some comments on de morgan, peirce, and the logic of relations - r.m. martin

7/29/2019 Some Comments on de Morgan, Peirce, And the Logic of Relations - R.M. Martin

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Some Comments on DeMorgan, Peirce, and the Logic of RelationsAuthor(s): R. M. MartinReviewed work(s):Source: Transactions of the Charles S. Peirce Society, Vol. 12, No. 3 (Summer, 1976), pp. 223-230Published by: Indiana University PressStable URL: http://www.jstor.org/stable/40319775 .

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R. M. MartinSomeCommentsnDeMorgan,Peirce,ndtheLogicof Relations

AugustusDe Morgan's "On the Syllogism,No. IV, and on theLogicofRelations" of 18591 s surely landmark n thehistory flogic. In thispapera theory fdyadicrelationsmaybe said to have been formulatedorthefirstime.Peirce was awareof thecontents fthispaper apparentlyn1866 or 1867 and expressed gain and again his indebtedness o it.2 Evenso, it is remarkablehatPeircefailedto follow De Morgan until1883 inone important espect. t was in his paperof thatyearthat Peirceforthefirstimegainedthefulleffectfvariables verrelations nd over ndivid-uals and admittedrelations s entitiesfullyon a par with classes.3ButDe Morgan did so rightfrom hebeginning. n thisrespecthis work ssuperiorto anythingPeirce did on relationsuntil 1883. By that datePeirceat last had fullpossession of the quantifiers,n termsof whichofcoursethelogic ofrelations akeson its fullymodernform.4Let us look at De Morgan's paper n some detail.We will then be in apositionto see preciselyust whatPeirce could have gained from t andwhathe added to it.

"Anytwoobjectsofthought rought ogether ythemind, nd thoughttogethern one act ofthought, re n relationp. 339]. ... All ourpreposi-tionsexpressrelation . . but thepreposition f s theonlyword of whichwe can say that t is, or maybe made, a partof the expressionof everyrelation . . thoughthe same thingmaynearlybe said of theprepositionto.When relationcreates noun ubstantive,f s unavoidable: ifA by itsrelation o B be C, it is a C of B. A volumemightbe written bout theidiom of relation:but it would be of thematter, ot of the form, f thesubject." This subjectivistic ccount of what a relation s is all that isprovided,butnone of theformalwork n anywaydepends upon it. Andnote the interestingomment bout prepositions s expressing elations,whichhas an extraordinarily odernring.5However, De Morgan doesnothingmore with prepositionsand sets immediately o work on therelations heyhelp to express.

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224 R. M. MartinDe Morgan concerns himself nlywiththe "formal aws of relation"so far s is necessary or he treatmentfthesyllogism.Just s Peircehashis Procrustean Boolian" equations,so De Morgan has his Procrusteansyllogisms."Let the names X, Y, 2, be singular ... I do not use themathematical ymbols of functionalrelation


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SomeCommentsnDeMorgan, eirce,nd theLogicofRelations 25here, ut t s clear hatDe Morgan ntendsll ofthese ormulaen thesense f 3')-Ofcourse3') is logicallyquivalentot(EW)(XTw-WcMtY)>,whichs closer oDe Morgan'siteraleadinghatX is oneofthe or s amemberfthe)Ls ofoneof theMs of Y. TheY ofclassmembershipsread is one of. ('(M'Y)' maybe defineds in Principia athematica,*30.01, nd MT' is characterizedy*32.13.)There s no doubtbutthatDe Morganhasdiscovered ere henotionof therelativeroductf tworelationsnessentiallytsmodernorm. ealsohasthenotion ftheordinaryogical umof tworelations utdoes"not atpresentind tnecessaryouse" it. Thus(4) 'X..(L,M)Y'expresseshat X is either ne oftheLs of Y or one ofthe Ms [ofY],orboth."Two furthernterestingotions resymbolizedy LM/f nd 'L/M\The use ofthe ittle uperiorndinferiorccents yno means ccordswith he mportancef thenotions ymbolizedymeansofthem.Thefirsts to "signifyn L ofevery [ofY]" andthe econd an L ofnonebutMs [ofY]." Clearlyhese otionsmay ewritten,n termsfrelationalabstraction,s

