IN the present volume are contained some of the earliestand some of the latest contributions of the author to thescience of logic. The several works and papers that havebeen included fall into two distinctly marked groups , corresponding on the whole to the difference in time of theircomposition , and presenting at first sight no very clear indication of common aim or principle . The first group consistsof independently published works , the Pure Logic and the

Substi tuti on of Simi lcw'

s,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 , thePrincip les of Sci ence and the Studi es in Defluctive Logi c, are

carried out in certain special directions . In the secondgroup appear such portions of a general exam ination 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 ina 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 theyextend to certain fundamental and characteristic ideas inthe general V iew of logic contained in them . They are

PREP/1CE

of the Pure Logic an alteration was"

introduced in his ow n

hand on 14 6 . For the expressions there given of De

Morgan ’s two types of propositional form,viz .

E verything is ei therA or BSome thing s are nei therA nor B

There are substituted

It has not been thought desirable to introduce into the textthese alterations

The arrangement of the four essays is chronological .Pure Logic or the Logi c of Quali ty Apart from Quanti ty,w i th Remarks on Boole

’

s System,and on the Relati on of Logic

to Mathemati cs was published in 18 64 . The Substi tution ofSimi lars, the True Princip le of Reasoning , Derived from a

Modification of Aristotle’

s Dictum, was published inThe memoir On the Mechani cal Performance of Logical

Inference was received by the Royal Society l 6th O ctober1 8 6 9

,and read 20th January 1 8 70. The paper On a

General System of Numeri cally Defini te Reasoning Was com

municated in 1 8 70 to the Literary and Philosophical Societyof Manchester.1

There have not been included in this collection one ortwo contributions of less extent to logical science, the con

tents of which have been reproduced in more completefashion in other works of the author. A communication tothe Manchester Literary and Philosophical Society on 3d

with the Logi cal Abacus, which is elaborately explained in the later work, Substi tution of Simi tars.

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 Princip les of Science, secondedition,

p. 172 .

PREFACE

Manchester, 18 7 1 is incorporated with no substantivealteration in the Princip les of Sci ence, 18 7 7 , pp wA short paper entitled t o Di scovered the Quantification of

the Predi cate ? appeared in the Contemp orary Revi ew , May1 8 7 3 , pp . 8 2 1-8 24 . It emphasises strongly the claims ofMr. George Bentham to recognition as the first Englishwriter on Logic who had stated and applied the principleof Quantification . The paper is a comment on an articleby Professor T . S. Baynes in the same journal, and as itformed part of a somewhat extended controversy , it hasbeen judged needless to reprint it here .

The author himself attached so much weight to hiscritical exam ination of J . S. Mill’s doctrines, and the labourbestowed on it played so large a part in the last ten or

twelve years of his life, that the editors greatly regret theirinability to add much to the four papers published duringthe author’s lifetime in theM M From the

large mass'

ot MS. material for the further portions of the

w ork , a general idea may b e gathered as to the special linesof criticism that would have been followed ; but with one

or two exceptions no portion seemed to b e in such astate that the editors without hesitation could print itas expressing the author’s views . The four papers published by the author himself appeared in the Oontemp orargRevi ew ,

Dec. 18 7 7 , Jan . 1 8 7 8 ,April 18 7 8 ,

and Nov.

under the title John Stuart Ill i ll’

s Phi losophg _7k ste_d . The

fragment from the author’s MSS. on The Method of Difi’

erenec

deals with the point referred to by him in the third of thesearticles . It is proposed here to indicate the relation inwhich the criticism of Mill stands to the author’s logicaltheory, and to give a brief connected account of the lines itwas to have followed .

The distinguishing feature of the doctrine first expounded

viii PREFACE

in the Pure Log ic is the restriction of logical treatment to thequalitative aspect of thoughts . It was in this that theauthor found his difference from the system of Boole to consist ; and from thi s follow the main features that characterisehis logical method as a whole . There are certain conse

quences fairly involved in it, moreover, which though notformally stated by the author himself

,are yet implied in his

general theory of logic, and from which the precise ground ofhis emphatic dissent from M ill may b e made tolerably clear.There is so much of agreement between the views of Milland those expressed by the author in regard to the ul timatefoundation of knowledge, both seeming to follow the lines ofEnglish empirical philosophy, that the strenuous di ssent ofthe author from M ill’s whole doctrine of logic at first presents a problem . But if the conception of a logic of qualityb e followed out, it will b e found to involve an oppositionbetween pure or formal logic and the theory of empiri calor concrete reasoning, which for Mill constitutes logic, sosharp as to b e irreconcil eab le. One or the other must yield .

Briefly, it is the first that has pre-eminence in Jevons’

s V iew,

the second is the all-important with Mill. Only within thelim its of the first is there, according to Jevons, reasoningcharacterised by cogency and universality, but wi thin itslimits falls no determ ination of concrete existence ; all itscontents are, in reference to concrete existence, hypothetical.According to Mi ll there is no other field for reasoning thanthat of concrete existence ; a pure logic is but a subordinateand relatively valueless abstraction from the actual processesof concrete reasoning.

To restrict reas oning, with i ts elements, the propositionand term ,

to qual ity is a position which seems to connectitself naturally with the fam ili ar logical distinction betweenthe comprehension and extension of notions . In his initial

But it may fairly b e argued, both from the nature of the

case and from many expressions of the author him self, thathis system is wholly independent of that distinction , andindeed proceeds from a point of v iew which renders thedistinction meaningless or inapplicable . The di stinction asordinarily expressed makes reference more or less explicitlyto classification , and whether the basis b e nom inalist or conceptualist, leads the logician to contrast terms or notionswhich possess both comprehension and extension with thosewhich are wanting in one or other aspect. It would seemimpossible to work out consistently from this point withoutassigning to extension a certain m easure of concrete exis tence , which , when closely investigated , has always beenfound to involve difficulty. Now it is abundantly evidentthat 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 h isdevelopment of logic it is w holly superfluous to recognisethe obscure and baffling differences that appear in the exposition of the various k inds of terms . Whatever b e theirorigin , whether grammatical or metaphysical, these differencesrest on concrete details of matter, and not on anything inthe pure form of thought . The familiar distinctions onwhich the logician dwells

,between Proper or Singular Term s

and Common Terms,are of necessity rejected on the ground

that quantitative difference is a secondary and derived

PREFACE

simi lar fashion, when the proposition is reached, it becomesevident that W hat it has to convey is altogether independentof the currently accepted distinctions of universal andparticul ar, and that if these b e retained at all , they must b einterpreted in accordance with the fundamental laws of

thinking that concerns itself with the identity or non-identityof the characterising qualities .In adopting such a point of view, the logician is by no

means prohi bited from using form s of expression that seemto imply reference to concrete existences, to classes or singleobjects marked by the possession of qualities ; and it is afurther question, on which the author has hardly touched,whether it is possible to formulate the fundamental axiomsof thinking in terms whi ch do not imply such hypotheticalobjects. What is fundamental in his view is that theobjects so referred to are hypothetical merely, are determinedby the qualities thought as characterising them, and thatthey are absolutely to b e distinguished from real concreteexistences. On this account he emphasises so strenuouslythe position he shares with other logicians who have advancedto the same conclusion from widely different points of view

,

that in logic there is no question of existence,

1 and insiststhat logically non—existence is equivalent to contradictori

ness . Such existence as may b e assigned in logic to an objector class of objects is invariably possible existence . Only onmaterial data then, and with reference to a defined universeof objects

,could it b e concluded that the universe possessed

one quality or combination of qual ities to the exclusion of itscontradictory.

That it is very difficult to retain the position thus takenmay at once b e admitted, and it is evident that the authorat various points of hi s exposition felt the difficulties . The

1 Stud ies in Ded uctive Logi c, pp. 141, 142 .

which is a legitimate and inevitable consequence of the

view (Pure Logi c, 15 9,note ; Stud i es, p . 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 apure logic

,the complete statement of the ultimate con

ditions to which apprehension of qualities,as the same or

different, is subject, and of the consequences that flownecessarily from such apprehension . Within the realm of

pure logic w e have absolu te security and certainty. The

laws to which thought necessarily conform s in dealing withidentical or non-identical contents presented to it are at oncethe primary and the ultimate tests of truth . From theselaws whatsoever absolute truth is attainable must b e derived ,and absolute certainty can only b e assigned to deductionsfrom them . Whenever w e pass from the region of purethought to regions of m ore concrete matter, w e find thatsuch measure of absolute certainty as may there b e attainable depends ou the completeness of conform ity betweenthe most general laws of such matter and the laws of purethought . The elementary axioms of abstract arithmeticexhibit a perfect correspondence with the laws of purethought

,and are indeed derivatives from these laws . Number

rests on logical discrim ination or discrim ination by purethought

,and its typical form s and laws are derivable from

the conditions of pure thought when applied to the abstract

x i i PRE PA CE

with such fulness as the nature of the question demands ,the connection between geom etrical axiom s and the laws of

thought,and the grounds on which are to b e rested the cer

tainty and universality of geom etrical reasoning .

There follows from the view thus described an immediateconsequence which, when expressed as a logical doctrine ,has become famili arly associated with the author’s name .

All reasoning, or, m ore exactly, all proof is deductive in character, and involves general propositions of absolute certainty .

Proof is the more exact and appropriate term , for it re

mains within the limits of logical science or pure thought ;whereas the term reasoning is vaguely extended to coverthe 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 notto b e construed after the concrete fashion , as an assertionfound to hold good about a number of concrete

,and possibly

not exhausted, particulars . It is either the quasi—concreteexpression of an established identity of qualities, or thesummary expression of exhaustive enumeration of a strictlylim ited collection of instances . It is either a qualitative ora numerical identity. In strictness , indeed, no concrete orsynthetical proposition which is more than a collectiveexpression for a determinate number of observed cases, canever possess certainty. Whenever w e pass beyond the purelaws of thought

,with their numerical derivatives , or beyond

the statement of actually observed fact, w e are in the regionof assumption and probability. The utmost that can b e

achieved is to determine with what degree of probability thegeneral assumptions w e make can b e held . The rules forestimating such probability are numerical in character, andpossess all the certainty of the laws of thought .

PRE FA CE x11

The range of assured knowledge is thus of narrow extentas compared w ith the indefinite expanse of. concrete existence. But within that range it possesses absolute certainty .

There and there only is proof possible. So soon as theconsideration of concrete existence enters in

,w e are

dependent on the hazardous data of intuitive experience ,as to which w e can never feel assured that they contain allthat 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 areassertions about concrete fact is hypothetical in character ,and the hypotheses made, however various in concrete ex

pression , are essentially the same in all cases, -v iz. assump

tions that the concrete propositions correspond perfectly tothe conditions of pure abstract thought. It is impossiblefor us to know accurately that they do correspond ; b ut ,knowing the types of valid inference , w e can gauge by therules of probability the extent to which they approximate tothe desired correspondence . Induction , then, as ordinarilydescribed, is not a special mode of reasoning . Psychologically

,w e may no doubt pass in thought without further query

from isolated particulars to a generality ; bu t particulars canin no way substantiate what is not contained in them .

There is only one method of reasoning— the deduct iv e ,and it is used in concrete material as in abstract thought .But in the former case w e have to note that our datainvolve assumptions that cannot b e completely justified , andthat are only m ore or less probable in a partially determ inabledegree .

Now it is the pointed opposition w hich the author makesbetw een the narrow but secure region of pure , abstractthought and the wide sphere of m ore or less probableassumptions regarding concrete existence , an opposition in

xiv PRE FACE

volved in and springing from his conception of pure logic,that constitutes the real difference between his views andthose of Mill . In the long run both agree in respect to thecharacter of what is vaguely called ‘ knowledge ’

of concreteexistence. Neither accorded absolute certainty to the propositions composing that ‘knowledge.

’ Though Mill um

doubtedly uses language in respect to so-called knowledgeof nature that implies the possibility of attaining certaintythere, yet it may fairly b e urged that such expressions are tob e taken with the qualifications necessitated by the generaldoctrine, that knowledge of nature can never amount to ab solute certainty. But

,in M ill ’s view, this knowledge of nature

constituted all our knowledge, and within the realm of con

crete existence lay the province and process of reasoning.

Apparently abstract propositions , such as those ofmathematics ,were only abstracted from the concrete particulars , and had noother foundation for their certainty than the concrete experi

ence from which they were drawn , no superior generalitythan followed from the relatively greater ease with whichthey could b e disentangled from concrete details . The lawsof pure thought, in which Jevons found the rules of absolutelycertain knowledge and reasoning, were by M ill regarded astrue but rather valueless prescripts defining the use of

language, and having no function of significance whendivorced from concrete fact. A pure logic, or logic of con

sistency in the employment of language, he did, indeed ,admit to b e possible , but in no way accorded to it specialimportance or viewed it as more than a result of abstractionfrom actual concrete reasoning. The possibility of an independent foundation for it he did not so much deny as ignore .

l

The difference is as complete as could well b e. To the

1 The most expl icit utterance of Mill on these points, w hich he rarely d iscusses, are in the Examinati on of Hami lto n (thi rd edition), pp.

‘

457 , 461.

PREFACE

one thinker, the theory of reasoning was a well - rounded ,independent whole , narrow it m ight b e in extent, but restingon principles of absolute certainty, not necessarily connectedwith any hypothesis as to the w ay in which knowledge of

concrete fact is obtained or increased , and supplying the finals tandard by which evidence in concrete matters is to b e tested .

To the other, the theory of reasoning was part of the generaldoctrine of the w ays in which an intelligence receiving itsdata as isolated empirical particulars gradually advanced toknowledge ,

to the establishm en t of general propositions,

and to criticism of the grounds on which these rested .

According to the first view ,reasoning, in the s trict sense of

the term,was confined to the region of analytical thinking ,

and w hat accompanies reasoning ,—viz . stringent cogency o f

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 w ere such as could b e obtained underthe conditions of synthetical think ing .

It was inevitable, then , that from the author’s point ofview in logical theory, the apparent attempt in Mill

’s Systemof Logic to show how general knowledge in concrete m aterialis 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 o

L

For by knowledge the authorunderstood w hat is perfectly conformable to the pure lawsof thought and is w arranted completely by them attainable

,

therefore , only in analytical thinking. To adm it,then as

Mill seemed to do,the ultimate uncertainty of w hat is called

knowledge of nature , and , while ignoring the pure laws ofthought

,to work out a theory of evidence on which know

ledge m ight rest,w as to occupy a w holly untenable position .

xvi PREPA CE

As regards the general point of view adopted in the

proposed exam ination of Mill ’s philosophy, the motives withwhich it was undertaken , and the end aimed at

, the authorhas given full explanation in the first of the series of articlescontributed to the Contemp orary Revi ew . Among his MSS.

are found portions of the sections intended to serve as thegeneral introduction to the whole and as a summary con

elusion . These contain in somewhat more detail the ex

planatory matter already given in the first article, and alsosome indication as to the arrangement of topics to b e followedin the completed work . It would appear that after thegeneral introduction the author purposed dealing in suc

cession with Mill’s E ssay on Religi on, with his views onFree Will and Necessity, with his peculiar reformation of the

Utilitarian Ethics, w ith his doctrine of Inseparable Association

,and then w ith 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 weregiven as illustrative confirmation of the estimate formedby the author, that M ill

’s m ind was essentially illogical.It cannot however b e determ ined , from the MSS.

,in what

way these sections would have been arranged, nor is thereany indication given as to the final order in which the

various topics falling under logical theory would have beentaken .

Taking first into consideration the sections not specificallylogical , w e find among the MSS. collected materials

,with

occasional written out portions of statement, relating to Mill’s

E ssay on Religion,to his view on Free Will

,and to his

doctrine of Inseparable Association . As regards the E ssayon Religi on, it is clear from the fragments, as also from areference to the topic in the first article in the Contemp oraryRevi ew ,

that the author’s intention was to dwell upon and

FE BEACE xvn

enforce the apparent inconsistency between the two passagesquoted from the essay in the said article .

1

The section on Free W i ll and Necessi ty begins by a fairlywritten out statement of Sir William Ham ilton ’s fam iliarposition

,that either conception, Liberty or Necessity, leads

when developed to an ultimate inconceivability, a positionwhich , on the whole, seems to have been accepted by theauthor. The remaining MSS. on the subject

,in fragmentary

remarks and references to passages in Mill’s writings , enableus to see the drift of his argument, that Mill , while professing the Determ inist view,

makes adm issions which are

wholly irreconcilable with it and practically nullify it. The

specific adm issions singled out for closer scrutiny are tw o innumber

,closely connected , the one m ore psychological , the

other more metaphysical in character. The first is the viewexpressed most definitely in the System of Logi c (Book v i

,

chap . ii) , that while the actions of the individual can only b eregarded as the outcome of his character

,desire to m odify

that character must b e recognised as a factor in its formation .

The second is contained in the distinction drawn in the

posthumous Essays 2 betw een Nature as the entire systemof things with all the ir attributes and Nature as that w hichis apart from human intervention. The author’s intentionseem s to have been to press the arguments that , accordingto the principles on which M ill proceeds, desire to modifycharacter must b e treated like all other desires as a deri vati ve

0' andand determined fact, not as original and determ inin

g ,

that, even i f for one purpose man b e severed from nature1 Th ree Essays on Rel ig ion,

p . 109 . The essence of religion is the strongand earnest d i rection of the emotions and desires tow ards an i deal obj ect

,

recognised as of the h ighest excellence , and as r ightfully paramount over allselfish ob jects of desi re .

’

Ib id . p . 103 Rel igion, as d istinguished from poetry,is the product of the craving to know w hether these imaginative conceptionshave real ities answ ering to them in some other w orld than ours . ’

3 P 46

xviii PRE FA CE

and regarded as intervening in it, on Mill’s general prin

ciples, his activity of intervention can b e contemplated inno other light than as a fact subject to natural law. If anyother significance b e assigned to the desire or interveningactivity, the only result is utter inconsistency of theory .

M ill ’s ethical theory is criticised , so far as the statement ofUtilitarian doctrine is concerned

,in the fourth of the articles

published in the Contemporary Revi ew . The MS. fragmentsentitled rlIorals indicate that the author purposed also discussing Mill ’s view on the relation between Intention andMotive

,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 exam inethe doctrine of Inseparable Association with special referenceto the fundamental axiom s of number, with a .view toexposing the inconsistency between the character allowedto these axioms by M ill and the theory of their empiricalorigin through inseparable association .

A very large portion of the MSS. is concerned withMill’s logical theory. The several sections indicate

,no

doubt,the arrangement of topics to b e m ade in the final

treatment. W e have been unable to find a definite statement as to this arrangement, and the order in which theyare here referred to rests only on occas ional 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 experimentalmethods . To the first group belong the sections headedSystem of Logic, Peti tio Princip i i , General Prop osi tions,

Parti culars to Parti culars. The fragments forming thesesections are evidently of various dates , and so far as any

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XX EKE FACE

the syllogism is to b e sought in his treatment of the generalproposition . From the passages selected for comment andfrom the fragments of criticism on them ,

it is evident thatthe author intended to insist that Mill regards the generalproposition in two aspects

,distinct and wholly irreconcilable ,

and that his inconsistency in at once rejecting the syllogismand allowing it to b e the only way in which w e can assureourselves of the correctness of inference arises from the factthat one aspect of the general proposition is brought to bearin the formal discussion of syllogism

,while the other pre

dom inates in the treatment of Induction and of the Deduc

tive Method . The general proposition , Mill insists , lookingat it from one point of view

,is no more than the particulars

it contains ; it is based on and summ arises particulars, andinto particulars it is resolvable. View ed in this light, thegeneral proposition as major premiss , whether its truth b eknown or assumed, can not b e the ground from which the truthof 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 hasanother aspect, not less comm on in Mill

’s treatment . The

general proposition , asji gglgIL QLJnfBLBnce , and in thisaspect it has its greatest importance for us, is more than asummary of observed particulars . It embodies ‘ inferencesand instructions for making innumerable inferences inunforeseen cases . ’ Wherever an inference to a particularcase is well founded , is valid , the conclusion is already ageneral proposition . Data sufficient to establish one instanceare sufficient to establish a class of instances .1

If,then , reasoning or proof cannot legitimately advance

1 The author evidently projected a special criticism on Mill ’s view of a

class,the not ion of w hich

,as defined b y Mill, seemed to him to involve the

same amb iguity and inconsistency as the t reatment of general propositions .The tw o questions are, indeed , one sub stant ially .

is fundam entally inference from particulars to particulars ;consideration of the second leads to a review of many salientpoints in Mill’s theory of Induction .

The discussion of the view that inference is from particulars to particulars is incomplete . From what remains

,

i t seem s that the author w ould have pressed the argumentthat though

,in point of fact

,w e may and do pass m entally

from particulars to particulars , the process is itself a ‘

com

pl icat-cd and precarious combination of induction anddeduction ,

’ indefinitely far removed, therefore , from satisfactory proof or evidence , and the rules of which , i f possibleat all

,must involve reference to considerations that go beyond

particulars . In other words , adm itting that reasoning fromparticulars to particulars may b e an actual, psychical process,the author m aintained that the identification of the conditionsof a natural process with the conditions of proof was notonly an error, but wholly inconsisten t with Mill

’s generalconception of logic as the science not of the ways in whichw e do reason but of the rules of valid reasoning .

Some portion of the criticism of the theory of Inductionis contained in the third of the articles from the Con

temp orary Rev iew ,and in the fragment , printed in this

volume , on the Method of D if erence. What remains wasarranged by the author under the heads Baconian Ill ethod

xxn PRE FA CE

ence on the necessity for general reasoning. In this,and

generally in all that Mill has described under the head of

the Deductive Method, the author finds what at once correctsand destroys the erroneous conceptions of reasoning asinference from particulars to particulars , and of Inductionas a process whereby w e arrive at general results fromparticular prem isses .1

The section on causation was evidently intended to takeup in succession the definition of cause and of the causalrelation offered by Mil l, the assumption of the universality ofcausat ion

,and the grounds on which its universality was rested .

In exam ining the definition of cause,the author draws atten

tion to the ambiguity of Mill’s language, which allows a twofold interpretation to b e easily put upon the all-importantterms, invariable antecedent and consequent. An invariablesequence asserted m ight mean, he points out, either that

cc invariably follows X,

or that while cc invariably follows I ,

a" is invariably preceded by X.

The language employed , he insists, leaves the ambiguity unresolved , and indeed tends to favour the second mode of

interpretation . It is only when , at a later stage, p lurali tyof causes is formally introduced that w e learn how abstractour interpretation of invariable antecedent must b e. The

author’s MSS. contain no further criticism of this point, buthis view on the subject can readily b e gathered from hispublished works .2

On the second point,the universality of causation

,the

author contrasts at length passages in which the certain tyof general knowledge is assumed , and is stated to depend on

1 See on thi s point Princip les of Sci ence (second edition), pp. 265, 508 .

See Pr incip les of Sci ence (second edition), pp. 222, 226 , 737 sqq.

PRE FA CE xx iii

the law of causation, W i th the passage 1 in which Mill asserts

that ‘the uniform ity in the succession of events, otherwise

called the law of causation,must b e received not as a law of

the universe,but of that portion of it only which is within

the range of our m eans of sure observation , with a reasonable degree of extension to adjacent cases . ’

As regards the third point, the treatment of the groundsfor the assumption of a universal law of causation coincidesthroughout with the criticism of Inductio p er cnumerati onem

simp li cem contained in the third of the articles on J . S. Mill ’sPhilosophy.

To this brief and necessarily inadequate summary of the

author’s projected work , it must b e added that only contactwith the MSS. can convey any fair idea of the painstakingand conscientious manner in which the scrutiny of Mill ’sw ritings had been carried out. “Thatev er Opinion may b e

formed of the value of the general objections taken to M ill ’slogical and philosophical doctrines , or of the appropriatenessof the lim its imposed by the author on his criticism ,

i t mustb e acknowledged that he took every precaution against oversight or hasty judgment , and that his every utterance w as

supported by the fullest evidence attainable . The inv estig.

tion of the fundamental principles of reasoning is a problemof such sub lety and complexity that exhaustive criticism of

one distinguished logician by another must always b e hailedwith satisfaction . It is not the least part of the severe losswhich science and philosophy incurred by the author’s un~

timely death,that he was prevented from utilising , as he

only could do, the m aterials he had collected .

1 System of Logi c, Book i i i , chap . xxi,sec. 4, conclud ing paragraph .

C O N T E N T S

PART I

W RITINGS ON THE THEO RY OF LOGICPAGE

1. PURE LOGIC on THE LO GIC OF Q UALITY APART FRO MQ UANTITY. ( 18 6 4 )

THE SUBSTITUTION O F SIMILARS. ( 18 6 9 )

ON THE MECHANICAL PERFO R M ANCE OF LOGICAL INFERENCE . (From the Phi losophi cal Transactions, 1 8 7 0)

ON A GENERAL SYSTE M O F NU M ERICALLY DEFINITEREASONING . (From the Memoi rs of the llIanchester

Li terary and Phi losophi cal Society, 18 70)

PART II

JOHN STUART M ILL’S PHILOSOPHY TESTED

(Port ions of an E xam ination of J. S. llIi ll’s Phi losophy)

1. ON GEO M ETRICAL REASONING11. ON RESE M BLANCE

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PU R E L O G I C

OR THE

OF QUALITY APART FROM QUANTITY

\VITH RE M ARKS ON BOOLE’

S SYSTE M

'N THE RELATIO N OF LOGIC AND M ATHE M ATICS

INTRODUCTION

urpose of this work to show that Logic assumes a E xtent andi ntent ofe of Sini plICIty, preci s i on , general i ty, and power, meanmg '

pari son In quali ty 18 treated apart from any3 quantity .

fam iliarly known to logicians that a term must'

cd with respect both to the individual things itd the qualities , circum stances

,or attributes it

implies as belonging to those things . The

individuals denoted form s the breadth or extent

aning of the term ; the qualities or attributesorni the depth , comprehension, or i ntent, of the

PURE LOGIC

2 . Logicians have generally thought that a propositionmust express the relations of extent and intent of the term sat one and the same time, and as regarded in the same

light. The systems of logic deduced from such a view,

when compared with the system which may otherwise b ehad

,seem to lack simplicity and generality.

Sep aration 3 . It is here held that a p rop osi tion exp resses the result ofa comp arison and judgment of the sameness or d if erence ofmeaning of terms

,ei ther in intent or extent of meaning . The

judgm ent in the one dimension of meaning,however, is not

independent of the judgment in the other dimension . It isonly then judgment and reasoning in one dimension whi chis properly expressed in a simple system . Judgment andreasoning in the other dimension will b e and must b eimplied. It may b e expressed in a numerical or quantitativ e system corresponding to “

the qualitative system,but

its expression in the same system destroys simpli city.

I do not wish to express any opinion here as to thenature of a system of logic in extent, nor as to its preciseconnection with the pure system of logic of quality .

4 . Reasoning in quality and quantity, in intent or extentof meaning, being considered apart

,it seem s obvious that

the comparison of things in quality, with respect to all theirpoints of sameness and difference, gives the primary andmost general system of reasoning. It even seem s likelythat such a system must comprehend all possible andconceivable kinds of reasoning, since it treats of any andevery way in which things may b e same or different. Allreasoning is probably founded on the laws of sameness anddifference which form the basis of the following system .

5 . My present task, however, is to show that all and

more than all the ordinary processes of log ic may be combined

in a system founded on comp arison of quali ty only, w i thout

reference to logi cal quanti ty.

Relation to 6 . Before proceeding I have to acknowledge that in a

2321

1? considerable degree this system is founded on that of

Professor Boole, as stated in his admirable and highly

Logic. All the interesting analogies or samenesses of logicaland mathematical reasoning which may b e pointed out, aresurely reversed by making Logic the dependent of Mathematies .

1 Investi gati on of the Law s of Thought. By George Boole , LL.D. London,

1854. Frequent reference w ill b e made to th is w ork in the follow ing pages .

CHAPTER I

OF TERMS

7 . Pure logi c arises from a comp arison of things as to

the ir sameness or d if erence in any quali ty or ci rcumstance

w hatever.

In discourse w e refer to things by the aid of marks,

names,or terms

,which are also

,as it were, the handles by

which the m ind grasps and retains its thoughts about things .Thus correct thought about things becomes in di scourse thecorrect use of nam es . Logic, while treating only of names ,ascertaining the relations of sameness and difference of theirmeanings, treats indirectly, as alone it can,

of the samenessesand differences of things.8 . A term taken in intent has for its meani ng the whole

infinite series of qualities and circum stances which a thingpossesses . O f these qualities or circum stances some may b e

known and form the description or definition of the meaning ; the infinite remainder are unknown .

Among the circum stances, indeed , of a thing, is the factof its being denoted by a given name, but w e may speak of

a thing, of which only the name is know n , as having a name

of unknown meaning .

The meaning of every name, then , is either unknown ormore or less known. But w e may speak of a term that ismore or less known as being simply know n.

9 . Among the qualities and circumstances of a thing isto b e counted everything that may b e said of it, affirmatively

TERATS

or negatively. Any possible quality or circumstance thatcan b e thought Of either does or does not apply to any giventhing, and therefore form s, either affirmatively or negatively ,

a quality or circum stance of the thing. Concerning anything

,then , there may b e an infinite number of statements

made, or qualities predicated .

10. When w e assign a name to a thing, with knowledge Relation Of

of, and regard to, certain of its qualities or circum stances ,”wam ng”

that name is equally the name of anything else of exactlythe same known qualities and circum stances . For there isnothing in the nam e to determ ine it to the one thing ratherthan the other. Any nam e, then, must b e the name i n

extent of anything and of all things agreeing in the qualitiesor circum stances which form its known m eaning in intent,

and in this system .

11. Though it is well to point out that all our names or P resentterm s bear a universal quantity when regarded in extent

,it W W W “

must b e understood, and constantly borne in m ind,that

further reference to the meaning of a term in extent orquantity of individuals

,is excluded in these pages .

The p rimary and only p resent meani ng of a name or term

i s a certain set of quali ti es, attri butes, p roperti es, or circum

stances,of a thing unknow n or p artly know n.

12 . Term will b e used to mean name,or any combination Term

of names and words describing the qualities and circum defi ned

stances of a,thing.

13 . The term s of this system may b e made to express Generali tyany combination of samenesses and differences in quality, Z

’

m

’

iigkind

,attribute , circum stance

,number

,magnitude, degree,

quantity, opposition , or distance in time or space. A termmay thus represent the qualities of a thing or person in allthe complexity of real existence , so well and ful ly defined thatw e cannot suppose there are

,or are likely to b e, two things

the same in so many circum stances . Such a term wouldcorrespond to the singular, p rop er, non-attri buti ve, or non Prop erconnotative names of the old logic. Such names are

m m "

accordingly by no means excluded from this system ; and

PURE LO GIC

it is here held that the old distinction of connotative andnon-connotative names is wholly erroneous and unfounded.

If there is any distinction to b e drawn, it is that singular,proper, or so-called non-connotative terms, are more full of

connotation or meaning in intent or quality than others,in stead of being devoid of such meaning.

14 . As logic only considers the relations of meaning of

terms , as expressed within a piece of reasoning, the specialmeaning of any term is of no account, provided that thesame term have the same meaning throughout any one p i ece ofreasoning .

Thus,instead of the nouns and adjectives , to each of

which a special meaning is assigned in common discourse,

w e shall use certain letters, A, B ,C

,D

,U

,V

each standing for a special term , or a definite meaning,and for any tirm or meaning, always under the abovecondition .

15 . Our term s, A, B, C, like the term s of

common discourse,may b e either known or unknown in

meaning. It i s the w ork of logic to show w hat relati ons ofsameness and diference betw een unknow n and know n terms

may make the unknown terms know n.

Were it not to explain ignotum p er ignotius, w e mi ghtsay that logi c is the algebra of kin d or quali ty, the calculus ofknow n and unknow n quali ti es, as algebra (more strictlyspeaking uni versal ari thmeti c, which does not recogniseessentially negative quantities) is the calculus of known andunknown quantities .

16 . Let it b e borne in m ind that the letters A,B

,C

, etc.,

as well as the marks 0, and = ,afterwards to b e intro

duced , are in no way mysterious symbols . The term A,for

instance, is merely a convenient abbreviation for anyordinary term of language

,or set of terms, such as Red, or

the Lords Comm i ssi oners for executing the ofi ce of the Lord

High Admiral of England .

Again,is merely a mark substituted for the sake of

clearness, for the conjunctions and , ei ther, or, etc of common

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CHAPTER I I

OF PROPOSITIONS

17 . A p rop osi ti on i s a statement of the sameness or diy’erence

of meaning of tw o terms, that is, of the sameness or differenceof the qualities and circumstances connoted by each term .

18 . According as a proposition states sameness or d ifi erence

,it is called afi rmative or negati ve.

19 . It is the p urp ose or use of a p rop osi ti on to make

know n the meaning of a term that is otherw ise unknow n.

Truth and 20. A proposition is said to b e true when the meanings”M y

of its term s are same or different,as stated ; otherwise it is

called false or untrue. As logic deals with things onlythrough terms , it cannot ascertain whether a proposition istrue or false , but only whether two or more propositions areor are not true together, under the condition of m eaning ofterms

N otation of 21. W e denote by the Copula i s,or by the mark the

gfigfim sameness of meaning of the term s on the two sides of aproposition .

For the present w e shall speak only of affirmative propositions, which are of superior importance ; and when nototherwise specified, p rop osi ti on may b e taken to mean afi rma

tive p roposi ti on.

22 . A p rop osi tion i s simp ly converti ble. The propositions

fifth?” A=B and B=A are the same statement ; either of theterm s A and B is the same in meaning as the other, umdistinguishable except in name.

PROPOS]TIONS

This simple conversion comprehends both the simp le con

versi on,and conversio p er acci dens of the school logic.

23 . One p rop osi tion and one know n term may make know n One term

one unknow n term.

From A= B,so far as w e know B, that is, know its p yop osi

m eaning, w e can learn A ; so far as w e know A, w e can”on ”

learn B .

W e thus know samely of both sides of a propositionwhatever w e know of either. The same m ight b e said of

uncertain or obscure knowledge .

24 . A proposition between any two term s of which the Cseless and

meanings are otherwise known as same or different, isuseless . For it cannot serve the purpose of a proposition s i tions

(5 Such is any proposition between a term and itself,as A =A

,B= B These useless propositions are called

Identi cal. They state the condition of all reasoning, but w eknow it without the statem ent .A proposition repeated, or a converted proposition

is also useless , except for the m ere convenience of memory,

ready apprehension .

8 617716 72633 .

Meaning oflaw s oflog i c.

CHAPTER III

OF DIRECT INFERENCE

25 . It is in the nature of thought and things,that things

w hich are same as the same thing are the same as each other.

More b riefly—SA M E As SA M E ARE SAME.

Hence the first law of logic—that terms w hi ch are same in

meaning as the same term,are the same in meaning as each other.

This law, it is Obvious,is anal ogous to Euclid ’s firs t

axiom,or common notion

,that things w hi ch are equal to the

same thing, are equal to each other. Things are Called equalwhich are same in magni tude, but what is true of suchsameness, is also true of sameness in any way in whichthings may b e same or different . Euclid’s geometrical lawis but one case of the general law.

26 . Logic proceeds by laws,and is bound by them .

For logic must treat names as thought treats things. Andthe laws of logic state certain samenesses or uniform ities inour ways of thinking

,and are of self—evident truth .

27 . When tw o afi rmative prop osi tions are same in one

member of each,the other members may be stated to be same.

From A=B,B=C,

which are the same in the memberB ,

w e may form the new proposition , A= O . For A andC being each stated to b e the sam e as B

,may by the law

of sameness b e stated to b e the sam e as each other.A proposition got by the Law of Sameness is said to b e

got by d irect inference,and is called a di rect inferent, or, in

common language, a direct inference.

30. In inferring a new proposition from two prem ises E l iminaw e are said to elim inate or rem ove the m ember which is thefigd

d e'

same in the two prem ises .From tw o p rem i ses w e may elim inate only one term

,and

i nfer one new p rop osi ti on. By saying that w e may,it is not

meant that w e alw ays can.

31. Propositions are said to b e related to each other Related

which have a same or common m ember,or which are so

related to other propositions so related and so on .

In other words, any two propositions are related w hichform part of a series or chain of propositions

,in which each

proposition is related to the adjoining ones or one .

32 . Terms are said to b e related which occur in one Relatedtermssame

,or In any related pIOpos1tIons.

d efi ned .

33 . From tw o related premises and one know n term w e Use of

may learn two unknow n terms,and not more.

”Ham’

s“

From A =R and B= C,w e learn any two of A

,B

,C

,

when the third is known.

34 . From any seri es of related p remi ses, and one know n Ser i es ofterm w e may learn as many unknow n terms as there are p

remi ses '

prem ises. Thus from w e may learnany fi ve term s when the sixth is known . For each usefulproposition may render one unknown term knownBetween each two adj oining prem ises one term may b e

elimi nated, becom ing known in one prem ise, and renderinganother term known in the other. There must at last

PURE LO GIC

the number of d izferent terms. If it b e still less, the propositions cannot b e all related ; if it b e greater, some of the

prem ises must b e useless, because they must lie betweenterm s otherwise known to b e same by inference .

It will b e remarked that systems of mathematical propositions or equations with known and unknown quantitiesare perfectly analogous in their properties to logical propositions .

36 . When a related prem ise contains a term or member“m s and not relevant to the purpose of the reasoning, this term is

elim inated by neglecting the prem ise ; and for every suchprem ise neglected a term is elimi nated . In regard both torelated and unrelated prem ises and their term s, the neglectof all irrelevant term s and prem ises may b e considered aprocess of elim ination which accompanies all thought.37 . Inference i sjudgment of judgments, and ascertains the

sameness of samenesses.

When in comparing A with B,and the same B with

C,w e judge that A=B and B=C

,w e obtain sci ence

,or

reasoned knowledge of things , as distinguished from the

mere knowledge of sense or feeli ng. But when w e judgethe judgments A=B = C to b e the same, as regards A andC

,with the judgment A= O , w e obtain Sci ence of Sci ence.

Here is the true province Of logic, long call ed ScientiaSci entiarum . Hence it is that logic is concerned not withthe truth of prepositions p er se but only with thetruth of one as depending on others .

Sci ence.

SCIENCE OF SCIENCE !A B C} !A C} REASONINGSCIENCE A B B C JUDG M ENTTHINGS A B C APPREHENSION

38 . Here w e find the clear meaning of the distinctionform and matter of thought.

Sameness of Samenesses

Sameness of th ings Of thought.Th ings

CHAPTER IV

OF CO M BINATIO N OF TERMS

39 . In discourse, when several names are placed together A dd i tionside by side, the meaning of the joint term is sometimes thenf

f‘7l '

sum of the m eanings of the separate terms .1

So in our system ,w hen tw o or more terms are p laced

together, the joint term must have as i ts meaning the sum ofthe meanings of the sep arate terms. These must b e thoughtof together and in one .

40. Any term s placed together will b e said to form with Comb inarespect to any of those separate term s a combi nati on or eomjiggl ebined term. With respect to all other term s they may b e

called simply a term . For it must b e remembered that anysingle term ,

A,B

,C

, etc.,is not more single in meaning than

a combination.

1 I shall here consider only the cases of comb ination in w hich the com

b inod term means the added m ean ings of the separate terms. The same

forms of reasoning apply , as I b elieve, mutatis mutand i s, to any cases ofcomb ination under som e such w i der law as this

Same p arts samely related make same w holes.

Only b y some such extension can logic b e made to emb race the major partof all ord inary reasoning , w h ich has never yet b een emb raced b y i t , save so

far as th i s may have b een done in some of Professor De Morgan ’s latestw ritings . But to show h ow such an extension may b e grafted on to my

system must b e reserved for a future opportunity . In most relat ions i ti s Ob vious that the order of terms in relat ion is no longer ind ifferent41)Concerning some inferences b y comb ination, see Thomson’s Outl ines,87 , 88 .

PURE L OGIC

41. The meaning of a combinati on of terms is the same in

“on ind ifw hatever order the terms be combined .

Thus AB BA ; ABCD BAOD DOAB , and so on .

For the order of the terms can at most affect only theorder in which w e think of them ,

and in things themselvesthere is no such order

’

of qualities and circumstances

(Boole, p .

42 . A comb inati on of a term w i th i tself i s the same in

meaning w i th the term alone.

Thus AA A,AAA A

,and so on .

Also,a combination of terms is not altered by combina

tion with the whole or any part of itself. Thus ABCDABCD BCD =A BB CC DD = ABCD

,since BB B

,

CO C , DD D .

The coalescence of same term s in combination must b econstantly before the reader’s m ind .

This important and self-evident law of logic was firstbrought into proper notice by Professor Boole (p . whoremarks : ‘To say “ good , good,

” in relation to any subject,though a cumbrous and useless pleonasm , is the same as

to say good.

”

Professor Boole gave to this law the name Law ofDuali ty. But as this name, on the one hand , is notpeculiarly adapted to express the general fact

,AAAAA

A,and is peculiarly adapted to express the fact A AB+Ab

I have ventured to transfer the name, and substitutea new one .

43 . In the term s as used under the above law there isno reference to degree of quali ty. Wh en required, eachdegree of quality may b e treated in a separate term , con

taining as part of its meaning every less degree of the

quality. Two or more degrees of a same quality in logicalcombination therefore produce the greatest of those degrees .44 It is in the nature of thought and thi ngs that w hen

o

same quali ties are joined to’

same quali ti es the w holes are

same.

Hence the law of logic

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PURE LOGIC

should require the expression of all tacit prem ises connectedwith terms of specific meaning. It is only the severalbranches of science, however, that can undertake the

necessary investigations in detail . Our inference remainstrue, however compli cated b e the relations of sameness anddi fference of the terms introduced. But it is inference fromprem ises which are stated

,not from those which might be or

ought to be stated .

49 . When p remises contain terms only p artially the same,

the combination of each w i th the p art that i s d ifi erent in the

other w i ll p roduce a term comp letely the same in each. Suchprem ises may b e considered as relatedThus

,in A = O and B = CD

,the terms C and CD are

only partially the same. But the combination of D withA= O gives AD = OD , having one member completely the

same as one member of B CD . Hence w e may inferAD =CD =B and eliminate the te rm C, which wascomm on in the premises : thus, AD =B.

Again,to elim inate B from the prem ises A =BC and

E =BD,combine D with each side of the first

,and C with

each side of the second Hence , AD BCD CE,or

AD =CE ,in which B does not appear.

50. From prem ises which have no term in common , thisprocess will only give us the inferences whi ch m ight b e had4 6 ) by the direct combination of the respective term s of

the prem ises . Thus A =B and C =D give AD=BD ,and

BC=BD,whence AD =BO . And w e might sim ilarly get

AC BD.

51 . The following process may b e called substi tution , andwill b e seen to give the same inference as the two processesof form ing a common term 4 9 , and then elim inat

ing it.For any term

,or part

- term,in one premise, may be sub

sti tuted i ts exp ressi on 2 9 ) in other terms.

In short, the two members of any prem ise may b e usedindifferently

,one in place of the other

,wherever either occurs .

Thus,if A = BCD and BC =E

,w e may in the former

COMBINATION

premise substitute for BC its expression E, getting A= .DE .

The full process of inference consists in combining D withboth sides of BC =E

,and eliminating the complete common

term BCD thus obtained, so that A=BCD=DE .

52 . lVe may substi tute for any p art of one member of a

p rop osi ti on the w hole of the other.

Thus,in A = BCD

,w e may substitute for any one of

B,C,D

,BC

,BD and CD ,

parts of BCD the one member,the whole, A, of the other m ember, inferring the new

propositions

A =ACD A =ABOA =AD A =AG

The validity of this process depends on the Laws of

Simplicity and of Part and “Thole as is seenby combining each m ember of the prem ise with itself.Thus

,from A BOD w e have AA BODB OD BCD.D AD

by coalescence of same term s, and substitution for BCDof its express ion A.

The new proposition thus inferred w ill have one of itssides p leonasti c, that is, with som e part of its meaningrepeated . But it is obvious that w e cannot

,as a general

rule, substitute for part of one s ide less than the whole of

the other, because w e cannot from the prem ise alone knowthat the meaning of the part- term removed is quite suppliedin the part of the other m ember put for it .

The above process may b e called intrinsi c elim inati on,to

distinguish it from the former process of elimination betweentwo prem ises

,which may b e called extri ns ic elim inati on

,and

is seen to b e that case of intrinsic elim ination in which w e

substitute for the whole of one side the whole of the o ther .In a single prem ise, intrinsic elim ination of a whole m emberwould give only an identical and useless result.Intrinsic e lim ination gives no new knowledge , but is of

constant use in strik ing out or abstracting terms concerningwhich w e do not des ire know ledge , and which are thereforeworse than useless in our results .

PURE LO GIC

Professor Boole’s system of elim ination (p. is, Ibelieve, equivalent to the above, though the correspondencemay not at first sight b e apparent.

Failure of 53 . A term cannot b e intrinsically eliminated whichft i 'ni ‘m ' occurs in b oth members of a proposition . The presence ofteen .

such part- term may b e called a cond i ti on of the sameness Ofthe remainder of the terms .

Same ly re 54 Terms are said to b e samely related in a premisewhen their interchange does not alter the prem ise.

terms .

Thus,B and C are sam ely related in A= BC, because

the prem ise is the same , A = OB after their interchange. But A and B are not samely related, because theirinterchange alters the premise into B =AC .

In A = BCDE any two of B, C, Dsamely related and may b e interchanged .

Inf erence 55 . O f samely related term s, an expression for the one

23

33s is the same as the expression for another after the twoterms in question have been interchanged .

Concern ing 56 . When several term s are samely related, w e obtainsem al ‘

the expressions concerning the rest from the expression forany one by successively changing each term into the nextwhen the term s are kept in some fixed order.It is evident that w e may always interchange the terms

in any part of a problem,provided w e do so throughout the

problem (g And in those cases in which the prem isesremain unchanged thereby, w e evidently get several inferences from the same prem ises. This method of interchangesis fam iliar to mathematicians .

57 . It will b e obvious that a mathematical term orquantity of several factors is strictly analogous in its lawsto a logical combined term , excluding the Law of Simp li ci ty.

CHAPTER V

OF SEPARATION OF TERMS

58 . It is in the nature of thought and things that w hen Law of

from same sets of quali ti es same quali ti es are taken, the re

maining sets are the same or,more b riefly— Same parts

from same w holes leave same parts.

