vaguely called ‘knowledge’ of concrete existence. Neither accorded absolute certainty to the propositions composing that ‘knowledge.’ Though Mill undoubtedly uses language in respect to so-called ‘ knowledge’ of nature that implies.the possibility of attaining certainty there, yet it may fairly be urged that such expressions are to be taken with the qualifications necessitated by the general doctrine, that knowledge of nature can never amount to absolute certainty. But,in Mill’s view, this ‘ knowledge’ of nature constituted all our knowledge, and within the realm of concrete existence lay the province and process of reasoning. Apparently abstract propositions, such as those of mathematics, were only abstracted from the concrete particulars,.and had no other foundation for their certainty than the concrete experience from which they were drawn, no superior generality than followed from the relatively greater ease with which they could be disentangled from concrete details. The laws of pure thought, in which Jevons found the rules of absolutely certain knowledge and reasoning, were by Mill regarded as true but rather valueless prescripts defining the use of language, and having no function of significance when divorced from concrete fact. A pure logic, or logic of consistency in the employment of language, he did, indeed, admit to be possible, but in no way accorded to it special importance or viewed it as more than a result of abstraction from actual concrete reasoning. The possibility of an independent foundation for it he did not so much deny as ignore.

The difference is as complete as could well be. To the

1 The most explicit utterance of Mill on these points, which he rarely discusses, are in the Lxamination of Hamilton (third edition), pp. 457, 461.

PREFACE XV one thinker, the theory of reasoning was a well-rounded, independent whole, narrow it might be in extent, but resting on principles of absolute certainty, not necessarily connected with any hypothesis as to the way in which knowledge of concrete fact is obtained or increased, and supplying the final standard by which evidence in concrete matters is to be tested. To the other, the theory of reasoning was part of the general doctrine of the ways in which an intelligence receiving its data as isolated empirical particulars gradually advanced to knowledge, to the establishment of general propositions, and to criticism of the grounds on which these rested. According to the first view, reasoning, in the strict sense of the term, was confined to the region of analytical thinking, and what accompanies reasoning,—viz. stringent cogency of proof, absolute certainty of conclusion, was possible only within that region. According to the other view, reasoning had little or no significance save in synthetical thinking, and its characteristics were such as could be obtained under the conditions of synthetical thinking.

It was inevitable, then, that from the author’s point of view in logical theory, the apparent attempt in Mill’s System of Logic to show how general knowledge in concrete material is gained and established, should present itself as a logical fallacy, doomed from the outset to failure, and only securing the outer aspect of coherence by skilful concealment of underlying inconsistency. For by knowledge the author understood what is perfectly conformable to the pure laws of thought and is warranted completely by them—attainable, therefore, only in analytical thinking. To admit, then as Mill seemed to do, the ultimate uncertainty of what is called knowledge of nature, and, while ignoring the pure laws of thought, to work out a theory of evidence on which knowledge might rest, was to occupy a wholly untenable position.

XV1 PREFACE

As regards the general point of view adopted in the proposed examination of Miull’s philosophy, the motives with which it was undertaken, and the end aimed at, the author has given full explanation in the first of the series of articles contributed to the Contemporary Review. Among his MSS. are found portions of the sections intended to serve as the general introduction to the whole and as a summary conclusion. These contain in somewhat more detail the explanatory matter already given in the first article, and also some indication as to the arrangement of topics to be followed in the completed work. It would appear that after the general introduction the author purposed dealing in succession with Mill’s &ssay on Religion, with his views on Free Will and Necessity, with his peculiar reformation of the Utilitarian Ethics, with his doctrine of Inseparable Association, and then with the main object of attack, the Logical Theory. The criticism of the logic was the central, the fundamental portion of the work ; the other discussions were given as illustrative confirmation of the estimate formed by the author, that Muill’s mind was essentially illogical. It cannot however be determined, from the MSS., in what way these sections would have been arranged, nor is there any indication given as to the final order in which the various topics falling under logical theory would have been taken.

Taking first into consideration the sections not specifically logical, we find among the MSS. collected materials, with occasional written out portions of statement, relating to Mill’s Essay on eligion, to his view on Free Will, and to his doctrine of Inseparable Association. As regards the Hssay on Religion, it is clear from the fragments, as also from a reference to the topic in the first article in the Contemporary Review, that the author’s intention was to dwell upon and

PREFACE XvVil enforce the apparent inconsistency between the two passages quoted from the essay in the said article.

The section on Free Will and Necessity begins by a fairly written out statement of Sir William Hamilton’s familiar _ position, that either conception, Liberty or Necessity, leads when developed to an ultimate inconceivability, a position which, on the whole, seems to have been accepted by the author. The remaining MSS. on the subject, in fragmentary remarks and references to passages in Mill’s writings, enable us to see the drift of his argument, that Mill, while professing the Determinist view, makes admissions which are wholly irreconcilable with it and practically nullify it. The specific admissions singled out for closer scrutiny are two in number, closely connected, the one more psychological, the other more metaphysical in character. The first is the view expressed most definitely in the System of Logic (Book vi, chap. 11), that while the actions of the individual can only be regarded as the outcome of his character, desire to modify that character must be recognised as a factor in its formation. The second is contained in the distinction drawn in the posthumous Essays” between Nature as the entire system of things with all their attributes and Nature as that which is apart from human intervention. The author’s intention seems to have been to press the arguments that, according to the principles on which Mill proceeds, desire to modify character must be treated like all other desires as a derivative and determined fact, not as original and determining, and that, even if for one purpose man be severed from nature