'XY(Z)(Z MYDXLZ)1and 'XY(Z)(X LZDZMY)1respectively.e Morgan ives hese otions onames, utPeirce hristensthemn 1870 and 1902)as theordinaryorforwardrrelativerogressive)involutionndbackwardorrelativeegressive)nvolutionfL andM, respec-tively.6De Morgan ntroducesext he mportantotionof converse.TheconverseelationfL, L~\ is defineds usual: fX..LY, Y.J/^X: ifX beoneoftheLs ofY, Y is oneoftheL""^ ofX." Andconversely,fcourse.The contraryornegation fa] relations introduceds follows. If XbenotanyL ofY, X is toY insomenot-L elation."f wouldbe betterif not-L' nd relation' erewere nterchanged.husX.. not-LY if andonlyifX.LY.In true terativetyle, e Morganconsidersontrariesf contraries,contrariesfconverses,nd conversesfcontraries,nunciatinghefol-lowing rinciples,ithmerewhisps fproof:(5) not-Ls theconversef not-L)""1.



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226 R. M. Martin(6) L""1 nd (not-L)"1recontrariesfeach other.(7) Thecontraryf a converses theconversef thecontrary.(8) Ifa firstelationecontainedn a second, hen he onversefthefirsts containedn theconversefthesecond;but thecontraryf thesecondnthecontraryfthe irst.(9) The conversionf compoundelationonvertsoth omponents,and nvertsheir rder.(10) Where heres a signof nherentuantity' or/], feachcom-ponent e changednto tscontrary,ndthesignofquantitye shiftedfrom necomponento theother,here s no changenthemeaning fthe ymbol.In essentiallyhefamiliar otation fPrincipia athematica,hereULis the conversefL, -?-Ltscontrary,nd C' is thesignfor elationalinclusion,hese rinciplesre s follows:(50 -L = iM-s-L),(60 UL= ^u(-L),(70 -K"L) = u(-s-L),(80 LCMD(ULCUM- -M C -s-L),(90 u(L/M) = (UM/UL), a(100 X(Y)(Y6M3XLY) = X(Y)(-XLYD-YeM).Theproofs fall of these re mmediaten termsftheir uantificationalstructure.The next rinciples:(11) When compound as no inherentuantity,he ontrarys foundby takinghecontraryfeitheromponent,ndgivingnherentuantityto theother.Herethemost ikely enderings(110 -XZ(EY)(XL Y YMZ) = XZ(Y)(YMZD-XLY).Theproof,gain, s immediate. notherrinciples that(12) If a compound elation e containedn another elation.. thesamemaybe saidwhen itheromponents converted,ndthecontraryof theother omponentnd of thecompoundhange laces.In otherwords,(120 (L/M)CND ((UL/UN)C -M (-s-N/uM) -*-L))or(12") (X)(Y)(X (L/M) Y 3 X N Y) D ((X)(Y)(X (


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SomeCommentsnDeMorgan, eine, nd theLogicofRelations 27"A relations said tobe convertible. . when t s tsownconverse: henX..LY givesY..LX" for ll X and Y. Clearlywe have here hemodernnotion fsymmetry.e Morgannotesmmediatelyhat13) LL"1 s con-vertibleor ach L. Thus(13') (X)(Y)(X(L/uL)YY(L/uL)X).Alsohe comments14) thatLL"1)'andL/L""1reconversesfeachother,so that