Hence the logical law When from same combinations ofterms same terms are taken

,the remaining terms are the same.

This is the converse of the Law of Same Parts andWholes and is equally self-evident with it . But itis not equally useful with it ; and in Pure Logic, in fact, isof no use at all. The removal of terms with their knownmeanings is not equally possible with their combination

,

and in useful logical prem ises, is not possible at all. For,

in a useful prem ise a part at least of one membermust b e unknown , and this part may or may not containthe part w e desire to remove . Even supposing then that aterm occurs on either side of a prem ise, w e cannot removeit from the known side, because w e cannot know whetheror not w e can remove it from the unknown or partiallyknown side .

Thus in AB =AC,suppose A and C known , and B un

known . W e cannot infer B=C,because B may contain

part or the whole of the known meaning of A,in addition

to the known meaning of C ,by the Law of Simplicity

and in leaving B ,w e do not remove A from one

member of the prem ise.

logic and

mathe

ma tics .

Parts not

PURE L OGIC

59 . The logic of known and unknown terms,it has been

said (5 is analogous to the calculus of known and un

known numbers .So, a logic in which all terms were known would have

an analogue in common Arithm etic, a calculus in which allthe numbers employed are known . Combination of term shas an analogue in multiplication of numbers, and separationof term s in division of numbers. As in logic combinationis unrestricted, so in calculus is mul tiplication . As in logicof known term s only, separation of term s is unrestricted, soin a calculus of known numbers only, divi sion is unrestricted .

But,as in logic of known and unknow n terms separati on is

restricted ,so in calculus of know n and unknow n numbers

division is restricted .

60. It is well known that, in lik e manner, w e cannotdivide both sides of an equation by an unknown factor

,and

assert the resul ting equation to b e necessarily true, becausethe unknown factor may b e= 0. Thus

,from xy

=xz,w e

cannot remove x,and assert y= z

, because if x happen tob e = 0,

the equation xy=xz is true, whatever finite numbersb e the meanings of y and z.

The correspondence is thus shown

LOGICAL PROPOSITIONS. MATHE M ATICAL EQ UATIONS.

Terms known admi t Numbers known admi tComb ination Multipl icationSeparation D i vi sion

(unless either d ivi dend contain d ivisor) (unless d ivisor=0)Terms unknow n admi t Numbers unknow n admi t

Comb ination Multipl icationbut do no t admi t but do no t admi t

Separati on D i vision .

The above analogies did not escape the notice of Pro

fessor Boole (pp . 3 6 and I am therefore at a loss tounderstand on what ground he asserts that there is a breachin the correspondence of the laws of logic and m athematics .

61. From the meaning of a w hole term w e cannot learn

the meaning of a p art.

In A =BC,if w e know A w e learn BC as a whole ; but

separate in prem ises , although they may also occur in com

bination . O therwise, w e always treat any whole combination as a single term .

CHAPTER VI

OF PLURAL TERMS

63 . A p lural term has one of several meanings, but i t is not

known w hich.

Thus B or C is a plural te rm, or term of many meani ngs ,for its meaning is e ither that of B or that of C, but it is notknow n which .

A term not in form plural , may b e distinguished assingle such is A .

64. The separate terms expressing the several possiblemeanings of a plural term are call ed alternatives

,and are to

b e joined together by the sign placed between each twoadjoining terms .Al l that has been said of single terms applies to plural

terms , mutatis mutand is.

65 . The meaning of a p lural term i s the same w hatever be

the order of the alternati ves.

E i ther B or C is the same in meaning as ei ther C or B,

that is , For the order in which w e think of

the possible quali ties of a thing cannot alter those qualities ,and the order must not convey any intimation that one

meaning is m ore probable than another.66 . A term is combined w i th a p lural term by combining

i t w i th each of i ts alternatives.

For what is A and either B or C , if it is B, is AB ; if itis C , is AC , and it is therefore either AB or AC .

67 . Let a plural term enclosed in brackets

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A plural term obeys the Law of Unity 6 9 )

7 2 . For any alternative or part of an alternative may b e

substituted 5 1) its expression in other termsThus

,if A B+ CD and D E

,substitute, getting

A=B+CE .

Substi tu 73 . A plural term may b e substituted like a single term”0” of for any term ,

single or plural, of which it is the expression .

p lural0

terms When In comb inat i on , the several alternatl ves must b eseparately combined 6 6 ,

Conversely, for a plural term may b e subs tituted itsexpression in a single termThus

, if A =BC and C =D+E,for C substitute D+E,

and A=BOr from the prem ises A BD BE B(D E) and

C =D+E,w e m ight by substitution get back to A= BC .

Plural 74 . A p lural term is know n w hen each of i ts alterna tives

is know n.

Thus,in A=B+C ,

A is known when the meanings of

each of B and C are know n . But of course from know inga single meaning of A,

w e cannot learn e ither or both of Band C .

Numb er Of 7 5 . With reference to the relation between the numberof prem ises

,and the numbers of known and unknown term s

3 3 -3 w e must treat as a separate term each alternativeof a plural term .

A proposition with a plural term thus corresponds to anequation with several unknown quantities .

Iflum l and 7 6 . As plural terms obey the laws of single terms, and

2253 a term single in form may b e plural in meaning, it will not

b e necessary for the future to distinguish p lural and sing le

terms, any more than it has been to distinguish combined

and simp le terms.

There is some danger of mi sconception concerning pluralterms. Though a plural term has one of several meanings

,

i t cannot bear in this system more than one at the same

PLURAL TERMS

time,so to

,speak . Hence it still remains a uni t, the nam e

of a single set of qualities, one of several sets , but it is notknown which. The whole of this system in short is uni tary ,

and involves the same remarkable analogies to a calculus ofunity and 0 which have been brought forw ard so explicitlyin Professor Boole ’s system .

CHAPTER VII

or NEGATIVE PROPOSITIONS

g,or notTERMS may also b e known and stated as differin

being the same in meaning.

7 7 . It is in the nature of thought and things that a thingw hi ch difi

'

ers from another difers from everything the same as

that other.

More briefly stated—Same as d if ferent are difi‘erent.

Hence in logicA term w hich difi

'

ers from another term in meaning differsfrom every term which i s the same as that other.

If A is not the same as B,whi ch is the same as C, then

A is not the same as C . The inference arises in the sameness

of B and C,allow ing us to substi tute one for the other. Hence

w e learn nothing of the sameness or difference of any twoterms, D and E, each of which differs from a third, F ; forD and E may each have any of an indefinite variety of

meanings, and each may yet differ from F (578 . Hence a chain of related prem ises between any of

which inferences can b e drawn,must not contain more than

a single negative prem ise. Also any inference in which anegative premise is concerned must b e a negative inference.

79 . A negative prop osi tion i s simp ly convertib le. For A is

not the same as B,is the same statement as B is not the same

as A .

80. Wh en same terms are combined w i th d if erent terms ,the w holes may be difi

'

erent.

zessary further to consider negative propositheir inferences may b e obtained by use of

positions.

CHAPTER VIII

OF CONTRARY TERM S

N egati ve 82 . The known meaning of a negative term is the absence

of the quali ty, or set of qual iti es, w hi ch forms the know nmean

ing of a certain other,i ts p osi tive term .

Thus not-A is the negative term signifying the absenceof the quality or set of qualities A . If the known meaningof A b e only a single quality, not-A means its absence ; butif A mean several qualities, not-A means the absence of anyone or more .

Thus,if A =B.C

not-A =R not-C not-RC not-B not-C.

83 . The negative of a negative term i s the corresp onding” 69mm

p osi tive term .

What is note not-A is A .

Si mp le con 84 . Since the relation of a positive to a negative term istrary termsd efi ned .

the same as the relation of a negative to a positive,let each

b e called the simp le contrary term of the other.‘

ato non . 85 . For convenience let not-A b e written a . Then anylarge and its small letter denote a pair of simple contrariesand not-a is A . Al so

,the contrary of BC 8 2 ) is

Bc+bC+ bc,

which expresses the absence of one or more of B and C .

86 . All that has been said of a term applies samely toone as to the other of a pair of contraries .

CON TRARY TERMS‘Thus

,a obeys the several laws

aa = a a+a =a and so on .

87 . Let a combined term or a proposition b e said to ‘ Involve’

involve a term when it contains either that term or its defi ned"

contrary .

88 . The contrary of a plural term is a term containing Contra rya contrary of each alternative . gr135

W “

Thus the contrary of -A+B+ C is abc. If any alternative has more than one contrary

,for each there will b e a

contrary alternative . Thus,A+BC has the plural contrary

c +abC abc.

89 . Any combined term which contains the simple con Contrarytrary of another term may b e called a contrary, or contrary

comb ination of this,or of any combination containing this .

Thus,any combined term containing A is a contrary of

any term containing a,and it will seldom b e necessary to

distinguish by name simp le contrari es,such as A and a from

contraries, or contrary combinations in general , which m erelycontain A or a. (See, however, 9 9 ,

90. It is in the nature of thought and things that a Law of

thing cannot both have and not have the same quali ty. £222a91. Hence a term which means a collection of qualities Con tracti o

in which the sam e quality both i s and i s not,cannot m ean

the qualities of anything which is or ever w ill b e known .

Such a term then has no meaning, that is to say, no

possible, useful, or thinkable meaning ; b ut it may b e saidto m ean nothing . Let it b e called a self-contradi ctory, or

,

for sake of brevity,a contradi ctory term .

92 . Let us denote by the term or m ark 0, combined Use of O.

with any term,that this is contradictory

,and thus excluded

from thought . Then Aa =Aa .0,Bb =Bb .0

,and so on . For

brevity w e may write Aa = 0,Bb = 0. Such propositions are

tacit prem ises of all reasoning .

Any two contrary term s in combination give a contrad ictory term .

Term o.

PURE LOGICO

93 .Any term being combined with a contradictory, the

whole is contradictory.

For the whole then means a collection of qualities whichdoes and does not contain some same qual ity, and is therefore by definition a contradictory.

Thus,if A=Bb =Bb.0

AC BbC EbC.0.

94. The term 0, meaning excluded from thought, obeysthe laws of terms .

O+0= Q

otherwise expressed —What is excluded and excluded isexcluded—What is excluded or excluded is excluded.

95 . Any term not know n to be contrad ictory must be taken

as not contradictory.

Any term known to b e contradictory is excluded fromnotice, and any term concerning which w e are desiringknowledge must therefore b e assumed not contradictory.

96 . In a p lural term of w hi ch not all the alternati ves are

cont radictory, the contrad ictory alternative or alternatives must

be excluded from notice.

If for instance A= 0+B,w e may infer A=B,

because Ai f it b e 0 is excluded ; and if it b e such as w e can desireknowledge of

,it must b e the other alternative B .

97 . N 0 contradi ctory term i s to be eliminated in d irect

inference.

For all w e can require to know of a contradictory term isthat i t is contradictory, and elim ination of a contradictoryterm would prevent rather than give such knowledge.

Thus if A =Cc . 0, B=Cc . 0, all that w e can require toknow of A and B is known from these premises , and cannotb e known from the inference A =B got by elim inating thecontradictory Cc .0.

So, if the only useful inferences are those showing each of A , B,

C,D

,E

,F,to b e

contradictory.

So, also, obviously, of intrinsic elimination .

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CHAPTER IX

OF CONTRARY ALTERNATIVES

99. It is in the nature of thought and things that a

thing i s ei ther the same or not the same as another thing.

O therwiseA set of quali ti es ei ther does or does not contain a certain

quali ty.

Hence, in logic, a term must contain the meaning of oneof any pai r of simp le contrary terms. ThusA term i s not altered in meaning by combination w i th any

simp le contrary terms as alternatives.

A=A

For if A has meanings containing only B,then Ab is

contradictory, andIf A has meanings containing only b, then AB= 0 and

A= 0+Ab Ab .

If A has meanings of which some contain B and some b,the law is still true.

This Law of Duality is not the same as Professor Boole’slaw of duality . (See

100. Some apparent exceptions may occur to this law.

For instance, let A = virtue, B = black, and b = not -black.

Then the statementVirtue i s ei ther black or not - black

,seem s true according

to the above law,and yet absurd. This arises from B and

b not being simple contraries ; for B may b e decomposed

LA IV OF D UALITY

into black-coloured—say BC,and b into not-black-coloured

,or

not black and not coloured,or bC+bc. Now

,virtue is really

not coloured at all, or is Abe, and, therefore, neither BC norbC. Here

,again

,w e must observe that the combination

Bc is contradictory from the tacit premise black i s a colour

eO ther apparent inconsistencies may b e sim ilarly ex

plained .

Professor De Morgan has excellently said ,1 ‘ It is not forhuman reason to say what are the simple attributes intowhich an attribute may b e decomposed .

’ And for such areason it is that I have as far as possible abstained fromtreating any term as know n to be simp le.

101. Let a term,combined with simple contraries as Develop

alternatives , b e called a development of the term as regards 2231’

s“ .

the contraries .Thus

,AB+Ab is called a development of A as regards

B,or in term s of B

,or involving B .

102 . Any term i s same in meaning after combinati on w i th Continued

all the p ossi ble combinati ons of other terms,and thei r contrari es

as alternati ves.

Since A =AB+Ab , and , again,A =AC+Ac, w e may

substitute for A in AB+Ab (g 5 1 ) its expression in term sof C . Thus

,

Again,since A=AD+Ad , w e may substitute a second

time, getting

A =ABCD+ABCd+ +Abcd , and so on .

103 . Let any two alternatives, differing only by a single Dual termpart-term and its contrary, b e called a dual term .

defi ned

Thus,AB Ab is a dual term as regards B, and

ABC+ABc as regards C, and w e may speak of B+ b orC+c as the dual part.104 . A dual term may alw ays be reduced to a single term

1Syllabus, p. 60.

PURE LO GIC

by removal of the contrary terms,w i thout altering the

meaning .

For the term thus obtained is , by the Law of Duality,

the same in meaning as the former dual termThus

,from such a term as AB+Ab, w e may always

remove the dual part B+b, and the meaning of the term Awill still b e as before, since A =AB+Ab is a self-evident9 9 ) truth always in our knowledge.

CHAPTER X

or CONTRARY TERM S IN PROPOSITIONS

105 . From any afi rmative p remi se w e may infer a negative Afi rmouoe

p rop osi ti on by chang ing any term on one side only into i ts

contrary. p osi tion .

From A =R w e have A not b for evidently B is not = b,

and hence, by Law of Difference A=R not = b,or

A not = b .

From AB =AC,sim ilarly

,AB not =Ac.

106 . The tw o terms of a negative p rop osi ti on are con Terms of

trari es. 2322126

For the two term s of a negative proposition are different tion .

in meaning. Hence there must b e some quality or qualitiesin the meaning of one , and not in that of the other ; thus ,the combination of the two terms would mean both the

absence and presence of a certain quality or qualities,and

would b e a contradictory. The two term s then are con

trary

107 . A negati ve p rop osi ti on may be changed into an N egative

0

1into afi r

afi rmati ve,of w hi ch one term i s a term of the negati ve, and mam

-

w

the other term thi s term combined w i th the contrary of the p rop osiother term of the negati ve.

PURE LO GIC

not AC,then AB ABc. For AB ABC+ABc ABc

, sinceABC is contradictory. And w e see that

ABc AB (contrary of AC)AB ABc+0 0.

Since w e may now convert any negative proposition intoan affirmative, it will not b e further necessary to use

negative propositions in the process of inference (5Inference 108 . From any contradictory combinati on w e may infer

that any p art of the combination not i tself contradi ctory is

not the same in meaning as the remaind er or any greater p art.

That the two parts differ may b e expressed in a negativeproposition

,or its corresponding affirmative.

For if the other part b e contradictory,it cannot b e the

same as the first part, which is not contradictory. And ifneither of the parts is contradictory in itself, they cannot b esame in m eaning, else their combination would not producea contradiction .

The affi rmative inferences corresponding 107 ) to thenegative ones deduced under this rule may b e otherwi sehad, so that it seems unnecessary to consider the negativeinferences further in thi s place.

109. The following are the chief laws or conditions oflogic

Condi tion or p ostulate. The meaning of a term must b esame throughout any piece of reasoning ; so that A =A,

B=B,and so on .

Law of Sameness. (S,

A = B= C ; hence A= O .

w of Simp li ci ty .

AA =A,BBB =B

,and so on .

Law of Same Parts and Whales.

A =B ; hence AC=BC.

Law of Uni ty.

A+A =A,and so on .

A =A

A =A (B+ b) (C+c)and so on .

It seems likely that these are the primary and sufficientlaws of thought

,and others only corollaries of them . Logic

may treat only of know n samenesses of things ; and differencesof things need b e noticed, only for the exclusion from purelogical thought of all that is self-contradictory .

In pure number and its science, on the other hand,

differences of things only are noticed .

The Laws of Simplicity, Unity, Contradiction , andDuality furnish the universal prem ises of reasoning. The

Law of Sameness is of altogether a higher order, involvingInference, or the Judgment of Judgments .

velopment.

Examp le.

CHAPTER XI

OF INDIRECT INFERENCE

110. Taken by itself, the development of a term 101)gi ves us no new knowledge about it. But taken with theprem ises of a problem, w e may learn that some of the

alternatives of the development are contradictory and to b erejected. The remaining al ternatives then form a new andoften useful expression for the term .

111. In thus using a development w e are said to inferindirectly, because w e use the premise to Show what a termis, not directly by the Law of Sameness, but indirectly byshow ing w hat i t is not.

Indirect Inference is direct inference with the aid of selfevident premises derived from the Laws of Contradiction andDuality. But all Inference is still by the Law of Sameness .

112 . Let A=B : required expressions for A,B

,a,b,in

ferred from this premise. Develop these terms as follows(i 101)

A : AB+Ab B=AB+aB

a = aB +ab b =Ab +ab

Exam ine which of the alternatives AB,Ab

,aB

,

contradictory according to the prem ise A=B.

A combined with A B gives A ABB AB Ba Aa aB 0

b Ab Bb 0

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PURE LOGIC

Method Of 115 . The process of indirect inference may similarly b eapplied to drawing any possible inference or expression fromany series of premises, however numerous and compli cated .

The full process may b e abbreviated according to the

following series of rules , which may b e said to form THE

METHOD OF INDIRECT INFERENCE

1 . Any premises being given, form a combination con

taining every term involved therein Changesuccessively each simple term of this into its contrary

,so

as to form all the possible combinations of the s imple term sand their contraries . "

2 . Combine successively each such combination withboth members of a premise. When the combination formsa contradiction with neither side of a prem ise, call it an

subject.included subject of the premi se when it forms a contradic

Excl uded . tion with both sides , call it an excluded subject of the

prem ise ; when it forms a contradiction with one side only,

Contrad ic call it a contrad i ctory combinati on or subject, and strike it out.”on“

W e may call either an included or excluded subject aPoss ib le. p ossi ble subject, as distingui shed from a contradictory com

Imp ossib le. bination or imp ossib le subject.Rep ea ted 3 . Perform the same process with each premise. Then

jgzpm’ a combination is an included subject of a series of prem ises,when it is an included subject of any one ; it is a contradictorysubject when it is a contradi ctory of any one it is an excludedsubject when it is an excluded subject of every prem ise.

Selection. 4 . The expression for any term involved in the prem isesconsists of all the included and excluded subjects containingthe term

,treated as alternatives .

Reduction . 5 . Such expression may b e simplified by reducing alldual terms and by intrinsic elim ination 5 2 ) ofall term s not required in the express ion .

6 . W hen it is observed that the expression of a termcontains a combination which would not occur in the

expressiono

of any contrary of that term,w e may eliminate

the part of the combination common to the term and itsexpression . (See below,

son.

INDIRECT IIVFEREN CE

7 . Unless each term of the prem ises and the contrary of Contrad ic

each appear in one or other of the possible subjects, the mypremises mus t b e deemed inconsistent or contradictory.

Hence there must always remain at least two possiblesubjects

116 . Required by the above process the inferences of the Examp le.

prem ise A= BC .

The possible combinations of the terms A,B

,C

,and their Develop

contraries,are as given in the margin . Each of these being mm "

combined with both sides of the prem ise, w e have the Comp ar ifollowing results 30”

ABC ABC ABC included subjectABc ABCc 0 ABc contradictionAbC ABbC 0 AbC contradictionAbc ABc 0 Abc contradiction

0 AaBC aBC aBC contradiction0 Ac aBCc 0 c excluded subject0 AabC aBbC 0 abC excluded subject0 Aabc aBc 0 abc excluded subject

It appears, then, that the four combinations ABc to aBCare to b e struck out

,and only the rest retained as possible

subjects .Suppose w e now require an expression for the term b as Selection.

inferred from the premise A= BC . Select from the includedand excluded subjects such as contain b , namely abC and abc.

Then b = abC+abc,but as occurs only with b

,and not E limina

with B,its contrary, w e may, by Rule 6 , elim inate b from

“07“

abC ; hence b = aC+abc.

117 . The validity of this last elim ination is seen by H im -

M

drawing the expression for aC, which is abC. Then “

Z

OE “

;between b = abC+abc

,and abC= aC,

w e may elim inate abC1) mm

by substituting 5 1 ) its expression aC. And sim ilarly inall other cases to which the rule applies .W e m ight also reduce the expression for b by Rule 5 , as

followsb abC+abc ab (C+ c) ab.

Other in

Rela tion ofB and C.

PURE LO GIC

118 . To express a w e have

a c +abC+abc,

but observ ing that none of Be,bC,

be, occur wi th A,

so thatBc=c ,

bC=abC,bc=abc

, w e substitute these simpler terms ,elimi nating a whence an evident truth

0

119 . Similarly, w e may draw any of the followinginferences

A ABC AB ACB AC+cC AB+abC

c aB+abc ac

aB Bc

aC bC

ab abC+abc ab (no inference)ac c +abc ac (no inference) .

120. Observe that since B and C are samely related toA

,w e may get any inference concerning one of these terms

from the sim ilar inference concerning the other by interchanging B and C

,b and c

Before proceeding to further examples of indirect inference, w e may make the following observations.

121. When any term appears on both sides of a prem ise,as A in AB =AC

,any combination contai ning its contrary

,

a, is an excluded subject. Thus

,in combining any term

with both sides of a proposition, w e render any contrary of

the term an excluded subject.So, in mathematics w e introduce a new root into an

equation when w e multiply both sides by a factor.122 . An excluded subject

,though admi tting of inference

and admi tted into inferences,is of inferior and often of no

importance. As its name expresses,it is usually a com

bination concerning which w e do not desire knowledge .

The sphere of an argument,or the Uni verse of Thought,

contains all the included subjects . An excluded subject is

INDIRECT INFEREN CE

such as lies beyond this sphere or universe. But w e

are obliged to consider excluded subjects,because the

excluded subject of one prem ise may b e the includedsubject of other prem ises .

123 . When a premise is plural in one or both sides , an Pluralexcluded subject is a contrary of all the alternatives on both10mm“

sides, and a contradictory combination is a contrary of allon one side, and not of all on the other side.

124 . O f an identical proposition the term itself appearing Identicalon either side is the only included subject. All others areexcluded, and there are no contradictory combinations . Itsuseless nature is thus evident.

125 . Any subject of a preposition remains an included , Commonexcluded , or contradictory subject as before

,after combina “ We“

tion with any unrelated terms . Thus,i f the argument b e

restricted to a sphere or common subject, defined by certainterms, these do not need expression in each prem ise, b utmay b e retained as an exterior condition . Thus

,by ABCD

w e m ight mean that ABCD is to b e understood as combined with each term of any prem ises placedwithin the brackets . ABCD is then the common subjectof the prem ises, which must contain no contrary of this .And any contrary of ABCD is an excluded subject of thewhole .

126 . Any set of term s which always occur in the Of un

premises in unbroken combination may b e treated as a fgzggwsimple term. tions .

Thus,i f BC occur alw ays thus in combination , w e

may write for it,say D

,and then d or not -BC is

bC+Bc+bc.

127 . Any set of alternatives which always occur together Unbrokenin the prem ises as alternatives may b e treated as a single fe

’mg.’term .

Thus,if B and C occur always as alternatives , w e

may for B+ C write, say D ,and then d or nei ther B nor

C is be.

128 . Any proposition may b e treated under the form of

PURE L OGIC

A =B,so long as w e do not require to treat its part-terms

or alternatives separately. (By 1 2 6 ,

129 . Hence the convenience in every branch of knowledge of using technical terms to stand for every large set

of terms which usually occur together. But such termsbecome the source of error if w e do not carefully keepbefore us their definitions , those adopted prem ises in whi chw e express the set of combined or alternative terms forwhich w e substitute a technical term .

130. In that branch of knowledge, however, called FirstPhilosophy, which is analyti c, and aim s at resolving thi ngs,or our thoughts about them , into their simplest components ,the use Of technical term s is fallacious. Such terms cannotassist analysis, since each arises from the synthesis of manysimpler terms, form ing its definition . All reasoni ng, then,in Metaphysics or First Phil osophy, ought to b e carried onin the simplest and most vernacular elements of speech.

Analytic science should b e like a m ill which grinds downthe ordinary grains of thought into their small est andsimplest particles. It is in the bakehouse w e shouldcombine these particles again into leaves of a size andconsistency suitable for ordinary use. But most m etaphysical reasoners, it seems to m e, have m istaken the m illand the bakehouse .

131. It is not always necessary to carry out the processof inference exactly as in the rules. Each or any prem isemay b e treated as a separate one, i f desirable, and itspossible subjects afterwards combined with the possiblesubjects of other premises . W e may thus successively addpremises, or try the effect of supposed ones .For instance

,since AB and ab are the possible subjects

of A=B,and BC and be of B=C, the possible combinations

of these, namely ABC and abc,are the possible subjects of

the two prem ises combined, observing that ABbc and abBC

are contradictory.

132 . If premises b e related, the indirect inferenceswill include all possible direct inferences . From unre

INDIRECT INFEREN CE

lated premises w e shall also get such inferences as are

possible.

Thus,from the unrelated prem ises

A B,C D

,

w e haveA BCD Bed

d =ABc+abc, and so on .

133 . It does not seem possible to give any general proof Proof of

that the conclusions of the indirect m ethod must agree withthose of the direct m ethod, which will make its truth any

the m ore evident . Such proof could b e little less than ageneral recapitulation of the several Law s of Thought.134 . It hardly needs to b e pointed out that the method E ncl itl ’s

of indirect inference is equivalent to Euclid ’s indirect de Zigzagmonstration

,or reductio ad absurdum . Euclid assumes the tion .

development of alternatives, usually that of equal or greater

or less,and showing that two of these lead to a contradiction

,

establishes the truth of the third .

135 . Nor is this process of reasoning at all new or un 007m m,

common in any branch of knowledge save logic, which was hifeii’

“I“

supposed to b e the science of all reasoning . Simple instances method .

occur perhaps as frequently as instances of direct inference,

and complicated instances are only rendered scarce by thelim ited powers of human memory and attention . Amonginstances of indirect argument w e may place all those discourses in which a writer or speaker states several possiblealternatives or cases of his subject, and, after showing some

of them to b e impossible, concludes the rest to b e necessary ,or else proceeds further to develop and consider these withregard to other prem ises A good instance is foundin Paley

’

s Argument on the Divine Benevolence (IIoralPhi l

,Book II

,chap . v) . The old logical process called

absci ssio infi ni ti has a close relation to indirect inference .

136 . Even brute animals, it w ould seem ,may reason by Quotation .

the indirect methodThis creature, saith Chrysippus (of the dog) , is not void

48 PURE

of Logick : for, when in following any beast he cometh tothree several ways, he smelleth to the one, and then to thesecond ; and if he find that the beast which he pursueth b enot fled one of these two ways, he presently, without smelling any further to it, taketh the third way ; which , saith thesame Philosopher, is as i f he reasoned thus : the Beast mustb e gone either this , or this , or the other way ; but neitherthis nor this ; E rgo, the third : so away he runneth .

’

SIR W. RALEIGH.

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PURE LO GIC

m isrepresented the ordinary course of reasoning. Not only,

as a fact, do the several sciences establish multitudes of

propositions of which the two term s are equivalent anduniversal, but all definitions are propositions of this kind ,and the definitions requisite in connecting the meanings ofmore and less complex term s , must always form a large partof our data in reasoning . If w e further consider that evenAristotle’s negative propositions have a universal predicate ,that men Show a constant tendency to treat the predicate of

the proposition A as universal, whence several common kindsof fallacy, and that reason ing from same to same things may

be detected as the fundamental p rincip le of all the sci ences, w e

need have no hesitation in treating the equation as the trueproposition

,and Aristotle ’s form as an imperfect preposition.

It is thus the Law of Sameness,not the d i ctum of Aris

totle, which governs reasoning .

Qaantifi ca 139 . It is only of very late years that the imperfectioni f?”

of ? ”of the ordinary proposition has been properly pointed out .It is the discovery of the se - called quantification of the

p redicate which has reduced the proposition to the form of aconvertible equation, and Opened out to logic an indefinitefield of improvement.

Boole’

s 140. Professor Boole ’s system,

first published in hisMathematical Analysis of Logi c, in 1 84 7 , involves thisnewly discovered quantification of the predicate . Accordingto Boole, the some

,which is the adjective of particular logical

quality,is an indefini te class symbol. Men are some mortals

is expressed b y ' him in the equation, x= vy, where x instructsus to select from the universe al l things that are men

,and

y to select all things that are mortal . The proposition theninforms us that the things which are men consist of an indefinite selection from among the things which are m ortal

,v

being the symbol of this indefinite quantity or class selected .

Further 141 . One more step seem s to me necessary. It is to

$12 separate completely the qualitative and quantitative meanings of all logical

’

terms,including the word some. In the

qualitative form of the proposition man i s some mortal—or

BO OLE ’

S S YSTE III

more correctly speaking, man i s some kind of mortal—w e

interpret some or some kind as meaning an indefinite andperhaps unknown collection of qualities

,which being added

to the qualities mortal,give the known qualities of man.

In the quantitative form men are some mortals,w e have the

equivalent statement that the collection of individuals in theclass some mortals is the collection of individuals in the classmen.

142 . I t is strange that the purely qualitative form of Quali tative

proposition man i s some kind of mortal,which is the most fi i

’

f’ (

218

2

’

s

distinct form of statement,and is perhaps the m ost prevalent

,regarded .

both in science and ordinary thought , was totally disregardedby logicians

,at least as the foundation of a system of logic.

‘The Logicians, until our day

,

’ says Professor De Morgan,

1

have considered the extent of a term as the only object oflogic

,under the nam e of the logi cal w hole : the intent was

called by them the metap hysical w hole,and was excluded

from logic.

’

143 . It will b e seen that this w ord some or some kind,

‘Soni e

’

the source of so much difficulty and error,must in our Zgigr

so’ne

system b e treated as a term of indefinite and unknownmeaning. It is an unknown term ,

not only at the beginning of a problem ,

b ut throughout it . In no tw o prem isesthen can the term some or some kind b e taken to mean the

same set of qualities. Thus w e cannot argue through orelim inate a term with some

,while at least it retains this

unknown term : that is to say,w e can never use it as a

common term 2 7 ) in direct inference . Thus , if A is someB

,and some B is some C

,w e cannot elim inate some B getting

A i s some C,because some being of unknown meaning

,the

some B is not necessarily the sam e in both cases . This isstill more plain in the form A i s some kind of B,

and some

kind of B i s some kind of C,for it is obvious that the one

kind of B is not necessarily the same as the other.144 . Since the term some or some kind is not only nu Symbol for

known but remam s unknown throughout any argument,w e

" m’e'

1 Syllabus, p . 6 1 .

PURE LO GIC

m ight conveniently appropriate to it some symbol such asU

,to rem ind us of its special conditions . Thus no term U

is to b e taken as same with any other term U , or U =U isnot known to b e tr ue . But in the propositions A and E itis always open to us, and is best to eliminate U by writingfor it the other member of the proposition Thus,A=UB

,m eaning that A is some kind of B,

involves threeterms . It is much better written as A=AB

,involving only

A and B,and yet perfectly expressing that the quali ties of

B are among those of A, but not necessarily those of A allamong those of B .

145 . The four prepositions of the old logic may thusfind expression in our system

E very A is R A=UB or A =AB

No A is B A=U b or A=Ab

Some A i s B UA=UB or CA=DESome A i s not B UA = U b or CA='Db .

146 . Two new propositions of De Morgan’s system are

thus expressed

E verything is ei ther A or R

Some things are nei ther A nor B

147 . All these propositions, and as many more as may

b e proposed, can b e brought and partially treated (g 12 8 )under the form A =B

,which I believe to b e the simple form

of all reasoning. The existence of doubly - universal propositions of this kind was far from being unknown to manyof the School Logicians

,but out of deference to the Aristo

teli an system ,such propositions were neglected. The present

Archbishop of York first embodi ed this preposition in asystem of logic

,giving it the name U. (Thomson

’s Outlines, passim) .

CHAPTER XIII

EXA M PLES OF THE M ETHOD

IN this chapter I shall place some m iscellaneous examplesof inference according to the system of the foregoingchapters , suited to Show the power of its method

,or i ts

relation to the old logic.

148 . Let us take a syllogism in FELAPTO N .

No A is B A = U b =Ab

Every A is C A = UC=AC

Some C is not B

From A =Ab w e m ight by combination (g 4 5 ) infer D irectAC =AbC, and from A=AG

,Ab =AbC whence AC=AbC= ”Wren“

Ab,or AC =Ab

,w hi ch i s a more preci se statement of UC= U b ,

or some C i s not B,the Aristotelian conclusion .

W e may , however, obtain this conclusion , as well as allother possible ones , by indirect inference .

O f the possible combinations of A,B

,C

,a,b, c, ABC and A60

ABc are contradicted by the first prem ise, and Abc (as wellas ABc) is contradicted by the second prem ise . AbC,

aBC,abC

c,abC

,abc

,are the remaining combinations in which w e

abc

find there is no relation between B and C p er se,since B

occurs with C and c,and C occurs with B and b . But

AC=AbC,and Ab =AbC

,whence, by elimination

,AC=Ab ,

the sam e conclusion as before .

The following conclusions may also b e drawn

[ U R L L V U J L/

a a a (B+ b ) (C+ c)a (no inference)

B= aB

b =AC +abC +abc AC+ab

C Ab +aBC+abC Ab + aC

c c +abc ac

ab abC +abc ab (C+c) ab (no inference)aC aBC+abC aC(B+ b) aC(no inference)be abc.

149 . The premise AB = CD is of some interest. Itcontradicts the combinations ABCd, ABcD,

ABcd,which

are AB and not CD,and AbCD

,aBCD

,abCD, which are

CD and not AB. From the remainder w e easily draw the

inferences.

A BCD+Ad +AbcD+Abcda BCd +BcD Bed ad +abcD+abcd .

AB_—_CD

Observing that A and B enter samely into the premise ,w e may easily deduce the expressions for B and b by interchanging A and B,

a and b in the ab ove ; thus 54 - 5 6 )

B ACD aBCd+c D+c d .

And since A,B enter samely with C, D ,

w e m ightdeduce the corresponding expressions for C ,

D,and c

,a,by

interchanging at once A with C , and B with D , or A withD

,and B with C .

From the expression for a w e thus get

d CBa+CbA+Cba+c a+dc +dcba.

Observe , that if the expression for A b e combined withthat for a

,nothing but contradictory term s will b e the

result, verifying Aa = 0. And,if w e combine the expres

sions for any term s not contrary, as B and d , w e get the

same result as w e m ight have drawn by the separate appli

cation of the process.

Thus, Bd AbCD+aBCd+c d 0+aBd .

In comparing the eight combinations of A,B

,C

,a,b,c,

w ith the prem ise, any one is contradictory which containsA w ithout containing e ither B or C ; or, again , which con

tains either B or C w ithou t containing A . Thus,ABC

,

ABc, and AbC, are the included subjects , abc is an excluded

subject, and the rest are contradictory .

W e may draw the inferences

A BC Bc+ bC

ABC

a = bc

B=ABC+Ac =AR

b =AbC +a

C z ABC+Ab =AC

c z ABc +a

Observe that B and C enter samely,so that their expres

sions may b e mutually derived by interchange .

151 . The premise A =Bc+ bC differs from the last in the Examp le :very important point that A cannot at once b e B and C .

A

i féIt has the included subjects ABc and AbC, and the ABC

excluded subjects aBC and abc. The following expressions 21

125are seen to b e s imple and symmetrical , and it is instructive a bc

to form their combinations .

PURE LOGIC

remains true when the negative propositions are convertedinto their corresponding affirmatives

Let us take the premises

A is not the same as B,

C is not the same as B .

These may b e expressed by the affi rmative propositionsA =Ab

, C= bC.

AbC If w e go through the process of indirect inference, and

£ 6

" attempt to express A and C in terms of each other, w e shallabC obtain 2a’”

=Ab

C=AbC+abC=bC

These are the prem ises over again, and there can b e nonew inference, except B=c .

Solutions153 . The proposition A =B being the simplest form of

statement, its full solution is given below, and the solutionsA=B

of the sim ilar propositions A = b,a =B

,a =b

,are inferred by

interchanging A and a,B and b .

Premise A=B A= b a =B a = b

Included subject AB Ab aB ab

Excluded subject ab aB Ab ABContradiction Ab AB AB aB

Contradiction aB ab ab Ab.

154 . Let us take A ABC,

B C BD CD .

W e have, by direct inference from the second premise,

BC=BCD 4 5 )Hence A=ABC=ABCD

The indirect process gives four included and two ex

cluded subjects,as in the margin.

Hence not only the above inference, but the following,among other possible ones

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Destruc ti ve 157 . The following is known as a Destructive Condit ional Syllogism .

If A is B,C is D ; but C is not D ; therefore, A is

not B .

Supplying the suppressed term , say E, expressing the

circumstances in which A is not B,the foll owing is the

statement of this syllogism in our system

AB ABCDCE CdE .

t ional

By direct inference

ABE ABD.CE ABD.CdE 0.

Hence ABE is known to b e contradictory ; thereforeAE is not ABE, or in the circumstances E,

A is not B.

158 . The forms of the old logic being comprehended inthis system along with an indefinite multitude of otherforms , logicians can onl y properly accept this generalisation ,due to Boole , by throwing off as dead encumbrances theuseless distinctions of the Aristoteli an system . The pasthistory of the Science must not, as hithert o, bar its progress .And Logic will b e developed almost like Mathematics, whenLogicians like Mathematicians discrim inate between the Studyof Thought and the Study of Antiquarian Lore .

I will now give a few complex problems,more suited to

show the power of the method .

159 . Let the premises b ep roblem.

A= B+CB= c +d

c= cD

AD =BCD.

And let it b e required to infer the description of anyterm , say a . By the indirect process , w e Shall find thatthe only combination uncontradicted by one or other prem iseis ABCd .

Thus, w e find there cannot b e any a at all, w ithout

EXAIIIPLE S

contradiction,whatever may b e the meaning of this result.1

It means , doubtless , that the prem ises are contradictory

W e also easily infer any of the following '

A BCd AB Cd ABC ABCd

B ACd AC Brl ABd ABCd

C ABd B0 Al l etc.

d ABC .

160. The following prem ises are such as m ight easily l i ming” ,

occur in physical science

A : ABC +AbC

B BDE+BdeC= CDa

1 The follow ing law , b eing of a less evident character than the rest,has Law of

b een placed apart. infini te/A

Every log ica l term must have i ts contrary .

That is to say t atever qual ity w e treat as p resent w e may also treat asabsent.

There is thus no b oundary to the universe of log ic. No term can b e pro Uni verse ofposed w ide enough to cove r i ts w hole sphere ; for the contrary of any term 309 50 nu

must add a sphere of indefinite magnitude . Let U b e the universe then ubounded .

i s not included in U . N or w ill special terms l im it the universe .

Th ing existi ng has i ts cont rary in thing not existi ng .

Th ing thinkab le has i ts contrary in thing not thinkab le.

Even th ing, the w idest noun in the language , has a contrary in that w hichi s not a thing .

O f course the ab ove is only t rue speak ing in the strictest logical sense, andusing all term s in the most perfect general ity .

If the ab ove b e g ranted as true , every proposition of the form A=B+ b Contrad ic

must b e regard ed as contradictory of a law of thought. For the contrary of to} ?

A from the ab ove is Bb , a contrad iction,or A is used as having no contrary, 3mm"

and form ing the universe .

Also every system of prem ises must b e rej ected w h ich altogether contradicts Contrad icany term or term s

. Thus in the ind i rect process w e must alw ays have at least p re

tw o comb inations remaining possib le, one of w h ich must contain the contrary mw es '

of each simple term in the other. In th is vi ew the peculiar prem isesA : B+C

B=c + d

159) contain sub tle contradictions . For a according to the fi rst premisemust b e be

,and b eing c

,i t must b y the second premise b e B,

and hence b yfirst premise also

,A

,or b oth A and a

,B and b .

But this sub ject needs more considerat ion.

PURE LO GIC

The series of possible combinations in the margin givesby inspection perhaps the most useful information, but thefollowing are a few formal inferences .

A ABcDE+ABcde+AbCD‘

e

BcD BeDE

abd abcd

ed ABcde+c de+abcd (E —e)Bcde+abd

bCD AbCDe+abCDe

bCDe.

There is no relation between abc and D and E.

161. I conclude with the solution of a still more com

plicated system of premises .

A+O+ E =B +D+F

Bc+bC=De+dE

AD =ADfD = e

C=Cd .

The possible combinations are

ABcDefABc F

ABc fAd EF.

Whence the following,among many inferences,

may b e drawn :

A BcDef+ABc +Ad EF

Be ADef +BcD ABcefcd BEe ABc

dE ABcd+ bCF c Bed bCF

d bCF CdE

AF ABc F+Ad EF

aF c dEF ad EF

b d EF

C d EF b .

Whence the remarkable and unexpected relation C= b ,which it would not b e easy to detect in the prem ises .

162 . Inferences may b e verified by combining the ex

pressions of two or more terms, and comparing the resultwith the expression of the combined term as drawn fromthe series of possible combinations . For instance

,in the

problem last given w e may combine the expressionfor A with that for dE ,

as follows

A . ( lE (BcDef+ABc oer)e ABc 0+0 Ad EF

,

the contradictory combinations be ing struck out . But theexpressions thus obtained may not always b e in theirsimplest term s .

163 . The reduction of inferences to their simplest terms , redaction

it may b e remarked, is in no way essential to their truth ; gffifeces '

it only renders them more pregnant with information . I tis

,perhaps, the only part of the process in which there is

any difficulty.

164 . III w orking these logical problems, it has been W orking offound very convenient to have a series of combinations ofterms beginning with those of A

,B

,and proceeding up to

those of A,B

,C

,D

,E

,F

,or more

,engraved upon a common

writing slate . In any given problem ,the series is chosen

w hich just furnishes sufficient letters for the distinct terms .The contradictory combinations may then b e rapidly struckout

,and the remain ing combinations lie ready before the eye .

CHAPTER XIV

COMPARISON W ITH BOOLE’

S SYSTE M

165 . To Show the power and facility of this m ethod, ascompared with that of Professer Boole, it will b e sufficient,as regards those already acquainted with Professor Boole’ssystem

,to present the solution of one of his complex ex

amples . Thus,let us follow Professor Boole’s investigation

of Seni or’s definition of wealth, namelyl— that w ealth i s

w hat is transferable, limi ted in supp ly, and ei ther p roductive

of p leasure or p reventi ve of p ain (Boole, p .

Let A NVealth

B TransferableC Lim ited in supplyD Productive of pleasureE Preventive of pain .

The definition in question is expressed by the proposition

A

which includes all the combinations of D ,E

,d,e, except dc.

Striking out the dual term (E+e) from BCD w e

may state the definition in the more concise form

A BCD BCdE .

W e may pass over Professor Boole’s expression for A

,after

1 Here, as usually elsew here, I take w ords in intent of meaning, and transform most statements accordingly .

s ooLE’

s EXAMPLE

intrinsic elim ination of E as being sufficiently obvious .

166 . Required C in terms of A,B

,D (Boole, p . E xp ression

Forming all the possible combinations of A,B

,C

,D

,E for C’

ABCDE

and their contraries, and comparing them with the prem ise,

ABCDO

w e shall find all the combinations from ABCde to aBCdEABCdE

inclusive contradicted . The remaining subjects are as inthe margin .

Selecting the terms containing C , w e have

C ABCDE+ABCDe+ABCdE+aBCdc

+ abCDE +abCDe +ad E +ad e.

Striking out the dual terms and intrinsically clim inating remaining E ’s or e’s by substitution of C

,w e have

C ABCD+ABCd+aBCd+abCD+ad .

Elim inating C from ABCD because ABD =ABCD,

and striking out the dual term s (A+ a) and w e haveeither of the expressions

C ABD+BCd +abC

C ABC +aBCd+ abC.

From the latter w e read, What is lim i ted in supp ly i s ei ther

w ealth, transferable (and ei ther p roducti ve of p leasure or not,

ABC ) , or else some kind of w hat i s not w ealth,but i s ei ther

not transferable (abC) , or, if transferable, i s not p roducti ve of

p leasure (aBCd ) .This conclusion is exactly equivalent to that of Professor

Boole , on p . 108 .

167 . H is se - called secondary propositions , nam ely,

‘1 . .vegati i

-e

Wealth that is intransferable and productive of pleasure,does Egg

”

not exist and 2 . Wealth that is intransferable and notproductive of pleasure does not exist,

’

are negative conclusions implied in the striking out of the contradictory com

himations AbCDE ,AbCDe, AbcDE

,AbcDe

,and Ad E

,

Ad e,Abc ,

Abcde,which are easily reducible to

AbD (C+c) (E+ e) 0 AbD 0

Abd (C+ c) Abd = 0.

for D.

O ther in

PURE L OGIC

The expression does not exist ’ is Open to exception .

168 . Again, required an expression for productive of

pleasure (D) , in term s of wealth (A) , and preventive of pain

(E) (Boole, p .

The complete collection of combinations containing D is

ABCDE abCDE

ABCDe abCDe

c DE abcDE

c Dc abcDe.

W e may then write D as follows

D

But w e may observe al so that

ADE =ABCDE and Ae=ABCDe.

Hence w e may substitute ADE and Ae for the two firstterm s of the expression for D . W e may also strike out thedual term (E+e) in the third term ,

and elim inate the pluralterm intrinsically 5 2 ) by substitution of D .

