calculus of relations

8/3/2019 Calculus of Relations

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On the Calculus of RelationsAuthor(s): Alfred TarskiSource: The Journal of Symbolic Logic, Vol. 6, No. 3 (Sep., 1941), pp. 73-89Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2268577 .

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74 ALFRED TARSKIfartheonly exhaustive ccountof thecalculusof relations. At the sametime,this book contains a wealthof unsolved problems, nd seemsto indicatethedirection or future nvestigations.It is therefore ather mazing that Peirce and Schr6derdid not have manyfollowers. It is true thatA. N. Whitehead nd B. Russell, n Principia mathe-matica, ncluded he theory f relationsn thewholeof ogic,made thistheorycentralpart of their ogical system, nd introducedmanynew and importantconcepts onnectedwith he conceptofrelation. Most of theseconceptsdo notbelong,however, o thetheory frelations roperbut rather stablishrelationsbetween histheory nd otherpartsof ogic: Principiamathematicsontributedbut slightly o the intrinsic evelopment f the theory f relations s an inde-pendent deductive discipline. In general, t must be said that-though thesignificance f the theoryof relations s universallyrecognized today-thistheory,specially he calculus of relations,s now npractically he same stage ofdevelopment s that nwhich t was forty-fiveears go.

The fact ust mentionedwas one ofthe chiefmotives fmyselecting he theoryofrelations s the subjectfor his talk. I shall confinemyself ereentirely othetheory fbinaryrelations, ince this s theonlybranchofthegeneral heorywhich s at all developed. Moreover shall be interested xclusivelynthat partof the theoryofbinary relationswhich s knownas the calculusof relations;and I shall indeedbe concernedmerelywiththe calculus of finite perations nrelations, s theywere ntroduced y Peirce. I should ike to acquaint you withtwo different ethods fsetting p the foundations fthiselementary alculusin a rigorously eductiveway, and I should ike to discuss,or at least to formu-late, some metalogicalproblems oncerning his calculus.

The firstmethod am going to considerhere consists n constructing hecalculus of relations s a part of a morecomprehensiveogical theory,whichcorresponds pproximatelyo the restricted unctional alculusas it was given,forexample,by D. Hilbert and W. Ackermann.2In this more comprehensiveogical theorywe have two kinds of variables,individualvariables and relationvariables; as individual variables we use thesmall ettersx',Iy 'z', **, and as relationvariablesthecapital letters RW,S),'Ty * . We have furthern our theory ertain onstants: first he connectivesof the sentential alculus, namely the negation sign I', the implication ign', the equivalence ign ', the disjunction ign V , and the conjunction ign'A'; secondly he twoquantifiers,he universalquantifierII' and the existen-tial quantifierE'.From these variables and constantswemayform ariousexpressions; mongthese we distinguishertain xpressionswhichwe maycall sentences, r rathersententialfunctions. Expressionsof the formxRy' (read 'x has the relationR to y') are called elementary entences, nd we form ompound entences asusual) by puttingnfront fa sentence henegation ign,ora quantifier ithsubscript ndividual variable-e.g., III