(14') (X) Y) ((Z) (Z ULY D X LZ) m (Z) (Y L Z D Z uLX))."A relations transitivehen relativef a relatives a relativef thesamekind," hat s,L is transitiveustwhere(15') (X)(Y)(Z)((XLY YLZ) D XLZ).De Morgan otes hat16) "a transitiveelation asa transitiveonverse."(160 (X)(Y)(Z)((X LY-YLZ)DXLZ)D (X)(Y)(Z)((X


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228 R. M. Martindescendantof none but non-descendants 18'], a non-descendantof allnon-ancestors19']> nda descendant fnone but ancestors 20'].A descen-dant s alwaysan ancestor fnone but descendants 21/], non-ancestor fall descendants 22'], a non-descendant f none but non-ancestors 23']>and a descendant of all ancestors 24']. A non-ancestors alwaysa non-ancestorof all ancestors 25'], and an ancestorof none but non-ancestors[26'].A non-descendants a descendant fnone butnon-descendants27;],and a non-descendant f all descendants 28']. Amongnon-ancestors recontained all descendants of non-ancestors, nd all non-ancestorsofdescendants 29']. Amongnon-descendants re containedall ancestorsofnon-descendants,nd all non-descendants f ancestors 30']."After hisperoration principia texempla, e Morgan returns o hisold stampingground,thesyllogism, nrichednow with theadmissionofrelationalformulae.His workhere is of considerable nterestn its ownright, ut liesbeyondtheconfines f thepresent iscussion.However,thewhole of his theoryof relational yllogisms eems reducibleto just oneform. The universal nd all containingformof syllogism s seen in thestatement f X..LMZ is [as?] the necessaryconsequence of X..LY andY..MZ." In terms fthetruth unctionsndquantifiers e clearly avethat(31') (X)(Y)(2)((X L Y Y M Z) D X (L/M) Z).It is from histhat ll othervalidrelational, yllogistic orms re derivable.It is of nterest o observe hat, lthoughDe Morganhas fullpossessionof the notions of relativeproduct,relativeprogressive nvolution,andrelative egressivenvolution,he makesno mention of the relative umoftworelations, hat s, of

XY(Z)(XLZvZMY).This notion remains o be introducedbyPeirce n his 1870 paperand the"Note B" paperof1883. And neither eircenorDe Morgan mention herelative imultaneousnvolutionofL and M,

XY(Z)(XLZ = ZM Y).X would be said to bearthesimultaneousnvolute ofL and M to Y, justwhereX bearsL to all and onlythethings hatbearM to Y - loverof alland onlythebenefactors f,or servant f all and onlythe loversof,andso on.

It is interestinglso thatDe Morgan does not introduce xplicitly helogicalproductf tworelations, ortheuniversalr//relation.he reasonis no doubt that his Procrustean ed is thatof the syllogismrather han



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SomeComments nDeMorgan,Peirce, nd theLogic ofRelations 229that of Boolean algebra.These notions also must await the 1870 paperofPeirce.

Also De Morgan lacksthenotionof a definiteRussellian) description,and thus cannot handle such important otions as (L'X), the one objector ndividualbearingL to X, as implicitly otedabove. Nor does he intro-duce the class of referentsf a givenrelation, 'X, theclass of all objectsbearingL to X, nor thatofthe class ofall relataof a givenreferent,'Y.Nor does he introduce he notionsofthedomainand conversedomain ofa given relation.Nor that of the plural descriptivefunction,L"K, theclass of all objectsthatbearL to some member f K.Peirce likewise failsto introducethese notions. The difference,ow-ever, s thattheywere all withinDe Morgan's grasp. He was clear thatobjects, classes, and relations re differentinds of entities, nd thus tobe handled appropriately. eirce never satisfactorily istinguishes ndi-viduals (or singulars)from lasses,and it took himmanyyearsclearly oseparateout relationsfromrelativesr classes of objectsstanding n givenrelationsto such and such. Both logicians had available the notion ofidentity, ut it is inconceivablethatPeirce could everhave hit upon thenotionsmentionedn theprecedingparagraph, specially hatof a definitedescription, rior o 1883,whereasDe Morgan had all thenecessary oolsimplicitlyt hiscommand.Of course, fter 883,withquantifiersxplicitlyavailable to Peirce, the whole of relational ogic takes on new wings,including hatoftriadic elations.The pity s that, n theyears fter 885,Peirce became a Daedalus preoccupiedwiththe existentialgraphs,andlost hiswingsaccordingly.But that s another tory.