Thus w e get the expression in the required term s :

which may b e“ translated into these words What i s p ro

ducti ve of happ iness i s ei ther some kind of w ealth p reventive

of p ain,or any kind of w ealth not p reventive of pa in,

or so me

kind of w hat i s not w ealth (Boole, p . 1 1 1 ad fi n .)169 . For the expression of d w e easily select

of which the meaning is What i s not productive of p leasurei s ei ther some kind of w ealth p reventive of p ain, or some kind

of w hat is not w ealth (Boole, p .

170. These are the chief inferences furnished by Mr.Boole. From the list of possible combinations w e couldeasily add a great many more inferences

,in fact as many

more, as may b e drawn concerning any of the five termsA

,B

,C

,D , E, and their contraries .

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PURE LOGIC

plexi ty of the problem increases , and a considerable liabilityto m istake arises . But this is in the nature of things

,and

the process of inference, consisting in the mere comparisonof terms as to their sameness or difference, seems to me the

simplest process that can b e conceived .

173 . Compared with Professor Boole ’s system ,in its

mathematical dress , this system shows the following ad

vantages1 . E very process is of self- evident nature and force, and

governed by laws as simple and primary as those of Euclid ’saxioms .

2 . The process is infallible, and gives no uninterpretableor anomalous results .

3 . The inferences may b e drawn with far less labourthan in Professor Boole’s system ,

which generally requiresa separate computation and development for each inference .

CHAPTER XV

RE M ARKS ON BOOLE’

S SYSTE M,AND ON THE RELATIO N O F

LOGIC AND M ATHE M ATICS

174 . SO long as Professor Boole ’s system of mathematical logicw as capable o f giving results beyond the pow er of any othersystem

,it had in this fact an impregnable stronghold . Those

w ho w ere not prepared to draw the sam e inference s in som e

other manner could not quarrel w ith the manner of ProfessorBoole . But if it b e true that the system of the foregoingchapters is of equal pow er w ith Profe ssor Boole ’s system ,

the

case is altered . There are new tw o systems of notation, giv ing

the sam e formal results , one of w hich giv e s them w i th selfe v ident force and m eaning, the other by dark and symb olicprocesse s . The burden of proof is shifted , and it must b e forthe author or supporte rs of the dark system to show that it isin som e w ay superior to the evident system .

17 5 . It is not to b e denied that Boole ’s system is consistentand perfect w ithin itself. It is, perhaps, one of the mo stmarvellous and adm irable piece s of reasoning ever put together .Indeed

,if Professor Ferrier, in his Insti tutes of Ill etaphys ics, is

right in holding that the chief e x cellence of a system is in beingreasoned and consistent w ithin itself

,then Professor Boole ’ s is

nearly or quite the m o st perfect system ever struck out by a

single w r iter.17 6 . But a system perfect w ithin its elf may not b e a perfect

representation of the natural system of human thought . The

law s and conditions of thought as laid dow n in the system mayno t correspond to the law s and conditions of thought in reality.

If so,the system w ill not b e one o f Pure and Natural Logic .

Such is,I believe

,the case . Profe ssor Boole ’s system is Pure

Logic fettered w ith a condition w hich converts it from a pure ly

68 PURE LOGIC

logical into a numerical system . His inferences are not logicalinferences ; hence they require

.

to b e interpreted, or translatedback into logical inferences, w hich might have been had w ithoutever quitting the self-evident processes of pure logic.

Among various objections w hich I m ight urge to Boole’

s

system,regarded as purely logical in purp ose, are four chief ones to

w hich I shall here confine my attention.

First Objection

17 7 . Boole’

s symbols are essentially difierent from the names or

symbols of common discourse—his logic is not the logic of common

thought.

Professor Boole uses the symbol to join term s together, onthe understanding that they are logical contraries, w hich cannotb e predicated of the same thing or combined together withoutcontradiction. He says (p. 3 2)—

‘In strictness,the w ords

“and

,

”

or,

” interposed betw een the te rm s descriptive of tw o or

more classes of obj ects, imply that these classes are quite distinct,so that no member of one is found in another. ’

178 . This I altogether dispute . In the ordinary use of theseconjunctions

,w e do not necessarily join logical contraries only ;

and w hen term s so joined do prove to b e logically contrary,

it is by virtue of a taci t premise, something in the meaningof the names and our knowledge of them

,w hich teaches us

they are contrary . And w hen our know ledge of the meaningsof the w ords joined is defective, it will often b e impossibleto decide w hether terms joined by conjunctions are contraryor not .

179 . Take, for instance, the proposition— ‘A peer is either aduke

, or a marquis, or an earl,or a V iscount

,or a baron .

’

If

expressed in Professor Boole’s symbols,it would b e implied that

a peer cannot b e at once a duke and marquis , or marquis andearl. Yet many peers do possess tw o or more titles

,and the

Prince of Wales is Duke of Cornw all,Earl of Chester, Baron

Renfrew ,etc . If it w ere enacted by parliament that no peer

should have more than one title, this w ould b e the tacit premisew hich Professor Boole assume s to exist.Again, Academic graduate s are either bachelors, masters

,

or doctors, ’ does not imply that a graduate can b e only one of

these ; the higher degree does not annul the low er.Shakespeare

’s lines

REMARKS ON BO OLE’

S S YSTEM

Beauty , truth , and rarity,G race in all simpl icity,Here inclosed in cinders l ie .

To th is urn let those repai rThat are either true or fai r, ’

certainly do not imply that beauty, truth, rarity, grace, and the

true and fair are incompatib le notions , so that no instance of one

is an instance of another.

In the sentence Repentance is no t a single act,b ut a habit

or virtue it cannot b e im plied that a virtue is not a habit ; byAristotle s definition it is.Milton has the e xpression in one of his sonnets—J Unstain

’

d

by gold or fee ,’

w here it is obvious that if the fee is not alw aysgold , the gold is a fee or brib e .

Tennyson has the e xpression w reath or anadem .

’ Mostreaders w ould b e quite uncertain w hether a w reath may b e an

anadem,or an anadem a w reath

,or w hether they are quite

distinct or quite the same .

From Darw in ’ s Origin, I take the expression,

‘W hen w e see

any part or organ d eveloped in a remarkable degree or manner.

’

In this,or is used tw ice

,and neither tim e disjunctively . For if

part and organ are not synonymous,at any rate an organ i s a

part. And it is obv ious that a part may b e developed at the

same time both in an e x trao rdinary degree and manner, althoughsuch cases may b e comparatively rare .

180. F rom a careful examination of ordinary w ritings, it w illb e found that the meanings of term s j o ined by ‘

and’

or varyfrom absolute identity up to absolute contrariety. There is nological condition of contrariety at all

,and w hen w e do choose

contrary e xpre ssions, it is because our subj ect demands it . The

matter,not the form of an e x pression

,points out w hether terms

are exclusive. And if there is one point on w hich logicians areagreed, it is that logic is formal

,and pays no regard to anything

not formally e x pres sed . (See181 . And if a further proof were wanted that Professor

Boole ’s symbols do no t correspond to those of language , w e haveonly to turn to his ow n work . He actually translates one samesentence into different sets of symbols, according to the view he

takes of the matter in hand . For instance (p . 5 9 ) he interp rets‘Either productive of pleasure or preventive of pain ’

so as not

to e xclude things both productive of pleasure and pre ventive of

pain.

‘ It is plain,

’

he remarks,from the nature of the subj ect,

70 PURE LOGIC

that the expression “either productive of pleasure or preventive

of pain,in the above definition, is meant to b e equivalent to

“either productive of pleasure or

,if not productive of pleasure

,

preventive of pain.

”

And in remarking upon other possible interpretations, hesays

,

‘That before attempting to translate our data into the

rigorous language of symbols, it is above all things nece ssary toascertain the intended import of the w ords w e are using (p .

This simply amounts to consulting the matter, and ProfessorBoole ’s symbols thus constantly imply restrictions not expressedin the forms of language, but existing, if at all, as tacit or understood premises .182 . In my system , ou the contrary

,I take A + B not to

imply at all that A may not b e B, but if this b e the case,it must

b e ow ing to an expressed premise A Ab or A b.

183 . How e ssential Professor Boole’

s restriction on his symbols is to the stability of his system ,

one instance w ill show . Takehis proposition on p. 35

mz y + a

and give the follow ing meanings to x, y, z

a: Caesary Conqueror of the Gaulsz First emperor of Rome.

Now , there is nothing logically absurd in saying ‘Caesar is theconqueror of the Gauls or the first emperor of Rome .

’

It is quite conceivable that a person should remember justenough of hi story to make this statement and nothing more .

And there is nothing in the logical character of the terms todecide whether the conqueror could or could not b e the same

person as the first emperor.But now take Professor Boole ’s inference from the proposition

x=y + z, namely a—z =y got by subtracting z from either side

of a: y 2 . Then w e have the strange inferenceCaesar, provided he is not the first emperor of Rome, is the con

queror of the Gauls.

This leads me to my second objection to Professor Boole ’ssystem .

Second Objection

184 . There are no such op erations as addition and subtraction in

p ure logic.

REMARKS ON E O OLE’

S SYSTEM

The operations of logic are the combination and separation ofterm s, or their meanings, corresponding to multiplication and

division in mathematics . I cannot support this statement w ithout going at once to the gist of the whole matter .185 . Number, then, and the science of number

,ari se out of

logic, and the conditions of number are defined by logic. It hasbeen thought that units

’

are units inasmuch as they are perfectlysimilar. For instance, three apples are three units

,inasmuch as

each has e xactly the sam e qualitie s as the other in being an

apple . The truth is exactly opposite to this . Uni ts are uni ts

inasmuch as they are logically contrary. In so far as three applesare exactly like each other, one could not b e distinguished fromthe other. W ere there three apples

,or any three things, so

perfectly similar in every w ay that w e could not tell thedifference, they w ould b e b ut one thing, just as

,by the law of

unity before stated,A A A A. But then w e mus t re

member that among the logical character s of a thing is itspo sition in space w ith relation to other things, not to speak of

its position in time . Now,w hen w e speak of three apples

,w e

mean three things, w hich, how ever perfectly same they may b e

in all other qualities, occupy different place s , and are thereforedistinct things . In so f ar as they are same they are one in that

they are difi'

erent they are three.

186 . The meaning of an abstract unit is something only known

as logically distinct from or contrary to other things. The meaningof a concrete unit is the abstract unit w i th certain qualities known or

defined .

For instance,in A ( l

'

1 l A' A A the meaning

of the units l ',

is that each is something logically distinctfrom the other, and w hen w e predicate of each of these that it isA

,say an apple, w e get three distinct A

’ s,A’ A A

’ ”

So in multiplication,tw ice two isfour

The logical significance of the process is that if w e have tw o

logically distinct notions, and w e divide each into tw o logicallydistinct notions

,w e get four logically distinct notions . In

logical formulae (A a) (B b) AB Ab aB ab,w here A

and a,B and b

,expre ss logical contraries .

187 . Now addition,subtraction

,multiplication, and division,

are alike true as modes of reasoning in numbers, w here w e havethe logical condition of a unit as a constant restriction. But

addition and subtraction do not exist,and do not give true

7 2 PURE LOGIC

results, in a system of pure logi c, free from the condition of

number.For instance, take the logical propo sition

Meaning what is eithe r A or B or C is ei ther A or D or E ,and

vice versa.

There being no exterior restrictions of meaning whatever,except that the same term must alw ays have the same meaning

w e do not know w hich of A, D,E,is B, nor which is C

nor, conversely, do w e know which of A,B,C

,is D

,nor which

is E. The proposition alone gives us no such information .

In these circumstances,the action of subtraction does not

apply. It is not necessarily true that, i f from same (equal)things w e take same (equal) things the remainders are same(equal) . It is not allowable for us to subtract the same thing(A) from bo th sides of the above proposition, and thence infer

B + C=D + E

This is not true if, for instance , each of B and C is the same

as E, and D is the same as A,which has been taken away.

Yet the equivalent inference by combination w i ll b e valid .

W e may combine a with both sides of the proposition,and w e

have

or,striking out the contradictory term s aA,

w e have

aB + aO =aD + aE

188 . But subtraction is valid under the logical restriction thatthe several alternatives of a term shall b e mutually exclusive orcontrary. Let

(1) AMN BMn OmN AMN DMn EmN

in w hich it is obviously impossible that AMN can b e eitherDMn or F/mN, contraries of AMN

,or any one of the three

alternatives any other. Then w e may freely subtract AMNfrom both sides, getting the necessary inference

(2) EMn OmN DMn EmN .

This subtraction, however, is merely equivalent to the com

bination wi th both sides of the proposition ( l ) of the term(Mn + mN) for the combination being performed

,and contra

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74 PURE LO GIC

instead of C ’

+ C + 0 + 0 It is by the Law of

Unity that C C C or the same coin counted tw ice is butone coin in number. In this case no attention is paid to differences of time but in many cases, things otherw ise perfectly thesame , like the beats of a pendulum

,are distinguished and made

into different units by one being before or after the other in time.

Third Objection193 . My third obj ection to Professor Boole’s system is, that

it is inconsistent w i th the self-evident law of thought, the Law of Unity

ProfessorBoolehaving assumedas a condition of his system thateach tw o term s must b e logically distinct, is unable to recognisethe Law of Unity. It is contradictory of the basis of his system .

The term a,in his system ,

means all things w i th the quali ty a,de

noting the things in e x tent, while connoting the quality in intent .If by 1 w e denote all things of every quality, and then subtract,as in numbers, all those things w hich have the quality cc, the re

mainder must consist of all things of the quality not-x. Thusa: (1 cc) means in his system all 93

’

s w i th all not-sc’

s,w hich, taken

together, must make up all things, or 1. But let us now attemptby multiplication w ith cc

,to select all cc’s from this expression for

all things.

—at.

Professor Boole would here cross out one x against one x,

leaving one cc,the required exp ression for all x

’

s. It is surelyself-evident, how ever, that x a: is equivalent to a: alone, whetherw e regard it in extent of meaning, as all the 93

’

s added to all the

cc’

s,w hich is simply all the 90

’

s, or in intent of m eaning , as ei ther a:

or x,w hich is surely a. Thus

,a cc a; is really 0, and not cc

,the

required result, and it is apparent that the process of subtractionin logic is inconsistent w ith the self-evident Law of Unity.

194 It is probable, indeed, that Professor Boole w ould altogether refuse to recognise such an expression as a: a: a

,on the

ground that it does not obey the condition of his symbols thateach tw o alternatives shall b e distinct and contrary

,x + x not

being so. It may b e answ ered, that the expression has beenarrived at by operations enounced as universally valid

,w hich

ought to give true results . And if it b e simply said thata: :c a; is not interpretab le in Professor Boole

’s system,it may

b e again answ ered, that w hen translated into its equivalent inw ord s, the expression 11: x a: has a very plain meaning. It is

REMARKS 0 1V E O OLE ’

S SYSTEM

either cc or so, p rovided i t be not a,’

and this,I must hold

,i s simply

not a,although it ought to b e at

,according to the mode in w hich

it w as got .

195 . In founding his system,Boole assumed that there cannot

b e tw o term s A B,the same in meaning or name s of the same

thing ; the law s of thought requi re nothing of the kind,and

cannot require it, because among know n and unknow n te rm s,

any tw o such as A B may prove to b e name s of the same thingAB . Thought m erely reduce s the m eaning of tw o same term sAB AB, by the Law of Unity

,to b e the same as that of one term

AB . And w hen it is once know n that all term s in que stion are

contrarie s of each other,or naturally e x clusive and distinct

,then

Boole ’s system and the w hole science of numbers apply .

196 . It is on this account that my objections have no bearingagainst Professor Boole

’s system as applied to the Calculus of

Probabilities,so far as I can understand the subj ect. For it is

a high advantage to that calculus to have to t reat only eventsmutually exclusive

,probabilities being then capable of simple

addition and substraction.

1 It seem s likely,indeed

,that this

distinction o f e xclusive and une xclusive alternative s is the Gord ian knot in w hich all the ab stract logical sciences m eet and are

entangled .

Fourth Objection

197 . The last obj ection that I shall at pre sent urge againstProfe ssor Boole ’s sy stem is , that the symbols g, L

} ,establish for

themselves no logical meaning, and only bear a meaning deri red fromsome method of reasoning not contained in the symboli c system. The

meanings, in short , are those reached in the self- ev ident indirectmethod of the pre sent w o rk.

198 . Professor Boole e xpressly allow s, as regards one of thesesymbols at least

, 8 , that it is not his m e thod w hich give s anymeaning to the symbol. It is the peculiarity of his system

,that

he bestow s a m eaning on his symbols b y interp retation. The

interpretation of g is e x plained on pp . 8 9,90, and he says

,

‘Although the abo ve determination of the significance of the

symbol g is founded upon the e xam ination of a particular case,

yet the principle involved in the d emonstration is gene ral, andthere are no circum stances under w hich the symb ol can presentitself to w hich the same mode of analysis is inapplicable .

’

Again (p.

‘Its actual interpretation,how ever, as an inde

finite class symbol,cannot

,I conceive

,ex cept upon the ground

1 Sec De Morgan’s Syllabus, § 243 .

76 PURE LO GIC

of analogy, b e deduced from i ts arithmetical properties, but mustb e established e x perimentally .

’

199 . If I understand this aright, i t sim ply means,that w here

eve r a term appears in a conclusion w ith the symbol affixed,

w e may, by a mode of analysis , by some process of pure reasoning apart from the symbolic process from which 3 emerged ,ascertain that the meaning o f 3 is some, an indefinite class term.

The symbol g is unknow n until w e give it a meaning. Before,

therefore, w e can know w hat meaning to give, and b e sure thatthis meaning is right, it seem s to me w e must have another distinct and intuitive system by which to get that meaning . Pro

fe sser Boole ’s system ,then

,as regards the symbol g, is not the

system bestow ing certain know ledge ; it is, at most,a system

pointing out truths which, by another intuitive system of reasoning, w e may know to b e certainly true .

200. It is suffi cient to show this with regard to a singlesymbol f} , because the incapacity of a sys tem ,

even in a singleinstance

,proves the nece ssity for another system to support it .

I believe that the other symbols, 11

7 , are open to exactly thesame remarks

,but from the w ay in w hich Mr. Boole treats them ,

involving the whole conditions of his system,it would b e a

lengthy matter to explain.

201. The obscure symbols 61, b y g, have the followingcorrespondence w ith the form s of the present system . % ap

pearing as the coeffi cient of a term means that the term is anincluded subject of the premise, so that

,if combined w ith both

m embers of the premise,it produces a self-contradictory term

w ith neither sideSimilarly

,means that the term is an excluded subject of the

prem ise , producing a self-contradictory w i th both side s of thepremise .

And either 11; or 11’ means that the term is a self-contrad ictory

or impossible term, producing a self-contradictory term w ith one

side only o f the premise .

202 . The correspondence of these obscure forms with the

self-evident inferences of the pre sent system is so close andobvious

,as to suggest irresistib ly that Professor Boole

’s operations w ith his abstract calculus of 1 and O

,are a mere counter

part oi self-evident operations w i th the intelligib le symbols of

pure logic. Professor Boole starts from logical notions, and

self-evident law s of thought ; he suddenly transmutes his formulac into obscure mathematical counterparts, and after variousintricate manoeuvres

,arrives at certain forms, corresponding to

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THE SUBST ITUT ION OF SIMILARS,THE

TRUE PRIN C I PLE O F REASO N ING

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8 2 THE SUBSTITUTION OF SIMILARS

possibly push beyond the stronger, and it is willinglyallowed that among us moderns can few or none b e foundto equal in individual strength of intellect the great men

of old . But Tim e is on our side . Though w e reverencethem as the ancients, they really lived in the childhood of

the human race, and these times are, as Bacon would havesaid

,the ancient times. 1 W e enjoy not only the best in

tell ectual riches of the Greeks and Romans, but also thewonderful additions to the physical and mathematical sciencesmade s ince the revival of letters . In our time w e possessan almost complete comprehension of many parts of physicalscience whi ch seemed to Socrates, the wisest of men

,beyond

the powers of the human m ind . W e have before us anabundance of examples of the modes in which soli d andundoubted truths may b e attained, and it is absurd tosuppose that am ong such successful exertions of the humanintellect w e can find no materials for a newer analytic of

the mental operations .3 . The mathematics especially present the example of a

great branch of abstract science, evolved almost wholly fromthe m ind itself, in which the Greeks indeed excelled, but inwhi ch m odern knowledge passes almost infinitely beyondtheir highest efforts. Intellects so lofty and acute as thoseof Euclid or D iophantus or Archimedes reached but the fewfirst steps on the way to the widening generalisations of

modern mathematicians ; and what reason is there to suppose that Aristotle , however great, should at a single bound

1 ‘De antiquitate autem, opinio quam homines de ipsa fovent , negl igensomnino est

, et vix verb o ipsi congrua. Mundi enim senium et grandaev i tas

pro ant iquitate vere hab enda sunt ; quae temporibus nostris tribui deb ent,nonjuniori aetati mundi , qualis apud ant iques fuit. Illa enim aetas respectunostri , antiqua et major ; respectumundi ipsius, nova et minor fuit. Atquerevera quemadmodum majorem rerum humanarum notit iam

,et maturius

judi cium,ab hom ine sene expectamus, quam ajuvene, propter experient iam ,

et rerum quas vid it et aud ivi t, et cogitav i t , vari etatem et copiam ; cedem

modo et a nostra aetate (si v i res suas nesset, et experiri et intendere vellet)majora multo quam a priscis temporib us expectari par est ; utpote aetate

mundi grandiore, et infinit is experimentis, et ob servat ionibus aneta et cumu

lata.

’—Nooum Organum, Lib . i Aphor. 84.

THE TRUE PRHVCIPLE OF REASONIZVG

have reached the highest generalisations of a closely kindredsci ence of human thought ?4 . K ant indeed was no intellectual slave , and it m ight

well seem discouraging to logical speculators that he con

sidered logic unimproved in his day since the time of

Aristotle,and indeed declared that it could not b e improved

except in perspicuity. But his opinions have not preventedthe improvement of logical doctrine , and are now effectuallydisproved . A succession of eminent men

,—Jeremy Bentham

,

George Bentham, Sir“

Ti lliam Ham ilton,Professor De Morgan

,

Archbishop Thomson,and the late Dr. Boole, have shown

that in the operations and the laws of thought there is awide and fertile area of investigation . Bentham did morethan assert our freedom of inquiry ; in his uncouth logicalwritings are to b e found m ost original hints , and in editinghis papers his nephew George Bentham pointed out the allimportant key to a thorough logical reform ,

the quantificationof the p redi cate.

1Sir W illiam Ham ilton

,Archbishop Thom

son,and Professor De Morgan , rediscovered and developed

the same new idea. Dr. Boole,lastly

, employing this fundam ental idea as his starting-point, worked out a mathematicalsystem of logical inference of extraordinary originality

5 . O f the logical system of Mr. Boole Professor De

Morgan has said in his Budget of Paradoxes ” 2 I m ightlegitimately have entered it am ong my p aradoxes, or thingscounter to general opinion : but it is a paradox which, likethat of Copernicus

, excited adm iration from its first appearance . That the symbolic processes of algebra, invented astools of num erical calculation , should b e competent to ex

press every act of thought,and to furnish the grammar and

d ictionary of an all-containing system of logic, would nothave been believed until it w as proved . When Hobbes, inthe time of the Commonwealth , published his Computati onor Logique, he had a remote

'

glimpse of some of the points

1 See Outl ine of a N ew System of Log ic, by George Bentham ,Esq . ,

London, 182 7 , p. 133 et seq.

2 No . xx i i i , A thcnazum .

84 THE SUBSTITUTION OF SIAIILARS

which are placed in the li ght of day by Mr. Boole. The

unity of the form s of thought in all the applications of

reason, however rem otely separated, will one day b e

matter of notoriety and common wonder ; and Boole’s

name will b e remembered in connection with one of

the m ost important steps towards the attainment of thisknowledge.

’

6 . I need hardly name Mr. Mill, because he has ex

pressly disputed the utility and even the truthfulness of

the reform s which I am considering,and has evolved m ost

divergent opinions of his own in a wholly different directionfrom the em inent men just mentioned .

7 . In the li fetime of a generation still living the dulland ancient rule of authority has thus been shaken , and theimmediate result is a perfect chaos of diverse and originalspeculations . Each logician has invented a logic of hisown, so marked by pecul iarities of his individual m ind, andhis customary studies, that no reader would at first supposethe same subject to b e treated by all. Yet they ‘ treat of

the same science , and, with the exception of Mr. Mill, theystart from almost the same discovery in that science .

Modern logic has thus becom e mystified by the diversityof views, and by the complication and profuseness of the

formulae invented by the different authors named . The

quasi-mathematical methods of Dr. Boole especially are somystical and abstruse, that they appear to pass beyond thecomprehension and criticism of most other writers, and arecalmly ignored . No inconsiderable part of a lifetime isindeed needed to master thoroughly the genius and tendencyof all the recent English writings on Logic, and w e can

scarcely wonder that the plain and scanty outline of Aldrich ,

or the sensible but unoriginal elements of Whately, continueto b e the guides of a logical student , while the works of

De Morgan or of Boole are sealed books .8 . The nature of the great discovery alluded to, the

quantification of the p redicate, cannot b e explained withoutintroducing the technical terms of the science. A preposi

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86 THE SUBSTITUTION OF SIAIILARS

of science hitherto distinct, and in showing the unity of

principles pervading them .

1

9 . And yet any one acquainted with the systems of the

m odern logicians must feel that something is still wanting.

So much diversity and obscurity are no usual marks of

truth,and it is almost incredible that the true general

system of inference should b e beyond the comprehension of

nearly every one , and therefore incapable of affecting ordinarythinkers . I am thus led to believe that the true clue tothe analogy of m athematics and logic has not hitherto beenseized, and I write this tract to subm it to the reader

’s judgment whether or not I have been able to detect this clue .

10. During the last two or three years the thought hasconstantly forced itself upon my m ind

,that the modern

logicians have altered the form of Aristotle ’s propositionwithout making any corresponding alteration in the dictumor self- evident principle which formed the fundamentalpostulate of his system . They have thus got the rightform of the proposition , but not the right way of using it .Aristotle regarded the prepositi on as stating the inclusionof one term or class within another ; and his axi om wasperfectly adapted to this V iew.

The se - called Dictum de omni is, in Latin phrase, asfollowsQui cguid dc omni valet, valet etiam dc gui busdam et

singuli s.

And the corresponding Dictum dc nullo is sim ilarlyQui cquid de nullo valet, nec de gni busdam nee dc singuli s

valet.

In English these di cta are usually stated somewhat asfollows

VVhatever is p redi cated afi rmati vcly or negatively of a

w hole class may be pred i cated of anything contained in that

class. Or,as Sir W. Ham ilton more briefly expresses them ,

t at p ertains to the higher class p ertains also to the lo wer.

2

1 Baden Pow ell,Uni ty of the Sciences , p . 41.

2 Lectures on Logi c, vol . i , p . 303 .

THE TRUE PRIN CIPLE OF REASONIN G'

These di cta, then, enable us to pass from the predicateto the subject

,and to affi rm of the subject whatever w e

know or can affirm of the predicate . But w e are notauthorised to pass in the other direction, from the subjectto the predicate, because the proposition states the inclusionof the subject in the predicate, and not of the predicate inthe subject.

The proposition ,

A ll metals are elements,

taken in connection with the d ictum dc omni authorises usto apply to all metals whatever knowledge w e may have of

the n ature of elements,because metals are but a subordinate

class included among the e lem ents ; and , therefore ,possess

ing all the properties of elem ents . But w e comm it anobvious fallacy if w e argue in the opposite direction

,and

infer of elements what w e know only of m etals . This isneither authorised by Aristotle ’s di ctum

,nor would it b e in

accordance with fact . Aristotle ’s postulate is thus perfectlyadapted to his view of the nature of a proposition

,and

his system of the syllogism was admirably w orked out inaccordance w ith the sam e idea .

11 . But recent reform ers of logic have profoundly alteredour V iew of the proposition. They teach us to regard it asan equation of two term s

,formerly called the subject and

predicate, but which , in becom ing equal to each other,cease

to b e distinguishable as such, and become convertible .

Should not logicians have altered,at the sam e time and in

a corresponding manner, the pos tulate according to whichthe proposition is to b e employed ? Ought w e not now tosay that whatever is known of either term of the propositionis known and may b e asserted of the other ? Does not thed i ctum

,in short

,apply in both directions , now that the two

term s are indifferently subject and predicate ?12 . To illustrate this w e may first quantify the predicate

of our own former example,getting the proposition,

88 THE SUBSTITUTION OF SHl/[l LARS

A ll metals are some elements,

where the cepul a are means no longer are contained among,

but are id enti cal w i th ; or availi ng ourselves of the signin a m eaning closely analogous to that which it bears inmathematics

,w e may express the preposition more clearly as

,

All metals some elements.

It is now evident that whatever w e know of a certainindefinite part of the elements w e know of all metals

,and

whatever w e know of all metals w e know of a certain indefinite part of the elements. W e seem to have gained noadvantage by the change ; and if w e are asked to definemore exactly what part of the elements w e are speaki ng of

,

w e can only answer, Those w hich are metals. The formula

A ll metals all metallic elements

is a more clear statement of the same proposition with thepredicate quantified ; for while it asserts an identity it implies the inclusion of metals among elements . But it isan accidental peculi arity of this form that the d ictum onlyapplies usefully in one direction

,since if w e already know

what metals are w e must know them to b e metallic elements ,the adjective metalli c including in its meaning all that canb e known of metals ; and from knowing that metals are

metallic elements w e gain no clue as to w hat part of the

properties of metals belong to elements . But it is hardlytoo much to say that Aristotle commi tted the greatest andmost lamentable of all m istakes in the history of sciencewhen he took this kind of proposition as the true type of

all propositions,and founded thereon his system . It was

by a mere fallacy of accident that he was m isled ; but thefallacy once comm itted by a master-m ind became so rootedin the m inds of all succeeding logicians, by the influence of

authority, that twenty centuries have thereby been rendereda blank in the history of logi c.

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THE SUBST1TUT10N OF S1MTEARS

itself was om itted, and the true system is to b e created bysupplying thi s om ission, and re-erecting the edifice from the

very foundation .

14 . I am thus led to take the equation as the fundamental form of reasoning, and to modify Aristotle

’s dictumin accordance therewith. It may then b e formulated somewhat as foll ows

Whatever is know n of a term may be stated of i ts equal or

equivalent.

Or in other words ,Wh atever i s true of a thing i s true of i ts like.

I must b eg of the reader not to prejudge the value of

this very evident axiom . It is derived from Aristotle’sd i ctum by om itting the distinction of the subject and predicate ; and it may seem to have become thereby even amore transparent truism than the origi nal , which has beencondemned as such by Mr. J. S. Mill and some others . Butthe value of the formula must b e judged by its results andI do not hes itate to assert that it not only brings into harmonyall the branches of logical doctrine, but that it unites themin close analogy to the corresponding parts of mathematicalmethod. All acts of mathematical reasoning may, I believe,he considered but as appli cations of a corresponding axiomof quantity ; and the force of the axiom may best b e i llustrated in the first place by looking at it in its mathematicalaspect.

15 . The axiom indeed with which Euclid begins to buildpresents at first sight little or no resemblance to the modifieddictum . The axiom asserts that

Things equal to the same thing are equal to each other.

In symbols,

Here two equations are apparently necessary in orderthat an inference may b e evolved ; and there is something

peculiar about the threefold symmetrical character of the

formula which attracts the attention , and prevents the truenature of the process of m ind from be ing discovered . W

’

e

get hold of“

the true secret by considering that an inferenceis equally possible by the use of a single equation , b ut thatwhen there is no equation no inference at all can b e draw n .

Thus if w e use the sign an to denote the existence of aninequality or difference , then one equality and one inequality ,as in

a b m 0,

enable us to infer an i nequal ity

( i f 0 .

Two inequalities , on the other hand, as in

a m b w e,

do not enable us to make any inference concerning therelation of a and c for i f these quantities are equal , theymay both differ from b

,and so they may if they are unequal .

The axiom of Euclid thus requires to b e supplemented bytwo other axiom s , which can only b e expressed in somewhatawkw ard language

,as follows

If thefirst of three things be equal to the second,but the

second be unequal to the th ird,the first i s unequal to the

third .

And again

If tw o things be both unequal to a third common thing ,

they may or may not be equal to each other.

16 . Reflection upon the force of these axioms and theirrelations to each other w ill show ,

I think , that the deductivepower always resides in an equality, and that difference assuch is incapable of affording any inference . My m eaningwill b e m ore plainly exhibited by placing th e symbols inthe following form

9 2 THE SUBSTITUTION OF Si d/[LARS

a b a

u hencec c.

Here the inference is seen to b e obtained by substitutinga for b by virtue of their equality as expressed in the firstequation a =b

, the second equation b = c being that in whichsubstitution is effected. One equation is active and the

other is p assive, and it is a pure accident of this form of

inference that either equation may b e indifferently chosenas the active one. Precisely the same result happens inthis case to b e obtained by a sim ilar act of reasoning inwhich b = c is ’ the active equation

,as shown below

b c c

II hence II

a a .

My warrant for this view of the m atter is to b e found inthe fact that the negative form of the axiom is now easilybrought into complete harmony with the affirmative form ,

except that, since it has only one equation to work by,there

can b e only one active equation and one form in which theinference can b e exhibited as below

a b a

S hence S

c c.

Inference is seen to take place in exactly the same

manner as before by the substitution of a for b,and the

negative equation or difference b m c is the part in whichsubstitution takes place, but which has itself

‘no substitutivepower. Accordingly w e shall in vain threw two differencesinto the same form ,

as in

a m b b m c

S or S

c a,

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THE SUBSTITU TfON OF STMJLARS

on algebra, and it is self-evident that a greater of a greater

is a greater, and w hat i s less than a less is less. Thus w e

certainly seem to have in the two formulae,

a > b > c hence a > o,

a < b < c hence a < c,

two valid modes of reasoning otherwise than by equations .But it is apparent

,in the first place, that the use of these

signs and demands some precautions which do notattach to the cepula the formulae,

a > b < a

a < b > a

do not establish any relation between a and c ; and I thinkthe reader will not find it easy to explain why these do notand the former do, without implying the use of an equationor identity. The truth is, that the formulae,

involve not only two differences, but also one identity inthe direction of those differences, whereas the formulae,

appear to fail in givi ng any inference because they involveonly differences both of direction and quantity.

Strength is added to this view of the matter by observ

ing that all reasoning by inequal ities can b e representedwith equal or superior clearness and precision in the formof equali ti es, whil e the contrary is by no means always true.

Thus the inequality

is represented by the equality

( 1 ) a= b+p ,

i s sim ilarly represented by the equality

<2 ) b = c+£c

in which q is again a positive quantity greaterBy substi tuti ng for b in ( 1) its value as givenobtain the equation

w hich , owing to the like signs of p and q, is a representationin a m ore exact and clear manner of the conclusion

a > c.

On the other hand, the formula

a > b < e

would evidently lead to the equation

a e+p r,

in which p is the excess of a over b , and r the excess of e

over b . Now this equation, taken in connection with theformer one , seem s to give much clearer information as tothe conditions under which inference is possible than do theformulae of inequalities, and I entertain no doubt at all that ,even when an inference seem s to b e obtained without theuse of an equation, a disguised substitution is really performed by the m ind

,exactly such as represented in the

equations . But I can only assert my belief of this fromthe exam ination of the process in my own m ind, and I mustsubm it to the reader’s judgment whether there are exceptionsor not to the rule , that w e always reason by means of

identities or equalities .19 . Turning now to apply these considerations to the

forms of logical inference , my proposed simplification of the

96 THE SUBSTITUTION OF SIR/[LARS

rul es of logic is founded upon an obvious extension of the

one great process of substitution to all kinds of identity.

The Latin word aequalis, whi ch is the original of our equal,was not restricted in signification to simil arity of quantities,but was often applied to anything which was unvaried orsim ilar when compared with another. W e have but tointerpret the word equal in the older and wider sense of lik eor equivalent, in order to effect the long-des ired un ion of

logical and mathematical reasoning. For it is not difficultto show that all form s of reasoning consist in repeatedemployment of the universal process of the substi tution

of equals, or, if the phrase b e preferred, substi tution ofs imi lars.

20. To prevent a confusion of mathematical and logicalapplications of the formul a

,it w ill b e desirable to use large

capital letters to denote the things compared in a logicalsense, but the copula or sign of identity may remain asbefore. Thus the symbols .

A =B

denote the identity of the things represented by the

indefinite term s or names A and B . Thus A may b e takenin one case to mean Iron,

when B m ight mean the cheap est

of the metal s, or the most useful of the metals. In anotherexample which w e have used A woul d mean element

,and B

that w hi ch cannot be decomp osed, and so on . The fundamentalprinciple of reasoning authorises us to substitute the termon one side of an identity for the other term ,

w herever this

may be encountered ,so that in whatever relation B stands to

a third thing C, in the same relation A must stand to C .

Or, using the sign cm to denote any possible or conceivablekind of relation , the formula

A B Ahence

C C

represents a self-evident inference . Thus,

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98 THE SUBSTITUTION OF SIAIILARS

By sub stituting in ( 2 ) by means of ( 1) w e have

The most useful metal m the metal most early used by

p rimi tive na tions .

23 . But two negative propositions will of course give noresult . Thus the two propositions ,

Snow don m the highest mountain i n Great Bri tain,

Snow don an the highest mountain in the w orld ,

do not allow of any substitution,and therefore do not give

any means of inferring whether or not the highest mountainin Great Britain is the highest mountain in the world .

24. Postponing to a later part of this tract 3 6 ) the

consideration of negative form s of inference, I will nownotice some inferences which involve combinations of terms .However many nouns, substantive or adjective, may b e

joined together, w e may substitute for each its equivalent.Thus , if w e have the propositions,

Square equilateral rectangle,

Equi lateral equal-si ded

,

Rectangle right-angled quadri lateral,

Quadri lateral=four-si ded figure,

w e may by evident sub stitutions obtain

Square= equal-sid ed

,right

-angled , four-si ded figure.

25 . It is desirable at this point to draw attention to thefact that the order in which nouns adjective are stated isa matter of indifference . A four-sid ed

,equal

-si ded figureis identically the same as an equal

-si ded,four-si ded figure

and even when it sometimes seems inelegant or difficult toalter the order of names descri bing a thing, it is grammatical usage

,not logical necessity, which stands in the way .

Hence, i f A and B represent any two names or terms, theirjunction as in AB will b e taken to indicate anything whichunites the qualities of both A and B,

and then it followsthat

AB BA.

s not equivalent to w ood of table but if w e

of w ood as equivalent to the adjective w ooden ,

tab le of w ood is the same as a w ooden table.

v new proceed to consider the ordinary pro:orm

A =AB,

he identity of the class A with a particulars B

,nam ely the part which has the proper

may seem when s tated in this way to b e as not

,because it really states in the form of

inclusion of A in a wider class B . Aristotleat it in the latter aspect only, and the ex

e ness of his syllogistic system is due to thisIt is only by treating the proposition as anrelation to the other forms of reasoning b e

the simplest and by far the mos t common>nt in which the proposition of the abovethe mood of the syllogism known by the

ple, w e may take the follow ing

Iron i s a metal,

A metal i s an element,therefore

Iron i s an element.

i ons thus expressed in the ordinary manneri ctly logical form

(1) Iron metallic i ron,

( 2 JlIetal elementary metal.

100 THE SUBSTITUTION OF SIAIILARS

Now for metal or metalli c in (1) w e may substitute its equivaleut in (2 ) and w e obtain

(3 ) Iron= elementary ,metal

,iron

which in the elliptical expression of ordinary convers ationbecomes Iron is an element

,or Iron is some kind of element,

the words an or some kind being indefinite substitutes for amore exact description

The form of this mode of inference must b e stated insymbols on account of its great importance . If w e take

A iron,

B metal,

C element,

premises obviously,

( 1 ) A=AB,

Q ) B=BQ

and substituting for B in (1) its description in (2 ) w e havethe conclusion

A ABC,

which is the symbolic expression of

28 . The mood Dari i,which is distinguished from Bar

bara in the doctrine of the syllogism by its particular m inorpremise and conclusion , cannot b e considered an essentiallydifferent form . For if

,instead of taking A in the previous

example= i ron

,w e had taken it

A some native minerals

B and C remaining as before, w e should then have theconclusion

A=ABO,

denotingsome nati ve m inerals are elements

which affords an instance of the syllogism Darii exhibited inexactly the same form as Barbara.

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10 2 THE SUBSTITUTION OF SIAIILARS

Making

the premises become( 1) B = BC,(2 ) B=BA.

Hence, by obvi ous substitution , either by ( 1) in (2 ) or by(2 ) in w e get

(3 ) BA= BC.

Precisely interpreted this means that gas w hich is oxygen iselement w hich is oaygen ; but when this full interpretation isunnecessary, w e may substitute the indefinite adj ective some

for the more particular description , getting,

Some gas is some element,

or,in the still more vague form of common language,

Some gas i s an element.

31. The mood Datisi may thus b e illustrated

1) Some metals are inflammable,

(2 ) A ll metals are elements,

( 3 ) Some elements are inflammable.

A elements,

C inflammable,B metals, D some

,

w e may represent the premises in the forms

(1) DB=DBCB=BA.

Substitution,in the second side of of the description

B given in (2 ) produces the conclusion

(3 ) DB=DBCA,

or,in words

,

Some metal= some metal element inflammable.

b eg of the reader not to judge the validity of my form s byany single instance only

, b ut rather by the wide embracingpowers of the principle involved . Even common thoughtmust b e condemned as loose and imperfect i f it should b efound in certain cases to b e inconsistent with a generalisation which holds true throughout the exact sciences as wellas the greater part of the ordinary acts of reasoning.

32 . Certain forms of se - called immediate inference, chieflybrought in to notice in recent times by Dr. Thomson

,are

readily derived from our principle .

Immedi ate inference by added determi nantlconsists in

joining a determining or quali fying adj ective , or som e equivalent phrase , to each m ember of a proposition, a new proposition being thus inferred . Dr. Thom son ’s ow n exampleis as follows

A negro is a fellow -creature

whence w e infer immediately,

A negro in sufi eri ng i s a fellow - creature in sufi eri ny .

To explain accurately the m ode in w hich this inferenceseem s to b e made according to our principle , let us take

A negro,

B =fellow - creature,

The prem ise may b e represented as

A AB .

104 THE SUBSTITUTION OF SI/lIILARS

be ing a fact which some may think to b e somewhat unnecessarily laid down in the first of the primary laws of

thought (seeIn the symbolic express ion of this fact,

AC AC ,

w e can substitute for A in the second member its equivalentAB, getting

AC ABC .

This may b e interpreted in ordinary words as ,

A s ufiering negro i s a sufi’

ering negro fellow -creature,

which differs only from the conclusion as. stated by Dr .Thomson by containing the qualification negro in the secondmember.33 . Immediate inference by comp lex concep tion closely

resembles the preceding, and is of exceedingly frequentoccurrence in common thought and language , although ithas never had a properly recognised place in logical doctrineuntil lately.

1

Its nature is best learnt from such an example as thefollowing

Oxygen i s an element,

Therefore a p ound w eight of oxygen is a

pound w eight of an element.

This is a very plain case of substitution ; for if w e make

0 oxygen,

P p ound w eight,

Q element,

w e may represent the prem ise as

O OQ .

Now it is self-evident that

r of o r of o,

1 Thomson’s Outl ine, 88.

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106 THE SUBSTITUTION OF SI/VILARS

is hypothetical , but exactly equivalent to the categoricalproposition ,

Iron, containing phosphorus, i s bri ttle ;

which is of the symbolic form,

AB ABC .

But propositions such as,

If the barometer falls, a storm i s coming ,

cannot b e reduced but by some such mode of expression as

the following

The circumstances of a falling barometer are the

ci rcumstances of a storm coming .

Nevertheless, sufficient freedom in the alteration of

expression being granted, they readily come under ourformulae.

36 . I have as yet introduced few examples of negativepropositions

,because , though they may b e treated in their

purely negative form , it is usually more convenient toconvert them into affirmative prepositions. This conversionis effected by the use of negative terms

,a practice not um

known to the old logic , but not nearly so much employedas it should have been . Thus the negative proposition,

A is not B

A w B,

is much more conveniently represented by the affi rmativeproposition or equation ,

A Ab,

in which w e denote by b the quali ty or fact of differingfrom B . The term b is in fact the name of the whole classof things , or any of them

,which differ from B ,

so that iti s a matter of indifference whether w e say that A differsfrom B and is excluded from the class B

,or that it agrees

A metal,

B element,

C transmutable,

c untransmutable.

Then the premises are

O ) A = AB

(2 ) B Bc.

Substituting in ( 1) by m eans of (2 ) w e get

( 3 ) A = ABc ;

or,metals=metals

,elementary,

untransmutable.

38 . Before proceeding to other examples of the syllogism,

1 It may seem to the reader contradictory to condemn the negative proposition as ster ile and incapab le of afford ing inferences, and shortly afterw ardsto convert i t into an affi rmative preposit ion of ferti le or inferential pow er .But on trial i t w i ll b e found that the propositions thus ob tained yield no

conclusions inconsistent w ith my theory . Thus the negative prem ises,A i s not B

,

B i s not C,

yield the affirmative proposit ions or equations,A z Ab

B=Be.

And w hen these prem ises are tested , w hether on the logical slate , ab acus , orlogical mach ine referred to in a later page , they are found to g ive no con

elusion concerning the relation of A and C. The description of A is gi ven inthe equation

,

A AbC Abe,

from w h ich it appears that A may indifl'

erently occur w ith or w i thout C.

108 THE SUBSTITUTIOIV OF SIZIIILARS

it w ill be well to point out that every affi rmative propositionor equation gives rise to a corresponding equation betweenthe negatives of the term s of the original. The generalpreposi tion of the form

A=B,

treated by the fundamental principle of reasoning, informsus that in whatever relation anything stands to A

,in the

same relation it stands to B,and sim ilarly vice verset. Hence ,

whatever differs from A differs from B ,and whatever differs

from B differs from A Now the term b denotes what di ffersfrom B

,and a denotes what diffe rs from A ; so that from

the single original proposition w e may draw the two propositions

But as these prepositions have an identical second member,w e can m ake a substitution, getting

a= b .

This form of inference , though l ittle i f at all noticed inthe traditional logic, is of frequent occurrence and of greatimportance. It may b e illustrated by such examples as

happy content ed,

hence unhappy not-contented

or again,

triangle= three~si ded recti linear figure,

hencew ha t i s not a triangle= w ha t i s not a three-sid ed

recti linear figure.

