Logic, Part I

8/13/2019 Logic, Part I

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,LOGI CPART I

BYw. E. JOHNSON,M.A.FELLOW OF XING'S COL EGE, CAMBRIDGE,

SIDGWICtr LECTURER IN UORAL SCIBNCE IN TIIEUNIVERSITY OF CAMBRIDGE.

DOVER

ndnra pelel mA d Alxedc

PUBLICATIONS, INC.NEW YORK


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r72 CHAPTERXarchitect, r in the classhypocrite? If this questionsanswered n the negative it must be on the groundthat, in somesenseof the term'exist'which is not ap -propriate to classes,Mr Pecksniffdoes not and neverdid exist, and hencehe cannot count as one when weare enumeratinghe memberscomprisedn any givenclass. Furthermore, ince he numericalpredication'atleast one'is highly indeterminate nd could be in thisor that casereplacedby the relatively determinate atleast n' where z stands for this or that number, theaffirmationthat 'the class C exists' s only a specialandlessdeterminatecaseof the affirmation hat 'the classCcomprisesat least z items,'and the number n cannotbe counted as such unless all the n items erist, Theconclusion herefore ollowsthat the senseof the word'exist'when predicatedof a class s dependentuponthat of the word 'exist'when predicatedof an item orindividual indicatedby a proper or uniquely descriptivenamet.

I This contention is directed against the position held in thePrinciy'ia Matlumatica, where E is ultimately defined in terms of f,,whereas n my view tr is to be ultimately defined n terms of E

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CHAPTER XITHE DETERMINABI,E \

$ r. Irs this chapter we propose o discussa certaincharacteristicof the adjective as such, which perhapsthrows the strongest ight upon the antithesisbetweenit and the substantive. Here it will be apposite o con-sider the traditional accountof the principlesof logicaldivision where a class of substantives) s representedas consistingof sub-classes.This process s governedby the following rules: (r) the sub-classesmust bemutually exclusive;(z) they must be collectivelyex -haustiveof the class o be divided; (S)division of theclass nto its co-ordinatesub-classes ustbe baseduponsolneone 'fundamentumdivisionis.' The first two ofthese rules may be said to be purely formal, and donot raiseany problemrof mmediate nterest; bur th etechnical term fundamentum divisionis-though per-haps readily understoodby the learnel-i5 ssgu4lly n-troducedwithoutexplicitaccountof its connection ith,or its bearing upon, ideas which have entered nto theprevious logical exposition. To illustrate the notionwe are told, for instance, hat, when a classof things isto be divided according o colour,or to size,or to someother aspect in which they can be compared, hen thecolour, size, or other aspectconstitutes he fundamen-tum divisionis. Now although,grammaticallyspeaking,words like colour and sizeare substantival, hey are in


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r74 CHAPTERXIfact abstract nameswhich stand for adjectives; so thatthe fundamentum divisionis is, in the first place, anadjective,and in the second,an adjective of the par-ticular kind illustrated by 'colour' when considered nits relation to red, blue, green, etc. Superficially thisrelation appears o be the sameas hat of a singleobjectto some classof which it is a member: thus two suchproposit ions s'Red is a colour'and'Plato is a man'appear o be identical in form; in both, the subject ap-pears as definite and singular,and, in both, the notionof a class o which thesesingular subjectsare referredappears o be involved. Our immediate purpose s toadmit the analogy, but to emphasise he differencesbetween hese wo kinds of propositions,n which com-mon logic would have said we refer a certain object toa class.I propose to call such terms as colour and shapedeterrninables n relation to such terms as red and cir-cular which will be called determinates;and, in intro-ducing his new terminology, o examine he distinctionbetween the relation of red to colour and the relationof Plato to n an. To predicate colour or shay'eof anobject obviously characterisest lessdeterminately hanto predicate of it red or circular; hence the formeradjectivesmay be said negatively to be indeterminarecomparedwith the latter. But, to supplement his nega-tive accountof the determinable,we may point out thatany one determinable uchas colour s distinctl yotherthan such a determinableas shape or tone; i.e. colouris not adequatelydescribedas indeterminate,since itis, metaphorically peaking, hat from which the specificdeterminates, ed, yellow, green, etc.,emanate; while