2 GrundzUge der theoretischen Logik, Berlin 1938 (second edition), p. 45 ff.


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ON THE CALCULUS OF RELATIONS 75-Or else by combining wo simpler entences y means ofone of the signs a',I', V , ' AIn a well-knownashionwe single ut from mongall sentences certain lassofsentenceswhichwe call axioms,weformulateurtherertain ulesof nferencesuch as the rulesofsubstitutionnd detachmentnd rulesconcerning he use ofquantifiers,nd all sentencesobtained from xiomsby applyingour rulesofinferencenynumber ftimeswe call theorems.We arenowgoing o subjectthistheory o a certainmodification,r, strictlyspeaking, o an extension, ntroducing ertain constantswhichare specific othecalculusofrelations, ltogether leven n number. These include first ourrelation onstants, amely hesymbol 1' for he universalrelation, he symbol'0, forthenullrelation, he symbol 1" forthe identity elation between ndi-viduals) and the symbol 0', forthe diversity elation. Then we have furthersixoperation igns; namely wosymbols orunaryoperations on relations), hesignofthe complement-' and the signofthe converse "; and four ymbolsforbinary perations, he signsofaddition +', multiplication. ', relative ddi-tion j-', and relativemultiplication;'. Finally we have the dentity ign = ',whichdenotes dentitybetween relations. We shall refer o the symbols 1','0o,y', '+', '.' and to the conceptsdenotedby thesesymbols s the absolute(or Boolean) constants nd concepts;thesymbols 1", 'V', 'tw, 'J', ';' and thecorrespondingonceptswill be called the relative or Peircean) constants ndconcepts.Fromrelationvariables,relation onstants, nd operation igns we constructexpressions fa newkind,which re called relationdesignations. Elementaryrelationdesignations re relationvariables and relation onstants. Compoundrelation esignations re formed rom impler nes by putting ymbols orunaryoperations bove them or by joining themby means of symbolsforbinaryoperations. We obtainin thisway suchexpressionss 'R' (read 'the converseofR'), 'R;S' (read the relative roduct fR and S'), and so on.The notionof a sentencewhich appeared in our originaltheoryreceivesacertain extension. As elementary entenceswe now take expressions f theformxRy' and expressions f theformR= Sy,where x' and 'y' standfor nyindividualvariables nd 'R' and 'S' for ny relation esignations. The ways nwhich ompound entences re constructed rom impler nes remain nchanged.To theaxiomsofour original heorywe add as newaxiomscertain entenceswhich are intendedto explainthe meanings f our new constants, nd whichcould forthe most part be regarded s definitionsf our original heorywereprovidedwith ppropriate ulesofdefinition. Thus we have thefollowing ewaxioms:

1. II 11xly.X Y/2. II -IDcx0y.x v3. II xl'x.4.- llll[(xRy A yl'z)--xRz].1: Y Z


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76 ALFRED TARSKI5. lil [xy' (txly)].x y6- 1r [ [x~y (d-,Ry)]x V7. 11 11 Axy yRx]8.1 H[x R+ S y -(xRy V xSy)].9. 1 1[x RSy *+(xRy A xSy)].

10.11 II [x RtSy+- (xRz V zSy)].X v z11. II I [X R;S yE-(xRz A zSy)].X z12. R=S< -*H1(xRy*--*xSy).2 yThe rulesofinferenceemain ssentially nchanged,with the exception hattheruleofsubstitution ow allows the replacementfrelation ariablesnotonly

byother elation ariablesbut also by anyrelation esignations.The theorythus outlinedmay be called the elementaryheory f (binary)relations. If weconfineurselves o those entences nd theorems hichcontainno individualvariables,we obtainthefragmentf thistheoryn whichwe arehere nterested, amely hecalculusofrelations.Let megivehere omeexamples f theorems f thiscalculus:I. (R= S A R = T) - S=T.

II. R= S -*(R+T=S+T A R.T = S.T).III. R+S = S+R A R.S = S.R.IV. (R+S).T = (R-T)+(S.T) A (R.S)+T (R+T).(S+T).V. R+O=R AR.1=R.

VI. R+R = 1 A R.R =0.VII. - I = O.

VIII. R = R.IX. R;S = S;R.X. R;(S;T) = (R;S);T.

XI. R;l' = R.XII. R;1 = 1 V 1;R= 1.


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ON THE CALCULUS OF RELATIONS 77XIII. (R;S). T = 0 -* (S;T) .R = 0.XIV. 0' = 1'.XV. R?S= R;S.

The proofs f thesetheorems resentno difficulties.The simplestway is totransformhe theoremson the basis of axioms 1-12) intoequivalentsentenceswhich ontainno specific onstants fthe calculusof relations nd thento provethe resultingentences ytheusualmethods f the restrictedunctional alculus.As the result f such a transformationf theoremXII, for nstance,we obtain asimple heorem fthefunctional alculus,( E xRy) V (II ZxRy),

which s morefamiliarn the followingorm:(EHIxRy) - (H ExRy).