The new proposition thus obtained may b e called the

contrap ositive of the one from which it was derived , thisbeing a name long appli ed to a sim ilar inference from the

old form of proposition.

39 . Though the details of this new view of logic maynot yet have been perfectly worked out, much evidence of

the truth of the system is to b e found in the s implicity ,

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1 10 THE SUBSTITUTION OF SI/lIILARS

means of the principle of substitution, w e find that anunlim ited system of forms of indirect reasoning developsitself spontaneously. O f this indirect system there is hardlya vestige in the old logic, nor does any writer previous toDr. Boole appear to have conceived its existence, though itmust no doubt have been often unconsciously employed inparticular cases . This indirect or negative method is closelyanalogous to the ind irect proof,or reductio ad absurdum

,so

frequently used by Euclid and other mathematicians , and asim ilar method is employed by the old logicians in the

treatment of the syllogisms called Baroho and Bohardo, bythe reducti o ad imp ossi bi le. But the incidental examples ofthe indirect logical method which can b e found in any bookprevious to the Mathematical Analysis of Logic of Dr. Boolegive no idea whatever of its all—commandi ng power ; for itis not only capable of proving all the results obtained alreadyby a direct method of inference

,b ut it gives an unlimited

number of other inferences which could not b e arrived at inany other than a negative or indirect manner. In a previ ous little work 1 I have given a complete, but somewhattedious , demonstration of the nature and results of thismethod , freed from the difficulties and occasional errors inwhich Dr. Boole left it involved . I will now give a briefoutline of its principles .41. The indirect method is founded upon the law of the

substitution of sim ilars as applied with the aid of the fundamental laws of thought. These laws are not to b e found inmost textbooks of logic

,but yet they are necessarily the

basis of all reasoning,since they enounce the very nature of

similarity or iden tity. Their existence is assumed or implied ,therefore, in the complicated rules of the syllogism

,whereas

my system is founded upon an immediate appli cation of the

laws themselves . The first of these laws , which I havealready referred to in an earlier part of this tract (p. is

1 Pure Logic, or the Logi c qf Quali ty apa rtf rom Quanti ty w i th Remarks onBoole

’

s System , and on the Relation of Logic and Mat hemat ics. By W . StanleyJevens, M.A. London Edw ard Stanford, 1864.

THE TRUE PRIN CIPLE OF REASONIN G

the LAW OF IDENTITY, that w hatever i s, is, or a thing i s i dentical w i th i tself ; or, in symbols ,

A=A.

The second law ,THE LAW or NON-CO NTRADICTIO N

,is that

a thing cannot both be and not be,or that nothing can com

b ine contrad i ctory attri butes or,in symbols

,

Aa = 0,

—that is to say,what is both A and not A does not exist

,

and cannot b e conceived .

The third law,that of excluded m i ddle

,or

,as I prefer to

call it,the LAW OF DUALITY

,asserts the self—evident truth

that a thing ei ther exists or does not exi st, or that everythingei ther p ossesses a given attri bute or does not p ossess i t.

Symbolically the law of duality is shown by

A AB Ab,

in w hich the sign -I’ indicates alternation,and is equivalent

to the true meaning of the disjunctive conjunction or.

Hence the symbols may b e interpreted as , A i s either B or

not B .

These laws may seem truism s , and they w ere ridiculedas such by Locke ; but, since they describe the very natureof identity in its three aspects, they must b e assumed astrue, consciously or unconsciously , and if w e can build asystem of inference upon them , their self- evidence is surelyin our favour.

42 . The nature of the system will b e best learn t fromexamples , and I will firs t apply it to several moods of the

old syllogism . Camestres may thus b e proved and illustrated

( 1 ) A sun is self- luminous,

(2 ) A p lanet i s not self- lum inous,

( 3 ) A p lanet, therefore, i s not a sun .

Now it is apparent that a planet is either a sun or it is nota sun ,

by the law of duality. But if it b e a sun,it is self

1 12 THE SUBSTITUTION OF SIAIILARS

lum inous by whereas by ( 2 ) it is not self- lum inous ; itwould

,i f a sun, combine contradictory attributes . By the

law of non -contradiction it could not exist, therefore, as asun

,and it consequently is not a sun .

To represent this reasoning in symbols take

A sun.

B p lanet.

C self-luminous.

Then the premises are

fl ) A =AC

@) B=Ba

By the law of duality w e have

E =BA °I~Ba,

and substituting this value in the second side of (2) w e have

B=BcA -IBca.

But for A in the above w e may substitute its expressionin getting

B= BcAC °l° Bea ;

and striking out one of these alternatives whi ch is contradictery w e finally obtain

B=Bca .

The meaning of this formula is that a p lanet is a p lanet

not self- luminous,and not a sun

,which only differs from the

Aristotelian conclusion in being more ful l and precise.

43 . The syllogism Camenes may b e ill ustrated by thefollowing example

(1) A ll monarchs are human beings,(2 ) No human beings are infallib le

,

(3 ) No infalli ble beings, therefore, are monarchs.

This is proved by considering that every infallible beingis either a monarch or not a monarch ; but i f a monarch,

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B element,from the premise

A ll metals are elements

w e conclude th at all substances which are not-elements are not

metals which is proved at once by the consideration, symhelically expressed above

,that i f they were metals they

would b e elements,or at once elements and not-elements

,which

is impossible .

45 . It is the peculiar character of this m ethod of in

direct inference that it is capable of solving and explaining,in the most complete manner, arguments of any degree of

complexity. It furnishes , in fact, a complete solution of

the problem first propounded and obscurely solved by Dr.Boole

Given any number of p rep osi tions involving any number ofd istinct terms, required the descrip tion of any of those terms orany combination of those terms as expressed in the other terms,

under cond i tion of the prem ises remaining true.

This method always commences by developing all thepossible combinations of the terms involved accordi ng to thelaw of duality. Thus

,if there are three terms, represented

by A,B

,C,then the possible combinations in which A can

present itself will not exceed four, as follows

(1) A =ABO . !-ABc -AbC -Abc.

If w e have any premises or statements concerning the natureof A, B,

and C,that is

,the combinations in which they can

present themselves,w e proceed to inquire how many of the

above combinations are consistent with the prem ises . Thus ,if A is never found with B,

but B is always found with C,

the two first of the combinations become contradictory, andw e have

A =AbC Abe,

or, A is never found with B,but may or may not b e found

with C .

THE TRUE PRIN CIPLE OF REASONIN G

This conclusion may b e proved symbolically by expressing the premises thus

and then substituting the values of A and B wherever theyoccur on the second side of

46 . As a simple example of the process , let us take the

foll owing prem ises,and investigate the consequences which

flow from them .

1

‘From A follows B,and from C follows D ; but B and

D are inconsistent with each other. ’

The possible combinations in which A,B

,C

,and D may

present themselves are sixteen in number, as follows

Each of these combinations is to b e compared with the

premises in order to ascertain whether it is possible underthe condition of those prem ises. This comparison will reallyconsist in substituting for each letter its description as givenin the premises, which may thus b e symbolically ex

pressedU) A =AR(2 ) C CD

,

( 3 ) B : Bd

The combination A b C D is contradicted by (1 )ing for A its value ; A B C d by c D

1 See De Morgan ’s Formal Log ic, p. 123 .

1 16 THE SUBSTITUTION OF SIAIILARS

so on . There will b e found to remain only four possiblecombinations

Now,i f w e wish to ascertain the nature of the term A

,

w e learn at once that it can only exist in the presence of Band the absence of both C and D .

W e ascertain also that D can only appear in the absenceof both A and B,

but that C may or may not b e presentwith D . Where D is absent, C must also b e absent, andso on.

47 . Objections m ight b e raised against this process of

indirect inference, that it is a long and tedious one ; and soit is

,when thus performed. Tedium indeed is no argument

against truth ; and if, as I confidently assert,thi s method

gives us the m eans of solving an infinite number of

problems, and arriving at an infinite number of conclusions,

which are often demonstrable in no simpler way, and in factin no other way whatever, no such objections would b e of

any weight. The fact however is, that almost all thetediousness and li ability to m istake may b e removed fromthe process by the use of mechanical aids, which are of

several kinds and degrees . While practising myself in theuse of the process, I was at once led to the use of the

logi cal slate, which consists of a common writing slate,with

several series of the combinations of letters engraved uponit, thus

A B A B C A B C D A B C D E A B C D E FA b A B c A B C d A B CD e A B CD E fa B A b C A B cD A BC d E A B CD e F

a b A b c A B c d A B C d e A B CD e fa BC A b CD A B c DE A B C d E F

c

a b C

a b c a b c d a b c d e a b c d e f

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1 18 THE SUBSTITUTION OF SIAIILARS

combinations of the letters A,B

,C

,D

,a,b, c, cl

,and

A,B

, O ,D

,E

,a,b,0,cl

,e.

‘The slips are furnished with little pins, so that, whenplaced upon the ledges of the board, those marked by anygiven letter may b e readily picked out by means of astraight-edged ruler, and removed to another ledge .

50.

‘The use of the abacus will b e best shown by anexample. Take the syllogism in Barbara

Man i s mortal,

Socrates i s man,

Therefore Socrates is mortal .

A Socrates,

B man,

C mortal.

The corresponding small italic letters then indicate thenegatives,

a ri ot-Socrates,

b = not-man,

c not-mortal,

and the premises may b e stated as

A is B,

B is C .

‘Now take the second set of slips containing all thepossible combinations of A, B,

C, a

,b, c

,and ascertain

which of the combinations are possible under the conditionsof the prem ises .

Select all the slips marked A ; and as all these oughtto b e B ’s, select again those which are not-B or b

,and re

jcet them . Unite the remainder, and, selecting the B’s,

reject those which are not-C or 0. There will now remainonly four slips or combinations

51.

‘Precisely the same obvious system of analysis isapplicable to arguments however complicated . As an ex

ample, take the prem ises treated in Boole’s Law s of Thought,

p . 12 5

Sim i lar figures consist of all w hose corresp onding

angles are equal, and w hose corresp onding si des are propor

ti onal.

Triangles w hose corresp onding angles are equal haveA ”

thei r corresp onding si des prop ortional, and vice versa .

Let

A simi lar,

B triangle,

C= hat 'ing correspond ing angles egaal,

D= hcw ing corresp onding sides p rop orti onal.

‘

The prem ises may then b e expressed in QualitativeLogic, as follows

‘ Take the set of 1 6 slips : out of the A’s reject thosewhich are not CD ; out of the CD

’

s reject those which are

not A ; out of the BC’

s reject those which are not BD ; andout of the BD’

s reject those which are not BC . There willremain only six slips , as follows

From these w e may at once read off all the conclusions

1 2 0 THE SUBSTITUTION OF SIMILARS

laboriously deduced by Boole in his obscure processes . W e

at once see, for instance, that the class a, or “ dissimi lar

figures, consist of all triangles (B) w hi ch have not their cor

respond ing angles equa l (0) and sides p rop ortional (d) , and ofall figures not being tri angles (b) w hi ch have ei ther their anglesequal (0) and sides notproportional (d) , or thei r corresp ond ingsi des p roportional (D) and angles not equal, or nei ther their

corresponding angles equal nor corresponding sides p rop ortional.”

(Boole, p .

52 . The selections as made upon the abacus are of

course subject to m istake, but only one easy step is requiredto a logical machine, in which the selections shall b e mademechanically and faul tlessly by the mere reading down of

the premises upon a set of keys, or handl es, representingthe several positive and negative terms, the copula, conjunctions

,and stops of a proposition .

’

53 . In the last paragraph I alluded to a further mechani

cal contrivance, in which the combination- slips of the abacusshould not require to b e moved by hand, but could b e placedin proper order by the successive pressure of a series of keysor handles. I have since made a successful working modelof this contrivance

,which may b e considered a machine cap

able of reasoning ,or of replacing alm ost entirely the action

of the mi nd in drawing inferences . When I have an oppor

tunity of describing the details of its construction, I thinkit will b e found to afford a physical proof, apparent to theeyes

,of the extreme incompleteness of the Aristotelian logic.

Not onl y are the syllogisms and other old forms of argumentcapable of being worked upon the machine

,but an indefinite

number of other form s of reasoning can b e represented by thesimple regular action of levers and spindles .

54 . The most unfortunate feature of the long history of

our present traditional logic has been the divorce existingbetween the logic of the schools and the logic of commonlife. There has been no apparent connection whatever b etween the formal strictness of the syllogistic art and the

more loose but useful suggestions of analogy from particu

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THE SUBSTITUTION OF SIMILARS

when drawing inferences from our personal experience, andnot from maxims handed down to us by books or tradition,w e much oftener conclude from particulars to particul arsdirectly, than through the intermediate agency of any generalproposition . W e are constantly reasoning from ourselvesto other people, or from one person to another, withoutgiving ourselves the trouble to erect our observations intogeneral maxims of human or external nature. When w e

conclude that some person will, on some given occasion, feelor act so and so

, w e sometimes judge from an enlarged cons ideration of the manner in whi ch human beings in general,or persons of some particular character, are accustomed tofeel and act ; but much oftener from merely recollecting thefeelings and conduct of the same person in some previousinstance, or from considering how w e should feel or act

ourselves . It is not only the vi llage matron who, whencall ed to a consultation upon the case of a neighbour’s chil d,pronounces on the evil and its remedy simply on the recol

lection and authority of what she accounts the simi lar caseof her Lucy.

’

56 . Mr. M ill expresses as clearly as it is well possiblethat w e argue in common li fe, as he thinks, not by thesyllogism

,but directly from instance to instance by the

sim ilarity observed between the instances. But thi s argument from sim ilars to simi lars is the identical process whichI have called the substitution of similars, and which I haveshown to b e capable of explaining the syllogism itself, andmuch more . In fact, w e find Mr. Mill enunciating thisprinciple himself in another chapter

,where he is treating

of argument from analogy or resemblance. After noticingthe stricter meaning of analogy as a resemblance of relations,

he continues 1

It is on the whole more usual, however, to extend thenam e of analogical evidence to arguments from any sort ofresemblance

,prov ided they do not amount to a complete

induction : without peculiarly di stinguishi ng resemblance of

1 System of Logic, vol . i i , p. 86 .

THE TRUE PRIN CIPLE OF REASONIN G 3

relations . Analogical reasoning, in this sense, may b e re

duced to the following formula : Two things resemble eachother in one or more respects ; a certain proposition is trueof the one ; therefore it is true of the other. But w e havenothing here by which to discrim inate analogy from induction, since this type will serve for all reasoning from experi

ence . In the m ost rigid induction , equally with the faintestanalogy, w e conclude, because A resembles B in one or m oreproperties, that it does so in a certain other property.

’

57 . If this b e, as Mr. M ill soclearly states, the type of

all reasoning from experience,it follows that the principle

of inductive reasoning is actually identical with that whichI have shown to b e sufficient to explain the form s of deductive reasoning . The only difference I apprehend is, that indeductive reasoning w e know or assume a sim ilarity oridentity to b e certainly known

,and the conclusion from it

is therefore equally certain ; but in inductive argumentsfrom one instance to another w e never can b e sure that thesimilarity of the i nstance is so deep and perfect as to warrant our substitution of one for the other. Hence the con

elusion is never certain,and possesses only a degree of

probability,greater or less according to the circum stances

of the case and the theory of probabilities is our onlyresource for ascertaining this degree of probability

,i f ascer

tainab le at all .58 . It is instructive to contrast m athematical induction

with the induction as employed in the experimental sciences .The process by which w e arrive at a general proof of aproblem in Euclid’s E lements of Geometry is really a process of generalisation presenting a striking illustration of

our principle . To prove that the square on the hypothenuseof a right-angled triangle is equal to the sum of the squareson the sides containing the right angle , Euclid takes only asingle example of such a triangle , and proves this to b e true .

He then trusts to the reader perce iving of his own accordthat all other right-angled triangles resemble the one accidentally adopted in the points material to the proof, so that

1 2 4 THE SUBSTITUTION OF SIMILARS

any one right-angled triangle may b e indifferently substitutedfor any other. Here the process from one case to anotheris certain, because w e know that one case exactly resemblesanother. In physical science it is not so, and the distinotion has been expressed

,as it seems to m e, with adm irable

insight by Professor Bowen in his well -known Treatise on

Logic, or the Law s of Pure Thought.1 He says of mathe

matical figures : ‘The same measure of certainty whichthe student of nature obtains by intuition respecting asingle real object, the mathematician acquires respecting awhole class of imaginary objects, because the latter has theassurance

,which the form er can never attain, that the single

object which he is contemplating in thought is a p erfectrepresentative of its whole class : he has this assurance,because the w hole class exists only in thought, and are therefore all actually before him,

or present to consciousness .For example : this bit of iron, I find by di rect observation ,melts at a certain temperature ; but it may well happenthat another piece of iron

,quite sim ilar to it in external

appearance, may b e fusible only at a much higher temperature, owing to the unsuspected presence with it of a l ittlemore or a little less carbon in composition . But i f the

angles at the base of this triangle are equal to each other,I know that a corresponding equality must exi st in the caseof every other figure which conforms to the definition of anisosceles triangle ; for that definition excludes every disturbing element. The conclusion in thi s latter case

,then, is

uni versal,while in the form er it can b e only singular or

particular.’

This passage perfectly supports my view that all reasoning consists in taki ng one thing as a rep resentative, that isto say

,as a substi tute, for another, and the only difficul ty

is to estimate rightly the degree of certainty, or of mereprobability, with which w e make the substitution . The

forms and methods of induction and the calculus of prob ab ilities are necessary to guide us rightly in this ; but to

1 Camb ridge, United States, 1866 , p. 354.

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1 2 6 THE SUBSTITUTION OF SIA/ILARS

3worth studying and mastering for any practical purpose .

In these and many other places Mr. Mill shows a lamentablewant of power of appreciating the principles involved in thequantification of the predicate . As regards the most originaldiscoveries of Dr. Boole, there is not , so far as I have beenable to discover

,a single word in Mr. M ill ’s edition of his

Logi c published in 1 8 6 2 , to indicate that he w as consciousof the publication of Mr. Boole ’s Mathemati cal Analysis in1 84 7 , and of his great work

,The Law s of Thought, in 1 8 54 .

Although accounted a disciple and potent supporter of the

doctrines of Jeremy Bentham ,he appears unaware that the

doctrine of the quantification of the predicate is traceableto his great master, or at all events to the work of a nephewfounded upon the manuscripts of Bentham .

60. I ought not to om it to notice that Dr. Thom sonsubstantially adopts the principle of substitution in treatingof what he calls the syllogism of analogy. He states thecanon in the following manner : 1 The same attributesmay b e assigned to distinct but sim ilar thi ngs , providedthey can b e shown to accompany the points of resemblancein the things and not the points of difference.

’ This meansthat one thing may b e substituted for another l ike to it,provided that their li keness really extends to the point inquestion , which can often only b e ascertained with more orless probabili ty by inductive inquiry . He adds, that theexpression of the agreement must consist of a qualified judgment of identity, or a proposition of the form U,

by whichsymbol he indicates a proposition denoted in this tract bythe expression A=B. This exactly agrees with my viewof the matter.

61. The principle of substitution of similars seems tothrow a clear light upon the infinite importance of classifi

cation. For classification consists in arranging things, eitherin the mind or in cabinets of specimens, according to theirresemblances, and the best classification is that which exhibitsthe most numerous and extensive resemblances . The pur

1 Law s of Thought, fi fth ed . p . 251.

THE TRUE PRIN CIPLE OF REASONIN G

pose and effect of such arrangement evidently is , that w emay apply to all members of a class whatever w e know of

any member, so far as i t is a member.

-Al l the members of

a class are mutual substitutes for each other as regards theircommon characteristics , and a natural classification is thatwhich gives the greatest probabil ity that characters as yetu nexam ined will exhibit agreements corresponding to thosewhich are exam ined . Classification is thus the infinitelyuseful mode of multiplying knowledge, by rendering knowledge of particulars as general as possible , or of indicatingthe greatest possible number of substitutions which may

give rise to acts of inference.

62 . I need hardly point out that not only in our reasonings

,but in our acts in common life , w e observe the principle

of sim ilarity. Any new kind of action or work is performedwith doubt and difficulty, because w e have no knowledgederived from a similar case to guide us . But no sooner hasthe work been performed once or twice with success thanmuch of the difficulty vanishes, because w e have acquiredall the knowledge which will guide us in sim ilar cases .Mankind

,too

,have an instinctive respect for precedents

,

feeling that, however w e act in one particular case, w e oughtto act similarly in all sim ilar cases

,until strong reason or

necessity obliges us to make a new precedent. The wholepractice of law in English courts, if not in all others, consistsin deciding all new causes according to the rule establishedin the most nearly sim ilar former causes

,provided any can

b e found suffi ciently sim ilar. No ruler, too, but an absolutetyrant can perform any public act b ut under the responsib ility of being call ed upon to perform a sim ilar act

,or make

a sim ilar concession,in sim ilar circum stances .

63 . At the present day, for instance, the Government iscalled upon to take charge of the telegraphs and railways

,

because great benefit has resulted from their managementof the post-office. It is implied in this dem and that thetelegraphs and railways resemble or are even identical withthe post - office

,in those points which render G overnment

THE SUBSTITUTION OF SIAIILARS

control beneficial, and the public m ind inevitably leaps fromone thing to anything whi ch appears sim ilar. The wholequestion turns, of course, upon the degree and particularnature of the sim il arity. G ranting that there is sufficientanal ogy between the te legraph and the post-offi ce to renderthe Government purchase of the form er desirable, w e mustnot favour so gigantic an enterprise as the purchase of the

railways until it is clearly made out that their successfulmanagement depends upon principles of economy exactlysim il ar to the case of the post-office .

64 . The great immediate question of the day is the

D isestablishment of the Irish Church . The opponents ofthe measure argue against i t by the indirect argument

,that

i f the Irish Church ought to b e disestablished, so ought theEnglish Church ; but as this ought not, neither ought theIrish Church . They are answered by pointing out that theIrish and English Churches

‘

are not sim i larly si tuated ; the

one possesses the sympathy of the great body of the people,and the other does not. This is an all- important point

,

which prevents our applying to one what w e apply to theother. But on either side it is unconsciously

,if not ex

pressly, allowed that similars must b e sim ilarly treated.

Almost the whole of our difficulties in the government of

Ireland arise from the different national characters of the

Irish and Engli sh, whi ch renders laws and institutionssuited to the one inapplicable to the other. Yet such isthe tendency of indi scriminating public Opinion to run inthe groove of simil arity, that it requires a bold legislator torepeal laws for Ireland which it is not intended or desiredto repeal for England.

65 . Before closing, I should notice that at some periodin the obscurity of the Middle Ages an attempt seems tohave been made to assimilate in some degree the logicaland mathematical sciences, by inventing a logical canonanalogous to the first axiom of Euclid . Between the d i ctumde omni et nullo of Aristotle, which had so long beenesteemed the primary and perfect rule of reason, and the

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1 30 THE SUBSTITUTION OF SIMILARS

Postulatum mathema ticum,ut quce eidem tertio cequa lia

sunt, etiam inter se sint cequalia,conforme est cum fabrica

syllogismi in logica : qui uni t ea quoe conueniunt in media.

67 . It is a truly curious fact in the history of Logic, thatthese canons shoul d so long have been adopted, and yet thatthe only form of proposition to which they correctly applyshould have been almost wholly ignored until the presentcentury.

It is only when applied to propositions of the form A=B

that these canons prevent us from falling into error,but

when used with the propositions of the old Aristoteliansystem they allow the free commi ssion of fallacies of undistributed m iddle . It has been well pointed out by Mr.Mansel

,

1 that ‘ these canons are an attempt to reduce al l

the three figures of syllogism directly to a single principle ;the di ctum dc omni et nullo of Aristotle , which wasuniversally adopted by the scholastic logicians, beingdirectly applicable to the first figure only. This reduction,so long as the predicate of propositions has no expressedquantity

,is illegitimate ; the term s not being equal, but

contained one within another, as is denoted by the namesmajor and m inor. Hence, as applied to the first figure, theword conceniunt has to express, at one and the same time,the relation of a greater to a less, and of a less to a greater,—of a predicate to a subject, and of a subject to apredicate.

’

Thus in the syll ogism of the m ood Barbara,

Metals are elements,

Iron i s a metal,

Iron,therefore, i s an element,

the terms elements and i ron are both said to agree withmetals

,the third common term , although elements is a wider

term ,and i ron a much narrower one, than metals. Nothing

can b e more unscientific and fallacious than such an applica1 A rtis Logicee Rud imenta , p. 65 .

fallacious conclusion. Doub tless this ab surdity may b e

aimed away by pointing out that metals and o.cygen do

really agree with the same part of the class elements, so

there is no really common third term ; but the so - calledem e canon of syllogism is unable to indicate when this18 case and when it is not . O ther rules have to b e

med in order to overrule the supreme rule , and theselve the principle of quantification

,because they depend

1 the inquiry as to what parts of the m iddle term are

tical respectively with the major and m inor terms . Y et

enturies logicians failed to acknowledge that identity isre bottom of the question .

58 . To sum up , w e may say that the logicians attemptedeconcile logical w ith mathematical form s of reasoning ,assum ing a canon w hich is true when applied to

i tified propositions ; but, as they applied the canon toi ant ified propositions , they failed in producing anythinga fallacious appearance of conformity. In the presentury logicians have abundantly recognised the importanceuanti fying the proposition b ut they have either adheredi e old form of canon , or th ey have om itted altogether totire into the axiom s which must b e adopted as the

ndw ork of the reasoning process . I have long feltuaded of the truth enounced by that most clear thinker

,

dillac that ‘

équations, propositions, jugem ens, sont au

l la meme chose, et que , par consequent, on raisonne de

132 THE SUBSTITUTION OF SIZIIILARS

la meme maniere dans toutes les sciences ;’ 1

and it hasbeen my endeavour at once to transform the preposition intothe equation, and to employ it with an axiom of adequatesimplicity and generality, not spoiling good new materialwith old tools .69 . I write this tract under the discouraging feeling that

the public is l ittle inclined to favour or to inquire into thevalue of anything of an abstract nature. There are numberless scientific j ournals and many learned societies, and

'

theyreadily welcome the m inutest details concerning a raremineral , or an undescribed species , the newest scientific toy,or the latest observations concerni ng a change in the

weather. All these things are in public favour becausethey come under the head of physical science. Mathematicians, again, are in favour because they help the

physical philosophers : accordingly the m ost incomprehensible speculations concerning a quintic, or a resolven t, or a

'

new theory of groups, are readily (and deservedly) printed ,al though not a score of men in England can understandthem . But Logic is under the ban of metaphysics . It isfalsely supposed to - lead to no useful w orks— to b e merespeculation ; and , accordingly, there is no j ournal

,and no

society whatever, devoted to its study. Hardly can a paperon a logical subject b e edged into the Proceedings of anylearned society except under false pretences . This state of

things is doubtless due to an excessive reaction against theformer pre-em inence of logical studies . Bacon

,in protesting

against the absurdities of the scholastic logicians,and the

deference paid to an ancient author, placed himself at thehead of this reaction . Were he living now

,he would

probably see that the slow pendulum of public opinion hasswung to the opposite extreme, and would employ his greatintellect in showing how absurd it is to cultivate the

branches of the tree of knowledge , and neglect the rootwhich root is undoubtedly to b e found in a true comprehen

sion of logical method .

1 La Logique : (E'

uvres de Omzd i llac,vol . xxn,

p. 173 .

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134 APPENDIX

w ith a different combination of letters, printed upon white paperand pasted on the face of the slip.

1 The nature of the combinations w ill b e readily gathered from pp. 115

, 116 , and 117 , and

a set of s ixteen of the slips is shown in the figure at restingupon one of the ledges in the usual manner.

In the face of each wooden slip are fixed pins of thin brassor steel w ire, projecting from the wood about 1 inch in an

inclined direction . Every slip has a pin near to its upper end,as at but the positions of the other pins are varied according to the combination of letters represented on the slip. Eachlarge capital or po sitive letter is furnished with a pin in theupper part of the space allotted to it

,as at (19 , 20, 2 1,

w hile ’

each small or negative letter has a pin in the low er partof its space

,as at (2 2 , 2 3 , 24, etc. ) At the low er end of each

slip and at the front is fixed, as at (9 , 11, 13 , 15 , a thicksquare piece of sheet lead, weighing from 1 oz . to 1 oz . , so

adjusted that each slip will hang in stable equilibrium and in anupright position w hen lifted by any of the pins. The lead mayb e covered at the front side w ith white paper.The only other requisite is a flat straight-edge or ruler of

hard wood about 16 inches long, 11 inch wide, and 1 inch thick .

It is show n in the figure at and an enlarged section at

w here the sharp edge will b e seen to b e strengthened w ith a slipof brass plate It will b e desirable to have a b ox made tohold all the slips in proper order

,arranged in trays, so that any

set may readily b e taken out by the aid of the straight-edge,inserted under the row of top pins .In using the abacus, one or other of the series of slips i s

taken out, according as the logical problem to b e solved contains

more or less term s . If there b e only tw o term s, the set of fouris used ; if three, the set of eight ; if four, the set of sixteen ;and if five

,the set of thirty - tw o slips . Thus the syllogism

Barbara would require the set of eight slips (see p. 117 , etc.)The se must b e set side by side upon the topmost ledge, as at

the order in which they are placed is not of any es sentialimportance

,b ut it is generally convenient for the sake of clear

ne ss that every positive combination should b e placed on the leftof the corresponding negative, and that the order

’

show n at (7and at pp. 115

,116

,and 117 , should b e as much as possible

maintained . W hen a seri es of the slips i s resting on one of the

ledges, it is evident that w e may separate out those marked wi th1 I shall b e happy to send a set of the printed letters to any person w ho may

d esire to have an ab acus constructed .

APPENDIX 135

A or any capital letter,by inserting the straight-edge horizontally

beneath the proper row of pins,and then raising the slips and

removing them to another ledge. The corresponding negativeslips w ill b e left w here they w ere

, ow ing to the absence of pinsat the point w here the straight-edge is placed . W e have alw aysthe O ption, too, of removing either the A’s or the a

’s,the B ’s or

the b’

s,and so on. Successive movements w ill enable us to

select any class or group out of the series : thus,if w e took the

series of sixteen , and removed first the a’

s and then the b ’s, w eshould have left the class of AB S

,four in number. Dr. Boole

based his logical notation upon the succe ssive selection of classes,and it is this O peration of thought w hich is represented in a

concrete manner upon the abacus .The examples given in the text (pp . 117 - 119 ) w ill partly serve

to illustrate the use of the abacus,but I w ill minutely describe

one more instance. Let the premises of a problem b e

(a) A is either B or C ; but(B) B is D ; and

(y) C is D .

Let it b e required to answ er any que stion concerning the

character of the things A,B

,C,D

,under the above cond itions .

(1) Take the set of six teen slips and place them on the topm ost or first ledge of the board .

Rem ove the A’s to the second ledge .

( 3 ) O ut of the A’s,remove the B ’s back to the first ledge .

O ut of w hat remain,remove the O

’

s back to the firstledge .

(5 ) “That still remain are combinations contradicted by thefirs t premise (a ), and they are to b e removed to the

low est ledge , and left there .

(6) The others having been joined together again on the firstledge , remove the B ’s to the second ledge .

( 7 ) O ut of the B ’s,remove the D ’s to the first ledge again,

and

(8) Reject to the low est ledge the B ’s w hich are not -D’

s,as

contradictory of (B) .

( 9) Similarly, in treating remove the C ’s to the secondledge , return those w hich are D’s

,and reject the C ’s

w hich are not-D’

s to the low est ledge .

The combinations w hich have escaped rejection are all w hich

136 APPENDIX

are possible under the conditions (a) and and they willb e found to b e the following

To obtain the de scription of any class,w e have now only to

pick out that class by the straight-edge, and observe their nature.

Thus,w hen the A’s are picked out

,w e find that they always

bring D with them ; that is to say, all the A’s are D ’s ; thisbeing the principal result of the problem . But w e may alsoselect any other class for examination . Thus the d ’

s are repre

sented by only one combination,which shows that w hat is

not-D is neither C, B, nor A.

Even when the common conclusion of an argument is selfevident

,it will b e found instructive to work it upon the abacus

,

because the whole character of the argument and the conditionsof the subject are then exhibited to the eye in the cleare stmanner ; and while the abacus gives all conclusions which can

b e obtained in any other way,it often give s negative conclusions

w hich cannot b e detected or proved but by the indirect method(see p . It also solve s with certainty problem s of such adegree of complexity that the mind could not comprehend themw ithout some mechanical aid. In my previous little w ork on

Pure Logic (see above, 40, p . 110) I have given a number ofexamples, the working of which may b e tested on the abacus,and other examples are to b e found in Dr . Boole ’s Laws ofThought.

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O N THE ME CHAN I CAL PERFORMANCE

LOG ICAL INFE RENCE

1 . IT is an interesting subject for reflection that from the

earliest times mechanical assistance has been required inmental operations . The word calculati on at once rem indsus of the employment of pebbles for m ark ing units

,and it

is asserted that the w ord (340 1.9a is also derived from the

like notion of a peb ble or material sign .

1 Even in the

time of Aristotle the wide extension of the decimal systemof numeration had been rem arked and referred to the use of

the fingers in reckoning ; and there can b e no doubt thatthe form of the m ost available arithmetical instrum ent

,the

human hand,has reacted upon the m ind and moulded our

numerical system into a form w hich w e should not otherwise have selected as the best.

2 . From early times , too,distinct m echanical instru

ments were devised to facilitate computation . The Greeksand Romans habitually employed the abacus or arithm eticalboard

,consisting

,in its m ost convenient form , of an oblong

frame w ith a series of cross wires , each bearing ten slidingbeads . The abacus thus supplied, as it w ere , an unlim itedseries of fingers , which furnished marks for successive higherunits and allowed of the representation of any number.

1 Professor Dc Morgan On the w ord ’

Api 9,uOs, Proceed ings of the Ph i lo

log ica l Society, p . 9 .

140 THE flIE CHANICAL PERF ORIVIAN CE

The Russians employ the abacus at the present day underthe name of the shtshob, and the Chinese have from timeimmem orial made use of an almost exactly similar instrument called the schw anp an.

3 . The introduction into Europe of the Arabic system of

numeration caused the abacus to b e generally superseded bya far more convenient system of written signs ; but mathematicians are well aware that their science, however muchit may advance, always requires a corresponding developm ent of material symbols for relieving the memory andguiding the thoughts . Almost every step accomplished inthe progress of the arts and sciences has produced some

m echanical device for facilitating calculation or representingi ts resul t. I may mention as tronom ical clocks , mechanicalglobes, planetarium s, slide rules, etc. The ingenious rodsknown as Napier’s Bones

,from the name of their inventor,

or the Promptuarium Mul tiplicationis of the same celeb rated mathematician ,1 are curious examples of the tendencyto the use of material instruments .4 . As early as the seventeenth century w e find that

m achinery was made to perform actual arithmetical cal culation . The arithmetical machine of Pascal was constructedin the years 1 6 4 2 -1 6 4 5 , and was an invention worthy of

that great genius . Into the peculiarities of the m achinessubsequently proposed or constructed by the Marquis of

Worcester,Sir Samuel Morland, Leibnitz, Gersten, Scheutz ,

Donkin,and others w e need not inquire ; but it is worthy

of notice that M . Thomas , of Colmar, has recently manufac

tured an arithmetical machine so perfect in constructionand so moderate in cost, that it is frequently employed withprofit in m ercantile

,engineering , and other calculations .

5 . It was reserved for the profound genius of Mr. Babbage to make the greatest advance in mechanical calculation , by embodying in a machine the principles of the

1 Rab dologiae seu numerationi s per v i rgulas lib ri duo : cum append ice deexpedi t issimo mult ipl icat ioni s Promptuario . Quibus accessit et Ari thmet icze

Localis Li b er Unus. Authore et Inventore Joanne N epero, Barone Merch is

toni i , etc. Lugduni , 1626 .

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142 THE MECHANICAL PERF OR/PIAN CE

single previous attempt to devise or construct a machinewhich should perform the operations of logical inference ; 1

and it is only I believe in the satirical writings of Swift thatan allusion to an actual reasoning machine is to b e found .

2

8 . The only reason which I can assign for this completeinability of logicians to devi se a real logical instrument, isthe great imperfection of the doctrines which they entertained . Until the present century logic has remainedsubstan tially as it was moulded by Aristotle 2 200 yearsago . Had the science of quantity thus remained stationarysince the days of Pythagoras or Euclid

,it is certain that

w e should not have heard of the arithmetical m achine of

Pascal, or the difference engine of Babbage . And I v en

ture to look upon the logical machine which I am about todescribe as equally a result and indication of a profoundreform and extension of logical science accomplished withinthe present century by a series of English writers

,of whom

I may specially nam e Jeremy Bentham, George Bentham ,

Professor De Morgan , Archbishop Thomson, Sir W .

Ham ilton , and the late distinguished Fellow of the RoyalSociety

,Dr. Boole . The result of their exertions has been

to effect a breach in the supremacy of the Aristotelian logic,and to furnish us, as I shall hope to show by visible proof,with a system of logical deduction almost infinitely moregeneral and powerful than anything to b e found in the oldwriters . The ancient syllogism was incapable of mechani

cal performance because of its extreme incompleteness andcrudeness , and it is only when w e found our system uponthe fundamental laws of thought them selves that w e arriveat a system of deduction which can b e embodied in amachine acting by simple and uniform movements.9 . To George Boole, even m ore than to any of the

1 See note at the end of this paper,p . 172 .

In the recent Lif e of Si r W . Ham i lton,b y Professor Ve itch , is gi ven an

account and figure of a w ooden instrument employed b y Sir W . Ham i lton inhis logical lectures to represent the comparat ive extent and intent of mean

ing of terms ; b ut i t w as merely of an il lustrative character,and does not

seem to have b een capab le of performing any mechanical operations .

OF LOGICAL INFEREN CE 143

logicians I have named, this great advance in logical doctrine is due . In his Mathemati cal Analysis of Logicand in his most remarkable work Of the Law s of Thought

(London, he first put forth the problem of logicalscience in its complete generality z— Given certain logi cal

p rem ises or condi tions, to determine the descrip tion of any class

of objects under those condi ti ons. Such was the general problem of which the ancient logic had solved b ut a fewisolated cases— the nineteen moods of the syllogism ,

the

sorites , the dilemma, the disjunctive syllogism ,and a few

other forms . Boole showed incontestably that it was possible

,by the aid of a system of mathematical signs , to

deduce the conclusions of all these ancient modes of reasoning

,and an indefinite number of other conclusions . Any

conclusion , in short, that it was poss ible to deduce from anyset of prem ises or conditions, however numerous and com

plicated , could b e calculated by this method .

10. Y et Boole ’s achievement was rather to poin t out theextent of the problem and the possibility of

'

solving it,than

him self to give a clear and final solution . As readers of hislogical w orks must b e well aware, he shrouded the simplestlogical processes in the mysterious operations of a m athemat ical calculus . The intricate trains of symbolic transformations

,by which many of the examples in the Law s

of Thought are solved, can b e followed only by highlyaccomplished mathematical m inds ; and even a mathematic ian would fail to find any demons trative force in acalculus which fearlessly employs unmeaning and incom

prehensible symbols , and attributes a signification to themby a subsequent process Of interpretation . It is surelysufficient to condemn the peculiar mathematical form of

Boole ’s method, that i f it were the true form of logicaldeduction, only well-trained m athematicians could ever comprehend the action of those laws of thought, on the habitualuse of which our existence as superior beings depends .

11.Having made Boole ’s logical works a subject of

study for many years past, I endeavoured to show in my

144 THE ME CIIANIGAL PE PE OPMAN GE

work on Pure Logic 1 that the mysterious mathematicalforms of Boole ’s logic are altogether superfluous , and that inone point of great importance , the employment of exclusiveinstead of unexclusive alternatives, he was deeply m istaken .

Rejecting the mathematical dress and the erroneous conditions of his symbols , w e arrive at a logical m ethod of the

utmost generality and simplicity. In a later work 2 I havegiven a more mature and clear view of the principles of thisCalculus of Logic, and of the processes of reasoning ingeneral , and to these works I must refer readers who may

b e interested in the speculative or theoretical views of thesubject . In the present paper my sole purpose is to bringforward a visible and tangible proof that a new system of

logical deduction has been attained . The logical machinewhich I am about to describe is no mere model illustrativeof the fixed form s of the syllogism . It is an analyticalengine of a very simple character, which perform s a com

plete analysis of any logical problem impressed upon it .By merely reading down the prem ises or data of an argument on a key board representing the terms, conjunctions

,

copula, and stops of a sentence, the machine is caused to

make such a comparison of those prem ises that it becomescapable of returning any answer which may b e logicallydeduced from them . It is charged

,as it were

,with a cer

tain amoun t of information which can b e drawn from itagain in any logical form which may b e desired . The

actual process of logical deduction is thus reduced to apurely mechanical form , and w e arrive at a machine em

bodying the Law s of Thought, which may almost b e saidto fulfil in a substantial manner the vague idea of an or

ganon or instrumental logic which has flitted during manycenturies before the m inds of logicians .

12 . As the ordinary views of logic and the doctrine of

1 Pure Logic, or the Logic of Qual i ty apartf ro m Quanti ty : w i th Remarkson Boole

’

s System, and on the Relati on of Logic and Mathemati cs . London,

1864 (Stanford ) .2 The Substi tution of Sim i ta rs, the True Princip le of Reaso ning : deri ved

f rom a Mod ifi cation of Aristotle’

s Di ctum. London, 1869 (Macm i llan) .

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146 THE MECHANICAL PERFORMAN CE

and it is required to arrive at the description of the class ofcompound or not-elementary bodies so far as affected by thisassertion . The process of thought is as followsBy the Law of Duality I develop the class not-element

into two possible parts , those which are metal and thosewhi ch are not metal

,thus

What is not element is ei ther metal or not-metal .

The given prem ise, however, enables me to assert that w hatis metal is element ; so that i f I allowed the first of thesealternatives to stand there would b e a not-element which isyet

’

an element. The law of contradiction directs me toexclude this alternative from further consideration , andthere remains the inference , commonly known as the con

tra-positive of the prem ise,that

What is not element is not metal.

Though thi s is a case of the utmost simplicity,the process

is capable of repeated application ad infi ni tum,and logical

problems of any degree of complication can thus b e solvedby the direct use of the most fundamental Laws of Thought.15 . To take an instance involving three instead of two

terms,let the premises b e

Iron is a metal (1)Metal is element (2)

W e can,by the Law of Duality, develop any of these terms

into four possible combinations . Thus

Iron is metal , elementor metal, not- elementor not-metal, elementor not-metal, not-element

But the first prem ise informs us that iron is a metal,and

thus excludes the combinations (7 ) and while the secondpremise informs us that metal must b e element, and thusfurther excludes the combination It follows that iron

quanti ti es i n d i fferent pro b lem s , and are rea lly used as DI‘1(marks to b e substituted for the full descriptions of thosquan tities . At the same time it is convenien t to substitutfor the corresponding negative term s small italic letters , t

b,c,etc. thus

i f A denote i ron,a denotes what is not-iron .

B metal,

b not-metal.

C element,

0 not-element.

When these general terms are combined side by side, as iA B C

, a B C ,they denote a term or thing combining th

properties of the separate term s . Thus A B C denotes i row hi ch i s metal and element ; a B C denotes metal w hich

element but not iron. These letter terms A,B, O ,

a,b,c,et

can,in short

, b e joined together in the manner of adjectiveand nouns .

17 . I must particularly insist upon the fact, how ev ethat there is nothing peculiar or mysterious in these lettesymbols . They have no force or m eaning b ut such as thederive from the nouns and adj ectives for which they stanas m ere abbreviations

,intended to save the labour of writing

and the want of clearness and conciseness attaching tolong clause or series of w ords . In the system put forth bBoole various symbols of obscure or even incomprehensib lmeaning were introduced ; and it was implied that th

inference came from Operations different from thosecommon thought and common language . I am particular]anxious to preven t the m isapprehension that the method

148 THE MECHAN ICAL PERFORMAN CE

inference embodied in the machine is at all‘ symbolic anddark

,or differs from what the unaided human m ind can

perform i n simple cases .18 . Great clearness and brevity are,

‘

how ev er,gained by

the use of letter terms ; for if w e take

A iron,

B metal,

C element

then the premises of the problem considered are simply

Iron is metalMetal is elemen t

The combinations in which A may manifestaccording to the Laws of Thought,

A B C

A B c

A b C

A b e

(ry) and (8) are con tradicted

A is identical with A B C,

and this term,A B C

, contains the full description of A oriron under the conditions (1) and

Similarly, w e may obtain the description of the term orclass of things not-element, denoted by c. For by the Law of

Duality c may b e developed into its alternatives or possiblecombinations .

(B) and (t) are con tradicted byexcluding these contradictory

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1 50 THE MECHANICAL PERF ORMAN CE

in connection with the prem ises, to el im inate the inconsistentcombinations, and afterwards to select from the remainingconsistent combinations such as may form any class of whichw e desire the description . Perform ing these processes inthe case of the prem ises (1) and w e find that of the

eight conceivable combinations only four remain consistentwith the premises, v iz.

(a )(6 )(n)

(9)

In this list of combinations the conditions (1) and (2 ) are,as it were , embodied and expressed , so that w e at once learnthat A according to those conditions consists of A B C only ;

B consists of (a ) or (e)b (n) 01

‘

(9)c (0)

(77) 01‘

(6)

20. It is easily seen that the solution of every problemwhich involves three terms A,

B, C wil l consist in making asim ilar selection of consistent combinations from the sameseries of eight conceivable combinations . Problems involving four distinct term s would sim ilarly require a series ofsixteen conceivable combinations, and if fi ve or six termsenter, there will b e thirty-two or sixty-four of such com

himations . These series of combinations appear to hold aposition in logical science at least as important as that ofthe multiplication table in arithmetic or the coeffi cients of

the binom ial theorem in the higher parts of mathematics . Ipropose to call any such complete series of combinations aLogi calAbecedarium,

but the number of combinations increasesso rapidly with the number of separate term s that I have notfound it convenient to go beyond the sixty-four combinationsof the six term s A,

B,C

,D

,E

,F and their negatives .