THE DETERMINABLE T75from shapeemanateanother completelydifferent seriesof determinates uchas riangular,square, ctagonal,etc.Thus our ideaof this or that determinablehasa distinctlypositive content which would be quite inadequately e-presentedby the word 'indeterminate.' Further, whathave been assumed o be determinables-e.g. colour,pitch, etc.-are ultimately diferent, in the importantsense hat they cannot be subsumedunder some onehigher determinable,with the result that they are in-comparablewith one another; w_hile_its the essenrialnature of determinatesunder any one determinable obe comparablewith one another. The familiar phrase'incomparable' s thus synonymouswith ,belongingtodifferentdeterminables,' nd 'comparable'with ,belong-ing to the samedeterminable'; not that this is the actualmeaning of the terms, but that enquiry into the reasonfor the comparabilityor incomparabilityof two qualitieswill elicit the fact tha they belongto the sameor to dif-ferentdeterminables especting to' is alsomore usuallyin relation to its class: ryesignificanceof the relation ''in one case o a determinatein the other to an individualand ts class. If it is askedwhy a numberof different individuals are said to belongto the sameclass, he answer s that all these differentindividuals are characterisedby some the same adjec-tive or combinationof adjectives. But can the samereason be given for grouping red, yellow and green(say) n one classunder he namecolour? What is mostprominently notableabout red,green and yellow s thatthey are different, and even,as we may say, opponent


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176 CHAPTERXIto oneanother; s there any (secondary) djectivewhichanalysis vouldrevealas characterisingall these differ-ent(prirnary)adjectives?n my view there s no such(secondary)adjective; in fact, the severalcolours areput into the same group and given the same namecolour, not on the ground of any partial agreement,buton the ground of the special kind of differencewhichdistinguish es ne colour rom another;whereas o suchdifference xistsbetween colouranda shape.Thus redand circular are adjectives etweenwhich there is norelationexceptthat of non-identityor otherness;whereasred and blue, besidesbeing related as non-identical,have a relation which can be properly called a relationof differencg where difference means more than mereotherness.(What s here true of colour s true of shape,pitch, feeling-tone, ressure, nd so on: the groundforgrouping determinatesunder one and rhe samedeter-minable is not any partial agreementbetween themthat could be revealedby analysis, ut the unique andpeculiar kind of difference hat subsistsbetween heseveraldeterminates nder the samedeterminable, ndwhich does not subsist between any one of.them andan adjectiveunder someorher determinable.) f this isgranted, he relationsasserted n the two prbpositions'Red is a colour' and Plato is a man,' thoughformallyequivalent,must yet be contrastedon the ground thatthe latter but not the former is basedupon an adjectivalpredication. For the lattLr is equivalent o preiicatingthe adjectivehuman'of 'Plato, 'while,withoutdenyingthat some adjectivesmay properly be predicatedof(the adjective) ed,yet the proposition Red is a colour,is not equivalent to predicating any adjectiveof red.

THE DETERMINABLE T77$ z. Bearing n mind this distinction , he questionariseswhether what are called abstract names can bedivided in the sameway as concretenames nto singularand general; in other words,whetheradjectives anbe

divided into these wo classes. The answerseems o bethat adjectivescan be divided into two classesmore orless analogous o the singular and general which dis-tinguish substantives, ut that the two different kinds ofadjectives are preferably distinguished as determinateand indeterminate. When, in consideringdifferentde -greesof determinateness,he predicationof oneadjectiveis found to imply another, but not conversely, hen theformer we shall call a super-determinateof the latterand the latter a sub-determinateof the former. Thusthe relation of super-determinate o sub-determinatemeansnot only that the former is more determinate hanthe latter, but also that the predication of the formerwould imply that of the latter. A simple example canbe taken from the determinable number': thus 7 issuper-determinateo 'greater than 3' ; the adjective'greater han 3,'though nbt itsel f asamnaana etermin-able, may be called determinable, nasmuchas it is notmerely ndeterminate ut capableof 9iqg furthel de -term_i-nsdinhe sense hat it generatesa definite seriesof determinates. To illustrate more preciselyrvhat istn."tti by ;g.n.t"tes'; let us take the determinablelessthan4' ; then less han4'generates3'and'z 'and'r 'in the sense hat the understandingof the meaningofthe former carrieswith it the notionof the latter. Nowno substantive lass-nameeneratests membersn thisway; take, or nstance,the apostles f Jesus,'he under-standing of this class-name arries with it the notion