In conclusion,some criticismfC. I. Lewis's commentson De Morganare in order.8 ewis says (p. 45) thatfor De Morgan "L or M, written yitself,will representhatwhichhas therelationL, or M..., and LY standsfor nyX whichhas therelation to Y...," and (p. 50) that he"introduc-tion of quantificationsnd the systematicmbiguity fL, M, etc.,whichare used to indicateboth the relationand thatwhich has the relation,hurry... De Morgan] into complicationsbefore the simple analysis ofrelations, nd typesofrelations,s readyfor hem." But it is not truethatDe Morgan exployshisrelational ymbols n thisambiguousway.On thecontrary, e is explicit p. 341) that X', 'Y', and 'Z' are singulartermsand he alwaysuses 'L' and 'M' as termsforrelations.Expressionsof thekind LY' do not occurexcept n the context X..LY' or X.LY'. Nor is ittruethatcomplicationsresultfrom hisalleged ambiguity.De Morgan's



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230 R. M. Martintreatmentf atomic ormulaeontainingelationalerms s surprisinglymodern, ore o than hat fPeirce pto1883, swehavenoted. urther,it s a positivemeritf De Morgan'swork hat he"quantifications"reintroduced,amely,herelativerogressivendregressivenvolutionsftwo relations.t is remarkablehathe handles hese o skillfully,ackingthequantifiers,s hedoes, ndgiven heclumsinessf his notation.Lewis concedesthat"it shouldalwaysbe rememberedhat t wasDe Morganwho aidthefoundationofthe ogicofrelations]."ndeed,towardsheend ofhispaper,De Morganhimself riteshat herethegeneraldea ofrelationmerges,ndfor he first ime n thehistoryfknowledge,henotions frelationnd relationfrelationre ymbolized."By relation frelation's meant, fcourse, ot relationsfhigherypebetween elations,utrelative roducts. ne cannotdissent erefromDe Morgan's wn estimatefwhathehad achieved.9ThenstituteorAdvancedtudyndNorthwesternniversityFOOTNOTES1. TransactionsftheCambridge hilosophicalociety (1864), 331-358.2. See especiallyEmilyMichael, "Peirce'sEarlyStudyof theLogic ofRelations,1865-1867,"TransactionsftheCharles . Peirce ociety (1974), 63-75.3. See theauthor's On Peirce's RelationalFormulae f 1883," to appear.4. See the author's"On Individuality nd Quantificationn Peirce's PublishedLogic Papers,1867-1885,"below pp. 231-45.5. Cf. the author's"On Some PrepositionalRelations," n TheLogical Enterprisein honor of FredericFitch (New Haven: Yale University ress, 1975), pp. 51-59;"On the Logic of Prepositions,"Papers Presented t the InternationalongressfLogic,Methodology,nd PhilosophyfScience,ondon, Ontario,1975, XI, pp. 15-16;and "On PrepositionalProtolinguistics,"n Konstruktionenersus ositionen,d. byK. Lorenz n honor of Paul Lorenzen Berlin:W. de Gruyter,o appear).6. Collectedapers, .77, 3.113,and 3.640.7. Collectedapers, .45 ff., .154 ff,,nd 3-323ff.8. In A SurveyfSymbolicogic Berkeley:Universityf California ress,1918),pp. 45-51.9. The author s grateful o the National Endowmentforthe HumanitiesforsupportunderGrantFC1O5O3.

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some comments on de morgan, peirce, and the logic of relations - r.m. martin

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