21. To a person who has once comprehended the

ole indirect process of inference becomes reducei etition of a few uniform operations of classifica

in,and elim ination of contradictories . Lo

ion becomes , in short, a matter of routine, andt of labour required the only impediment ton of any question . I have d irected much attenre, to reduce the labour required, and haw

i s publications described devices w hich part) lish this purpose . The Logical Slate consists of

te Abecedarium engraved upon a comm on w r

LIICl merely saves the labour of writing out theins.

1 The same purpose may b e effected by heof combinations printed ready upon separate sheea series of proper length being selected for

n of any problem ,and the inconsistent combina'

struck out with the pen as they are discovererl ation with the premises .A second step tow ards a mechanical logic w as

b e easy and desirable . The fixed order of thems in the w ritten abecedarium renders it necessaer them separately , and to pick out by repeatedutal attention those which fall into any partiConsiderable labour and risk of m istake thus

ogical Abacus was devised to avoid these objec ‘

ras constructed by placing the combinations o .

trium upon separate movable slips of wood,1

en b e easily classified,selected and arranged acco

conditions of the problem . The construction ant

s Abacus have , however, been sufficiently desc

152 THE !MECHAN ICAL PERFOR/VAN CE

abacus is peculiarly adapted for the logical class-room . Byi ts use the operations of classification and selection, on whichBoole ’s logic, and in fact any logic must b e founded, can b e

represented, and the clearest possible solution of any question can b e shown to a class of students

, each step in thesolution being made distinctly apparent .

23 . In proceeding to explain how the process of logicaldeduction by the use of the abecedarium can b e reduced toa purely mechanical form , I must first point out that certainsimple acts of classification are alone required for the purpose.

If w e take the eight conceivable combinations of the term sA, B,

C,and compare them with a proposition of the form

A is B ( 1)

w e find that the combinations fall apart into three distinctgroups

,which may b e thus indicated

Excluded combinations

Included combinations consistent with premise ( 1)

Included combinations inconsistentb b

W i th prem i se ( 1) C

The highest group contains those combinations whichare all a ’

s,and on account of the absence of A are unaffected

by the statement that A’s are B ’s ; they are thus excluded

from the Sphere of meaning of the prem ise,and their con

sistency with truth cannot b e affected by that premise . The

middl e group contains A-combinations,included within the

meaning of the prem ise, but which also are B-combinations,

and therefore comply with the condition expressed in the

prem ise . The lowest group consists of A-combinations also,

but such as are distinguished by the absence of B,and

which are therefore inconsistent with the premise requiringthat where A is

,there B shall b e likewise. This analysis

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154 THE IIIE CHANICAL PE RF ORIl/IAN CE

To effect the above classification, w e first move down toa lower rank the combinations inconsistent with (1) w e

then raise the b ’s, and out of the remaining B ’s lower theo’

s. But as all our operations are directed only to dist inguish the consistent and inconsistent combinations, w e

now join . the highest to the second rank , and the third tothe lowest, as follows

Combinations consistent with( 1) and (2 )

Combinations inconsistent with( 1) or or both

25 . The problem is now solved, and it only remains toput any question w e may desire . Thus i f w e want the

description of the class A,w e may raise out of the con

sistent combinations such as are a’

s, and the sole remaini ngcombination A B C gives the description required , agreeably to our former conclusion . To obtain the descriptionof B , w e unite the consistent combinations again and raisethe b

’

s ; there will remain two combinations A B C anda B C, showing that B is always C , b ut that , so far

as the conditions of the problem go, it may or may notb e A.

26 . In considering such other kinds of propositions asm ight occur, w e meet the case where two or more terms arecombined together to form the subject or predicate

,as in

the exampleA B is C,

meaning that whatever combinations contain both A and B ,

ought also to contain C . This case presents no difficulty ;and to obtain the included combinations it is only necessaryto raise out of the whole series of combinations the a

’

s and

b’

s, simultaneously or successivelv . The result, in whateverway w e do it

,is as follows

A AIncluded combinations B B

C e

W e may then remove such of the included combinations,

i .e. A B c only , as may b e inconsistent with the prem iseand proceed as before .

27 . Had the predicate instead of the subject con tainedtwo term s as in

11 IS B C,

w e should have required to raise the a’

s and then low er theb

’

s and the c’

s,in an exactly sim ilar manner.

28 . The only further complication to b e considered arisesfrom the occurrence of the disjunctive conjunction or in thesubject or predicate , as in the case

A is B or C (or both) .

To investigate th e proper m ode of treating this condition,

w e may take the same series of eight conce ivable combinations and raise those containing a

,in order to separate the

excluded combinations . But it is not now suffi cient simplyto lower such of the included combinations as contain b

,and

condemn these as inconsistent with the prem ise . For

though these combinations do not contain B they may con

tain C,and may require to b e adm itted as consistent on

account of the second alternative of the predicate . Whilethe AB

’

s are certainly to b e adm itted, the Ab’

s mus t b esubjected to a new process of selection . Now the simplestmode of preparing for this new selection is to join the AB’

s

to the a’

s or excluded combinations, to move up the Ab’

s

i nto the place last occupied by the A’s , to lower such of

the Ab’

s as do not contain C . The result will then b e asfollows

156 THE MECHANICAL PERFORMAN CE

Excluded combinations and in A Acluded combinations consis B Btent with l st alte rnative C c

Included combination inconsistent Awith 1st but consistent with 2d b

alternati ve

Included combination inconsistent withboth alternatives

It is only the lowest rank of combinations, in this casecontaining only A b c, which is inconsistent with the prem ise as a whole, and which is therefore to b e condemned ascontradictory ; and if w e join the two higher ranks w e haveeffected the requisite analysis .29 . It will b e apparent that should the subject of the

prem ise contain a disjunctive conjunction, as in

A or B is C,

a sim ilar series of operations would have to b e performed .

W e must not merely raise the a’

s and treat them as ex

cluded combinations, but must return them to undergo anew sifting, whereby the aB

’

s w ill b e recognised as includedin the m eaning of the subject

,and only the ab

’

s will b etreated as excluded. This analysis effected ,

the remainingoperations are exactly as before .

30. The reader will perhaps have remarked that in thecase of none of the prem ises considered has it been requisiteto separate the combinations of the abecedarium into morethan four groups or ranks

,and it may b e added that all

problem s involving simple logical relations only have beensuffi ciently represented by the examples used . The task of

constructing a mechanical logic is thus reduced to that ofclassifying a series of wooden rods representing the conceiv

able combinations of the abecedarium into certain definitegroups distinguished by their positions, and providing suchmechanical arrangements , that wherever a letter term occurs

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OF LOGICAL INFEREN CE

in the subj ect or predicate of a proposition,or a conjunction

copula or stop intervenes,the pressure of a corresponding

lever or key shall execute systematically the required movements of the combinations .31 . The principles upon which the logical machine is

based will now b e apparent to the reader ; and as the con

struction of the machine involves no mechanical difficultiesof any importance

,it only remains for me to give as clear a

description of its component parts and movements as theirsomewhat perplexing character adm its of.32 . The Machine which has been actually finished is

adapted to the solution of any problems not involving m orethan four distinct positive terms , indicated by A,

B,C

,D

,

w ith,of course

,their corresponding negatives , a

,b,c,d .

The requisite combinations of the abecedarium are,therefore ,

sixteen in number 2 0,p . and each combination is

represented by a pair of square rods of baywood (PlateII

,fig . united by a short piece of cord and slung over

two round horizontal bars of wood (d , d ,figs . 2 and so

as to balance each other and to slide freely and perpendicularly in wooden collars (b , b , figs . 2

, 3 ,and 4 ) closed by

plain wooden bars (c, c) . To each rod is attached a thinpiece of baywood

,81 inches long and 1 inch w ide (a,

a,

figs . 2 and bearing the letters of the combination re

presented . Each letter occupies a Space of 1 inch in height ,b ut is separated from the adjoining letter by a blank spaceof white paper 11 inch long. Both at the front and backof the m achine are pierced four horizontal slits , 11 inchapart

,extending the whole w idth of the case

,and 1 inch in

height , so placed that when the rods are in their normalposition each letter shall b e visible through a slit . The

machine thus exhibits on i ts tw o sides , when the rods are

in a certain position , the combinations of the abecedarium asshown in fig. 5 b ut should any of the rods b e moved upwardsor downwards through a certain lim ited distance the letterswill become invisible as at f, f (Plate Il I, fig.

33 . Externally the machine consists of a framework ,

158 THE IIIE CHANICAL PERF ORAIAN CE

seen in perpendicular section in fig. 3 (g , g) , and in horizontal section in fig. 4 (g , g) , which serves at once to supportand contain the moving parts . It is closed at the frontand back by large doors (h,

h,

fig. in the m iddle panelsof which are pierced the slits rendering the letters of the

abecedarium visible .

34 . The rods are moved upwards , and the opposite rodsof each pair are thus caused to fall downwards

,by a series

of long flat levers seen in section at l, l, l (figs . 2 , 3 , 6

These levers revolve on pivots inserted in the thickerpart

,and move in sockets attached to the inner side of the

framework . Brass arms (m ,m

,figs . 3 and connected

by copper wires (n,n) with the keys of the machine (0, o) ,

actuate the levers, which are caused to return,when the

key is released , by spiral brass springs (s, s) .35 . The levers communicate motion to the rods by

m eans of brass pins fixed in the inner side of the rods(fig. As it is upon the peculiar arrangement of thesepins that the whole action of the machine depends

,the

position of each of the 2 7 2 pins is shown by a dot in fig. 1 ,

in which are also indicated the function of each pin and thecombination represented by each pair of rods . It is seenthat certain pins are placed uniform ly in all the adj oiningrods , as in the rows opposite the words Finis

, Conjuncti on,

Gop ala,Full Stop . These may b e call ed Op erati on p ins, and

must b e distinguished from the letter p i ns, representing theterms of the combination , and varied in each pair of rods tocorrespond with the letters of the abecedarium . On exam in

ing fig. 1 , it will b e apparent that the pins are distributedin a negative manner ; that is to say, it is the absence of apin in the space A,

and its presence in the space a,which

constitutes the rod a representative of the term A. The

rods belonging to the combination A b C d,for instance

,

have pins in the spaces belonging to the letters a,B

,c, D .

36 . The key board of the instrument is shown in fig. 4 ,

where are seen two sets of term or letter keys, markedA

,a,B

,b,C

,c,D

,d ,separated by a key marked COPULA

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160 THE MECHANICAL PERF ORMAN CE

The back rods will at the same time fall , and the combinations containing b will disappear from the abecedarium

,bu t

in the Opposite direction .

39 . It is now necessary to explain that each rod hasfour possible positions ful ly indicated in the figs. 6 -1 3 .

The fi rst of these positions is the neutral or initial position,

in which the letters are visible in the ab ecedarium , and the

letter pins are Opposite letter levers so as to b e acted uponby them . The second position is that into which a rod isthrown by a subject key ; the third position lies in the

opposite direction, and is that into which a rod is thrownby a predicate key . The fourth position lies 1 inchbeyond the third . The four positions evidently correspondto the four classes into which combinations were classifiedin the previous part of the paper as follow s

Second position— Combinations excluded from the

sphere of the premise.

Firs t position— Combinations included, but consistentw ith the prem ise.

Third position— Temporary position of combinationscon tradicted by the premise : also temporary position of

combinations excluded from some of the alternatives of adisjunctive predicate.

Fourth position—Final position of contradictory or inconsistent combinations .40. Let us now follow out the motions produced by

impressing the simple proposition

A is B

upon the machine, all the rods being at first in the initialor first position . The keys to b e pressed in successionare

First. The subject key, A .

Second . The copula key.

Third. The predicate key, B .

Fourth . The full-stop key.

The subject key A has the effect of throwing all the a

OF L O GICAL INFEREN’

CE 1

rods from the first into the second position, the back rods

rising and the front rods falling 1 inch .

The c0pula key will in this case have no effect , for, asseen in fig. 9 (Plate IV) , it acts only on rods in the thirdposition, of which there are at present none .

The predicate key (fig . 1 1 ) does not act upon such of

the rods (those m arked a ) as are in the second position , butit acts upon those in the first position

,provided they have

pins opposite the lever. The effect thus far will b e thatthe a rods are in the second position

,the Ab rods in the

third , while the AB rods remain undisturbed in the firstpos ition . An analysis has been effected exactly similar tothat explained above 2 3

,p .

41 . The full- stop key being now pressed has a doubleeffect . I t acts only on a single lever at the front of the

machine (figs . 2 and but the front rods all have in thespace opposite to the lever two pins 1 inch apart (fig.

These pins w e may distinguish as the a and,8 pins, the a

pin being the upperm ost. W hile a rod is in the firstposition the lever passes between the pins and has no effect ;but if the rod b e lowered 1 inch into the second position

,

the lever will cause the rod to return to the first positionby means of the a pin ; b ut i f the rod b e raised into thethird position

,the B pin will come into gear, and the rod

w ill b e pushed 1 inch further into the fourth position .

Now in the case w e are examining, the AB’

s are in the

first position and will so remain ; the a’

s are in the secondand w ill return to the first

,the Ab

’

s are in the third , andwill therefore proceed onw ards to the fourth . The readerwill now see that w e have effected the classification of the

combinations as required into those consistent with the

prem ise A is B ,whether they b e included or not in the

term A,and those contradicted by the prem ise which have

been ej ected into the fourth position . An exam ination of

the figures 6 will show that only one lever (fig . 8 )moved by the finis key affects rods in the fourth position ,so that any combination rod once condemned as contradictory

M

162 THE IPIE CHAN ICAL PERF ORIIIAN CE

so remains until the close of the problem , and its letters areno more seen upon the abecedarium .

42 . Any other proposition , for instance, B is C, can nowb e impressed on the keys , and the effects are exactly sim ilar,except that the Ab combinations are out of reach of the

levers . The B subject key throws the b’

s into the second ,the C predicate key throws the Bc

’

s into the third, and thefull stop throws the latter into the fourth , where they jointhe Ab

’

s already in that place of exclusion , while the re

mainder all return to the first position .

The combinations now visible in the abecedarium willb e as follows

a a

b b b b

D d D d

They correspond exactly to those previously obtained fromthe same prem ises (see except that each combinationof A

,B, C, a,

b,c is repeated with D and d . If w e now

want a description of the term A,We press the subject key

A,and all disappear except

A n on A B C d

which contain the information that. A is always associatedwith B and w ith C , but that it may appear with D orw ithout D

, the conditions of the problem having given usno information on this point. The series of consistent comb inations is restored at any time by the full-stop key, thecontradictory ones remaining excluded .

43 . Any other subject key or succession of subject keysbeing pressed gives us the description of the correspondingterms . Thus the key c gives us two combinations, a b c D ,

a b c d,informing us that the absence of C is always

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164 THE ME CIIAN ICAL PERF ORIIIAN CE

over a pulley q and through a perforation r in the flatboard which form s the lever itself, in this case a lever ofthe second order. This broad lever sweeps the rods fromthe fourth position as well as any which may b e in the

third into the first, and together with the other lever (fig .

13 ) it reduces the whole of the rods to the neutral position,

and renders the machine, as it were, a tabula rasa,upon

which an entirely new set of conditions may b e impressedindependently of previous ones . Its offi ce thus is to obliterate the effects of former problem s .46 . When several of the letter keys on the subject side

only or the predicate side only are pressed in succession,the effect is to select the combinations possessing all the

letters marked on the keys . Thus if the keys A ,B , C b e

pressed there will remain in the abecedarium only the com

b inations A B C D and A B C d ; and i f the key D b e nowpressed , the latter combination will disappear, and A B C Dwill alone remain . The effect will b e exactly the same

whatever the order in which the keys are pressed , and i fthey b e pressed simultaneously there will b e no differencein the result . The machine thus perfectly represents thecommutative character of logical symbols which Mr. Boolehas dwelt upon i n pp . 2 9 -3 0 of the Law s of Thought.

What I have called the Law of Simplicity of logicalsymbols , expressed by the formula AA=A,

1 is also perfectly fulfilled in the machine ; for i f the same key b e

pressed two or more times in succession , there will b e nomore effect than when it is pressed once . Thus the succession of keys A A C B B A C would have merely the effectof A B C . This applies also to the predicate keys , but notof course to an alternation of subject and predicate keys .

47 . To impress upon the machine the condition

A B is C D,

or w hatever combines the p rop erties of A and B combines the

properti es of C D ,w e strike in succession the subject keys

1 Pure Logi c, p . 15 . Boole’s Law s of Thought, p . 31.

OF LO GICAL INFEREN CE

A and B,the copula , the predicate keys C and D and the

full stop . The subject keys throw into the second positionboth the a combinations and the b ’S ° the predicate keys , outof the remaining AB’

s, throw the e s and d ’

s into the thirdposition ; and the full stop completes the separation of the

consistent and contradictory combinations in the usualmanner.

48 . It yet remains for us to consider a proposition witha disjunctive term in subject, predicate, or both m embers .For such propositions the conjunction keys are requisite ,

that adjoining the subject keys (fig. 4 ) for the subject , andthe other for the predicate . These keys act in oppositionto each other

,and each is opposed , again, to its correspond

ing letter keys . Thus while the subj ect keys act on leversat the back of the machine (Plate 111

,fig . the subject

conjunction key acts on the lever r in fron t,while the

predicate conjunction key t is at the back . These leversare show n in their full size in figs . 10 and 1 2

,and are seen

to differ from all the o ther levers in having the edge v

moving on small wire hinges u in such a w ay that it can

exert force upwards b ut not downwards . The lever can

thus raise the rods ; b ut in case it should strike a pin inre turning, the edge yields and passes the pin without m ovingthe rod . In connection with these levers each rod has tw opins (figs . 1 and 2 ) at a distance of only 1 inch, and the

peculiar effect of these pins w ill b e gathered from figs. 10

and 1 2 (Plate IV) . Thus i f w e press in succession the

predicate keysA or B

,

the key A w ill throw the a’

s into the third position . The

conjunction key will now act upon the a pins of the A’

s

and m ove them into the second position, and at the sam e

time upon the,8 pins of the a

’

s and return them into thefirst position . The key B now selects from the a

’

s thosewhich are b

’

s,and puts them into the third position ready

for exclusion by the full stop, which will also join to the

166 THE MECHANICAL PERFORMAN CE

aB’

s still remaining in the first position the A’s whichwere temporarily put out of the way in the second position .

Should there b e, however, another alternative , as in the

termA or B or C ,

the conjunction key would b e again pressed, which gives theab

’

s a new chance by returning them to the first,and the

key C selects only the abc’s for exclusion . The action

would b e exactly sim ilar with a fourth alternative .

49 . The subject conjunction key is sim ilar but oppositein action . If the subject key A b e pressed it throws thea

’

s into the second position ; the conjunction key then actsupon the a pin of the a

’

s returning them to the first posit ion

,and also upon the

,8 pin of the A’s

,sending them to a

temporary seclusion in the third position. The key Bwould now select the a

’

b’

s for the second position ; the con

junction key again pressed would return them,and add the

aB’

s to those in the third , and so on . The final resultwould b e that those combinations excluded from all thealternatives would b e found in the second position , whilethose included in one or m ore alternatives would b e partlyin the first and partly in the third positions .In the progress of a proposition the Oopula key would

now have to b e pressed, and when the subject is a disjunetive term its action is essential. It has the effect (fig. 9 )of throwing any combinations which are in the third backinto the first . It thus joins together all the combinationsincluded in one or m ore alternatives of the subject, and prepares them for the due action of the predicate keys .

50. It must b e careful ly observed that any doubly universal proposition of the form

all A’s are all B ’s,

or,in another form of expression,

A= B,

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168 THE IlIE CHANICAL PERFORIIIAN CE

‘ 3d . That wherever the properties A and B are bothabsent, the propert ies C and D are both absent also ; andv ice verstt, where the properties C and D are both absent ,A and B are both absent also. ’

This somewhat complex problem is solved in Boole’swork by a very difficult and lengthy series of eliminations ,developments, and algebraic multiplications . Two or threepages are required to indicate the successive stages of the

solution; and the details of the algebraic work would probably occupy many more pages . Upon the machine the

problem is worked by the successive pressure of the following keys

l st . A,B

,Copula

,C

,d,Conjunction

,c,D

,Full stop .

2d . B ,C

,Copula

,A

,D

,Conjunction

,a,d,Full stop .

3d . a,b , Copula, c, d ,

Full stop .

c, d , Copula, a,b,Full stop .

There will then b e found to remain in the abecedariumthe following combinations

A B c D

A b C D

A d

A b c D

On pressing the subject key A, the A combinationsprinted above in the left-hand column will alone remain

,

and on exam ining them they yield the same conclusion asBoole’s equation (p . nam ely, ‘Wherever the propertyA is present, there e ither C is present and B absent

,or C

is absent . ’

Pressing the full- stop key to restore the a combinations,

and then the keys b , C, w e have the two combinations

A b c nA b c a

from which w e read Boole’s conclusion (p.

‘Whereever the property C is present, and the property B absent,

direct Process of Inference,as stated in my Pure

pp . 4 2,4 3 and equivalent processes are necessary in B

mathematical system . The m achine does not the )

supersede the use of mental agency altogether, b ut it ntheless supersedes it in most important steps of

process .53 . This m echanical process of inference proceeds it

continual selection and classification of the conce ivableb inat ions into three or four groups . It should b e nc

that in Boole ’s system the sanre groups are indicatecertain quasi-m athematical symbols as follows

The coeffi cient 8 indicates an excluded comb inatior

i included1

11 inconsistent

inconsistent

It is exceedingly questionable whether there isanaloo y at all between the significations of these syr

in m athematics and those which Boole imposed uponin logic. In reality the symbol 1 denotes in Boole ’sinclusion o f a combination under a term ,

and 0 excl i

According uly

-1 indicates that the combination is includ

the subject and not in the predicate , and is therefore i:sistent with the proposition , am 1

11 indicates inclusion i

predicate and exclusion from the subject of an equai

proposition or identity, from w hich also results incr

tency . Inclusion in both term s is indicated by 1,

exclusion from both 0

0 ,in which case the combinati

consistent with the proposition .

170 THE MECHANICAL PE RFORIPIAIVCE

54 . To the reader of the preceding paper it will b eevident that mechanism is capable of replacing for the mostpart the action of thought required in the performance of

logical deduction . Having once written down the conditions or prem ises of an argument in a clear and logical form ,

w e have but to press a succession of keys in the ordercorresponding to the terms, conjunctions , and other partsof the propositions , in order to effect a complete analysis of

the argum ent. Mental agency is required only in interpreting correctly the grammatical structure of the premises ,and in gathering from the letters of the abecedarium the

purport of the reply. The intermediate process of deduc

tion is effected in a material form . The parts of the

machine embody the conditions of correct thinking ; therods are j ust as numerous as the Law of Duality requiresin order that every conceivable union of qualities may haveits representative ; no rod breaks the Law of Contradictionby representing at the same time term s that are necessarilyinconsistent ; and it has been pointed out that the peculiarcharacters of logical symbols expressed in the Laws of

Simplicity and Commutativeness are also observed in theaction of the keys and levers . The m achine is thus theembodiment of a true symbolic method or Calculus . The

representative rods must b e classified, selected , or rejectedby the reading of a proposition in a manner exactly answering to that in which a reasoning mind should treat its ideas .At every step in the progress of a problem ,

therefore, theabecedarium necessarily indicates the proper condition of amind exempt from m istake .

55 . I may add a few words to deprecate the notion thatI attribute much practical utility to this mechanical device .

I believe, indeed, that it may b e used with much advantagein the logical class-room ,

for whi ch purpose it is more con

v enient than the logical abacus which I have alreadyemployed in this manner. The logical machine may b e

come a powerful means of instruction at some future time

by presenting to a body of students a clear and visible

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17 2 [MECHANICAL LO GICAL INFEREN CE

NOTE to 7 .

It has b een pointed out to me b y Mr. W hite, and has also b een noticed inNature (10th March 1870, vol . 1. p. that in the year 1851, Mr. AlfredSmee, the Surgeon of the Bank of England , pub lished a w ork called

The Process of Thought adap ted to w ords and language, together w ith a d e

scr ip tio nof the Relationa l and DifferentialMachines (Longmans) , w hich alludesto the mechanical performance of thought.After perusing this w ork , w hich w as unknow n to me w hen w riting the

paper, i t cannot b e doub ted that Mr. Smee contemplated the representationb y mechanism of certa in mental processes . His i deas on this sub j ect arecharacterised b y much of the ingenuity w hich he is w ell know n to have d isplayed in other b ranches of science . But i t w ill b e found on examinati onthat his designs have no connex ion w ith mine. His represent the mental

states or operations of memory and judgment, w hereas my machine performslogical inference . So far as I can ascertain from the ob scure descriptions andimperfect d raw ings given b y Mr. Smee, his Relational Mach ine is a kind of

Mechanical Dictionary, so constructed that i f one w ord he proposed i ts relations to all other w ords w i ll b e mechanically exhi b ited . The D ifferentialMachine w as to b e employed for comparing ideas and ascertaining thei ragreement and d ifference. It might b e roughly l ikened to a patent lock, theopening of w hich proves the agreement of the tumb lers and the key .

It does not appear, again, that the machines w ere ever constructed ,although Mr. Smee made some attempts to reduce his designs to practice .

Indeed he almost allow s that the Relational Machine is a purely vi sionaryexistence w hen he mentions that i t w ould, i f constructed , occupy an area as

large as London.—l oth O ctob er 1870.

ON A G ENERAL SYSTEM

NUMERICALLY DEFINITE REASONING

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1 76 A GENERAL S YSTE IPI OF

2 . The late Professor Boole has also treated this subject ,but under a differen t name, and in a very different form .

H is chapter on the subject 1 is entitled On Statistical Conditions

,

’ by which he evidently means,

‘Numerical Conditions . ’ It contains a most remarkable and powerful attemptto erect a general method for ascertaining the higher andlower lim its of logical classes, to b e employed as a suhsidiary portion of his general calculus of probabilities . Apaper on the same subject had been previously written byhim

,and entitled ‘ O f Propositions Numerically Definite ,

’

but was only published after his death , by Professor De

Morgan in the Transactions of the Cambridge Phi losophi cal

Society (vol. xi. part ii. O f these writings of Professor Boole I must say, what I have elsewhere said of otherportions of his writings, that they appear in them selves perfeet and almost inim itable . At the same time I must addthat Boole ’s extraordinary power of analysis , and his perfectcommand of symbolic methods , usually led him to overestimate the part they should play in reasoning, and to

under-estimate the value of a simple and intuitive comprehension of the subject. The very principle which he fearlessly adopts, that uni ntelligible symbols may give intelligib leand even demonstrative results, will probably b e rejected byfuture mathematicians, as it has been lately rejected in thestrongest terms by Mr. Sandeman .

2

3 . As Mr. Boole ’ s logical views were the basis fromwhich I started in form ing the simple but general systemof logical form s explained in my Pure Logic in the year1 8 6 4

,and

,more simply still, in my Substi tution of Sim i lars,

published in 1 8 6 9 ,so the numerically definite system of

reasoning which is here described arises from a simpl ifica

tion of the previous methods of De Morgan and Boole .

4 . 111 the qualitative system of logic,which I have given

1 Law s of Thought, chap. xix. p . 295 . The use of the w ord statisti calas equivalent to nume rical is erroneous, although sanctioned b y so h igh an

authority as SirJ. Herschel, w ho applied i t to the numb ering of the stars .Statistical means w hat refers to the State or People.

‘1 Pcl icotctics, 1868, Preface, pp. ix. and x .

N UMERICALL Y DEFINITE REASONIN G 7

“in the works referred to, a term is taken to mean the qualityor group of qualities which belong to and mark out a classof objects . Thus

,the general term A stands for any group

of qualities belonging to a class of objects .Let the term

,when enclosed in brackets , acquire a quan

titative meaning, so as to denote the number of individualsor objects which possess those quali ties . Then

(A) = numb er of objects possessing qualities of A ,or say

,

for the sake of brevity, the number of A’s. If

,for instance,

A= character and quality of being a Member of Parliament,(A) = numb er of existing Members of Parliament = 6 5 8 .

5 . Every logical proposition or equation now gives ri seto a corresponding numerical equation . Sam eness of qual ities occasions sameness of numbers. Hence if

A=B

denotes the identity of the qualities of A and B ,w e may

conclude that

It is evident that exactly those objects, and those objectsonly

,which are comprehended under A must b e compre

hended under B . It follows that wherever w e can draw anequation of qual ities, w e can draw a sim ilar equation of

numbers . Thus,from

A B Cw e infer

A Cand sim ilarly from

(A) (B)

meaning the number of A’s and C ’s are equal to the numberof B ’s

,w e can infer

But, curiously enough , this does not apply to negative pro

positions and inequalities . For if

178 ON A GENERAL SYSTEM OF

A=Bm D

means that A is identical with B , which differs from D ,it

does not follow that

(D)

Two classes of objects may differ in quali ties , and yet theymay agree in number. This is a point which stronglyconfirms me in the opinion I have already expressed, thatall inference reall y depends upon equations, not differences ;

1 and I shall therefore employ throughout this paperonly equations which may b e almost indifferently used inthe qualitative or quantitative meaning.

6 . I shall employ, as in logic, a joint term , such as A B(or A B C) , to mean the class possessing all the qualities ofA and B (or of A and B and C) . To every positive termthere corresponds a negative term , denoted by the corresponding small italic letter. Thus the negative of A is a, of

B b,and so on. If, then , A means man

,a means simply

not man. Hence a b wi ll mean the combination of quali tiesof not being A and not being B .

7 . The sign -I is used to stand for the disjunctive con

junction or, but in an unexclusive sense . Thus

A=B ~I C

means that whatever has the qualities of A, must haveeither the qualities of B or of C but it may have the qua liti es of both. This unexclusive character of the terms andsigns of logic, which creates a profound difference betweenmy system and that of Professor Boole, prevents me fromconverting alternatives into numbers as they stand . It doesnot follow from the statemen t that A is either B or C, thatthe number of A ’s is equal to the number of B ’s added tothe number of C ’s ; for some objects , or possibly all

,have

been counted twice in this addition . Thus,i f w e say An

elector is ei ther an elector for a borough,or for a county, or

for a uni versi ty, it does not foll ow that the total number of

1 Substi tution of Sim i lars, pp. 16, 17 .

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180 ON A GENERAL S YSTE IPI OF

A is B or not B ;

in symbolsA=AB 1 Ab .

If any third term C enters into a problem ,it is equally

certain thatA AC °I° Ac

and combining these two developments , w e have

A =ABO ABc ~I' AbC °IAbe.

The same process of subdivision can b e carried on ad

infini tum with respect to any term s that occur ; and thisIndirect Method of Inference, which I have described inthe books mentioned, consists in determining the possibleexistence of the various alternatives thus produced . The

nature and procedure of this method will, as far as possible,b e rendered apparent in the mode of treating numericalquestions . It has also been partially explained to the

Society, in connection with the logical abacus ,in which the

working of the method is m echanically represented (Proceedings of the Manch. Li t. and Phi l. Soc

,3d April 1 8 6 6 ,

p . 1 6 1 ; see also Phi losophi cal Transactions, 1 8 70, p.

10. The data of any problem in numerically definitelogic will b e of two kinds

1 . The logical conditions governing the combinationsof certain qualities or classes of things, expressed inpropositions .

2 . The numbers of individuals in certain logical classesexisting under those conditions.

The qucesi tum of the problem will b e to determ ine thenumbers of individuals in certain other logical classes existing under the same logical conditions

,so far as such

numbers are rendered determ inable by the data. The

usefulness of the method will, indeed, often consist in showing whether or not the magnitude of a class is determ inedor not

,or in indicating what further hypotheses or data are

required. I t will appear, too, that where an exact result

N UMERICALL Y DEFINITE REASONIN G

is not determ inable w e may yet assign limits within whichan unknown quantity must l ie .

11 . Let us suppose,as an instance, that in a certain

statistical investigation,among 100 A’s there are found

4 5 B ’s and 5 3 C ’s ; that is to say, in 4 5 out of 100

cases where A occurs B also occurs, and in 5 3 cases Coccurs . Suppose it to b e also known that wherever B is

,

C also necessarily exists . The data then are as follow

(A) 100

Numerical equations (B) 4 5

(C) 5 3

Logical equation B BC .

Let it b e required to determ ine( 1) The number of cases where C exists without B .

(2 ) The number of cases where neither B nor C exists .The logical equation asserts that the class B is identical

with the class BC, which is the true mode of asserting thatall B’ s are C ’s . Two distinct results follow from this

,

namely —l st, that the number of the class BC is identicalwith the number of the class B ; and 2d , that there are no

such things as B ’s which are not C ’s .The logical equation is thus exactly equivalent to tw o

additional numerical equations , namely,

(B) =<Bc>(Be) 0

W e have now ful l means of solving the problem ; for, bythe law of duality,

(0)

(B) + (bC)

5 3 4 5 (bC) ,whence

182 ON A GENERAL SYSTE IPI OF

To obtain the second, the number of Abe’

s,w e have

(A) (ABC) (ABc) (A210 (Abe)

Il ence

(Abe) 4 7 .

I now proceed to exempli fy the use of the method byapplying it to examples drawn chiefly from previous writers .12: Professor De Morgan suggests the foll owing as an

argument which cannot b e put into any ordinary form of

the syllogism .

1

‘For every man in the house there is a person who isaged ; some of the men are not aged . It follows

,that some

persons in the house are not men.

’

This argument proceeds, as I conceive, not by any for mof syllogism

,but by a pair of simple equations. Taking

A man,

B aged person,

and putting w,w

’for unknown and indefinite numbers, the

first prem ise gives the equation—w (1)

meaning that the number of aged persons equals or exceedsthe number of men . The second statement may b e put inthis form

,

( 2 )

which implies that there is a certain indefinite number of

men who are not aged.

Develop A and B in ( 1) by the law of duality, and w e

have

(AB)4(Ab) (AR)+ (aB) w .

Subtract (AB ) from both sides, and insert for (Ab)in and w e have

(3 )1 Syllabus of a prop osed System of Logi c, 1860, p. 29 .

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184 ON A GENERAL SYSTEM OF

Developing the logical terms on each side , w e have

(ABC) (ABc) (ABC) (aBC) (ABC) (ABc) (aBC)

Subtracting the common terms, there remains

(ABC)

The meaning of this conclusion is , that there must b esome C ’s which are A’s

,am ounting to at least the sum of

the quantities w and w ’, the unknown excesses beyond half

the B’s which are A’s and C ’s. The number (c ) is whollyundeterm ined by the prem ises, but it cannot b e negative ;in proportion as its amount is greater, so is the number ofthe ABC

’

s. The conclusion , in short, is that w +w’ is the

lower limit of (ABC) .14 . The above problem is only one case of a more

general problem ,whi ch may b e stated as follows —G iven

the numb ers of three classes of objects , A , B, and C, todeterm ine what circumstances or conditions will necessitatethe existence of a class ABC.

This may b e solved very simply

—(A)(ABC) (Abe) ,

(ABC)

It is evident that the number of ABC’

s is indeterm inate,because there is no condition to determine (Abe) . Butreducing this to its minimum,

zero , w e learn that the lowerlimi t of (ABC ) is the excess of the sum of B ’s and C ’sover A’s. As no result can b e negative, w e al so learn thatif (Abc) = 0, then (A) cannot exceed

15 . Thi s method gives us a clear view of the conditionsof any logical argument. Take a syll ogism in Barbara

,

thusEvery A is B in symbols A=AB.

Every B is C B= BC .

Every A is C A=AG.

determ ine the number of all the classes of objects con

cerned ?There are altogether e ight conceivable combinations of

A, B, C , and their negatives, a,b,c,according to the laws

of thought ; but of these, four combinations are renderedpossible by the prem ises , so that w e have four quantities

assigned(ABc) 0

, (Abe) O,

(AbC) 0, (aBe) 0.

There remain,then

,four unknown quantities ; and unless

w e have these assigned directly or indirectly,w e do not

really know the relative numbers of the classes . But the

numbers of any four existing classes may,by a proper

arrangement of equations , b e made to yield the number ofany other existing class . Thus

,i f

(A) 9 3

(aBC) 5

w e may draw the following conclusions

(A) 9 3

9 8 .

16 . It is interesting to compare my mode of treatingnumerically definite propositions with the earlier mode of

Professor De Morgan. Taking X,Y

,and Z to b e the

three term s of the syllogism ,he adopts 1 the foll owing no

tatiorr

u= w hole number of individuals in the universe of

the problem .

a= numb er of X ’s .y= numb er of Y ’s .

z=numb er of Z’

s.

Making m denote any positive number, mXY means thatm or more X’s are Y’s . Sim ilarly uYZ means that u or

1 Syllabus, p. 27 . Mr. De Morgan denotes negat ive terms b y smallRoman letters, for w hich Ihave sub st ituted ital ic letters .

186 ON A GENERAL S YSTEM OF

more Y’s are Z’

s. Smaller letters denote the negatives ofthe larger ones, somewhat as in my system . Thus mmeans that m or more X ’s are not Y’s

,and so on .

From the two premises

mXY and nYZ ,

Mr. De Morgan draws the two distinct conclusions

(m+n y)XZ , and (m+n+u cc y z);cy.

Let us consider what results are given by my own notation.

The premises may b e represented by the equations

where m and n are the same quantities as in Mr. DeMorgan ’s system , and m’ and n

’ two unknown but positivequantities, indicating that the number of XY

’

s is m or more,

and the number of YZ ’

s is n or more.

The possible combinations of the three term s X ,Y

,Z

,

and their negatives are eight in number, namely

XYZ,

xYZ,

XYz, cc

,

XyZ , :cyZ ,

Xyz,

and these altogether constitute the universe, of which thenumber is n . The problem is at once seen to b e indeterm inate in reality ; for there are eight classes, of which the

number would have to b e determ ined , and there are onlysix known quantities , namely, u, ac, y, z, m ,

and n,by which

to determ ine them . Accordingly w e find that Mr. DeMorgan’s conclusions, though not absolutely erroneous, havelittle or no meaning . From the prem ises he infers that(m+n—y) or more X ’s are Z ’

s. Now

m+n y= Y

(XYZ ) (c z) .

Thus Mr. De Morgan represents the number of the whole

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188 ON A GENERAL SYSTEM OF

(xZ ) (:cY Z ) (XYz)

that is to say, the lower lim it of the class M is a part ofitself, xYZ ,

diminished by the number of ano ther class XYz.

Whi l e believing, however, that Mr. De Morgan’s mode

of treating the subject adm its of improvement, it is impossible that I should undervalue the extraordinary acuteness and originality of his writings on this and many otherparts of formal logic. Time is required to reveal the wealthof thought which he has embodied in his Formal Logic, andin his Logi cal Memoirs publi shed by the Cambridge Philosophical Society.

18 . In Mr. De Morgan’s third paper on the syllogisml

he puts the syllogism in the following form If the

fractions ( 1. and B of the Y’s b e severally A’s and B ’s , andif a+ ,

8 b e greater than unity, it follows that some A’s areB ’s . The logician demands a = 1 or B= 1 , or both ; hecan then infer. ’ These arguments are readily representedin my notation as follows

The prem ises are a

B (Y ) (BY ) .Hence

(a+B)

(d + l3) (Y ) (Y ) (ABY ) (abY ).

(ABY ) (a+ ,8 1)

From this w e learn that the number of A’s which are B’s ,because they are Y

’

s,is the fraction (a+B 1) of the Y’s

together with the undeterm ined number (abY ), which cannotb e negative . Hence if a + ,

8 > 1 , the second side has apositive value, and there must b e some A’s which are B ’s .If a.= 1 , then this number is B (Y) , or if ,

8 = 1, it is a (Y) ,since (abY) then = 0. If a = 1 and then obviously

1 Camb r idge Phi l . Trans . vol . x , part i , p. 8.

tions , occurs a problem concerning the coexistence of twophenomena, in which he asserts the general proposition‘ that, i f A occurs in a larger proportion of the cases whereB is than of the cases where B is not, then will B also occurin a larger proportion of the cases where A is than of the

cases where A is not .

’

This proposition is not proved by Mr. Mill,nor do I

remember see ing any proof of it ; and it is not, to my m ind ,self- evident . The following

,however

,is a proof of its truth

,

and is the shortest proof I have been able to find .

The condition of the problem may b e expressed in theinequality

(B) (b )

or reciprocally in the inequality

(B)<(b )

(AB)<

(Ab )

Subtracting unity from each side , and simplifying, w e have

(d B)<(ab)

(AB) (Ab)

1.

1.

b (Ab)t i

'

Multiplyi ng each si de of t ns i nequa i ty j(aB)

w e 0 am

(Ab)<(ab)

(AB)<

(aB)

Restoring unity to each side, and simplifying

(A)<(a)

(AB)or reciprocally

(AB)>

(A )>

(d )

1 System (y Logic, fifth ed . vol . i i, p. 54 .

190 ON A GENERAL S YSTEM OF

which expresses the result to b e proved, namely, that Boccurs in a larger proportion of the cases where A is thanof the cases where A is not .

20. The examples hi therto considered have been mostlyfree from logical conditions ; that is to say, the classes ofobjects have been supposed capable of combination or coinci dence in all conceivable ways . W e will briefly examinethe effects of certain simple logical conditions .

If there b e two term s A and B,and one condition, all

A’

s are B’

s,symbolically expressed in the equation

A=AB,

then there will b e three possible classes to b e determined ,namely,

and w e shall require three assigned quantities . If w e have(U) = w hole number of objects, with (A) and (B) , then

(A)(B) .

21. If with two term s, A and B, the logical conditionb e A=B,

there will remain two classes only, AB and ab,

and two assigned quantities only will b e required. The

same would happen with any of the conditions A = b,

a=B,or a= b .

22 . In any problem involving three term s or classes ofthings

,say A

,B

,and C , there arise eight conceivable classes ,

the numbers of which may have to b e dete rm ined. Variouslogical conditions , however, greatly reduce the numbers .Thus the two conditions

A B C

leave only two possible classes , ABC, and abc.

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192 ON A GENERAL S YSTEM OF

26 . In Chapter XVIII of his Law s of Thought, Boole hasgiven several examples of the application of his very com

plicated General Method of Probabilities . O f these examplesmy notation will give a vastly simpler solution, as I proceedto show.

Boole ’s third example is as follows (p. 2 7 9 )‘The probabil ity that a witness, A, speaks the truth is

p ,the probability that another witness , B,

speaks the truthis q, and the probability that they disagree in a statementis r. What is the probabili ty that i f they agree i n a statement, their statement is true ?

’

This is solved in the simplest possible manner. Let

a = prob . of A and B both speaking truth .

B= prob . of A b ut not Bry=prob . of not A but B8=prob. of neither A nor B

Then w e have the follow ing data

Prob . of A speaking truth a+ ,8 p .

B a+ fy q.

Prob. that they disagree B+ «y=r.

As it is certain that one or other of the alternatives musthappen, w e have the condition

These four equations are suffi cient to determine all the fourunknown quantities by ordinary algebra. Thus

p +q~ r

2

Now,the probability required is, that i f A and B agree in

N OZMERICALL Y DEFINITE REASONIN G

a statement their statement is true . By the principles of

aprobability this is and inserting the above values of

a and 8 w e havea p+q

—r

a+ 8 —r)’

which is the sam e as the result which Boole obtained in amuch m ore complicated nranner. This verifies the anticipations both of Boole him self (p . 2 8 1 ) and of Mr. Wilbraham ,

1

in his criticism on Boole ’s Method of Probab i li ties, that thereally determ inate problems solved in the book , as 2 and 3of Chapter XVIII

,m ight b e more shortly solved .

’ Booleremarks, indeed, that they do not fall directly within the

scope of known m ethods ; but I conceive that my logicalsymbols and method furnish all that is required .

27 . In a sim ilar nranner w e may solve the second of

Boole ’s examples referred to by Mr. Wilbraham ; this is asfollows

The probability that one or both of two events happenis p , that one or both of them fail is q. What is the prohability that only one of these happensUsing a

, B,«

y, 8 to denote the probabilities of the fourobvious conjunctions of events , as before, w e have the data,

a+B+v=n

The probability required is B+ ry, and

B+v= g—5= q 1+ a+5 + fy

q 1+p .

This is Mr. Boole ’s result , obtained by him in a much morecomplicated manner.

28 . This simple substitution of the probability of anevent for its logical symbol cannot b e val id , however, i f

1 Phi losoph ical Ill agazinc, 4th series, vol . vri , p. 465 vol . v i i i , p . 91.

O

194 ON A GENERAL SYSTEAI OF

there b e any connection between the events which rendersone more or less likely to happen when the other happens .The probabilities of A and B being p and q, the probabilityof AB is pq, under the supposition that B is just as likelyto happen when A happens as when A does not happenand sim ilarly that A is just as likely to happen when Bdoes as when B does not ; in short, that they are inde

p endent events. As a case where w e are not to assume

logical independence , w e may take the following examplefrom Boole’s work (p . 2 7 6)Example 1 . The probability that it thunders upon a

given day is p ,the probability that it both thunders and

hails is q ; but of the connection of the two phenomena ofthunder and hail nothing further is supposed to b e known .

Required the probabili ty that it hails on the proposed day.

’

Let A mean that it thundersB hail s ;

Then there are four possible events, AB ,Ab, aB,

ab.

The probabilities given are

Prob . of A p ,

AB=q

The probability required is that of B,which is evidently

Prob . of AB+prob . of aB.

Now the probability of AB is given , but the probabili ty of

aB is not given, and w e cannot assume it to b e (1 p ) x (prob.

of B ) , because w e are told that nothing is known of the

connection of the phenomena, which implies that they may

have some connection by causation, so that the non-occurrenceof A will alter the probability of the occurrence of B . The

prob. of aB is therefore unknown, except that it is the prob .

of a multipli ed by the unknown prob . that, if a occurs,B

occurs with it, as Boole points out . Hence the only possibleanswer is the same as Boole’s

Prob. of B=q+ ( 1—p )c,

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196 S YSTEM OF N UMERICALL Y DEFINITE REASONIN G

scientific demonstration, and of all sound thought in the

common affairs of life ; yet w e find the most opposite andcontradictory Opinions held by different logicians as to thenature of the reasoni ng process . Metaphysical speculationwill never remedy the present deplorable condition of the

science ; for it is metaphysical speculation which has mystified the subject, and rendered it the laughing - stock of

scientific men. Antiquarian research into the errors of

earlier logicians, in which some logicians still exclusivelyemploy themselves

,will only add to the perplexi ty and

obscurity. I hold that logic can only b e regenerated bythose who will render themselves acquainted with the exactmethods of research which lead to undoubted truths in themathematical and physical sciences . Logic, in short, mustb e di ssociated from metaphysics, with which it has nonecessary connection, and must become an exact science .