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r78 CHAPTERXI'men summonedby Jesus o follow him,' but it doesnot generate Peter and John and Jamesand Matthewetc., 'and this fact constitutes ne important differencebetween he relationof sub-determinateo super-deter-minateadjectives nd that of general o singular sub-stantives. < /

$3. Another equallysignificant ifferences broughtout by considering hat aspectof substantive-classesnwhich-to use he terminology of formal ogic-increaseof intension s accompanied y decrease f extension.The phrase 'increaseof intension' conjures up thenotion of adding on one attributeafter another,by thelogical process alledconjunction; so hat, taking b,g, r,to be three adjectives,ncrease n intensionwould beillustrated by regarding y', q, r conjoined as giving agreater intension thanp, g; and?, q as giving greaterintension than P. We have now to point out that theincreased etermination f adjectivalpredicationwhichleads o a narrowing of extensionmay consist-not ina process f conjunctionof separate djectives-but inthe processof passing from a comparatively indeter-minate adjective o a comparativelymore determinateadjectiveunder the samedeterminable. Thus there isa genuinedifference etween hat process f increaseddeterminationwhich conjunctivally ntroduces oreignadjectives, nd that other process y which without in -creasing, o o speak, henumberof adjectives, edefinethem more determinately.In fact, he foreign adjectivewhich appears o beaddedon in the conjunctiveprocess,s really not intro-duced from outside,but is itself a determinateunderanother determinable,present from the start, though

THE DETERMINABLE T7gsuppressedn the explicit connotarionof the genus.We propose o use a capital letter to stand for a de-terminable,and the correspondingsmall letter withvarious dashes to stand for its determinates. Thus.in passing rom the genusy' to the species g, we arereallypassing rompQ to ?{; or again he apparent n-crease f intensionfrom ? to ?g to pqr is morecorrectlysymbolisedas a passing rom ?8R to pgR to ?qr. Inthe successiverocess f dividinga summumgenus nt othe next subordinatesub-genera, nd this again intosub-sub-genera,he summumgenusought to be repre-sentedby a conjunction f determinables,ayPQRST;thegeneranextsubordinateothis,bypQRST,p'QRS T,Pt'QRST, etc.,and the generanext subordinate o thefirst of these by y'gRST, pg'RST, pq"RST, and so ondown to the in_7tmapeciesepresented y determinates.Thus: PQRST-

Nfsr p'gEr p"Ahsr. . . .pqEsr p-4 sr-psCsr -...In this way we represent rom the outset he natureof the ultimate individuals under the summumgenus,as being characterisableointly by the determinablesPORST, while any genusor speciesis epresented ythese same determinables, ne or more of which arereplaced y determinates.This meets criticismwhichhas often been directedagainst he formal accountofthe inversevariation of extensionand intension, incewe seenow that t he samenunaber f adjectives houldbc used n giving the connotation f thewider as of then;rrrowerclass. To illustrate these symbols from abotanical lassification f plants: let the determinable