1 Three Essays on Religion, p. 109. ‘The essence of religion is the strong and earnest direction of the emotions and desires towards an ideal object, recognised as of the highest excellence, and as rightfully paramount over all selfish objects of desire.’ Jbid. p. 103—‘ Religion, as distinguished from poetry, is the product of the craving to know whether these imaginative conceptions have realities answering to them in some other world than ours.’

ll as LF

XViil PREFACE and regarded as intervening in it, on Mill’s general principles, his activity of intervention can be contemplated in no other light than as a fact subject to natural law. If any other significance be assigned to the desire or intervening activity, the only result is utter inconsistency of theory. Mill’s ethical theory is criticised, so far as the statement of Utilitarian doctrine is concerned, in the fourth of the articles published in the Contemporary Review. The MS. fragments entitled Morals indicate that the author purposed also discussing Mill’s view on the relation between Intention and Motive, but do not suffice to yield a fairly precise statement of the points aimed at.

Under the heading Metaphysics a criticism is projected of the explanation Mill offers of Necessary Truths. The fragments show the author’s intention to have been to examine the doctrine of Inseparable Association with special reference to the fundamental axioms of number, with a view to exposing the inconsistency between the character allowed to these axioms by Mill and the theory of their empirical origin through inseparable association.

A very large portion of the MSS. is concerned with Mill’s logical theory. The several sections indicate, no doubt, the arrangement of topics to be made in the final treatment. We have been unable to find a definite statement as to this arrangement, and the order in which they are here referred to rests only on occasional expressions of the author. On the whole they fall into two main groups, those dealing with Mill’s theory of the syllogism, those which criticise the doctrine of causation and the experimental methods. To the first group belong the sections headed Systen. of Logic, Petitio Princip, General Propositions, Particulars to Particulars. The fragments forming these sections are evidently of various dates, and so far as any

PREFACE xix fairly written out portions are concerned, they frequently overlap one another. The following seems to have been the main line of criticism :—

Beginning with a pointed reference to the extreme inconsistency between the admitted novelty of Mill’s view of ‘Syllogism and the disclaimer of any novelty made in the Preface to the System of Logic, the author dwells on the general conception of the function of Logic as expressed by Mill. ‘Logic is the Science of Proof or Evidence. In so far as belief professes to be founded on proof, the office of Logic is to supply a test for ascertaining whether or not the belief is well founded.’ ‘ Logic neither observes, nor invents, nor discovers; but judges.’ Agreeing on the whole with this conception of the function of logic, but indicating a doubt as to whether even within the limits of the introduction to the System of Logic, Mill does not use expressions irreconcilable with it, the author proceeds to say that he will begin by showing that ‘ Mill upholds at one and the same time the four following doctrines :—(1) Logic is the science of Proof or Evidence. (2) The Syllogism is the only mode of reasoning by which we can assure ourselves of the correctness of an inference. (3) The Syllogism is nevertheless entirely optional, and imposed by the “ arbitrary fiat” of logicians. (4) The Syllogism is at the same time necessarily a fallacy of the kind called Petitio Principii.’”

Turning to the fourth of these doctrines the author states fully Mill’s familiar position, and urges in opposition to it that the conclusion is not contained in the major premiss but only in the major and minor combined. At the same time he recognises that such an answer by no means exhausts the question, and that the real foundation for Mill’s view of

1 System of Logie (seventh edition), vol. i, pp. 8, 9. 2 From the author’s MS.

XX PREPACLE the syllogism is to be sought in his treatment of the general proposition. From the passages selected for comment and from the fragments of criticism on them, it is evident that the author intended to insist that Mill regards the general proposition in two aspects, distinct and wholly irreconcilable, and that his inconsistency in at once rejecting the syllogism and allowing it to be the only way in which we can assure ourselves of the correctness of inference arises from the fact that one aspect of the general proposition is brought to bear in the formal discussion of syllogism, while the other predominates in the treatment of Induction and of the Deductive Method. The general proposition, Mill insists, looking — at it from one point of view, is no more than the particulars it contains; it is based on and summarises particulars, and into particulars it is resolvable. Viewed in this light, the general proposition as major premiss, whether its truth be known or assumed, can not be the ground from which the truth of a conclusion contained in it is established. The form of reasoning in which it is so used, the Syllogism, is clearly guilty of petitio principii. But the general proposition has another aspect, not less common in Mill’s treatment. The general proposition, as a result of inference, and in this aspect it has its greatest importance for us, is more than a summary of observed particulars. It embodies ‘inferences and instructions for making innumerable inferences in unforeseen cases. Wherever an inference to a particular case is well founded, is valid, the conclusion is already a general proposition. Data sufficient to establish one instance are sufficient to establish a class of instances.’

If, then, reasoning or proof cannot legitimately advance

1 The author evidently projected a special criticism on Mill’s view of a class, the notion of which, as defined by Mill, seemed to him to involve the same ambiguity and inconsistency as the treatment of general propositions. The two questions are, indeed, one substantially.

PREFACE XX1 from a general proposition, but if a general proposition may be established by reasoning, it becomes necessary to ask what, according to Mill, is the common form of reasoning, and how is the validity of a general conclusion established? Consideration of the first question is, in brief, criticism of Mill’s well-known doctrine, that all reasoning is fundamentally inference from particulars to particulars ; consideration of the second leads to a review of many salient points in Mill’s theory of Induction.