W e must therefore seek in every way to connect it withthe other exact sciences. In thi s paper I have attemptedto show that questions do exist in whi ch logical andnumerical methods coalesce and lend mutual aid.

PART 11

JOHN STUART MILL’

S PHILOSOPHY TESTED

PORTIONS OF AN EXAMINATION OF

JOHN STUART MIL L’S PHILO SO PHY

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2 00 jOHN STUART MILL’S PHILOSOPHY TESTED

‘I do not like the expression scientific shuffling and intellectual dishonesty which G . S. B. has used, for fear it shouldimply that Mill know ingly misled his readers. It is impossibleto doubt that Mill’s mind w as sensitively honourable,” and,whatever may b e his errors of judgm ent

,w e cannot call in

question the perfect good faith and loftiness of his intentions .On the other hand , it i s equally difficult to accept what Mr.Malleson says as to the scrupulous accurateness of Mill’s Essayson Religion. He was scrupulous, but the term accurateness,” ifit means “ logical accurateness,

”cannot b e applied to his w orks

by any one w ho has subj ected them to minute logical criticism .

’

I then pointed out that, in pp . 109 and 103 of his E ssayson Religi on,

Mill gives two different definitions or descriptions of religion . In the first he says that

the essence of religion is the strong and earnest direction of theemotions and desires towards an ideal Object, recognised as of

the highest excellence, and as rightfully paramount over allselfish objects of desire .

’

In the second statement he says

Religion, as distinguished from poetry, is the product of thecraving to know whether these imaginative conceptions haverealities answering to them in some other world than ours.

’

A week afterwards Mr. Malleson made an ingen i ous attemptto explain away or to pall iate the obvious discrepancy byreference to the context. I do not think that any contextcan remove the discrepancy ; in the one case the object ofdesire is an ideal object ; in the other case the craving ,

which I presume means a strong desire, is towards reali tiesin some other world ; and the difference between ideal andreal is too wide for any context to bridge over. Besides

,I

will ultimately give reasons for holding that Mill ’s textcannot b e safely interpreted by the context, because there isno certainty that in his writings the same line of thought issteadily maintained for two sentences in succession .

Mill’s E ssays on Religion have been the source of per

plexity to numberless readers . His greatest adm irers havebeen compelled to admit that in these essays even Mill

GE O/IIE TRICAL REASON IN G

seems now and then to play with a word,or unconsciously

to m ix up two views of the same subject . It has beenurged, indeed, by many apologists, including Miss HelenTaylor, their editor, that Mill wrote these essays at wideintervals of time, and was deprived, by death, of the Oppor

tunity of giving them his usual careful revision . Thisabsence of revision , however, applies mainly to the thirdessay, while the discrepan t definitions of religion werequoted from the second essay. Moreover, lapse of time willnot account for inconsistency occurring between pages 103and 109 of the same essay. The fact simply is that theseessays, owing to the exciting nature of their subjects

,have

received a far m ore searching and hostile criticism than anyof his other writings . Thus inherent defects in his intellectual character, which it w as a matter of great difficultyto expose in so large a work as the System of Logic, werereadily detected in these brief, candid, but most ill-judgedessays .

But,for my part, I will no longer consent to live

silen tly under the incubus of bad logic and bad philosophywhich Mill’s Works have laid upon us. On almost everysubject of social importance— religion , morals , politicalphilosophy

,political economy , metaphysics , logic— he has

expressed unlresitating Opinions, and his sayings are quotedby his adm irers as i f they were the oracles of a perfectlywise and logical m ind . Nobody questions , or at least oughtto question , the force of Mill ’s style, the persuasive power ofhis words

,the candour of his discussions, and the perfect

goodness of his motives . If to all his other great qualitieshad been happily added logical accurateness,his w ritings wouldindeed have been a source of light for generations to come.

But in one way or another Mill ’s intellect was wrecked .

The cause of injury may have been the ruthless trainingwhich his father imposed upon him in tender years ; it may

have been Mill ’s own lifelong attempt to reconcile a falseempirical philosophy with conflicting truth . But howeverit arose, Mill

’s m ind was essentially illogical .

2 0 2 jOHN STUART MIIL’

S PHILOSOPHY TESTED

Such, indeed , is the intricate sophistry of M ill’s principal

writings, that it is a work of much mental effort to traceout the course of his fallacies. ‘

For about twenty years pastI have been a more or less constant student of his booksduring the last fourteen years I have been compelled , bythe traditional requirements of the University of London

,to

make those works at least partially my text - books inlecturing. Some ten years of study passed before I beganto detect their fundamental unsoundness . During the lastten years the convi ction has gradually grow n upon my mindthat Mill’s authority is doing immense injury to the causeof philosophy and good intell ectual training in England .

Nothi ng surely can do so much intellectual harm as a bodyof thoroughly illogical writings, whi ch are forced uponstudents and teachers by the weight of Mill’s reputation ,and the hold which his school has obtained upon the universities. If

,as I am certain, Mill

’s philosophy is sophisticaland false, it must b e an indispensable service to truth toshow that it is so. This weighty task I at length feelbound to undertake .

The mode of criticism to b e adopted is one which hasnot been suffi ciently used by any of his previous critics .Many able writers have defended what they thought thetruth against Mill’s errors ; but they confined themselvesfor the most part to skirm ishi ng round the outworks of theAssociationist Philosophy,firing in everyhere and there a wellaimed shot. But the ir shots have sunk harmlessly into thesand of his foundations. In order to have a fair chance of

success,different tactics must b e adopted ; the assault must

b e made directly against the citadel of his logical reputation .

H is magazines must b e reached and exploded ; he must b ehoist, like the engineer, with his own petard . Thus onlycan the disconnected and worthless character of his philosophyb e exposed.

I undertake to show that there is hardly one of his moreimportant and peculiar doctrines which he has not himselfamply refuted. It will b e shown that in many cases it is

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2 04 joHN STUART MILL ’S PHILOSOPHY TESTED

makes a careful statement to the opposite effect, and thisstatement, subversive as it

.

is of his whole system of induction

,has appeared in all edi tions from the first to the last .

On such fundamental questions as the meaning of propositions

,the nature of a class, the theory of probability

, etc. , he

is in error where he is not in direct conflict w ith himself.But the indictment is long enough already ; there is notspace in this article to complete it in detail. To sum up ,there is nothing in logic which he has not touched

,and he

has touched nothing without confoundi ng it.To establish charges of this all-comprehensive character

w ill of course require a large body of proof. It will not b esufficient to take a few of Mill ’s statements and show thatthey are m istaken or self- inconsistent. Any writer may

now and then fall into oversights, and it would b e manifestly unfair to pick a few unfortunate passages out of awork of considerable extent, and then hold them up asspecimens of the whole . On the other hand , in order tooverthrow a phi losopher’s system ,

it is not requisite to provehis every statement false. If this were so

,one large

treatise would require ten large ones to refute it . What isnecessary is to select a certain number of his more prom inentand peculiar doctrines, and to show that, in their treatment,M al . In this article, I am , of course, lim ited inspace and can apply only one test, and the subject which Iselect for treatment is M ill ’s doctrines concerning geom etricalreasoning.

The science of geometry is specially suited to form atest of the empirical philosophy. Mill certainly regardedit as a crucial instance, and devoted a considerable part of

hi s System of Logi c to proving that geometry is a stri ctlyp hysi cal sci ence, and can b e learnt by direct observation andinduction. The particular nature of his doctrine, or ratherdoctrines, on this subject will b e gathered as w e proceed .

O f course, in thi s inquiry I must not abstain from asearching or even a tedious analysis , when it is requisitefor the due investigation of Mill’s logical method ; but it

GE Oil/E TRICAL RE ASONIIVG

will rarely b e found necessary to go beyond elementarymathematical knowledge which almost all readers of the

Contemp orary Revi ew will possess.As a first test of Mill ’s phi losophy I propose this simple

question of fact— Are there in the m aterial universe suchthings as perfectly straight lines W e shall find that Millreturns to this question a categorical negative answer.There exist no such things as perfectly straight li nes. How

then can geometry exist, if the things about w hich it isconversant do not exist ? Mill ’s ingenuity seldom fails h im .

Geometry, in his opinion, treats not of things as they are

in reality, but as w e suppose them to b e . Though straightlines do not exist, w e can experim ent in our m inds uponstraight lines , as if they did exist. I t is a peculiarity of

geometrical science, he thinks, thus to allow of mental cap eri

mentation . Moreover, these m ental experiments are just asgood as real experiments

,because w e know that the imaginary

lines exactly resemble real ones, and that w e can concludefrom them to real ones with quite as much certainty as w econclude from one real line to another. If such b e Mill ’sdoctrines, w e are brought into the following position

1 . Perfectly straight lines do not really exist.2 . W e experiment in our m inds upon imaginary straight

lines .3 . These i maginary straight lines exactly resemble the

real ones .4 . If these imaginary straight lines are not perfectly

straight,they w ill not enable us to prove the truths of

geometry.

5 . If they are perfectly straight, then the real ones ,which exactly resemble them ,

must b e perfectly straightergo, perfectly straight lines do exist.I t would not b e right to attribute such reasoning t o Mill

without fully substantiating the statements . I must therefore ask the reader to bear w ith me while I give somewhatfull extracts from the fifth chapter of the second book of

the System of Logic.

2 06 jOHN STUART MILL ’

S PHILOSOPHY TESTED

Previous to the publication of this system,

’ it had beengenerally thought that the certainty of geometrical andother mathematical truths was a property not exclusivelyconfined to these truths, but nevertheless existent. Mill

,

however, at the commencement of the chapter, altogethercalls in question this supposed certainty

,and describes it as

an i llusion, in order to sustain which it is necessary tosuppose that those truths relate to, and express the

properties of, purely imaginary objects . He proceeds 1

‘It is acknowledged that the conclusions of geometry are

deduced,partly at least, from the so—called Definitions

, and thatthose definitions are assumed to b e correct descriptions, as far asthey go, of the objects wi th which geometry is conversant. Noww e have pointed out that, from a definition as such, no proposition,unless it b e one concerning the meaning of a word

,can ever

follow ,and that what apparently follow s from a definition

,follow s

in reality from an implied assumption that there exists a realthing conformable thereto. This assumption

,in the case of the

definitions of geometry, is false :2 there exist no real things

exactly conformable to the definitions. There exist no pointswithout magnitude ; no lines without breadth

,nor perfectly

straight ; no circles with all their radii exactly equal, nor squaresw ith all their angles perfectly right. It will perhaps b e saidthat the assumption does not extend to the actual, but only tothe possible

,existence of such things . I answer that, according

to any test w e have of possibility, they are not even pos sible .

Their existence, so far as w e can form any judgment, wouldseem to b e inconsistent with the physical constitution of ourplanet at least, i f not of the universe.

’

About the meaning of this statement no doubt can arise .

In the clearest possible language Mill denies the existenceof perfectly straight lines, so far as any judgment can b e

formed , and this denial extends, not only to the actual, butthe possible, existence of such lines. He thinks that they

1 Book 11, chap. v, sec. 1, near the commencement of the second

paragraph .

2 The w ord fa lse occurs in the editions up to at least the fi fth edit ion.

In the latest or ninth edition Ifind the w ords, not strictly true, sub stitutedfor false.

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2 08 [OH/V STUART JVILL’

S PHILOSOPHY TESTED‘It remains to inquire, W hat is the ground of our belief in

axioms—what is the evidence on which they rest ? I answer,they are experimental truths ; generali sations from observation.

The proposition, Tw o straight lines cannot enclose a space—or

in o ther w ords,Tw o straight lines which have once met

,do not

meet again, but continue to diverge—is an induction from the

evidence of our senses . ’

Thi s opinion,as Mill goes on to remark , runs counter to

a scientific prejudice of long standing and great force, andthere is probably no proposition enunciated in the wholetreatise for which a m ore unfavourable receptionwas to b eexpected. I think that the scientific prejudice ” stillprevails, but I am perfectly willing to agree with Mill’sdemand that the opinion is entitled to b e j udged, not by itsnovelty, but by the strength of the arguments which are

adduced in support of it. These arguments are the subjectof our inquiry. M ill proceeds to point out that the properties of parallel or intersecting straight lines are apparentto us in almost every instant of our lives . ‘W e cannotlook at any two straight lines which intersect one another,without seeing that from that point they continue to divergemore and more.

’ 1 Even Whewell, the chief opponent of

Mill ’s views,all owed that observation suggests the properties

of geometrical figures ; but M ill is not satisfied with this ,and proceeds to controvert the arguments by which Whewelland others have attempted to show that experience cannotp rove the axiom .

The chief difficulty is thi s : before w e can assure ourselves that two straight lines do not enclose space, w e mustfollow them to infinity. Mill faces the difficulty withboldness and candour

What says the axiom 2 That tw o straight lines cannot enclosea space 3 that after having once intersected, if they are prolongedto infinity they do not meet

, but continue to diverge from one

another. How can this,in any single case

, b e proved by actualobservation W e may follow the lines to any di stance w e please

1 Same sect ion,near the b eginning of fourth paragraph .

GE O l l/E T1?! CAL RE ASONING

but w e cannot follow them to infinity for aught our senses can

te stify,they may, immediately beyond the farthest point to

w hich w e have traced them ,begin to approach, and at last meet

Unless,therefore

,w e had some other proof of the impossibility

than observation afford s us,w e should have no ground for

believing the ax iom at all .‘To these arguments, w hi ch I trust I cannot b e accused of

understating, a satisfactory answ er w ill, I conceive , b e found, ifw e adve r t to one of the characteristic properties of geometricalform s— their capacity of b eing painted in the imagination w itha distinctne ss equal to reality : in other w o rds

,the e xact re

semblance of our ideas of form to the sensations w hich sugge stthem . This

,in the first place

,enable s us to make (at least w ith

a little practice) m ental pictures of all possible combinations oflines and angles, w hich re semble the realities quite as w ell asany w hich w e could make on paper 5 and in the ne x t place

,

make those pictures just as fit subjects of geometrical experimentation as the realitie s them selves inasmuch as p ictures

,if

suffi ciently accurate,e xhibit of course all the prope rtie s w hich

w ould b e manifested by the realitie s at one given instant, andon simple inspection : and in geom et ry w e are concerned onlyw ith such propertie s, and not w ith that w hich pictures could not

exhibit,the mutual action of bod i es one upon another. The

foundations of geometry w ould therefo re b e laid in direct ex

perience , even i f the expe rim ents (w hich in this case consis tmerely in attentive contemplation) w ere practised solely uponw hat w e call our ideas

,that is

,upon the diagram s in our m ind s ,

and no t upon outw ard objects . For in all system s of experi

mentation w e take some objects to serve as representative s of

all w hich resemble them and in the pre sent case the condi tionsw hich qualify a real obj ect to b e the rep re sentative of i ts class

,

are com pletely fulfilled by an object e x isting only in our fancy.

W ithout denying, therefore , the po ssibility of satisfying ourselve sthat tw o straight lines cannot enclose a space , by m erely thinkingof straight line s w ithout actually looking at them ; I contend ,that w e do not beli eve this truth on the ground of the imaginaryintuition simply

,b ut because w e know that the imaginary line s

exactly resemb le real ones, and that w e may conclude from themto real one s w ith quite as much certainty as w e could concludefrom one real line to another. The conclusion

,therefore, is still

an induction from observation.

’ 1

1 Book 11,chap. v , sec . 5 . The passage occurs in the second and th i rd

paragraphs.

2 10 jOHN STUART MILL ’

S PHILOSOPHY TESTED

I have been obliged to give this long extract in full ,because, unless the reader has it all freshly before him ,

he

wil l scarcely accept my analysis. In the first place,what

are w e to make of Mill ’s previous statement that the axiom sare inducti ons from the evi dence of our senses ? M il l adm itsthat

,for aught our senses can testify, two straight lines,

although they have once met,may again approach and

intersect beyond the range of our vision .

‘Unless, therefore, w e had some other proof of the impossibility thanobservation affords us , w e should have no ground for

believing the axiom at all. ’ 1 Probably it would not occurto most readers to inquire whether such a statement isconsistent with that made two or three pages before, but onexam ination w e find it entirely inconsisten t. Before , theaxiom s were inductions from the evidence of our senses ; now,

w e must have ‘ some other proof of the impossibility thanobservati on affords us . ’

This further proof, it appears, consists in the attentivecontemplation of mental pictures of straight lines and othergeometrical figures . Such pictures

,if sufficiently accurate ,

exhibit, of course, all the properties of the real objects , andin the present case the conditions which qualify a realobject to b e the representative of its class are completelyfulfilled . Such pictures, Mill adm its , must b e sufi ci ently

accurate ; but what, in geometry,is sufii cient accuracy ?

The expression is,to my m ind, a new and puzzling one .

Imagine, since Mill allows us to do so, two parallel straightlines . What is the suf ficient accuracy with which w e mustframe our mental pictures of such lines

,in order that they

shall not m eet ? If one of the l ines, instead of being reallystraight

,is a portion of a circle having a radius of a hundred

m iles,then the divergence from perfect straightness within

the length of one foot would b e of an order of magnitudealtogether imperceptible to our senses . Can w e , then ,detect in the men tal picture that which cannot b e detectedin the sensible object ? This can hardly b e held by Mill ,

1 End of the second paragraph .

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jOHN STUART IIIILL’

S PHILOSOPHY TESTED

one foot apart,but which a hundred m iles away are one

foot p lus the thousandth of an inch apart,they are not

parallel . Will my senses enable me to perceive the magnitude of the thousandth part of an inch placed a hundredm iles off ?But w e have had enough of this trifling. Any one who

has the least knowledge of geometry must know that astraight line m eans a p erfectly straight l ine ; the slightestcurvature renders it not straight. Parallel s traight linesmean p erfectly parallel straight lines ; i f they b e in the leastdegree not parallel, they will of course meet sooner or later,provided that they b e in the same plane . Now Mill saidthat w e get an impression on our senses of a straight line ;it is through this impression that w e are enabled to formimages of straight l ines in the m ind . W e are told

,

l moreover, that the imaginary lines exactly resemble real ones, andthat it is long- continued observation which teaches us this .It follows m ost plainly

,then

,that the impressions on our

senses must have been derived from really straigh t lines .Mill ’s philosophy is essentially and directly empirical ; heholds that w e learn the principles of geometry by directocular perception

,either of lines in nature

,or their images

in the m ind . Now if our observations had been confined tolines w hich are not parallel, w e could by no possib ility haveperceived , directly and ocularly, the character of lines whichare parallel. It follows , that w e must have p ercei ved p erfectlyp arallel lines and p erfectly straight lines, although Mi ll previously told us that he considered the existence of such things

to be ‘

inconsi stent w i th the physical consti tution of our p lanet,

at least, if not of the uni verse.

’

Perhaps it may b e replied that Mill simply made amistake in saying that no really straigh t lines exist, and ,correcting this blunder of fact

,the logical con tradiction

vanishes . Certainly he gives no proper reason for his confident denial o f their existence . But merely to strike out apage o f Mill’s Logic will not v indicate his logical character.1 Same section, ab out th ir teen lines from the end of the th ird paragraph .

GE OIlIE TR/CAL REASON ING

How came he to put a statem ent there which is in absoluteconflict with the rest of his arguments ? No interval o f

time, no want of revision , can excuse this inconsistency , forthe passage occurs in the first edition of the System of Logi c

(vol . i, p . and reappears unchanged (except as regardsone word ) in the last and ninth edition . The curious substitution of the words ‘

not s trictly true ’

for the wordfalse ’ shows that Mill’s attention had been directed to theparagraph ; and a good many remarks m ight b e made uponthis little change of words

,were there not other matters

claim ing prior attention .

W e have seen that Mill considers our knowledge of

geometry to b e founded to a great extent on mental exp eri

mentati on . I am not aw are that any philosopher everpreviously asserted

,with the same dist inctness and con

sciousness of his m eaning , that the observation of our ow n

ideas m ight b e substituted for the observation of things .

Philosophers have frequently spoken of their ideas or

notions,b ut it was usually a mere form o f speech

,and their

ideas meant their direct know ledge of things . Certainlythis was the case with Locke

,w ho w as always talking

ab out ideas . Descartes , no doubt,held that whatever w e

can clearly perce ive is true ; bu t he probably m eant that itw ould b e logically possib le . I do not think that Descartesin his geometry ever got to mental cap cri

fmentation. But

however this may b e ,Mill

,of all m en

,ough t not to have

recommended such a questionable scientific process , if w e

may judge from his statem ents in other parts of the Systemof Logic. The fact is that Mill

,before com ing to the sub

j cet of Geometry,had denounced the handling of ideas instead

of thi ngs as one of the most fatal errors— indeed, as thecardinal error of log ical phi losophy. In the chapter uponthe Nature and Import of Propos itions ,1 he says

‘The notion that w hat is of primary impo rtance to the

logician in a p ropo sition, is the relation betw een the tw o ideas

1 Book i,chap. v

,sec . 1

,fi fth paragraph .

2 14 jOHIV STUART [WILL ’

S PHILOSOPHY TESTED

corresponding to the subject and predicate (instead of the relation betw een the tw o phenomena which they respectively express) ,seem s to m e one of the most fatal errors ever introduced intothe philo sophy of Logic and the principal cause w hy the theoryof the science has made such inconsiderable progress during thelast tw o centuries . The treatise s on Logic, and on the branchesof Mental Philo sophy connected with Logic, which have beenproduced since the intrusion of this cardinal e rror

,though some

times w ritten by men oi extraordinary abilities and attainments ,almo st alw ays tacitly imply a theory that the investigation of

truth consists in contemplating and handling our ideas, or con

ceptions of things, instead of the things themselves : a doctrinetantamount to the assertion, that the only mode of acquiringknow ledge of nature is to study it at second hand

,as represented

in our ow n minds . ’

Mill here denounces the cardinal error of investigatingnature at second hand, as represented in our ow n m inds .Yet his words exactly describe that process of m ental ex

perimentat ion which he has unquestionably advocated ingeometry, the most perfect and certain of the sciences .It may b e urged, indeed, with some show of reason, that

the method which m ight b e erroneous in one science m ightb e correct in another. The mathematical sciences are calledthe exact sciences, and they may b e of peculiar character.But

,in the first place, M ill

’s denunciation of the handlingof ideas is not lim ited by any exceptions ; it is applied inthe most general way, and arises upon the general questionof the Import of Propositions . It is, therefore , in distinctconflict with Mill’s subsequent advocacy of mental experi

mentation .

In the second place, Mill is entirely precluded fromclaim ing the mathematical sciences as peculiar in theirmethod , because one of the principal points of his philosophy is to show that they are not peculiar. It i s theoutcome of his philosophy to show that they are founded ona directly empirical basis

,like the rest of the sciences . He

speaks 1 of geometry as a ‘ strictly physical science,

’

and

1 Book i i i, chap . xx i v

,sec. 7 , ab out the tenth line.

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2 16 jOHIV STUART IVILL’S PHILOSOPHY TESTED

planets) must b e the name of our ideas of straight lines .He prom ised expressly that nam es ‘ in this work

,

’ that is,

in the System of Logic, should alw ays b e spoken of as thenames of things them selves. It must have been by oversight, then, that he forgot this emphatic promise in a laterchapter of the same volume. W e may excuse an accidentallapsus memori es

,but a philosopher is unfortunate who makes

many such lapses in regard to the fundamental principles ofhis system .

But let. us overlook Mill’s breach of prom ise , and assumethat w e may properly employ ideal experiments . W e are

told 1 that, though it is impossible ocularly to follow lines‘ in their prolongation to infinity

,

’

yet this is not necessary .

‘Without doing so w e may know that i f they ever do meet,or i f, after diverging from one another

,they begin again to

approach, this must take place not at an infinite , but at afinite distance. Supposing, therefore, such to b e the case

,

w e can transport ourselves thither in imagination,and can

frame a mental image of the appearance which one or bothof the lines must present at that point, which w e may relyon as being precisely sim ilar to the reality .

’ Now, w e are

also told 2 that ‘ neither in nature nor in the human m inddo there exist any objects exactly corresponding to the

definitions of geometry .

’ Not only are there no perfectlystraight lines , but there are not even lines without breadth

,

Mill says,

3 W e cannot concei ve a line without breadth ; w e

can form no mental picture of such a line : all the lineswhich w e have in our minds are lines possessing breadth .

’

Now I want to know what Mill means by the prolongationof a line which has thickness and is not straight. Let usexam ine this question with some degree of care .

In the firs t place,i f the line, instead of being length

without breadth , according to Euclid’s definition,

has thickness

,it must b e a wire i f it had had two dim ensions without

1 Book 11, chap. v , sec. 5, b eginning of fourt h parag raph .

2 Book i i , chap. v, sec. 1, b eginning of th i rd paragraph .

3 Same section, second paragraph , eleven l ines from end .

GE O/IIE TRICAL REASON IN G

the third , it w ould surely have been described as a surface ,

not a line. But then I w ant to know how w e are to understand the p rolongati on of a w i re. Is the course of the wireto b e defined by its surface or by its central line

,or by a

l ine running deviously within it If w e take the last, then ,the line be ing devious and uncertain

,its prolongation must

b e undefined . If w e take a certain central line,then e ither

this line has breadth or it has no breadth ; i f the former,all

our difficulties recur ; i f the latter W ell , Mill deniedthat w e could form the idea of such a line . The same

difficulty applies to any line or lines upon the surface , or tothe surface itself regarded as a curved surface without thickness . Unless

,then , w e can get rid of thickness in some

way or other, I feel unable to understand what the prolongation of a line m eans .

But let us overlook this diffi culty , and assume that w e

have got Euclid’s line— length without breadth . In fact

,

Mill tells us 1 that w e can reason about a line as i f it had nobread th

,

’ because w e have ‘

the power,when a perception is

presen t to our senses,or a conception to our intellects , of

attending to a part only of that perception or conception,

instead o f the whole .

’ I believe that this sentence suppliesa good instance of a non sequi tur,

being in conflict with thesentence which immediately follows . Mill holds that w e

learn the properties of lines by experimentation on ideas inthe m ind ; these ideas must surely b e conceived , and theycannot b e conce ived without thickness . Unless , then , thereasoni ng about a line is quite a differen t process fromemperimenting ,

I fail to make the sentences hold together atall . If

,on the other hand , w e can reason about lines with

out breadth , b ut can only experiment on thick lines , wouldi t not b e much better to stick to the reasoning process ,whatever it may b e , and drop the m ental experimentationaltogether

But let that pass . Suppose that , in one w ay or other,w e manage to attend only to the direction of the line , not its

1 Sam e paragraph , seventeen lines from end .

2 18 jOHN STUART MILL’S PHILOSOPHY TESTED

thickness . Now,the line cannot b e a straight line, because

Mill te lls us that neither in nature nor in the human m ind isthere anything answering to the definitions of geometry

,and

the second definition of Euclid defines a straight line . If

not straight,what is it ? Crooked

,I presume. What

,then

,

are w e to understand by the prolongation of a crooked line ?If the crooked line is made up of various portions of linetending in differen t directions , i f, in short, i t b e a zigzagline, of course w e cannot prolong it in all those directionsat once, nor even in any tw o directions

,however sligh tly

divergent. Let us adopt , then , the last bit of line as ourguide . If this bit b e perfectly straight, there is no difficultyin saying what the prolongation will b e . But then Milldenied that there could b e such a bit of straight line ; forthe length of the bit could scarcely have any relevance in aquestion of this sort. If not a straight line , it may yet b e

a piece of an ellipse , parabola, cycloid , or some other mathemat ical curve . But i f a piece of an ellipse

,do w e mean a

piece of a perfect ellipse ? In that case one of the defini

tions of geometry has something answering to it in the m indat least ; and i f w e conceive the more complicated mathematical curves

,surely w e can conceive the straight line, the

most simple of curves . But i f these pieces of line are notperfect curves, that is , do not fulfi l definite mathematicallaws

,what are they ? If they also are crooked

,and made

up of fragments of other lines and curves , all the difficultycom es over again . Apparently, then, w e are driven to theconception of a line

,no portion of which

,however small ,

follows any definite mathematical law whatever. For i f anyportion has a definite law , the las t portion may as well b esupposed to b e that portion ; then w e can prolong it inaccordance with that law

,and the result is a perfect m athe

matical line or curve , of which Mill denied the existencee ither in nature or in the human m ind . W e are driven,

then , to the final result that no portion of any line followsany mathematical law whatever. Each line must follow i tsown sweet will . What then are w e to understand by the

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2 2 0 jOHN STUART IIIILL’

S PHILOSOPHY TESTED

this ratio to the extent of 7 07 places of decimals ,1 and it isa question of mere labour of computation to carry it to anygreater length . It is obvious that the result does not andcannot depend on m easurement at all

,or else it would b e

affected by the inaccuracy of that measurement. It isobviously impossible from inexact physical data to arrive atan exact result, and the computations of Mr. Shanks andother calculators are founded on a p ri ori considerations , infact upon considerations which have no necessary connectionwith geometry at all. The ratio in question occurs as anatural constant in various branches of mathematics , as forinstance in the theory of error, which has no necessary con

nection with the geometry of t he circle .

It is amusing to find,too

,that Mill himself happens to

Speak of this same ratio,in his E scamination of Hami lton.

2

and he there says , ‘ This attribute was discovered , and isnow known , as a result of reasoning.

’ He says nothingabout measurement and comparison . What has become, inthis critical case

,of the empirical character of geometry

which it was his great object to establish ? A few linesfurther on (p. 3 7 2 ) he says that mathematicians could nothave found the ratio in question ‘ until the long train of

diffi cult reasoning which culm inated in the discovery wascomplete .

’ Now, w e are certainly dealing with a theorem

of geometry, and if this could have been solved by comparison and measurement, why did mathematicians resort to thislong train of d ifi cult reasoning ?I need hardly weary the reader by pointing out that the

same is true, not merely of many other geome trical theorems ,but of all . That the square on the hypothenuse of a rightangled triangle is

'

exactly equal to the sum of the squareson the other sides ;

‘ that the area of a cycloid is exactlyequal to three times the area of the describing c ircle ; thatthe surface of a sphere is exactly four times that of anyof its great circles ; even that the three angles of a plane

1 Proceed ings of the Royal Society (1872 vol . xxi , p . 319 .

t 2 Second ed ition, p. 37 1.

GE OIlIE TRICAL REASONIN G

triangle are exactly equal to two right-angles these andthousands of other certain mathematical theorem s cannotpossibly b e proved by measurement and comparison . The

absolute certainty and accuracy of these truths can only b eproved deductively. Reasoning can carry a result to infinity

,

that is to say, w e can see that there is no possible lim ittheoretically to the endless repetition of a process . Thus itis found in the 1 l 7 th proposition of Euclid ’s tenth book

,

that the s ide and diagonal of a square are incommensurable .

N0 quan tity, however small, can b e a sub -multiple of both,

or, in other words , their greatest common measure is aninfinitely small quantity . I t has also been shown that thecircum ference and diameter of a circle are incomm ensurable .

Such results cannot possibly b e due to measurement .It may b e well to remark that the expression ‘ a false

empirical philosophy ,’ which has been used in this article,

is not intended to imply that all empirical philosophy isfalse . My m eaning is that the phase of empirical philosophyupheld by Mill and the well-known m embers of his school ,is false . Experience

,no doubt

,supplies the materials of

our knowledge, b ut in a far different manner from thatexpounded by Mill.Here this inquiry must for the presen t b e interrupted .

It has been shown that Mill undertakes to explain the originof our geom etrical know ledge on the ground of his se - calledEmp i ri cal Phi losophy, but that at every step he in volves himself in inextricable difficulties and self- contradict ions . Itmay b e urged ,

indeed,that the groundwork of geometry is a

very slippery subject, and form s a severe test for any k indof philosophy . This may b e quite true, b ut it is no excusefor the way in which Mill has treated the subj ect ; it is onething to fail in explaining a difficult matter : it is anotherthing to rush into subjects and offer reckless opinions andarguments

,which on m inute analysis are found to have no co

herence . This is what Mill has done , and he has done it , notin the case of geometry alone , b ut in almost every other pointof logical and metaphysical philosophy treated in his works .

ON RESE M BLANCE

IN the previous article on John Stuart Mill ’s Philosophy, Im ade the strange assertion that Mi ll ’s m ind w as essentially

i llogical. To those who have long looked upon him as theirguide

,philosopher

,and friend

,such a statem ent must of

course have seemed incredible and absurd , and it w i ll requirea great body of evidence to convince them that there is anyground for the assertion . My first test of his logicalnesswas derived from his writings on geometrical science . Ishowed by carefully authenticated ex tracts, that Mill hadput forth views which necessarily imply the existence of

perfectly straight lines ; yet he had at the same time dist inctly denied the existence of such lines . It was pointedout that he emphatically prom ised to use names alw ays asthe names of things, not as the names of our ideas of

things ; yet, as straight lines in his opinion do not exist,the name straight line is e ither the name of ‘ just nothingat all

,

’ as James Mill would have said,or else it is the

name of our ideas of what they are. It is by experimentingon these ideal straight lines in the m ind that w e learn theaxi oms and theorems of geometry according to Mill nevertheless Mill had denounced, as the card inal error of phi losophy,

the handling ideas instead of things,and had

,indeed, in the

earlier editions of the System of Logi c, asserted that not asingle truth ever had been arrived at by this method, excepttruths of psychology. Mill asserted that w e m ight ex

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2 2 4 jOHN STUART MILL ’

S PHILOSOPHY TESTED

for treatment in this second article a much broader andsimpler question—one which lies at the basis of the

philosophy of logic and knowledge . W e will endeavour togain a firm comprehension of Mill’s doctrine concerning thenature and imp ortance of the relation of Resemblance. Thisquestion touches the very nature of knowledge itself. Now

,

critics who are considered to b e quite competent to judge,have declared that Mill’s logic is peculiarly distinguished bythe thorough analysis which it presents of the cognitive andreasoning processes . i t icted imself to theem t form nd m ethods of ar ment b ut has pushed his1w m lg k ,

boldly i nto the psychology andhiloso h of reasoning. In the System of Logi c, then ,

w e shall find it clearly decided whether resemblance is,or is

not,the fundamental relation with which reasoning is con

cerned . It was the doctrine of Locke, as fully expoundedin the fourth book of his great essay, that knowledge is theperception of the agreement or disagreement of our ideas .

‘Know ledge then,’ says Locke , ‘ seems to me to b e nothing

b ut the perception of the connection and agreement, or disagreement and repugnancy, of any ofour ideas . In this alone it consists.

W here this perception is , there is know ledge ; and w here it isnot

,there , though w e may fancy, guess, or believe, yet w e alw ays

come short of know ledge .

’

Many other philosophers have likewise held that a certainagreement between things, variously described as resemblance,similarity

,identity, sameness, equality

,etc.

,really con

stituted the whole of reasoned know ledge, as distinguishedfrom the mere knowledge of sense. Condillac adopted thisview and stated it with adm irable breadth and brevity,saying , ‘L

’

é v idence de raison consiste uniquement dansl’

identité .

’

Mill has not failed to discuss this matter, and his Opinionon the subject is m ost expressly and clearly stated in the

chapter upon the Import of Propositions .1 He analyses thestate of m ind called Belief, and shows that it involves one

1 Book i , chap. v .

RE SE IllBLAIVCE

or more of five matters of fact, namely,Existence, Co

existence, Sequence, Causation , Resemblance . One or otherof these is asserted (or denied) in every proposition which isnot merely verbal . No doubt relations of the kinds mentionedform a large part of the matter of knowledge, and they mustb e expressed in propositions in som e way or other. I believethat they are expressed in the term s of propositions

,w hile

the copula always signifies agreement, or,‘as Condillac would

have said, i denti ty of the terms . But w e need not attemptto settle a question of this difficulty. W e are only con

cerned now with the position in his system which Millassigns to Resemblance . This comes last in the list, and itis with some expression of doubt that Mill assigns it a placeat all . He says 1

‘Besides propositions w hich assert a sequence or coexistencebetw een tw o phenomena

,there are therefore also propositions

w hich assert resemblance betw een them ; as,This colour is like

that colour —The heat of to-day is equal to the heat of yesterday.

It is true that such an assertion m ight w ith some plausibility b ebrought w ithin the description of an af firmation of sequence, byconsidering it as an assertion that the simultaneous contemplationof the tw o colours i s follow ed by a specific feeling termed the feelingof resemblance . But there would b e nothing gained by encum

bering ourselves,e specially in this place, w ith a generalisation

w hich may b e looked upon as strained . Logic does not undertaketo analyse m ental facts into their ultimate elem ents . Resemblancebetw een tw o phenomena is more intelligible in itself than any

explanation could make it,and under any classification must

remain specifically distinct from the ordinary cases of sequence andcoexistence .

’

It would seem ,then , that Mill had, to say the least , con

templated the possibility of resolving Resemblance intosomething simpler, namely, into a special case of sequenceand coexistence ; but he abstains , not apparently becauseit would b e plainly impossible, b ut because logic does not

undertake ultimate analysis . It would encumber us with astrained generalisation ,

’ whatever that may b e . He there1 Book i , chap. v , sec . 6 .

Q

2 2 6 jOHIV STUART IlIILL’S PHILOSOPHY TESTED

fore accords it provisionally a place among the matters of

fact which logic treats .Postponing further consideration of this passage, w e turn

to a later book of the System of Logi c, in which M ill expresses pretty clearly his Opinion

,that Resemblance is a

minor kind of relation to b e treated las t in the system of

Logic,as being of comparatively small importance. In the

chapter headed ‘O f the remaining Laws of Nature,’ 1

w e

find Mill distinctly stating that 2 ‘the propositions w hich

affi rm O rder in Time,in e ither of its two m odes , CO

existence and Succession, have formed, thus far, the subjectof the present Book. And w e have now concluded the

exposition,so far as it falls w ithin the lim its assigned to

thi s work,of the nature of the evidence on which these pre

positions rest,and the processes of investigation by which

they are ascertained and proved . There remain threeclasses of facts Existence, O rder in Place, and Resemblance,in regard to which the same questions are now to b eresolved .

’

From the above passage w e should gather that Resemblance has not been the subj ect treated

'

in the precedingchapters of the third book

,or certainly not the chief subject.

O f the remaining three classes of facts, Existence is dism issed very briefly. So far as relates to simple existence ,Mill thinks 3 that the inductive logic has no knots to untie,and he proceeds to the remaining two of the great classesinto which facts have been divided . His Opinion aboutResemblance is clearly stated in the second section of the

same chapter, as follows‘Resemblance and its Opposite, except in the case in which

they assume the names of Equality and Inequality, are seldomregarded as subjects of science ; they are supposed to b e perceived by simple apprehension ; by m erely applying our sensesor directing our attention to the tw o object s at once, or inimmediate succession.

’

1 Book i i i, chap. xx iv. 2 Fi rst section, near the b eginning .

3 Same section.

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2 2 8 jOH/V STUART MILL ’S PHILOSOPHY TESTED

of Logi c, then, w e have not been concerned with Resemblance. The subjects discussed have been contained inpropositions which affirm O rder in Time, in either of itsm odes

,Coexistence and Succession . Resemblance is another

matter Of fact, which has been postponed to the twenty- fourthchapter Of the third book , and there dismissed in one shortsection , as being . seldom regarded as a subject of sci ence.

Under these circumstances w e should hardly expect to

find that Mill ’s so- called Experimental Methods are whollyconcerned with resemblance. Certainly these celebratedm ethods are the subject of science ; they are

,according to

Mill , the great methods of scientific discovery and inductiveproof ; they form the main topic of the third book of the

Logic, indeed , they form the central pillars of the wholeSystem of Logi c. It is a little puzzling

,then

,to find that

the nam es of these methods seem to refer to Resemblance,or to something which much resembles resemblance . The

first is called the Method of Agreement ; the second is theMethod of D ifference ; the third is the Joint Method Of

Agreement and Difference ; and the remaining two methodsare confessedly developments Of these principal methods .Now

,does Agreement mean Resemblance or not ? If it

does, then the whole of the third book may b e said to treatof a relation w hich Mill has professedly postponed to thesecond section Of the twenty-fourth chapter.Let us see what these methods involve. The canon of

the first m ethod is stated in the following words,1 whichmany an anxious candidate for academic honours has com

m itted to memory

If tw o or more instances of the phenomenon under investigation have only one circum stance in common, the circumstancein w hich alone all the instances agree is the cause (or effect) ofthe given phenomenon.

’

Now,when two or more instances of the phenomenon

under investigation agree, do they, or do they not, resemble1 Book i i i , chap. vi i i , sec. 1, near the end .

not? ifit,

RE SE IPIBLAN CE

ch other ? Is agreement the same relation as resemblance,or is it something different ? If

,indeed, it b e a separate

kind of relation , it must b e matter of regret that Mill didnot describe this relation of agreemen t when treating of the

Import Of Propositions . ’ Surely the prepositions in w hichw e record our Observations of the phenomenon underinvestigation ’

must affirm agreem ent or difference, and asthe experimental m ethods are the all- important instrum entsof science, these propositions must have corresponding importance . Perhaps

,however

,w e shall derive some light

from the contex t ; reading on a few lines in the descriptionof the Method of Difference ,1 w e find Mill saying that

‘ In the Method of Agreem ent w e endeavoured to Obtaininstance s w hich agreed in the given circum stance b ut differed inevery other : in the present m ethod (i . .e the Method of Difl'

er

ence) w e require, on the contrary,tw o instances resembling one

another in every other respect,but differing in the presence or

absence of the phenomenon w e w ish to study .

’

It would really seem ,then ,

as i f the great ExperimentalMethod depends upon our discovering two instances resembling one another. Here resemblance is specifi ed by name .

W e seem to learn clearly that Agreem ent must b e the same

thing as l esemb lance ; if so ,Difference must b e its opposite.

Proceeding accordingly to consider the Method of Difference w e find its requirem ents described in these w ords zi

’

‘

The tw o instances which are to b e compared with one

another must b e exactly sim ilar, in all circum stances exceptthe one which w e are attempting to investigate .

’

This exact sim ilarity is not actual identity , of course,because the instances are tw o, not one. Is it then resemblance ? If so

,w e again find the principal subject of Mill ’s

logic to b e that which he relegated to section 2 of chapterxxiv . If w e proceed with our reading of Mill ’s chapter onthe ‘ Four Experimental Methods ,

’

w e still find sentenceafter sentence dealing with this relation of resemblance ,

1 Same chapter, second sect ion.

2 Same chapter, th i rd sect ion, th ird paragraph , fourth line.

530 joHN STUART MILL’

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sometimes under the very same name, sometimes under thenames of sim ilarity, agreem ent

,likeness

,etc. As to its

apparent opposite, difi’

erenee,it seem s to b e the theme of

the whole chapter. The Method of D ifference is thatwonderful method which can prove the most general law on

the ground of two instances ! But of this peculiarity of theMethod of Difference I shall treat on another occasion .

Perhaps,however, after all I may b e m isrepresenting

Mil l’s statements. It crosses my m ind that by Resemblancehe may mean something different from exact simi lari ty.

The Methods of Agreement and D ifference may requi re thatcomplete likeness which w e should call i denti ty of quali ty.

It is only fair to inquire then , whether he uses the wordResemblance in a broad or a narrow sense. On this pointMil l leaves us in no doubt ; for he says distinctly,1 ‘Thisresemblance may exist in all conceivable gradations , fromperfect undistinguishableness to something extremely slight . ’

Again on the next page, whi le distinguishing careful lybetween such different things as num erical identity andundisting uishable resemblance , he clearly countenances thewide use of the word resemblance, saying,2 ‘Resemblance ,when it exists in the highest degree of all, amounting toundistinguishableness, is Often called identity.

’ It seem sthen, that all grades Of likeness or sim ilarity, from um

distinguishable identity down to something extremely slight,

are all properly comprehended under resemblance and it isdifli cult to come to any other conclusion than that the agreement and sim ilarity and difference treated throughout theExperimental Methods are all cases of that m inor relation ,seldom considered the subject of science, which was postponedby Mi ll to the second section of the twenty-fourth chapter.But the fact is that I have only been playing with thi s

matter. I ought to have quoted at once a passage whichwas in my m ind all the time—one from the chapter onthe Functions and Value of the Syllogism . Mill sums

1 Book i, chap. i i i , sec. 11, paragraph 4.

‘1 Same section,fi fth paragraph , th i rd l ine .

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2 32 jOHN STUART IlIILL’S PHILOSOPHY TESTED

w ith the faintest analogy,w e conclude becauseA resembles B in oneor more properties

,that it does so in a certain other property.

’

It seems, then, that the universal type of the reasoningprocess wholly turns upon the pivot of resemblance. The

stone which was despised and slightingly treated in a briefsection of the twenty-fourth chapter, has become the cornerstone Of M ill ’s logical edifice . It would almost seem as i fMill were one of those persons who are said to think independently with the two halves of their brain. On the one sideof the great longitudinal fissure must b e held the doctri ne thatresemblance is seldom a subject of science ; on the otherside, M ill must have thought out the important place whichresemblance holds as the universal type of the reasoning andinductive processes . Double-mindedness, the Law of O b livi

scence, or some Deus ex machina, must b e called in ; for it

is absurd to contemplate the possibility of reconciling Mill ’sstatement Of the uni versal typ e of all reasoning with hisremarks upon Locke ’s doctrine. Locke

,he says in the

passage al ready quoted, unduly extended the imp ortance ofresemblance, w hen he made all reasoning a case of i t, andLocke’s definition of knowledge and of reasoning required tobe limi ted to our know ledge of and reasoning about resemblances.

Y et,according to Mill him self, the universal typ e of ALL

reasoning turns wholly on resemblance. Under such circumstances, it is impossible to discuss seriously the value of

Mill ’s analysis of knowledge . Which part of the analysisare w e to discuss ? That in which resemblance is treatedas the basis of all reasoning, or that in which it belongs tothe remaining ’ and m inor matters of fact

,

’ which had notbeen treated in the books of induction, and which thereforeremained to b e disposed of

W e have not yet done with this question of resemblanceit is the fundamental question as regards the theory of

knowledge and reasoning,and

,even at the risk of being very

tedious, I must show that in the deep of Mil l’s inconsistencythere is still a lower deep. I have to point out that some

RE SE JIBLAIVCE

of his Opinions concerning the import of prepositions mayb e thus formulated

1 . The names Of attributes are names for the resemblancesO f our sensations .