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r8o CHAPTERXIP stand for the number of cotyledons, for the dis-positionof the stamens,R for the form of the corolla,S for the attachmentof the petalsand sepals, nd 7 forthe divisibility of the calyx. Then PORST representsthe summum genus 'iblants'asdescribableunder thesefive heads,but otherwise undetermined in character.Then , /, ?", might stand respectively for having nocotyledons,having t, and having z, thus representingthe defining characteristicof eachof the threeclasses-acotyledon, monocotyledon,and dicotyledon-by thesymbols 'QRST, /QRST, y''tQRST Again {, y', 9",might stand respectively or the stamensbeing under,aroundor upon the carpels, hus representing he threesub-divisions-hypogynous, perigynous,epigynous-ofdicotyledons, sy'ttgRsT, /"7'RST ?" "RSZ Takingregular and irregular to be the two possible forms ofcorolla, hen the next sub-division undery'"y'RST willbe y'"gtrST and p"y'r'ST. Again s and.y' may standrespectively for separability and inseparabilit y of thecalyx and corolla,and yield the further sub-divisions,sayy't'g'rsT, p'tgtr(7. The calyx may contain only onepart or 3 or 4or 5 or 6, and if these are representedrespectivelyby t, t', t", t"', /"', a relatively determinatecharacterisations finally symbolisedby p"gtrstt"' t^y.There are cases for which a modification of thisgeneral scheme s required. The casesare those inwhich one particular sub-division is definable by theabsence f an element upon which the predication ofother determinables epend,while in the sub-divisionsco-ordinate i th this the element n question s present.For example, the class of plants called acotyledonsmight be definedby the absence f any corolla,etc.,and

THE DETERMINABLE T8Thencesuchvariations as that of the form of the corollaor the dispositionof the stamens,etc., are inapplicableto this particular sub-division. * v

$ +. Now adjectives under the same determinableare related to one another in various ways. One rela-tional characteristicholds in all cases; namely that, ifany determinate adjectiv.echaracterisesa given sub-sJfntive, then it is impossible that any other determi-nate under the same determinable should characterisetHe same substantive: e.g. the proposition that 'thissurface s red' is ihcornpatiblewith the proposition'this(same)surface s blue.' It has been usual to modify theabove statement by adding the qualification-at thesame time and at the same place; this qualificationapplies where the substantive extends through someperiod of time and over some region of space,n whichcasethe existent substantive,having temporal or spatialparts, may be said to be extended. For this reasonthe qualificationwould perhaps better be attached tothe substantive itself, and we should say that, whereopponentadjectivesarepredicated, eferences made odifferent substantives,since ny onepart of an extendedsubstantive is existentially other than any other part"

A secondqharacteriptig f many determinatesunderthe same determinable is that the di&renselhgt$nenaiflg5"n, "i{e-ofd,et*."{$rllggs: "orgtone another; so that if a, b, c, are three determinates,th'ere .te """et in which we may say that the differ-encebetweena and e is greater than that between aand b; e.g. the dif ference betwien red and yellow isgreater than that between red and orange. In thiscase the several determinates are to be conceived as


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IB2 CHAPTERXInecessarily ssuming a certain serial order, which de-velops fgg the ideaof what may be called 'adjectivalbetweenness.'lTheerm 'between' s usedhere n a fa-\miliar metaphortical ense erived rom spatial relations,and is figuratively magedmost naturally n spatial orm.)Thus if b is qualitatively betweena andc, and c qualita--tively betweenb andd,andso on, the wholeserieshas tsorder directly {gggrminedb;r he natureof the adjectivesthemselves. The further distinctionsamongst seriesasinterminableor as cyclic, and again of series of morethan one order of dimensions,ead o logicalcomplexi-ties which need not be entered nto here. Suffice t tosay that this characteristic,which holds of so manydeterminates, ives significanceo anotherwell-knownrule for logical division: diuisio nonifaciat saltum: onemeaning of which appears to be that we contemplatenot merely enumeratinga set of coordinatesub-classes,but enumerating hem in a certain order. The rule pre--s"q11b_qgbat the order in which.the sub-classes re enq-meratedshould correspond o the orderof 'betweenness'predicableof their differentiating characteristics.The order of betweenness hich characteriseshedeterminates ust consideredmay be either discreteorcontinuous. I n the caseof discreteseries here sonede-terminate that can be assignedas next after any givendeterminate;but, in the caseof a continuous eries,determinate can always be conceivedas between anytwo given determinates, o that there are no two deter-minateswhich can be said to be next to one another nthe serialorder. Itfollowsfrom his account f continuitythat, between any two determinateswhich may be saidto have a finite adjectiva ldifference,may be interpolated