2 . Certain propositions affirm the possession of properties ,or attributes

,or common peculiarities .

3 . Such prepositions do not, properly speaking , assertresemblance at all .Proceeding in the first place to prove that Mill has made

statements of the meaning attributed to him, w e find the

matter of the first in a note 1 written by Mill in answer toMr. Herbert Spencer, w ho had charged Mill with confounding exact likeness and literal identity. With the truth of

this charge w e w ill not concern ourselves now ; w e haveonly to notice the following distinct statement : ‘

\Vhat,

then , is the common something which gives a m eaning tothe general name ? Mr. Spencer can only say

,it is the

similari ty Of the feelings ; and I rejoin, the attribute is prec isely that sim ilarity. The names of attributes are in theirultimate analyses names for the resemblances of our sensations (or other feelings) . Every genera l name , whetherabstract or concrete, denotes or connotes one or more Of

those resemblances . ’ Mill ’s meaning evidently is that wheny ou apply a general nam e to a thing ,

as for instance incalling snow w hi te

, you m ean that there is a resemblancebetween snow and other things in respect of their whiteness .The general name w hi te connotes this resemblance ; the

abstract name w hi teness denotes i t .

Let us now consider a passage in the chapter on the

Import O f Propositions,which must b e quoted at som e

length .

2

‘It is sometime s said,that all prepositions w hatever, of w hich

the predicate is a general name,do, in point Of fact, affirm or

1 Book 11, chap. 11,sec . 3 , near the b eginning of the thi rd para

g raph of the footnote . Th is note does not occur in som e of the earlyed itions .

2 Book i, chap. v

,sec. 6 , second paragraph .

2 34 l HIV STUART IlIILL’S PHILOSOPHY TESTED

deny resemblance . Al l such propo sitions affirm that a thingbelongs to a class but things being classed together according totheir resemblance, everything is Of course classed with the thingsw hich it is supposed to resemble most ; and thence, it may b e

said,w hen w e affi rm that gold is a metal

,or that Socrates is a man,

the affirmation intended is,that gold re sembles other metals , and

Socrates other men,more nearly than they resemble the Objects

contained in any other of the classes coo rdinate with these .

’

O f this doctrine Mill goes on to speak in the followingcurious remarks,1 to which I particul arly invite the reader’sattention

‘There is some slight degree of foundation for this remark,but no more than a slight degree. The arrangement Of thingsinto classes

,such as the class metal

, or the class man,is grounded

indeed on a resemblance among the things which are placed inthe same class, b ut not on amere general resemblance the resemblance it is grounded on consists in the possession by all thosethings, Of certain common peculiarities and those peculiarities it i sw hich the terms connote, and which the propositions consequentlyassert ; not the resemblance . For though when I say, Gold isa metal, I say by implication that if there b e any other metalsit must resemble them, yet if there were no other metals Im ight still assert the proposition with the same meaning as at

present, namely, that gold has the various properties implied inthe word metal ; just as it might b e said, Christians are men,

even if there w ere no men w ho w ere not Christians. Propositions

,therefore

,in which obj ects are referred to a class, because

they possess the attributes constituting the class,are so far from

asserting nothing but resemblance, that they do not, properlyspeaking, assert resemblance at all.

’

I have long wondered at the confusion of ideas whichthis passage exhibits . W e are told that the arrangementof things in a class is founded on a resemblance betweenthe things, but not a mere general resemblance,

’ whateverthis may mean . It is grounded on the possession Of certain‘

common peculiarities . ’ I pass by the strangeness Of thisexpression ; I should have thought that common p eculiari tyis a self-contradictory expression in its ow n terms but here

1 Same section, thi rd paragraph .

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2 36 jOHN STUART flIILL’S PHILOSOPHY TESTED

The important passage quoted above is, as w e mightreadily expect

,inconsistent with various other statements in

the System of Log ic, as for instance m ost of the seventhsection of the chapter on Definition, where w e are told 1

that the philosopher,only gives the same name to things

which resemble one another in the same definite particulars ,’

and that the inquiry into a definition 2 ‘ is an inquiry intothe resemblances and differences among those things . ’ Elsewhere w e are told 3 that the general names given to Objectsimply attributes

,derive their whole meaning from attri

butes and are chiefly useful as the lang uage by m eans ofwhich w e predicate the attributes which they connote.

’

Again , in the chapter on the Requisites of a Phil osophicalLanguage

,he says 4

‘Now the meaning (as has so often been explained) of ageneral connotative name , resides in the connotation ; in theattribute on account of w hich, and to express which, the name

is given. Thu s,the name animal being given to all things which

posses s the attributes Of sensation and voluntary motion,the

w ord connotes these attributes exclusively, and they constitutethe whole Of its meaning .

’

Now,the attri bute

,as w e learned at starting ,

i s but another

name for a Resemblance,and yet a p rop osi tion of w hi ch the

predi cate i s a general name,does not p rop erly sp eaking assert

resemblance at all.

The inconsistency is still m ore striking when w e turnto another work, namely, J. S. Mil l ’s edition of his father’sAnalysis of the Human Mind . Here, in a note 5 on the

subject of classification,“ Mill Objects to his father’s ultra

nom inalist doctrine, that ‘

men were led to class solely forthe purpose of econom ising in the use of nam es . ’ Millproceeds to remark 6 that ‘

w e could not have dispensed1 Book i , chap. vi i i , sec. 7 , paragraph 4, ab out the seventeenth l ine.

This section is numb ered 8 in som e of the early editions.2 Same section, paragraph 8 , l ine 7 .

3 Book iv , chap . i i i , eight l ines from end O f chapter .4 Book iv

,chap. iv , sec. 2 , second l ine .

5 Vol . i , p. 260.

6 Page 261.

RESE IIIBLAN CE

with names to mark the points in which different individualsresemble one another : and these are class -names . ’ Referring to his father’s peculiar expression individual qualities,

’

he remarks very properly‘It is not individual qualitie s that w e ever have occasion to

predicate . lVe never have occasion to predicate of an

object the individual and instantaneous impre ssions w hich itproduces in us. The only meaning of predicating a quality at

all,is to affi rm a re semblance. “Then w e ascribe a quality to

an object, w e intend to asse r t that the Object affects us in a

manner similar to that in w hich w e are affected by a know n classof objects . ’

A few lines further down he proceeds‘Qualitie s , therefore , cannot b e predicated w ithout general

names ; nor,consequently, w ithout classification. W here ver

there is a general name there is a class : classification,and

general names,are things e xactly coextensive .

’

This is,no doubt

,quite the true doctrine ; b ut what b e

comes Of the paragraph already quoted,which appeared

in eight editions O f the System of Logic, during Mill’s life

t im e ? In that paragraph he asserted that propositionsreferring an obj ect to a class because they possess the

attributes constituting the class,do not

,properly speaking

,

assert resemblance at all . Now,when comm enting on his

father ’s doctrine, Mill says that the only meaning of p redi

cating a quali ty at all,i s to afi rm a resem blance.

In a later note in the same volume Mill is , i f possible ,

still m ore explicit in his assertion that the predication of

general names is a matter of attributes and resemblances .He begins thus 1

‘Rejecting the notion that classes and classification w ouldnot have existed b ut for the nece ssity O f economising names

,w e

may say that objects are formed into classes on account of

their resemblance .

’

On the next page he says , in the most distinct manner1 James Mill ’s Analysis of theHuman Mind . New ed ition,

vol . i , p. 288 .

2 38 joHN STUART MILL ’S PHILOSOPHY TESTED

‘Still, a class-name stands in a very different relation to the

definite resemblances which it is intended to mark,from that

in w hich it stands to the various accessory circums tances whichmay form part of the image it calls up. There are certainattribute s common to the entire class

,w hich the class-name w as

either deliberately selected as a mark of,or, at all events, w hich

guide us in the application of it . These attributes are the realmeaning Of the class -name—are w hat w e intend to ascrib e toan Object when w e call it by that name.

’

There can b e no possible m istake about Mill ’s meaningnow. The class-name is intended to mark defini te resem

blances. These resemblances must b e the attributes whichthe class-name was either deliberately selected as a mark of

,

or whi ch guide us in the application of it. These attributesare the real meaning of the class-name—are what w e intend

to ascribe to an object, when w e call it by that name. Y et

w e were told in the passage of the System of Log ic to whichI invited the reader’s special attention , that propositions inwhich objects are referred to a class, because they possessthe attributes constituting the class , are so far from assertingnothing but resemblance, that they do not, prop erly sp eaking ,

assert resemblance at all. A class-name is now spoken of asintended to mark defini te resemblances. Previously w e wereinformed that,in saying, Gold is a metal

,

’ I do not assertresemblance , forsooth , because there m ight b e no other metalbut gold . Y et metal is spoken of as a class

,so that the

word metal is a class-name, and the whole discussion refers

to propositions Of which the predicates are general names .The fact is, the passage contains more than one non

sequi tur it tacitly assumes that metal m ight continue tob e a class-name, while there was only one kind Of m etal

,so

that there woul d b e nothing else to resemble. Then thereis another non-sequi tur when Mill proceeds straightway toanother example, thus just as it m ight b e said

,Christians

are men,even i f there w ere no men who were not Chris

tians. ’ The words ‘ just as ’ here mean that this examplebears out the last ; but Christians and men being plural , the

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2 40 jOHN STUART MLLL’

S PHILOSOPHY TESTED

v enience in extending the boundaries of a class so as toinclude things whi ch possess in a very inferior degree, if inany, som e of the characteristic properties of the class

,pro

v ided that they resemble that class more than any other.He refers to the systems of classification of living things

,in

which al most every great fam ily Of plants or animals has afew anomalous genera or species on its borders, whi ch are

adm itted by a kind of courtesy . It is evident, howe ver,that a matter of this sort has nothi ng to do with the fundamental logical question whether propositions assert resemblance or not. This paragraph is due to the ambiguity of

the word resemblance , which here seem s to mean vague orslight resemblance, as distinguished from that incontestableresemblance which enables us to say that things have the

same attribute. In fact,a very careful reader of the sections

in which Mill treats of resemblance will find that there isfrequent confusion between definite resemblance, and something which M ill variously calls mere general resemblanceor vague resemblance,

’ which will usually refer to sim ilarities dependi ng on the degree Of qualities, or the form s ofobjects .There is

,however, a second case bearing out Mil l ’s

opinion that there is some slight @ ree Of foundation ’

for

the remark that propositions whose predicates are generalterms affirm resemblance. This is a m atter into whi ch w e

must inquire with some care, so that I give at full lengththe paragraph relating to it .

1

There is still another exceptional case, in which, though thepredicate is the name Of a class , yet in predicating it w e affirmnothing but resemblance, that class being founded not on resemblance in any given particular, but on general unanalysableresemblance . The classes in question are those into w hich our

simple sensations or rather simple feelings, are divided . SensationsOfwhite

,for instance, are classed together, not because w e can take

them to pieces,and say they are alike in this, and not alike in

that , but because w e feel them to b e alike altogether, though in

1 Book i, chap. v, sec. 6 , paragraph 5 .

RE SE IVBLAN CE

different degrees . W hen, therefore, I say, The colour I saw

yesterday w as a white colour, or, The sensation I feel is one of

tightness, in both case s the attribute I affirm of the colour or of

the sensation is mere resemblance— simple likene ss to sensationsw hich I have had before , and w hich have had tho se

,

names b estowed upon them . The names O i feelings , like other concretegeneral names, are connotative b ut they connote a mere re

semblance . W hen predicated O f any individual feeling, the information they convey is that O i its likeness to the other feelingsw hich w e have been accustomed to call by the same name . Thusmuchmay suffice in illustration O f the kind of propositions in w hichthe matter-of—fact asserted (or denied) is simple resemblance .

’

Such a paragraph as the above is likely to produce in

tellectual vertigo in the steadiest thinker . In an offhandmanner w e are told that this much may sufi cc in illustration of an excep ti onal case, in which resemblance happens tob e predicated . This resemblance is mentioned slightingly asmere resemblance , or general unanalysable resemblance. Yet

,

when w e come to inquire seriously what this resemblance is ,w e find it to b e that primary relation of sensation to sensation

,which lies at the basis of all thought and knowledge .

Professor Alexander Bain is supposed to b e , since Mill’s

death,a mainstay Of the empirical school, and , in his w orks

On Logic,he has unfOrtunately adopted far too much of

Mill’s views . But,in Professor Bain ’s own p1oper writings ,

there is a vigour and logical consistency of thought forwhich it is impossible not to feel the greatest respect .Now w e find Mr. Bain laying down, at the commence

ment of his writings on the Intellect,l that the PrimaryAttributes of Intellect are ( l ) Consciousness of Difference,( 2 ) Consciousness of Agreement, and (3 ) tetentiv eness . He

goes on to say with adm irable clearness that discrim inationor feeling of difference is an essential of intell igence .

The beginning of knowledge, or ideas , is the discrim ination of one thing from another. As w e can neither feel,

1i lIental and ill oral Science, a Comp end i um of Psychology and Ethics , 1868 ,

pp. 82 , 83 . The same doctrine O f the nature of know ledge is stated in the

treatise on the Senses and the Intellect , second ed i tion, pp . 325-331 in the

Deducti ve Logic, pp . 4, 5 , 9 , and elsew here .

R

2 42 jOHzV STUART IPIILL’S PHILOSOPHY TESTED

nor know,without a transition or change of state , -every

feeling, and every cogni tion, must b e viewed as in relationto some other feeling, or cognition . There cannot b e asingle or absolute cognition . Then, again , Mr. Bain proceeds to say that the conscious state arising from Agreementin the m idst of difference is equally marked and equallyfundamental

Supposing us to e x perience, for the first time,a certain

sensation, as redne ss and after being engaged with other sensations

,to encounter redness again w e are struck w ith the feeling

of identity or recognition the O ld state is recalled at the instanceof the new

,by the fact Of agreem ent, and w e have the sensation

of red,together w ith a new and peculiar consciousness, the con

sciousness Of agreement in diversity. As the di versity is greater,the shock of agreem ent is more lively.

’

Then Professor Bain adds emphatical ly‘All knowledge finally resolves itself into differences and

agreements . To define anything, as a circle, is to state i ts agreem ents w ith some things (genus) and its d ifl

’

erence from otherthings

Professor Bain then treats as the fundamental act of

intellect the recognition of redness as identical with rednesspreviously experienced. This is

,changing red for white,

exactly the same illustration as Mill used,in the example

‘

The colour I saw yesterday was a white colour. ’ NowMr.

‘ Bain says , and says trul y, that all knowledge finallyresolves itself into differences and agreements . Propositionsaccordingly, which affirm these elementary relations

,must

really b e the most important of all classes of prepositions .

They must be the elementary propositions which are pre

supposed or summed up in more compli cated ones . Y e t

such is the class Of propositions which Mill dism isses in anoffhand manner in one paragraph

,as still another excep

tional case .

’

If w e look into the details of M ill’s paragraph, perplexityonly can b e the result. He speaks of

‘

the clas s beingfounded not on resemblance in any given particular

,but on

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2 44 jOHIV STUART IIIILL’S PHILOSOPHY TESTED

to b e desired . But M ill , strange to say, has treatedall-fundamental relation among the Remaining Laws of

Nature ,’ ‘Minor Matters Of Fact

,

’ or Exceptional Cases . ’

It is usuall y impossible to trace the causes which led toMill’s perversities

,but, in this important case

,it is easy to

explain the peculiarity of his views on Resemblance. Hewas labouring under heredi tary p rejudice. His father, JamesMill

,in his m ost acute , but usually wrong-headed book, the

Analysi s of the Phenomena of the Human Mind,had made

still more strange m istakes . In several curious passages theson argues that w e cannot resolve resemblance into anythingsimpler. These needless arguments are evidently suggestedby parts Of the Analysis in which the father professed toresolve resemblances into cases of sequence

!

Thus,when James Mill is discussing 1 the Association of

Ideas,he objects to Hum e specifying Resemblance as one of

the grounds of association . He says

Re semblance only remains,as an alleged principle of associa

tion,and it is nece ssary to inquire whether it is included in the

law s w hich have been above expounded . I believe it w ill b efound that w e are accustomed to see like things together . W henw e see a tree, w e generally see more trees than one ; w hen w esee an ox, w e generally see more oxen than one ; a sheep ,more sheep than one ; a man

,more men than one. From this

Observation, I think, w e may refer resemblance to the law of

frequency, of which it seems to form a particular case.

’

I cannot help regarding the m isapprehension containedin this passage , as perhaps the m ost extraordinary one whichcould b e adduced in the whole range Of philosophical literature . Resemblance is reduced to a particular case of the law

of frequency, that is, to the frequent recurrence of the same

thing,as when , in place of one man

,I see many men. But

how do I know that they are men, unless I observe thatthey resemble each other ? It is impossible even to speakof men without implying that there are various things calledmen which resemble each other sufficiently to b e classed

1 Analysis, fi rst edition, v ol . 1, p . 79 . Second edition, v ol . i , p . 111.

RE SE IPIBLAN CE

d called by the same name . Nevertheless , Jamesto have been actually under the impression thatrid of resemblance

e same work,

1 indeed, w e have the follow

easy to see,among the principles of association,

w hatprinciple it is, w hich is mainly concerned in Classificaby w hich w e are rendered capable of that mighty

Operation on w hich,as its basis

,the w hole of our intellectual

structure is reared . That principle i s Resemblance . It seem sto b e sim ilarity or resemblance w hich

,w hen w e have applied a

name to one individual,leads us to apply it to another

,and

another, till the w hole form s an aggregate, connected together bythe common relation O f every part of the aggregate to one and

the same name . Sim ilarity,or Resemblance

,w e must regard as

an Idea familiar and suffi ciently understood for the illustrationat presen t required. It w ill itself b e strictly analysed, at a

subsequent part O f this Inquiry.

’

In writing this passage, James Mill seems to have forgotten

,quite in the manner of his son

,that he had before

treated Resemblance as an alleged principle of association ,and had referred it to a particular case of the law of fre

quency . Here it reappears as the principle on which thewhole of our intellectual structure is reared . I t is strangethat so important a principle should elsewhere b e call ed an‘ alleged principle

,

’ and equally strange that it should afterwards b e ‘ strictly analysed .

’ Before w e get down to thebasis of our intellectual structure it m ight b e supposed thatanalysis had exhausted itself.James Mill gives no reference to the subsequent part of

the inquiry where this analysis is carried out, nor do I findthat J . S. Mill, or the other editors of the second edition ,have supplied the reference . Doubtless , however, the

analysis is given in the second section Of Chapter XIV ,

where,in treating of Relative Terms ,2 he inquires into the

1 Ana lysis, fi rst edition,vol . i , pp. 212 , 213 . Second ed ition

,vol . i

, pp .

270, 271.

2 Fi rst edition,vol . i i , p. 10. Second ed ition,

vol . i i , pp. 11, 12 .

246 jOHN STUART MILL ’S PHILOSOPHY TESTED

meaning of Same, D ifferent, Like, or Unlike, and comes to theconclusion that the resemblance between sensation andsensation is, after all, only sensation . He says

Having two sensations, therefore , is not only having sensation,but the only thing which can

,in strictness, b e called having

sensation ; and the having tw o and knowing they are tw o, whichare not tw o things , but one and the same thing, is not only sensation

,and nothing else than sensation

,but the only thing which

can, in strictness, b e called sensation. The having a new sensation

,and know ing that it is new ,

are not tw o things, but oneand the same thing .

’

This is,no doubt, a wonderfully acute piece of sophi sti

cal reasoning ; but I have no need to occupy space inrefuting it

,because J . S. Mill has already refuted it in

several passages which evidently refer to his father’s fallacy .

Thus,I have already quoted , at the commencement of this

article, a statement in whi ch J. S. Mil l argues that re

semblance between two phenomena is m ore intelligible thanany explanation could make it. Again

,in editing his

father’s Analysis, Mill comments at some length upon thissection ,1 showing that it does not explain anything, norleave the likenesses and unlikenesses Of our simple feelingsless ultimate facts than they were before.

But though Mill thus refuses to dissolve resemblanceaway altogether, his thoughts were probably warped inyouth by the perverse doctrines which his father sounsparingly forced upon his intellect. Too early the

brain -fib res received a decided set,from which they could

not recover, and all the power and acuteness of Mill ’sintellect were wasted in trying to make things fit, whichcoul d not fit

,because m istakes had been made in the very

comm encement of the structure.

This m isapprehension Of the Mil ls, pere et fi ls, concerningresemblance, is certainly one of the m ost extraordinaryinstances of perversity of thought in the hi story of philosophy. That which is the summum genus Of reasoned

1 Vol . i i , pp. 17 -20.

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2 48 jOHIV STUART IP/ILL’

S PHILOSOPHY TESTED

the common marks or names applied to them . The connection Of resemblance is denied existence. This ul tra-nominali sm of the father is one of the strangest perversities Ofthought which coul d b e adduced ; and though John StuartMill disclaims such an absurd doctrine in an apologetic sortof way, yet he never, as I shall

’

now and again have toshow, really shook himself free from the perplexities of

thought due to hi s father’s errors .It may seem to many readers that these are tedious

matters to discuss at such length . After all , the Import ofPropositions and the Relation of Resemblance are matterswhich concern metaphysicians only

,or those who chop

logic. But this is a m istake . A system of phil osophy—a

school of metaphysical doctrines— is the foundation onwhich is erected a structure of rules and inferences, touching our interests in the most vital points . John StuartMill, in his remarkable Autobiograp hy, has expressly statedthat a principal object of his System of Logi c was to overthrow deep- seated prejudices, and to storm the stronghold inwhi ch they sheltered themselves . These are his words 1

Whatever may b e the practical value Of a true philosophy Ofthese matters, it is hardly possible to exaggerate the mischiefs ofa false one . The notion that truths external to the mind mayb e known by intuition or consciousness

,independently Of ob ser

vation and experience, is, I am persuaded,in these times, the

great intellectual support Of false doctrines and bad institutions .By the aid of this theory

,every inveterate belief and every

intense feeling, of w hich the origin is not remembered, isenabled to dispense with the Obligation of justifying itself byreason, and is erected into its ow n all-sufficient voucher andjustification. There never w as such an instrument devised forconsecrating all deep seated prejudices. And the chief strengthOf this false philosophy in morals, politics, and religion, lies inthe appeal w hich it is accustomed to make to the evidence of

mathematics and of the cognate branches of physical science. TO

expel it from these, is to drive it from its stronghold : and

because this had never been effectually done, the intuitive school ,

1 Autob iography, pp. 225-227 .

RE SE IVBLAN CE 9

even after w hat my father had w ritten in his Analysis of theMind, had in appearance, and as far as published w ritings wereconcerned, on the w hole the best O f the argument. In attempting to clear up the real nature of the evidence of mathematicaland physical truths

,the System of Logic met the intuitive

philosophers on ground on w hich they had previously beendeemed unassailable and gave its ow n ex planation , fromexperience and association, of that peculiar character of w hat arecalled necessary truths, w hich is adduced as proof that theire vidence must come from a deeper source than e x perience.

t ether this has been done effectually,is still subjud ice ; and

even then, to deprive a mode O f thought so st rongly rooted inhuman prejudice s and partialities

,Of its more speculative support,

goes b ut a very little w ay tow ard s overcoming it ; but thoughonly a step , it is a quite indispensable one for since

,after all

,

prejudice can only b e successfully combated by philo sophy, now ay can really b e made against it permanently until it has beenshow n not to have philosophy on its side .

’

This is at least a candid statement of motives, means,and expected results . Whether Mill ’s exposition of the

philosophy of the mathematical sciences is satisfactory ornot

,w e partially inquired in the previous article ; and in

one place or another the inquiry will b e further prosecutedin a pretty exhaustive manner. Mill allowed that thecharacter Of his solution was still sub g

’

udi ee, and it must

remain in that position for some time longer. But of the

importance of the matter it is impossible to entertain adoubt . If Mill ’s own philosophy b e yet more false thanwas

,in his Opinion, the philosophy which he undertook to

destroy, w e may well adopt his own estim ate of the results .

Whatever,

’

he says , ‘

may be the p ractical value of a true

phi losop hy of these matters,i t i s hard ly p ossi b le to exaggerate

the mi schi efs of a false one.

’ Intensely believing , as I do,that the philosophy of the Mills , both father and son , is afalse one

,I claim ,

almost as a right, the attention of thosewho haVe sufficiently studied the matters in dispute tojudge the arduous work of criticism which I have felt it myduty to undertake .

THE EXPERIMENTAL M ETHODS

MY last art icle on Mill’s Philosophy treated of what oughtto b e, or rather necessarily is, the basis of all reasoningprocesses— the Relation of Resemblance. It was shownthat Mill first of all dism isses this relation as a m inor

,or

even a doubtful matter of fact, or as still another exceptionalcase that he then unintentional ly makes it the pivot, nay,almost the substance Of the reasoning processes, as treated inthe book on Induction ; yet that, in a later chapter of thatbook, he returns to the subject Of Resemblance as if it had sofar been passed over, and finally comes to the conclusion thatResemblance is seldom regarded as the subject of science.

From the base let us proceed to the pill ars of Mill ’slogical edi fice. These are the celebrated Methods of

Experimental Inquiry— the Method of Agreement, the

Method of D ifference, the Method of Residues, and the

Method Of Concomitant Variations ; to which may b e

added, as a kind of corollary, the Joint Method of Agreement and D ifference. Mill ’s exposition of these methods isconsidered perhaps the most valuable part of his treatise,and much of the celebrity of the book is due to this part.Many people, indeed , whose reading in logic has not beenextensive, think that these are Mill ’s own methods, that heinvented them . Any one at all acquainted with the historyof logical science knows , of course, that this is not the case,nor did M ill ever claim that it was . Francis Bacon set

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2 52 jOHIV STUART IlfIILL’S PHILOSOPHY TESTED

in which these methods are set up . W e must inquire whatis the warrant for’

,their validity, and it wi ll b e my duty to

prove that in this point M ill has fallen into a completeei reulus in probando. These methods are the only means Ofproving the connection of cause and effect ; yet the methodsdepend for their val idity upon our assurance of the cer

tainty and universality of that connection, that is , upon theuni versal law of causation .

To students of Mill’s logic it is so familiarly knownthat he bases induction upon the notion of causation

,that it

may seem superfluous to prove the position . I must nevertheless refer to the chapter treating Of the Law of Universal Causation

,

’ 1 where he speaks of‘the notion of Cause

being the root of the whole theory of Induction .

’

Observethe comprehensive force of the expression, the wholetheory.

’ Elsewhere,2 the universal fact ’

of the uniformcourse of nature is parenthetically described as our warrantfor all inference from experience,

’ again an unlim ited andmost comprehensive remark . The fourth paragraph of the

same chapter commences thus : ‘Whatever b e the mostproper mode of expressing it, the proposition that the courseof nature is uni form is the fundamental principle, or generalaxiom ,

of Induction . It is true that Mill sometim es distinguishes between the Uniform ity of Nature and the Lawof Causation

,and gets into perplexities which I have not

space to unravel here . It will therefore b e better to referto a later chapter,3 where M ill places the matter beyonddoubt, saying

‘As w e recognised in the commencement, and have beenenabled to see more clearly in the progress of the investigation,the basis of all these logical Operations is the law of causation .

The validity of all the Inductive Methods depends on the

1 Chap. v, b eginning of second secti on. As alm ost all the quotations inthi s article are taken from the th i rd b ook of the System of Logic, i t w i ll b eunnecessary again to cite the numb er of the b ook, w h ich , unless otherw isespecified, w i ll alw ays b e the Th i rd Book, t reating Of Induction.

’

2 Chap. i i i , b eginning of th i rd paragraph .

3 Chap. xxi , first paragraph .

THE EXPERIMEN TAL METHODS 2 53

assumption that every event,or the beginning of every pheno

menon, must have some cause some antecedent, on the existence

Of w hich it i s invariably and unconditionally consequent . In

the Method of Agreement this is obvious thatMethod avow edlyproceeding on the supposition

,that w e have found the true

cause as soon as w e have negatived every other. The assertionis equally true of the Method of Difference . That methodauthorises us to infer a general law from tw o instances 1

one,in

which A exists together w ith a multitude of other circum stances,

and B follow s another, in which, A being removed,and all

other circumstances remaining the same, B is p revented.

how ever, does this pro ve ? It proves that B,in the particular

instance,cannot have had any other cause than A ; but to con

clude from this that A w as the cause,or that A w ill on other

occasions b e follow ed b y B,is only allow able on the assumption

that B must have some cause that among its antecedents in anysingle instance in w hich it occurs

,there must b e one w hich has

the capacity O f producing it at other tim e s . This being admitted,

it is seen that in the case in question that antecedent can b e no

other than A but,that if it b e no other than A

,it must b e A

,

is not proved,by these instances at least, but taken for granted .

There is no need to spend tim e in proving that the same thingis true of the other Inductive Methods . The universality Of the

law of Causation is assumed in them all.’

It would b e easy to show that this passage is in substance all wrong and unscientific. The idea that w e mustassume each phenomenon to have one anteceden t , and onlyone , which has the capacity of producing it at (all othertimes

,is quite inconsistent with the scientific idea of causa

tion,as well as with Mill’s own statements in other places .

It is to a conjunction of causes , joined to all kinds of

negative conditions— that is to say , the absence Of counteracting causes— that the production of an effect is due ; andthis fact alone is enough to disperse Mill ’s extraordinaryassertion that tw o instances can prove a general law. But

the point with w hich w e are concerned now is the completedependence of the Inductive Methods 011 the Law Of Causation ; not merely the occasional truth of that law ,

but itsUniversali ty is assumed in all the m ethods .

1 This is the ab surd statement alluded to on the preceding page.

2 54 jOHN STUART Il/ILL’S PHILOSOPHY TESTED

Th_e_four_great_pillars of M ill’s l ogical fledifice n rest

,t hen ,

upon the universal law Of causation . Upon what does thislaw

"

rest ? An ancient system of cosm ogony representedthe world as resting on an elephant, and the elephant on atortoise ; w e want something to correspond to the tortoise.

Now it is quite certain that Mill would not derive the lawof causation from intuition, consciousness, or any manner ofinnate source. It was the avowed purpose Of his Systemof Logi c to show that an appeal to intuition , independentlyOf Observation and experience

,was the great intellectual

support of false doctrines and bad institutions . It is fromexperience, then , that w e must learn the universali ty of the

law of causation . But here the great di fficul ty Of Mill ’sposition begins to b e felt. He allows that w e do not seethis law of nature writ up in plain figures, neither inmaterial nature nor in the m ind . The law was quite un

known,he adm its, in the earliest ages . It is an induction

by no means of the most Obvious kind. But M ill ’s ownwords must b e carefully quoted . Speaking of the fundam ental principle, or general axiom of induction, he says 1

I hold it to b e itself an instance of induction, and inductionby no means Of the most Obvious kind. Far from being the

first induction w e make,it is one of the last

,or at all events one

O f those w hich are latest in attaining strict philosophicalaccuracy .

’

But here comes the rub. If the inductive method, bywhich w e ascertain the connection of causes and effects,presuppose the general law Of causation

,and this law of

causation is one of the latest results of inductive inquiry,how could w e ever begin methods areof no vag lity, until w e have proved a most general, in factan uni versal law

,which can only b e proved by those

m ethods . Logic,let it b e always remembered

,is

,according

to M ill, the Science of Proof,and

,in such a matter, as the

methods of inductive proof,w e cannot b e supposed to deal

with mere surm ise. W e have now got into this position .

1 Chap. i i i , fourth paragraph .

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2 56 jOHIV STUART IlIILL’

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p er enumerationem simp lieem; and the law of universal causation,

being collected from results so Obtained,cannot itself rest on any

better foundation.

‘It w ould seem,therefore

,that induction p er enumerationem

simp lieem not only is not necessarily an illicit logical process,b ut is in reality the only kind of induction possible ; since themore elaborate process depend s for i ts validity on a law

,itself

Obtained in that inartificial mode . Is there not then an incon

sistency in contrasting the loo senes s of one method w i th the

rigidity of another, when that other is indebted to the loosermethod for its ow n foundation

The inconsistency, how ever, is only apparent. Assuredly, i finduction by simple enumeration were an invalid process

,no

process grounded on it could b e valid ; just as no reliance couldb e placed on telescopes, if w e could not trust our eyes . But

though a valid process, it is a fallible one, and fallible in verydifferent degrees : if therefore w e can substitute for the morefallib le form s of the process

,an Operation grounded on the same

proce ss in a less fallible form , w e shall have effected a verymaterial improvement . And this is what scientific inductiondoes. ’

Various reflections are suggested by this unfortunatepassage. Mill here discovers that the law of causationcould not have been derived from rigid induction ; he eveninserts the words of course,

’ as i f no one could have failedto see this . It must therefore b e derived from ‘

the looseand uncertain mode Of induction,

’ with which w e shall havemore to do . But

,in the first place , this treatment of the

matter does not square with that in Chapter III, where hetreats of the same subject The G round of Induction .

’

Here he told us , as already quoted, that the uniform ity of

the course of nature is our warrant for all inferences fromexperience.

’ Now even ‘ a loose and uncertain mode of

induction ’

must b e a case of inference from experience .

Again,Mill distinctly says : 1 ‘The statement that the

uniform ity of the course of nature is the ultimate majorprem ise in all cases of induction , may b e thought to requiresome explanation .

’ Here he speaks without qualification1 Chap. i i i , b eginning of fi fth paragraph .

THE EXPERIIUEN TAL METHODS

of induction,

’ which must include even theloose induction of the ancients . In writing this chapterMill had not yet discovered that, as induction is based uponc_a_usat ipn, causation would have to b e based upon somethingelse . Accordingly, though in the third paragraph of the

second section of the chapter he mentions the ‘ loose ’

in

duction of the ancients , it is only to depreciate and almostderide it . He thinks it was above all by poin ting out the

insufficiency of this rude and loose conception of induction,

that Bacon m erited the title so generally awarded to himof Founder of the Inductive Pl1ilos0phy.

1 It is curious,

then,that Mill in the later chapter finds it necessary to

make this loose , uncertain,and insufficient m ethod the

basis of his system,inasmuch as it is represented to b e our

means of learning the universality O f the law of causation,

On which the validity of the rigid inductive processes depends . Now

,in a footnote to Chapter III w e are referred

to Chapters XXI and XXII ; and in Chapter XXI w e are

sim ilarly referred back to Chapter III. Nevertheless,as I

have said , the doctrine O f the early chapter fails to squarewith that of the later one . But there is so much else tocome, that I need not dwell upon this discrepancy .

The next reflection that suggests itself is the apparen tincongruity of basing the whole of our inductive know ledgeOf nature upon a loose and uncertain and insuy

'iei ent kind ofinduction. In severa l places Mill speaks of this k ind of

induction with unm itigated scorn . He says 2

‘The Induction Of the ancients has been w ell described byBacon

,under the nam e Of

“ Induct io per enumerationem simplicem ,

ub i non reperitur instant ia contradicto ria .

”

It consists in ascrib

ing the character of gene ral truths to all propo sitions w hich are

true in e very instance that w e happen to know of. This is thekind of induction 3 w hich is natural to the mind w hen unaccustomed

1 Same section ,fifth paragraph .

2 Chap. i i i , sec . 2 , th i rd parag raph .

3 In the first and second ed itions w e here find the significant w o rdsi f it deserves the narne, ’ that is , of induction thus w e find the great em

S

2 58 jOHN STUART IlfIILL’S PHILOSOPHY TESTED

to scientific methods. The tendency, which some call an instinct,

and which others account for by association,to infer the future

from the pas t, the know n from the unknow n,is simply a habit

O f expecting that what has been found true once or several times,

and never yet found false, wi ll b e found true again.

Popular notions are usually founded on induction by simpleenumeration in science it carries us but a little w ay. W e are

forced to begin w ith it w e must Often rely on it provi sionally,

in the absence of means of more searching inve stigation. But,

for the accurate study of nature, w e require a surer and a morepotent instrument. ’

He proceeds, in the next paragraph, still more stronglyto denounce this loose method of induction . Speaking of

moral and political inquiries,he says

The current and approved mode s O f reasoning on these sub

jects are still of the same vicious description against w hichBacon protested ; the method almost exclusively employed bythose professing to treat such matters inductively, is the veryinductio p er enumerationem simp licem w hich he condemns and theexperience w hich w e hear so confidently appealed to by all sects,parties

,and interests, is still

,in his ow n emphatic w ords

, mera

palpatio.

’

An Obvious difficulty presents itself ; if rigid inductiondepends upon the experimental methods ; i f these dependupon the law of causation, and this law depends upon inductio p er enumerationem simpli cem ; then the valid i ty of all

our inductions dep ends on a loose and uncertain foundation .

The upper parts of the logical edifice cannot b e fi rmer thanits base. Mill

,when he comes to the point, shows a credit

able consciousness Of this diffi culty,and accordingly dis

covers for the occasion that this loose method of inductionis not always loose . In the third chapter 1 he remarks

pirieal ph ilosopher, w hose w ork i t w as to“ show the inductive b asis o f all

mathemat ical and other science , accidentally questioning the propriety of

allow ing the nam e‘ induction ’

to that process upon w h ich he ultimately

b ases our know ledge of the unive rsal law of causation,as w ell as the ax ioms

of geometry. W hen b e inserted these unlucky w ords he must have forgottenthat i t w as the b asis o f his system ,

or else he had not yet d iscovered the fact.1 Sec. 2

,fourth paragraph .

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2 60 jOH/V STUART AIILL ’S PHILOSOPHY TESTED

induction as against the unscientific,notwithstanding that

the scientific ultimately rests on the unscientific, the preceding considerations may suffi ce.

’

But Mill,though he appears to have explained the

inconsistency successfully,has not really cleared himself.

He is yet in a coil of difficulties . I now want , to knowprecisely what this loose kind of induction is. Logic, asMill clearly stated in his Introduction , is the Science of

Proof. In so far as belief professes to b e founded on proof,the office of logic is to supply a test for ascertaining whe theror not the belief is well grounded. The purpose of M ill ’streatise is thus concisely set forth 1

Our obj ect, then,w ill b e

,to attem pt a correct analysis of the

intellectual process called Reasoning or Inference , and of suchother mental operations as are intended to facilitate this as w ellas

,on the foundation of this analysis

,and pam

'

passu w ith it,to

bring together or frame a set of rules or canons for testing thesufficiency of any given evidence to prove any given proposition .

’

Now I want to know where , in Mill’s treatise, is to b e

found the analysis of this process of induction p er enumera

tionem simp li cem ? And where is the set of rules and

canons for perform ing it ? On this process, as w e havefound, ultimately rests the proof of all truths, both of mathemat ical science

,and of causation ; whatever w e prove by the

four experimental methods is really proved by the underlying inductive process on which their validity depends .Mill’s logic is supposed to present the m ost thorough analysisof the foundations of our knowledge, and he him self put itforth professedly as intended to clear up the real nature of

the evidence of mathematical and physical truths .2

for a general t ruth is resolvab le into the part icular ones on w h ich it isfounded

,so that Mill ’s new position amounts to saying that certain past acts

of induction are a w arrant for future acts . But w here w as the w arrant for thepast acts ? It is ab solutely impossi b le to meet all Mi l l ’s arguments, b ecause,as each new d ifficulty presents i tsel f, b e invents a new explanation

,regardless

or rather ob liv ious, of consistency w ith h is old ones.1 Introduction,

sec. 7 , second paragraph .

2 Autob iography , p. 226, quoted ab ove,pp. 248, 249 .

Tl i E EXPERIAIEN TAL IVE THODS

It was above all things necessary that Mill should haveanalysed and described this process of ‘ simple enumeration ’

with care and comple teness , because it is the basis of hiswhole empirical system . Where is the analysis ? W hereare the rules of the method ? If w e search the treatise, w efind the process men tioned here and there, but, strange tosay , almost always in a depreciatory and scornful manner.It is a loose, rude , uncertain , insufficient, fallible , nu

scientific, precarious , —even a vicious process . Such are the

epithets which Mill applies to the basis of his empiricalphilosophy

, except in the section or two written w hen hehappened to remember that it was the basis . Then , again ,

where are the rules o f this me thod of induction ? If it b eusually so insufficient and fallible a support , surely itall the more requisite that w e should have precise ruleswhereby to judge when it is precarious and when it is no t .But the rules and canons given in the treatise are those of

the four Experimental Methods , and these rules cannotpossib ly help us , because the me thods themselves derivetheir validity from the underlying law of causation , which isestablished by inductio p er enumerationem simp li ccm . I say ,

then,that just where Mill ’s analysis should have been most

careful,and his canons most explicit, there is nothing of the

sort,and i f w e seek for a description of this fundamental

kind of inductive reasoning, w e find i t called by a Latinphrase

,and treated with impatience and contempt .

But let us make the best of such descriptions of the

fundamental process of his system ’

as Mill favours us with .

I have already quoted one passage in which he says that thekind of induction in question consists in ascribing the

character of general truths to all propositions w hich are truein every instance that w e happen to know of.

’

Elsewhere Mill,1 in reference to coexistence s independen tof causation , says

‘In the absence,then, of any universal law of coexistence ,

similar to the universal law of causation w h i ch regulates sequence ,Chap. xxu, sec. 4, las t paragraph .

2 6 2 jOHN STUART AIILL ’S PHILOSOPHY TESTED

w e are throw n back upon the unscientific induction of the

ancients, per enumerationem simplicem,

ubi non rep eri tur instantia

contradictoria . The reason w e have for believing that all crow sare black, is simply that w e have seen and heard of many blackcrow s , and never one of any other colour. It remains to b e con

sidered how far thi s evidence can reach,and how w e are to

measure its strength in any given case.

’

I t is true that in the sections which follow w e have some

vague discussions as to the circum stances under which w e

may trust an empirical induction . But in w riting thesesections Mill seems again to have forgotten that the law of

causation is itself founded on the same basis. In the passage quoted above w e are told that in the absence of auniversal law similar to the universal law of causation, w e

are thrown back upon the unscientific induction of the

ancients. But surely in the case of causation also w e are

similarly thrown back on this unscientific induction . i f w e

W ish to know the ultimate warrant for our inferences . Inthese sections Mill professes to treat only of Coexistencesindependent of Causation ,

’ such being the title and subjectof the whole chapter. He gi ves no indication how w e are

to apply the sam e process to prove the law of causationitsel f

,which is always by h im sharply distinguished from

the cases treated in the chapter named. In fact,he tells us

in the first paragraph of the fourth section , that the application of a system of rigorous scientific induction is precludedin the cases here treated .

‘The basis of such a system iswanting : there is no general axiom standing in the same

relation to the uniformities of coexistence as the law of

causation does to those of succession.

’ In fact , Mill writesthroughout this chapter as i f the law of causation had

nothing to do with induction by simple enumeration,upon

whi ch w e are thrown back in other cases .Turning again then to the most distinct account w hich

w e get of this method , w e find that induction by simpleenumeration consists in ascribing the character of generaltruths to all propositions which are true in every instance

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2 64 jOHN STUART MILL ’

S PHILOSOPHY TESTED

The different antecedents and consequents, being, then, supposed to b e, so far as the case require s , ascertained and d iscrim in

ated from one another w e are to inquire w hich is connected withw hich . In every instance w hich come s under our observation

,

there are many antecedents and many consequents . If thoseantecedents could not b e severed from one another except inthought, or if those consequents never were found apart, it w ouldbe impossible for us to distinguish (aposterimi at least) the reallaw s

,or to assign to any cause its effect

,or to any effect its

1

cause .

He goes on to explain that, to effect this analysis, w emust b e able to mee t with some of the antecedents apartfrom the rest , and observe what follows from them . W e

must follow the Baconian rule of varying the circumstances,

and it is this rule which is developed into the four Experimental Methods . But here w e are in a most palpabledifficulty. W e cannot assign any sequent to a particularantecedent without going through the elaborate investigations referred to above, without in fact employing the

experimental methods, explicitly or implicitly. Even the

first of those methods, that of Agreement, is insuffi cient forthe purpose, because w e are told 1 that it has the defect ofnot proving causation , and can, therefore, only b e employedfor the ascertainment of empirical laws . Now w e are in a

perfect vici ous circle. Causation is proved only by the

Method of Di fference . That Method derives its validityfrom the universality of the law of causation . The universal ity of this law is ascertained by induction by simpleenumeration, . w hich requires that w e shal l have ascertainedthe truth of the law in every particular case, a thing whichclearly could not b e done without the Method of Difference .

The whole edifice of Mill’s inductive logic, elaboratelydescribed in the twenty-five long chapters of the third book,collapses . The basis disappears altogether, and the fourpillars, the four Experimental Methods

,are left supporting

themselves in a logical void .

If another stroke were needed for the overthrow of M ill’s1 Chap . xvi i, second paragraph .

THE EXPERIAIEN TAL AIE TIIODS 2 65

vaunted system,it could easily b e given in the form of one

question . Are there really no apparent exceptions to theuniversal law of causation ? Mill grounds this law uponinduction by simple enumeration , ubi non rep eritur instantia

contradictoria,

‘ when no contradictory instance is encoun

tered .

’ Applied to the case of causation, this processwould require that in all our experience w e had nevernoticed, or at least investigated , a case without ascertainingthat the law of causation was verified . What a monstrousassumption is this ! Will any one deny that there are

whole regions of facts fam iliarly known to us where w e

cannot detect the action of causation ? What determ inesthe sex of young animals ? What produces unexpectedform s and diseases , monstrous births, lusus naturae, as theyare significantly called ? All kinds of tumours

,ulcers

,and

local diseases , spring up in various parts of the humanbody

,and medical science can usually give no explanation

of them . I t is astonishing how statements made in a workof repute are allow ed to pass unquestioned , althoughdirectly contrary to the most obvious facts . O f course w e

may expect or believe that all such phenomena w ill sooneror later b e explained as the effects of undiscovered causes ,but such expectation must b e ap riori in its origin, i f Mill

’sown accoun t of the way in which w e ascertain the law of

causation empirically is true . It is useless to say that w e

can p rove the law of causation empirically, when apparen texceptions to its truth are endless in number. A certainprobability no doubt may b e given to the law empirically,but this does not help Mill , w ho frequently implies that thelaw of causation is certainly and uni versally true, and that ,as soon as the principle of causation makes its appearance ,

the precarious inferences derived from simple enumerationare superseded and disappear from the fi eld.