THE DETERMINABLE I83an indefinitenumber of determinates aving a finite dif-ference, nd this number becomes nfinite as the differ-ences ecome nfinitesimal.Amongst continuous eriesfurther differencesbetween the interminable and thecyclic,and again between those of one or more orderof dimensions, old as n discrete eries.The reference here to determinablesof higher orIower dimension equiresexplanation. Our familiar ex-ample of colour will explain the point: a colour mayvary according o its hue, brightnessand saturation; sothat the precise determinationof a colour requires usto define three variables which are more or less nde-pendentof oneanother n theircapacity f co-variation;but in one important sense hey are not independentofone ano ther, since they could not be manifested n se-paration. The determinablecolour is thereforesingle,though complex, n the sense hat the several consti-tuent charactersupon whose variations its variabilitydepends re inseparable. 'x

$ S. Returning to the conceptionof the absolutelydeterminate adjective, we have to note an importantdistinctionbetween absolutely determinateand com-paratively indeterminatepredications. The distinctionmay thus be formulated: If, of two substantiveshesamedeterminate adjective can be predicated, then allthe adjectivesand relations definable in terms of thedeterminable, hat can be predicatedof the one, couldbe predicatedof the other. But i{, of two substantivesthe same ndeterminate adjectivecanbe predicated, henonly certain of the adjectivesand relationsdefinable nterms of the determinable, hat can be predicated ofthe one, can be predicated of the other. To illustrate


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r84 CHAPTERXIfirst the case of an indeterminate predication; Iet ustake the numerical adjective greater than 7'; then ofany collectionof which this numerical adjectivecouldbe predicated,other adjectivessuch as 'greater than 5'and greater han3'could alsobe predicated;but somecollectionshat are rgreater than 7'such as the apostles,are greater than r r and divisible by 4 for instance,whereas ther collectionshat are'greater han 7,'suchas the muses,are less han r r and are not divisible by4: hence t is only someof the numericaladjectives hatare predicableof the muses hat are alsopredicableofthe apostles,although the adjective greater than 7' ispredicableof them both. Turning now to the caseofdeterminatepredication; i{, instead of defining a col-lection by the indeterminateadjective greater than 7we had defined t by the determinateadjective'rwelve,'then any numericaladjective that is predicable of onecollectionof twelve, say the apostles,would be predic-able of any other collection of twelve, say the monthsof the year or the sonsof Israel; for example, greaterthan r r,' 'divisibleby 4,' 'a factor of 96.' What wehave here seen o hold of determinate and indetermi-nate numberholdsof any otherdeterminable. The caseof colour lends tself easily or illustrationon accountofthe specific names which have been assigned to itsdeterminates:hus, f thecolours f two different bjectsare the sameshade gf yellow, then though these twoobjectsmay differ in any number of other respects uchas shapeand size, yet we may say that any colour-property of the one object will agreewith the colour-property of the other; if the colour of one is morebrilliant or less saturated han the colour of an orange,

THE DETERMINABLE I85then the same will hold for the colour of the other.In fact, whatever sensationaldeterminable we take,whether t be colour, or sound,or smell, he determinatecharacterisationsunder any such determinablewouldlead to the same orms of generalisation hat have beendeveloped by scienceonly in the sphere of quantity.It is agreed hat in the sphere of senseperception,dif-ferences of quality strictly speaking hold only of themental or sensational, nd that the physical can onlybe defined in quantitative terms. Thus in the Weber-Fechner experiments the experient judges of equiva-lence or difference n the intensity or quality of hissensations,with which are correlatedquantitative dif-ferences n the stimuli. The attempts that psycholo-gists have made to discover formulae of correlationbetween he stimuli on the onehand and the sensationson the other hand show hat determinatengssn a.quali-tative or intensive scalecarriGivJtii it the same ogicalconsequences s doesdeter.rninatenessf magnitude orphysically measurablequ-angr-t-ies.urthermore deter-minatenessn either case s,only approximatelyattain-able,whether we rely upon the immediate udgmentsofperception or are able to utilize instrumentsof measure-rnent. The practical impossibilityof literally determi.nate characterisationmust be contrastedwith the uni-versally adoptedpostulate hat the charactersof thingswhich we can only characterisemore or less ndetermi-nately, are, in actual fact, absolutelydeterminate'.

I The notion of the Determinable will U" ,t ot{fn hter chaptersto have mportance n a large number of applications.

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Logic, Part I