1

No doubt a skilful controversialist m ight find in Mill ’sbook many Openings for a plausible reply, but none of themw ould bear cross- examination . It m ight b e pointed out

1 Chap. xvi i i , sec. 4, end of th i rd paragraph .

2 66 jOHN STUA l i ’T JUILL’S PHILOSOPHY TESTED

that M ill,in the twenty-first chapter, shows his conscious

ness of the precarious nature of induction by simpleenumeration , but urges that, by widening the sphere of

induction , w e may indefinitelv increase the certainty of the

inference . This argument, how ever, would show completem isapprehension of the theory of probability

,and the

principles of evidence . Mill , though urging this view of

the matter, had never formed any clear ideas on the subject.O bserve that the method of simple enum eration consists inascribing the character of general truths to all propositionswhich are true in every instance that w e happen to knowof. Uncertainty enters in a double manner : there is theuncertainty whether what is true of certain particular casesis true of all other cases . This uncertainty would b egradually rem oved by increasing the number of casesexam ined . The other uncertainty depends upon the ditticulty of showing that any one consequent follows from aninvariable antecedent. Now Mill makes it abundantlyplain that only the Method of D ifference can establish thisfact of sequence with certainty . It follows that every othermethod of ascertaining the connection can give it only witha degree of probability. Then every particular case towhich w e apply simple enum eration is more or less uncertain ; and so far as this uncertain ty is common to all thecases , no multiplication of new cases can remove the nuoertainty. In fact it comes to this, that the degree of certainty( that is, m ore properly speaking , probability) which w e can

give to the universal law of causation cannot exceed , andmay fall short of, the certainty of the process by which w e

discover the connection of causes and effects,prior to the

establishment of the great experimental m ethods . Now asthose m ethods are expounded as the modes of discoveringcausation

,and are often described as the only modes of

proving causation,w e are again left by Mill without any

analysis of the real base of his system . There is an evidentvicious circle . The Method of Difference proves causation ;it reposes on the universal law of causation ; which is

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UTILITARIANISM

IN some respects Mill’s Essays , published under the titleUtilitarianism

,

’

are among his best writings . They h ave ,in the fi rst place

,the excellence of brevity . Ninety-six

pages, printed in handsome type, make but a light task forthe student who wishes to enter into the intricacies of

moral doctrine. Moreover, the last Essay consists of adigression concerning the nature and origin of the idea ofJustice

,and it occupies nearly one-third of the whole book .

Thus Mill managed to compress his discussion of so important a subject as the foundations of Moral Right andWrong into some sixty pleasant pages .And pleasant pages they certainly are, for they are

written in Mill ’s very best style . Now Mill, even whenhe is most prolix, when he is pursuing the intricacies of

the most involved points of logic and philosophy, can

seldom or never b e charged with dulness and heaviness .His language is too easy, polished , and apparently lucid .

In these Essays on‘ Utilitarianism,he reaches his own

highest standard of s tyle. There is hardly any o ther bookin the range of philosophy

,so far as my reading has gone,

which can b e read with less effort . There is somethingenticing in the easy flow of sentences and ideas

,and with

out apparent diffi culty the reader finds himself agreeablyborne into the m idst of the m ost profound questions of

e thical philosophy, questions which have been the battle

UTILITARIANISM 2 69

ground of the human intellect for two thousand five hundredyears .

Partly to this excellence of style , partly to Mill’snnmense reputation , acquired by other works and in otherways , must w e attribute the importance which has beengenerally attached to these ninety-six pages . Probably no

other modern work of the same small typographical extenthas been equally discussed , criticised

,and adm ired

,unless

,

indeed , it b e the Essay on Liberty of the same author.The result is

,that Mill has been generally regarded as the

latest and best expounder of the great U tilitarian Doctrine— that doctrine which is, by one and no doubt the pre

ponderating school , regarded as the foundation of all moraland legislative progress . Many there are who think that

,

what Hume and Paley and Jeremy Ben tham began,Mill has

carried nearly to perfection in these agreeable Essays .Nothing can b e m ore plain

,too

,than that Mill himself

believed he was dutifully expounding the doctrines of hisfather

,of his father’s friend

,the great Bentham , and of the

other unquestionable U tilitarians among whom he grew up .

Mill seem s to pride him self upon having been the firs t,not

indeed to invent , but to bring into general acceptance thename of the school to w hich he supposed him self to belong .

He says 1 ‘

The author of this essay has reason for b el ieving himself to b e the first person w ho brought the wordutilitarian into use. He d id not invent it, b ut adoptedit from a passing expression in Mr. Galt ’s Annals ofthe Parish. After using it as a designation for severalyears , he and others abandoned it from a grow ing disliketo anything resem b ling a badge or watchw ord of sectariandistinction. But as a name for one single opinion ,

not

a set of opinions— to denote the recogni tion of utilityas a standard

,not any particular w ay of applying i t

the term supplies a w ant in the language, and offers , in1 Uti li ta riani sm ,

fi fth ed it ion ,p. 9

,footnote . Except w here otherw ise

speci fied , the refe rences throughout th is article w ill b e to the pages o f thefi fth edition of Uti l i ta r iani sm .

2 70 jOHN STUART MILL ’S PHILOSOPHY TESTED

many cases , a convenient mode of avoiding tiresome circumlocution .

’

In the Autobiograp hy (p. Mill makes a statementto the same effect

,saying

‘I did not invent the w ord,but found it in one of Galt’s

novels, the Annals of the Parish, in w hich the Scotch clerg yman,of w hom the book is a supposed autobiography, is represented as

w arning his parishioners not to leave the Go spel and becomeutilitarians . W ith a boy’s fondness for a name and a b anner Iseized on the word, and for some years called myself and othersby it as a sectarian appellation ; and it came to b e occas ionally usedby some others holding the Opinions it w as intended to designate .

As those opinions attracted more notice, the term w as repeatedb y strangers and opponents, and got into rather common use justab out the time w hen those w ho had originally assumed it

,laid

dow n that along with other sectarian characteristics . ’

It is pointed out, however, by Mr. Sidgwick in hisarticle on Bentham ism ,

1 that Bentham himself suggested thename ‘ Utilitarian

,

’ in a letter to Dumont,as far back as

June 18 02 .

Mill explicitly states that it was his purpose in theseEssays on Utilitarianism to expound a previously receiveddoctrine of utility . Towards the close of his first chapter

,

containing General Remarks , he says (p.

‘ On the pre

sen t occasion , I shall , without further discussion of the

other theories , attempt to contribute something towards theunderstanding and appreciation of the Utilitarian or Happiness theory, and towards such proof as it is susceptible of.

’

He proceeds to explain that a preliminary condition of the

rational acceptance or rejection of a doctrine is that itsformula should b e correctly understood. The very imperfeet notion ordinarily formed of the U tilitarian formula wasthe chief Obstacle which impeded its reception ; the mainwork to b e done , therefore, by a U til itarian writer w as toclear the doctrine from the grosser m isconceptions . Thusthe question would b e greatly simplified ,

and a large proportion of its difficulties removed . His Essays purport

1 Fortnightly Review ,May 1877, vol. xxi , p. 6 48.

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2 72 jOH/V STUART MILL ’S PHILOSOPHY TESTED

Eminent men of the most different schools and tones of

thought—such as the Rev . Dr. Martineau, Mr. Sidgwick, Dr.Ward , Professor Birks

,the late Professor Grote— have

criticised and refuted M ill time after time .

Since commencing my analysis of Mill’s Philosophy, Ihave been surprised to find

,too

,that some who were

supposed to support Mill’s school through thick and thin,

have long since discovered the inconsistencies which I wouldnow expose at such wearisome length, as i f they were new

discoveries . Such is the ground which my friend , ProfessorCroom Robertson

,takes in his quarterly review , Mind , which

must b e considered our best authori ty on philosophical questions . As to this matter of U tilitarianism

,a very em inent

author, formerly a friend of Mill himself, assures me that thesubject is quite threshed out, and implies that there is noneed for me to trouble the public any more about it. Infact, it would seem to be allowed within philosophical circlesthat Mill’s works are often wrongheaded and unphilosophical . Y et these works are supposed to have done so muchgood that obloquy attaches to any one who would seek todiminish the respect paid to them by the public at large.

Philosophers,and teachers of the last generation at least,

have done their best to give Mill’s groundless philosophy 3.

hold upon all the schools and all the press, and yet w e of

this generation are to wait calmly until this influence dissolves of its own accord . W e are to do nothing to lessenthe natural respect paid to the m emory of the dead , especially of the dead who have unquestionably laboured withsingle-m inded purpose for w hat they considered the good oftheir fellow-creatures . But in nothing is it m ore true thanin philosophy, that the evil that men do lives after them ;

the good is oft interred with their bones .’ Words and falsearguments cannot b e recalled . Throw a stone into the surface of the s till sea

,and you are powerless to prevent the

circle of disturb ance from spreading more and more widely.

True it is , that one dis turbance may b e overcome andapparently obliterated by other deeper disturbances ; but

UTILITARIANISAI 2 73

Mill’s works and Opinions were dissem inated by the immenseformer influence of the united band of Bentham ist philosophers . He is criticised and discussed and repeated , inalmost every philosophical work of the last thirty or fortyyears . He is taken throughout the world as the representative of British philosophy, and it is not sufficient for a feweminent thinkers in O xford , or Cambridge, or London

,or

Edinburgh , or Aberdeen, to acknowledge in a tacit sort of

way that this doctrine and that doctrine is w rong . Eventu

ally,no doubt

,the opinion O f the Lecture Halls and Com

bination Room s will guide the public Opinion ; b ut it maytake a generation for tacit opinions to permeate society . IVe

must have them distinctly and boldly expressed . It is

especially to b e remembered that the public press throughout the English- speaking countries is m ostly conducted bymen educated in the tim e w hen Mi ll ’s works w ere entirelypredom inant . These m en are now for th e most part cut off

,

by geographical or professional Obstacles, from the direct influence of O xford or Cambridge . The circle of disturbancehas spread beyond the immediate reach of those centres ofthought. To b e brief, I do not believe that Mill

’s immensephilosophical influence, founded as it is on confusion of

thought,will readily collapse . I fear that it may remain as a

permanent obstacle in the way O f sound thinking . Ci tius emer

yi t veri tas ex errore, quam ca: confusi one. Had Mill simply erredas did Hobbes about elementary geom etry, and Berkeley aboutinfinitesimals, it would b e necessary m erely to point out theerrors and consign them to m erciful oblivion . But it is notso easy to consign to Oblivion ponderous works so full ofconfusion of thought that every inexperienced and unwarn edreader is sure to lose his way in them , and to take for pro

found philosophy that which is really a kind of kaleidoscopicpresentation of ph i los0phic ideas and phrases , in a successionof various but usually inconsistent combinations . To thepublic at large

,Mill ’s works still undoubtedly remain as the

standard of accurate thinking, and the most esteemed repertory of philosophy. I cannot therefore consider my criticism

2 74'

jOHIV STUART MILL ’S PHILOSOPHY TESTED

superfluous , and at the risk of repeating much that has beensaid by the eminent critics already m entioned

,or by others

,

I must show that M ill has thrown ethical philosophy intoconfusion as far as could well b e done in ninety-six pages .

The nature of the Utili tarian doctrine is explai ned by Millwith sufficient accuracy in pp . 9 and 10, where he says

‘The creed which accept s as the foundation of morals,U tility

,

or the Greatest Happiness Principle, hold s that actions are rightin proportion as they tend to promote happiness, w rong as theytend to produce the reverse of happiness . By happines s is intended pleasure

,and the absence of pain ; by unhappiness, pain,

and the privation of pleasure . To give a clear view of the moralstandard set up by the theory, much more requires to b e saidin particular, what thing s it include s in the ideas of pain andpleasure and to what extent this is left an Open question . But

these supplementary explanations do not affect the theory of lifeon which this theory of morality is grounded—namely, thatpleasure

,and freedom from pain

, are the only things desirable asends and that all desirable things (which are as numerous in theutilitarian as any other scheme) are desirable either for the pleasure inherent in them selves

,or as means to the promotion of

pleasure and the prevention of pain.

’

Mill proceeds to say that such a theory of life excitesinveterate di se in many m inds

,and among them some of

the m ost estimable in feel ing and purpose. To hold forthno better and than pleasure is felt to b e utterly mean andgrovelling—a doctrine worthy only of sw ine . Mill accordingly proceeds to inquire whether there is anything reallygrovelling in the doctrine—whether, on the contrary, w e

may not include under pleasure, feel ings and motives whi chare in the highest degree noble and elevating. The wholeinquiry turns upon this question—Do pleasures differ inqual ity as well as in quantity ? Can a small amount ofpleasure O f very elevated character outweigh a large amountof pleasure of low quality ? W e should never think of

estimating pictures by their size. The productions of Westand Fuseli, which were the wonder and admiration of ourgrandparents , can now b e bought by the square yard, to

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2 76 jOHN STUART MILL ’

S PHILOSOPHY TESTED

ness, grossness, and sensuality of others ; because I hold that

pleasure s differ in nothing, but in continuance and intensityfrom a just computation of w hich

,confirmed by what

'

w e observeof the apparent cheerfulness

,tranquillity, and contentment

, of

men of different tastes, tempers, stations,‘

and pursuits, every

question concerning human happines s must receive its decision.

’ 1

Bentham,it need hardly b e said , adopted the same idea

as the basis of his ethical and legislative theories . In hisuncomprom is ing style he tells us 2 that

‘Nature has placed mankind under the governance of tw o

sovereign masters, pain and p leasure. It is for them alone to

point out what w e ought to do, as w ell as to determine w hat w eshall do . On the one hand the standard of right and w rong, onthe other the chain of cause s and effects

,are fastened to their

throne . They govern us in all w e do, in all w e say, in all w e

think : every effort w e can make to throw off our subjection willserve b ut to demonstrate and confirm it. In w ord s a man maypretend to abjure their empire : b ut in reality he will remainsubject to it all the w hile . The princip le of uti lity recognises thissubjection, and assume s it for the foundation of that system ,

the

Object of which is to rear the fabric of felicity by the hand s ofreason and of law . Systems which attempt to question it , dealin sounds instead of sense, in caprice instead of reason , in darkness instead of light .

’

Elsewhere Bentham proceeds to show how w e may estimate the values oi

' pleasures and pains , meaning obviouslyby values the quantities or forces . As these feelings are

both the ends and the instruments of the moralist and legislator, it especially behoves us to learn how to estimate thesevalues aright , and Bentham tells us most distinctly.

3

‘To a person,he says, considered by himself, the value of a

pleasure or pain considered by i tself, w ill b e greater or less,

according to the four follow ing circum stances : (1) Its intensity.

(2) Its duration. (3 ) Its certainty or uncertainty. (4) Itsprop inquity1 The Pri ncip les of Moral and Pol i tical Ph i losogohy, Book i , chap. v i , second

paragraph .

2 An Introducti on to the Princip les of Morals and Legi s lat i on, p. 1.

3 Princip les, etc . chap . iv , secs. 2 -5. The statement is not a verb at imextract but an ab ridgement of the sections named .

or remoteness. But when the value of any pleasure or pain is tob e considered for the purpose of e stimating the general tendencyof the act , w e have to take into account also : (5 ) The fec undi ty,or the chance it has of being follow ed by sensations of the samekind

,that is

,pleasure s

,if it b e a pleasure pains

,if i t b e a pain .

(6 ) Its puri ty, or the chance it has of not being follow ed by sensations of the Opposite kind : that is , pains, if it b e a pleasure ;pleasures, if it b e a pain . Finally, w hen w e consider the interestsof a number of persons

,w e must also e stimate a pleasure or pain

w ith reference to Its ertent that i s,the number of persons

to w hom it e x tend s, or w ho are affected by it. ’

Thus did Bentham clearly and explicitly lay the foundations of the moral and political sciences , and to impressthese fundamental propositions on the memory he fram edthe following curious mnemonic lines , which may b e quotedfor the sake of their quaintness

Intense,long, certain, speedy, fruitful, p ure

Such marks in pleasures and in pains endure .

Such pleasures seek, if private b e thy end

If it b e public,w ide let them eaten/l .

Such pains avoid,whichever b e thy V iew

If pains must com e, let them at tend to few .

’

In all that Bentham says about pleasure and pain , thereis not a word about the intrinsic superiority of one pleasureto another. He advocates our seeking pure pleasures ; b utwith him a pure pleasure was clearly defined as one not

likely to b e followed by feelings of the Opposite kind ; thepleasure of opium - eating , for instance , would b e calledimpure , simply because it is likely to lead to bad healthand consequent pain ; if not so followed by evil consequences ,the pleasure would b e as pure as any other pleasure . WithBentham m orality becam e

,as it w ere, a question of the

ledger and the balance- sheet ; all feelings were reduced to

the sam e denom ination of value, and whenever w e indulgein a little enjoyment, or endure a pain , the consequencesin regard to subsequent enjoyment or suffering are to b e

inexorably scored for or against us,as the case may b e .

Our conduct must b e judged wise or foolish according as ,

2 78 jOHIV STUART IlIILL ’S PHILOSOPHY TESTED

in the long-run, w e find a favourable ‘ hedonic ’ balancesheet.What Mill in his earlier life thought about these founda

tions of the util itarian doctrine, and the elaborate structurereared therefrom by Bentham ,

he has told us in his Autobiography, pp . 6 4 to 7 O. Subsequently Mill revolted , asw e all know

,against the narrowness of the Bentham ist

creed. While wishing to retain 1 the precision of expression ,the definiteness of meaning

,the contempt of declamatory

phrases and vague generalities, which were so honourablycharacteristic both of Bentham and of his own father

,

James Mill, John Stuart decided to give a wider basisand a more free and ‘ genial ’ character to the utilitarianspeculations .

Let us consider how Mill proceeded to give this ‘ genial ’

character to the utilitarian philosophy. It must b e adm itted ,he says,2 that utilitarian writers in general have placed thesuperiority of mental over bodily pleasures chiefly in the

greater permanency, safety, uncostliness, etc., of the former

— that is,in their circum stantial advantages rather than in

their intrinsic nature . As regards Bentham ,at least, Mill

m ight have om itted the word chi efly. But according to M ill ,there is no need why they should have taken such a ground .

They might have taken the other,and, as it may b e called

,

higher ground, w ith entire consistency. It is quite compatiblew ith the principle of utility to recognise the fact, that some kinds

of pleasure are more desirable and more valuable than others .It w ould b e absurd

,that while, in estimating all other things,

quality is considered as well as quantity, the estimation of pleasures should b e suppo sed to depend on quantity alone .

’

Then Mill proceeds to point out,w ith all the persuasive

ness of his best style, that there are higher feelings whichw e woul d not sacrifice for any quantity of a lower feeling .

Few human creatures,he holds

,would consent to b e changed

into any of the lower animals for a promise of the fullestallowance of a beast’s pleasures ; no intelligent human being

1 Autob iography, p . 214.

2 Uti li tari anism,p . 11 .

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2 80 jOHN STUART MILL ’S PHILOSOPHY TESTED

is,will in the majority of cases—bring happiness . He must

aim at something which is capable of being reached. M illtells us (p . 1 8 ) that if by happiness b e meant a continuityof highly pleasurable excitement, it is evi dent enough thatthis is impossible to attain .

‘A state O f exalted pleasure las ts only moments,or in some

cases,and w i th some interm issions

,hours or days, and is the

occasional brilliant flash of enjoyment, not its permanent and

steady flame . Of this the philo sophers w ho have taught thathappiness is the end of life w ere as fully aware as those w ho

taunt them . The happines s which they meant w as not a life of

rapture ; b ut moments of such,in an existence made up of few

and transitory pains, many and various pleasures, with a decidedpredominance of the active over the passive, and having as the

foundation of the whole, not to exp ect more from life than i t is capableof bestow ing.

1 A life thus composed, to those w ho have beenfortunate enough to obtain it

,has always appeared worthy of the

name of happines s . ’

Then Mil l goes on to point out what he considers hasbeen sufficient to satisfy great numbers of manki nd (p . 1 9 )

‘The main constituents of a satisfied life appear to b e tw o,either of which by itself is often found sufficient for the purposetranquillity, and excitement. With much tranquillity, many findthat they can b e content with very little pleasure : with muchexcitement

,many can reconcile themselves to a considerable

quantity of pain. There is assuredly no inherent impo ssibilityin enabling even the mass of mankind to unite both.

’

From these passages w e must gather that at any rate themass of mankind will attain happiness i f they are satisfiedwith these main constituents, and w e are especially told thatthe foundation of the whole utilitarian philosophy (Mill doesnot specify the substantive to which the adjective w hole

applies in the above quotation , but it must from the contextb e either ‘ utilitarian philosophy

,

’ search for happiness ,’ or

some closely equivalent idea) is not to exp ect from life more

than i t is cap able of bestow i ng .

The question , then, may fairly arise whether upon a fair1 Ital icised b y the present w ri ter .

UTILITARIAIVISIII 2 8 1

calculation of probabilities they are not wise,upon Mill ’s

own showing, who aim at moderate achievements in life , sothat in accomplishing these they may insure a satisfied l ife.

This seems the m ore reasonable, i f, as Mill elsewhere tellsus, the nobler feelings are very apt to b e killed off by thechilly realities of life .

‘Many, ’ he says (p . 14) w ho begin w ith youthful enthusiasmfor everything noble, as they advance in years sink into indolenceand selfishness. But I do not believe that those w ho undergothis very common change , voluntarily choose the low er description of pleasure in preference to the highe r, I believe that beforethey devote them selves exclusively to the one

,they have already

become incapable of the other. Capacity for the nobler feelingsis in most natures a very tender plant

,easily killed

,not only by

hostile influences, b ut by mere w ant of sustenance ; and in themajority of young persons it speedily dies aw ay if the occupationsto w hich their po sition in life has devoted them ,

and the societyinto w hich it has throw n them ,

are not favourable to keepingthat higher capacity in e x ercise. Men lose their high aspirationsas they lose their intellectual tastes

,because they have not time

or Opportunity for indulging them ; and they addict them selve sto inferior pleasures, no t because they deliberately prefer them ,

b ut because they are either the only one s to w hich they haveaccess, or the only ones w hich they are any longer capable O f

enj oying . It may b e questioned w hether any one w ho has

remained equally susceptible to both classe s of pleasure, everknow ingly and calmly preferred the low er though many, in all

age s , have broken dow n in an ineffectual attempt to combine both.

’

It would seem,then , that for the m ass of mankind there

is small prospect indeed of achieving happiness through highaspirations . They will not have time nor opportunity forindulging them . If they look for happiness solely to suchaspirations they must b e disappointed , and cannot have asatisfied life ; i f they attempt to combine the higher andlower lives they are likely to ‘ break down in the ineffectualattempt . ’ Now

,I subm it that

,under these circum stances ,

it is folly, according to Mill’s scheme of morality, to aim

high ; it is equivalent to going into a life- lottery, in whichthere are no doubt high prizes to b e gained, but few and far

2 82 jOHN STUART MILL ’S PHILOSOPHY TESTED

between . It is simply gambling with hedonic stakes ; preferring a small chance of high enjoyment to comparativecertainty of moderate pleasures . Mill clearly adm its thiswhen he says (p .

‘ It is indisputable that the beingwhose capacities of enj oyment are low has the greatest chanceof having them full y satisfied ; and a highly endowed beingwill always feel that any happiness which he can look for

,as

the world is constituted,is imperfect. ’

Although,then, the foundation of the whole is not to

expect from life more than it is capable of bestowing, w e are

actually to prefer becom ing highly endowed, although w e

cannot expect life to satisfy the corresponding aspirations .That is to say

,although seeking for happiness, w e are to

prefer the course in which w e are approximately certain of

not obtaining it.But Mill goes on to give some explanations . He says

that the highly endowed being can learn to bear the imperfections of his happiness , ‘ i f they are at all bearable (p .

This is small comfort i f they happen to -b e not at all bearable,

an alternative which is not further pursued by Mill . Andwill not this intolerable fate b e most li kely to befall thosewhose aspirations have been pitched most highly ? ButMill goes on

They (that is, the imperfections of life or happines s ?) w ill notmake him envy the being w ho i s indeed unconscious of the

imperfections,b ut only because he feels not at all the good w hich

those imperfections qualify. It is better to b e a human beingdissatisfied

,than a pig satisfied better to b e Socrates dissatisfied,

than a fool satisfied . And if the fool, or the pig, is of a differentopinion, it is because they only know their ow n side of the

question. The other party to the compari son know s both sides . ’

Concerning this position of affairs the most appositeremark I can make is contained in the somewhat trite andlgar saying, Where ignorance is bli ss,

’tis folly to b e wise .

’

If Socrates is pretty sure to b e dissatisfied, and yet, owingto his wisdom , cannot help wishing to b e Socrates , he seemsto have no chance of that individual happiness which depends

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2 84 jOHIV STUART MILL ’S PHILOSOPHY TESTED

on the question of quantity. W hat means are there of determining w hich is the acutest of tw o pains

,or the intensest of tw o

pleasurable sensations, except the general suffrage of those w ho

are fam iliar with both ?

Now,were w e dealing with a writer of average logical

accuracy there would b e considerable presumption that whenhe adduces evidence and claims a result in his own favourin this confident way, there woul d b e some ground for the

claim . But my scrutiny of Mill’s System of Log ic has taught

m e caution in adm itting such presumptions in respect of hiswritings

,and here is a case in point. He claim s that the

suffrage of the maj ority is in favour of Socrates ’ life, althoughhe has adm itted that the vast m ajority of men somehow orother elect not to b e Socrates . He assumes, indeed, that thisi s because their aspirations have been first killed off byunfavourable circumstances ; his only residuum of fact iscontained in this somewhat hesitating conclusion alreadyquoted

‘It may b e questioned w hether any one w ho has remainedequally susceptible to both classes of pleasures, ever knowinglyand calmly preferred the lower though many, in all ages, havebroken down in an ineffectual attempt to combine both.

’

Although,then

,m illions and m illions are continually

deciding against Socrates ’ life,for one reason or another (and

many in all ages who make the ineffectual attempt at acombination break down) , Mill gratuitously assumes thatthey are none of them competent witnesses

,because they

must have lost their higher feelings before they coul d havedescended to the lower level ; then the comparatively fewwho do choose the higher life and succeed in attaining it areadduced as giving a large majority, or even a unanimous votein favour of their own choice . I subm it that this is a fallacyprobably to b e best classed as ap eti tiop ri nc ip i i Mill entirelybegs the question when he assumes that every witnessagainst him is an incapacitated witness , because he musthave lost hi s capacity for the nobler feelings before he couldhave decided in favour of the lower.

UTILITARIANISM 2 85

The verdict which Mill takes in favour of his high-qualitypleasures is entirely that of a packed jury . It is on a parwith the verdict which would b e given by vegetarians infavour of a vegetable diet. No doubt

,those who call them

selves vegetarians would almost unanimously say that it isthe best and highest diet ; but then , all those w ho have triedsuch diet and found it impracticable have disappeared fromthe jury, together with all those whose common sense ,

or

scientific knowledge, or weak state of health , or other circumstances, have prevented them from attempting the experiment .By the same m ethod of decision , w e m ight all b e required toget up at fi ve O

’

clock in the m orning and do four hoursof head—work before breakfast , because the few hard-headedand hard-bodied individuals who do this sort of thing are

unanimously of opinion that it is a healthy and profitablew ay of beginning the day .

O f course , it will b e understood that I am not denyingthe moral superiority of some pleasures and courses of lifeover others . I am only showing that Mill’s attempt toreconcile his ideas on the subject with the Utilitarian theoryhopelessly fails . The few pleasant pages in which he m akesthis attempt (Uti li tari anism, pp . form

,in fact

,a most

notable piece of sophistical reasoning . Much of the interestof these undoubtedly interesting passages arises from the

kaleidoscopic way in which the standing difficulties of ethicalscience are woven together, as i f they were logically coherentin Mill’s mode of presentation . The ideas involved are asold as Plato and Aristotle. The high aspirations correspondto ‘

TO a bv Of Plato . The superior man w ho can judgeboth sides of the question is the Be

’

k rw roq cusp of Aristotle .

The U tilitarian doctrine is that of Epicurus . Now,Mill

managed to persuade him self that he could in twenty pagesreconcile the controversies of ages .Nor is it to b e supposed that Bentham , in making his

analysis of the conditions of pleasure , overlooked the difference of high and low ; he did not overlook it at all— he

analysed it. A pleasure to b e high must have the marks of

2 86 jOHN STUART .lIILL’S PHILOSOPHY TESTED

intensity, length, certainty, fruitfulness , and purity, or of

some of these at least ; and when w e take Altrui sm intoaccount, the feelings must b e of wide extent—that is'

,fruit

ful of pleasure and devoid of evil to great numbers of people .

It is a higher pleasure to build a Free Library than toestablish a new Race Course ; not because there is a FreeLi brary

- bui lding emotion,which is essentially better than a

Race-Course- establishing emotion,each being a simple nu

analysable feeling ; but because w e may,after the model

of inquiry given by Bentham ,resolve into its elements

the effect of one action and the other upon the happiness ofthe community. Thus

,w e should find that M ill proposed

to give geniality to the Utilitarian philosophy by throwing into confusion what it was the verym erit of Bentham tohave distinguished and arranged scientifically. W e musthold to the dry old Jeremy, i f w e are to have any chance Of

progress in Ethics. Mill, at some crisis in his mentalhistory

,

’ decided in favour of a genial instead of a logical andscientific Ethics, and the result is the m ixture of sentimentand Sophistry contained in the attractive pages under review.

In order to treat adequately of Mi ll’s ethical doctrines itwould no doubt b e necessary to go on to other parts of theEssays , and to inquire how he treats other moral elements ,such as the Social or Altruistic Feelings . The existence of

such feelings is adm itted on p . 4 6 , and, indeed , insisted onas a basis of powerful natural sentiment , constituting thestrength of the Utilitarian m orality. But it woul d b e anendless work to exam ine all phases of Mill’s doctrines , and toshow whether or not they are logically consistent inter se.

They are really not worth the trouble. Just let us notice,however, how he treats the question whether moral feelingsare innate or not. On this point Mill gives (p. 4 5 ) thefollowing characteristic deliverance If

,as is my own

belief, the moral feel ings are not innate, but acquired, theyare not for that reason the less natural . It is natural toman to speak , to reason , to build cities , to cultivate the

ground, though' these are acquired faculties . The moral

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2 88 jOHIV STUART IlIILL ’S PHILOSOPHY TESTED

in tracking out his course of thought. The word nature may

b e Mill’s key to a profound philosophy ; but I rather thi nkit is the key to m any of his fallacies .I often amuse myself by trying to imagine what Bentham

would have said of Bentham ism expounded by Mill .Especially would it b e interesting to hear Bentham on Mill’suse of the word natural . ’ No passage in which Benthamanalyses the meaning of nature

,

’ or natural,

’ occurs to '

m e

but the following is his treatment of the w ord unnatural,

as employed in Ethics

Unnatural, w hen it means anything , means unfrequent andthere it means something ; although nothing to the present purpose .

But here it means no such thing for the frequency of such actsis perhaps the great com plaint . It therefore means nothing ;nothing, I mean

,which there °is in the act itself. All it can serve

to express is, the disposition of the person w ho is talking of i t

the disposition he is in to b e angry at the thoughts of it .’ 1

Would that the grand old man,as he stil l sits benignl y

pondering in his own proper bones and clothes,in the upper

regions of a well-known institution, could b e got to deliverhim self in like style about feelings which are not the less

natural because they are acquired .

Before passing on , however, I must point out, in the

extract from p . 4 5, the characteristic habit which Mill has

Of m inim ising things which he is obliged to adm it . Insteadof denying straightforwardly that w e have moral feelings ,he says they are not present in all of us in any ‘perceptibledegree .

’

The moral faculty is capable of springing upspontaneously ‘ in a certain small degree.

’ This willrem ind every reader of the way in which

,in his E ssays on

Religi on,instead of flatly adopting Atheism or Theism ,

whichare clear logical negatives each of the other

,he concludes

that though God is almost proved not to exist,He may

possibly exist,and w e must ‘ imagine ’ this chance to b e

as large as w e can, though it belongs only ‘to one of the

l ower degrees of probability.

’ Exactly the sam e manner1 Pr incip les of Morals and Legislation, ed . 1823 , vol . i , p . 31.

UTILITARIAIVISIU

of meeting a weighty question will b e discovered again inhis demonstration of the non-existence of necessary truths .I shall hope to examine carefully his treatment of this important part of philosophy on a future occasion . W e shallthen find

,I believe, that his argument proves non-existence

of such things as necessary truths,because those truths

which cannot b e explained on the association principle are

very few indeed . I b eg pardon for introducing an incon

gruous illustration , but Mill’s manner of m inim ising an all

important admission often irresistibly reminds me of the

young woman who, being taxed with having borne a child,

replied that it was only a very small one .

Such are the intricacies and wide extent of ethicalquestions, that it is not practicable to pursue the analysisof Mill’s doctrine in at all a full manner. W e cannotdetect the fallacious reasoning with the same precision asin matters of geometric and logical science . This analysisis the less needful too, because , since Mill

’s Essays appeared ,Moral Philosophy has undergone a revolution . I do notso much allude to the reform effected by Mr. Sidgwick ’sMethods of E thi cs, though that is a great one, introducing asit does a precision of thought and nom enclature which waspreviously wanting. I allude , Of course , to the establishment of the Spencerian Theory of Morals, which has madea new era in philosophy1. Mill has been singularly unfor

tunate from this point of V iew . He might b e defined asthe last great philos0phic writer conspicuous for his ignorance of the principles of evolution . He brought to confusionthe philosophy of his master, Bentham ; he ignored thatwhich was partly to replace , partly to complete it .

1 A very important art icle b y Dr. E . L. Y oumans upon Mr. Spencer’sph i losophy has just appeared in the N orth Amer ican Review for O ctob er1879 . Dr. Y oumans traces the h istory of the Evolution doctrines, and provesthe original ity and independence ofMr. Spencer as regards the closely relatedv iew s of Mr. Darw in,

Mr. W allace , and Pro fessor Huxley . The em inent men

in question are no doub t in perfect agreem ent ; b ut Dr. Youmans seems toth ink that readers in general do not properly understand the singular o riginal ity and b oldness ofMr. Spencer's vast and partial ly accompl ished enterprisein ph ilosophy .

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I am aware that, in her Introductory Notice to theEssays on Religion (p . viii) , Miss Helen Taylor apologisesfor Mil l having om itted any references to the works of

Mr. Darw in and Sir Henry Maine ‘ in passages wherethere is coincidence of thought with those writers

,or

where subjects are treated which they have since discussedin a manner to which the Author of these Essays wouldcertainly have referred had their works been publishedbefore these were written.

’ 1 Here it is implied that Millanticipated the authors of the Evolution philosophy in some

of their thoughts,and it is a most am iable and pardonable

bias which leads Miss Taylor to find in the works of one sodear to her that which is not there. The fact is that thewhole tone of Mill’s m oral and political writings is totallyopposed to the teaching of Darwin and Spencer, Tylor andMaine . Mill’s idea of human nature was that w e came

into the world like lumps of soft clay, to b e shaped by theaccidents of life, or the care of those who educate us . Austininsisted on the evidence which history and daily experienceafford of

‘

the extraordinary pliability of human nature ,’

and Mill borrowed the phrase from him .

2 No phrase couldbetter express the m isapprehensions of human nature which ,it is to b e hoped, wil l cease for ever with the last generationof writers . Human nature is one of the last things whichcan b e called ‘ pliable .

’

G ranite rocks can b e more easil ymoulded than the poor savages that hide am ong them . W e

are all of us full of deep springs of unconquerable character,which education may in some degree soften or develop , butcan neither create nor destroy. The m ind can b e shapedabout as much as the body ; it may b e starved into feebleness

,or fed and exercised into vigour and fulness ; but w e

start always with inherent hereditary powers of growth .

The non -recognition of this fact is the great defect in themoral system of Bentham . The great Jeremy was accus

1 Mr. Morley does not seem to countenance any such claims. On the con

trary, he remarks in his Cr itica l Miscel lan ies, p. 324, that Mi ll ’s Essays losein interest b y no t deal ing w ith the Darw inian hypothesis.

2 Autob iograp hy, p. 187.

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2 92 jOHN STUART MILL ’S PHILOSOPHY TESTED

to discuss the grounds or results of the Spencerian philosophy.

TO me it presents itself, in its main features, as unquestionablytrue ; indeed, it is already difficult to look back and imaginehow philosophers could have denied of the human m ind andactions what is so Obviously true of the animal races generally. As a reaction from the old views about innate ideas

,

the philosophers of the eighteenth century w ished to believethat the human m ind was a kind of tabula rasa

,or carte

blanche,upon which education could impress any character.

But i f so,why not harness the lion, and teach the sheep to

drive away the wolf ? If the moral , not to speak of the

physical characteristics of the lower animals, are so distinct,why should there not b e moral and mental differences amongourselves, descending, as w e obviously do, from differentstocks with different physical characteristics ? Notice whatMr. Darwin says on this point

‘Mr. J. S. Mill speaks, in his celebrated work , Utilitarianism(18 64, p . of the social feelings as a “ powerful natural sentiment

,

”

and as“the natural basis of sentiment for utilitarian

morality but on the previous page he says,

“ if,as is my ow n

belief,the moral feelings are not innate

,but acquired, they are

not for that reason le ss natural. It is with hesitation that Iventure to differ from so profound a thinker

,b ut it -

can hardlyb e disputed that the social feelings are instinctive or innate inthe low er animals ; and w hy should they not b e so in man ?

Mr. Bain and others believe that the moral sense is acquired byeach individual during his lifetime .

‘ On the general theory of

evolution this is at leas t extremely improbable .

’ 1

Many persons may b e inclined to like the philosophy of

Spencer no better than that of M ill. But,i f the one b e

true and the other false,liking and d isliking have no place

in the matter. There may b e many things which w e cannotpossibly like ; b ut i f they are , they are . It is possible thatthe Principles of Evolution

,as expounded by Mr. Herbert

1 The Descent of Man, and Selection in Relation to Sex, 1871, v ol . i , p . 71.

I cannot help think ing that M r . Darw in felt the inconsi stency and confusiono f ideas in the passages quoted , although he does not so express h imself. O tw ise, w hy does he quote from tw o pages ?

UTILITARIANISIVI 2 93

pencer, may seem as w anting in geniality as the formulasf Bentham . There is nothing genial

,it must b e confessed ,

bout the mollusca and other cold-blooded organism s withrhich Mr. Spencer perpetually illustrates his principles .Ieaven forbid that any one should try to give geniality toIr. Spencer’s views of ethics by any operation comparable0 that which Mill performed upon Benthamism .

Nevertheless , I fully believe that all which is sinister.nd ungenial in the Philosophy of Evolution is eitherhe expression of unquestionable facts

,or else it is the

putcome of m isinterpretation . It is impossible to see howMr. Spencer, any more than other people, can explain away.he existence of pain and evil . Nobody has done this ;) erhaps nobody ever shall do it certainly system s of

l‘

heology will not do it. A true philosopher wil l not expect;0 solve every thing. But i f w e admit the patent fact thatJain exists , let us Observe also the tendency which Spencerand Darwin establish towards its m inimisati on. Evolutionis a striving ever towards the better and the happier. Theremay b e almost infinite powers agains t us , b ut at least thereis a deep- laid scheme working towards goodness and happiness . So profound and widespread is this confederacy O f

the powers of good , that no failure and no series of failurescan disconcert it. Let mankind b e thrown back a hundredtimes , and a hundred times the better tendencies of evolutionw ill reassert them selves . Paley pointed out how manybeautiful contrivances there are in the human form ,

tendingto our benefit. Spencer has pointed out that the Universeis one deep - laid framework for the production of such beneficent contrivances . Paley called upon us to adm ire suchexquisite inventions as a hand or an eye . Spencer callsupon us to admire a machine which is the most comprehensive of all machines, because it is ever engaged in inventing beneficial inventions ad infi ni tum . Such at least ismy way of regarding his Philosophy.

Darwin,indeed, cautions us against supposing that

natural selection always leads towards the production of

2 94 jOHIV STUART MILL ’S PHILOSOPHY TESTED

higher and happier types of life . Retrogression may resultas well as progression . But I apprehend that retrogressioncan only occur where the environment of a li ving species isaltered to its detriment . Mankind degenerates when forced,like the Esquimaux , to inhabit the Arctic regions . Still inretrograding, in a sense, the being becomes more suited toits circum stances—more capable therefore of happiness .The inventing machine of Evolution would b e workingbadly if it worked otherwise. But

,however this may b e,

w e must accept the philosophy if i t b e true, and , for mypart

,I do so without reluctance .

According to Mill, w e are little self- dependent gods,fighting with a malignant and murderous power calledNature

,sure, one would think, to b e worsted in the struggle.

According to Spencer, as I venture to interpret his theory,w e are the latest manifestation of an all-prevailing tendencytowards the good— the happy. Creation is not yet con

cluded, and there is no one of us who may not becomeconscious in his heart that he is no Automaton , no merelump of Protoplasm

,but the Creature of a Creator.

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[PIE TH'

OD OF DIFFEREN CE

‘In the Method of Agreement w e endeavoured to obtaininstances which agreed in the given circum stances but differedin every other : in the present method w e require, on the con

trary, tw o instances resembling one another in every otherrespect, b ut differing in the presence or absence of the phenomenon w e wish to study. If our object b e to discover theeffects of an agent A, w e must procure A in some set of asoertained circum stances

,as A B C

,and having noted the effects

produced,compare them with the effect of the remaining circum

stances B 0 , w hen A is absent. If the effect Of A B C is a b c,

and the effect of B C, b c, it is evident that the effect of A is a.

’

After pointing out how w e may ,b egin at the other end

and selecting an instance in which it occurs , such as a b c,

must look out for another instance in which the remainingcircumstances b 6 occurwithout a,

Mill proceeds to illustratethe use of the method in these words 1

It is scarcely nece ssary to give exam ples of a logical processto w hich w e ow e almost all the inductive conclusions w e drawin early life. When a man is shot through the heart, it is bythis method w e know that it w as the gun

- shot which killedhim : for he w as in the fulness of life immediately before, all

circumstances b eing the same, except the wound .

’

Briefly described then, this method requires that allcircumstances remain exactly the same except one particulat circumstance, which being altered may b e regarded asthe cause of whatever alteration appears in the consequents .A man is in the fulness of health ; a shot passes throughhis heart ; he falls dead ; ergo the gun—shot is the cause of

his falling dead ; this is approximately certain , because it isvery unlikely that any other circumstance should have beenaltered

,that any other cause but the shot shoul d have come

into Op eration just at the same moment. But then there isno generalisation in thi s experiment ; it is the mere observation of a man in one state followed by a shot and the man

in an altered state. Let us see what M ill thinks may b e

learnt from one experiment.In a number of places he asserts in the most unequivocal

1 Next paragraph .

IlIE TIIOD OF DIFFERENCE 2

way that two instances are sufficient to give a general lawprovided they conform to the requirements of the Method of

Difference . I do not, indeed, find this explicitly stated

where it ought to b e stated , nam ely,in the formal descrip

tion of the Method ; but later on w e find such passages asthe following 1

‘Plurality of causes, therefore , not only does not diminish thereliance due to the Method of Difference

,but does not render a

greater number of observations or e xperiments necessary : tw oinstances

,the one positive and the other negative , are still suffi

cient for the most complete and rigorous induction .

’

In the twenty-first chapter of the third book,

2 speakingof the Me thod of Difference

,he says

That Method authorises us to infer a general law from tw o

instances ; one in w hich A e x ists together w ith a multitude of

other circum stances,and B follow s another

,in w hich

,A being

removed, and all o ther circum stance s remaining the same,B is

prevented . \Vhat,how ever

,does this p rove ? It proves that B

,

in the particular instance, cannot have had any other cause thanA ; b ut to conclude from this that A w as the cause

,or that A

w ill on o ther occasions b e follow ed by B,is only allow able on

the assumption that B must have some cause ; that among itsantecedents in any single instance in w hich it occurs

,there

must b e one w hich has the capacity of producing it at othertimes . This being adm i tted

,it is seen that in the case in

question that antecedent can b e no o ther than A but,that if it

b e no other than A it must b e A,is no t proved

,by these

instances at least,b ut taken for granted.

These remarks can bear no other construction than thatwhat takes place in the instances exam ined w ill take placealw ays ; that is, w here A is present, a w ill follow . The

induction,M ill says , is rigorous and comple te , and induction,

w e remember, w as inference from known to unknown cases .It follows

,for instance , that because one man w as shot

through the heart and died, therefore all men shot through

1 Book i ii,chap. ix, sec . 2 , m idd le of th is paragraph .

2 Midd le of first paragraph .

2 98 METHOD OF DIFFERENCE

the heart will die. Surely this is an obvious inference andit gives a general law—All m en shot through the heartd ie . Does the reader think that the Method Of D ifferenceauthorises such a conclusion ? If so, the results are alarming. Because one man pricks his finger and dies as a resultof it, all men pricking their fingers must die ? Because one

man eats an egg and is immediately ill, all men must b emade ill by an egg

? Because one man goes to sea andsuffers from nausea, all men must suffer likewise ? Suchridiculous results are the outcome of Mill ’s Method. Aman is in the fulness of health ; he steps on a boat whichpushes off into a rough sea ; he suffers from nausea ; nocircum stances are changed except the dry land for the

heaving boat ; therefore A the heaving boat must b e the

cause of a the nausea. So far good ; but Mill says w e havea rigorous and complete induction, and may draw a generalconclusion ; this can b e no other than that all men steppingon to a heaving boat will suffer the effect a .

There is no difficulty in seeing where Mill’s blunder lies .He has overlooked the multiplicity of circum stances, and

that a cause is usually i f not always a conjunction of manycircum stances . Nothing can b e more opposed to fact andscience than that one definite cause A is joined with one

definite effect a. A man steps on to a heaving boat ; nodoubt this is a definite antecedent which may b e symbolisedby A

,i f you like

,and there is a definite result a ; b ut then

the other antecedents are almost innum erable— the man ’sconstitution

,his habit of travelling at sea or otherwise, his

state of health at the time, the accidental condition of hisstomach

,his posture

,or even the way in which his thoughts

are engaged.

The fact obviously is,that from a single experiment all

w e can infer is,that the like result will follow in like cir

cum stances. If the same man with the same state of healthand other circum stances steps on to a heaving boat, he wil lb e ill. If Thomas Jones dies from the prick of a pin , thenWilliam Roberts will also die from a like cause , provided he

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