(Onlythesymbol1" cannotbe eliminated n thisway;but theonlytheoremsnwhichthis symboloccurs,namely heoremsXI and XIV, follow lmost mme-diatelyfrom xioms3-5.)The above way of constructingheelementary heory f relationswillprob-ably seemquitenatural to any onewhois familiarwithmodernmathematicallogic. If, however,we are interested ot in the whole theory f relationsbutmerelyn the calculusof relations,we must admit thatthis methodhas certaindefects rom hepointofviewof implicityndelegance. We obtainthe calculusof relationsn a veryroundaboutway,and in proving heorems f this calculuswe are forced o make use of conceptsand statementswhich are outside thecalculus. It is forthisreasonthat I am going to outlineanothermethod ofdeveloping hiscalculus.In constructinghe calculus of relations ccording o the secondmethodweuse onlyone kind ofvariables,namelyrelationvariables, nd we use the sameconstants s in the firstmethod,with the exceptionof the quantifiers. Fromthese constantsand variables we constructrelationdesignations xactly asbefore. In the construction f sentences,however,certain modifications renecessaryon account of the absence of individualvariables and quantifiers.As elementary entenceswe take only sentencesof the form R = S', where'R' and 'S' stand forrelationdesignations; nd we form ompound entencesfrom impler nes by meansofthe connectives f the sentential alculus.Moreoverwe singleout certain entenceswhich we call axioms. These canbe divided ntothreegroups.The axioms of the first roup characterize,o to speak, the meanings f thesentential onnectives: lthough ll the constants ccur n them, t is only thesentential onnectiveswhich occur in them n an essentialway. In order toformulate heseaxioms,we take any axiomsystemfor the sentential alculuswhich contains as primitive erms all the sentential connectivespreviously


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78 ALFRED TARSKIenumerated,nd substitute orthe variablestherein rbitraryentences fourcalculus (in all possibleways).The axioms of the secondgroup serve to characterize he meaningsof theabsolute constants. We obtaintheseaxiomsby takingany set of axiomsforBoolean algebra nd replacing lass variablesby relation ariables;e.g.,wemayfor hispurposeuse theorems-VII given bove.3 Thus thepartofthe calculusof relationswhich nvolvesonlythe absolute conceptscoincides n its formalstructurewithBoolean algebra.The axioms of the thirdgroup are specific o the calculusof relations, ndexpressfundamental roperties f the relativeconcepts. As these axioms wemay take theoremsVIII-XV. The first ourofthese involveexclusively hesymbols1", '"', and ';', the next two establish omeconnections etween heabsolute nd relative oncepts, nd the ast twomaybe considered s definitionsof thesymbols0" and 'J'.

Since all the above axioms are theorems f the calculusofrelations s con-structed your firstmethod, hey reundoubtedlyrue entences rom hepointof view of the intuitivemeaningwhichwe ascribeto the symbols ccurringnthem.We may, however,make it plausible n anotherwaythatthesesentences retrue,namelybymeans ofa geometric epresentation.4 et us supposethat ourrelationvariablesdenoterelations etweenreal numbers, nd let us considerrectangular oordinate ystem n the plane. Every relationR may thenberepresenteds a certainpoint set in theplane,namely as the set of all points(x, y) suchthatx has therelationR to y; and converselyverypoint set in theplanerepresents certain elation, amely he relationwhichholdsbetween wonumbers andy if and only fthe point x, y) belongs o theset. The formula'R= S' is valid in thisgeometrical epresentationf and only fthe sets corre-sponding oR and S are identical. The relations ,0, 1', 0', arerepresentedycertainparticular ointsets: viz., 1 by thewholeplane, 0 by the emptypointset, the identityrelationby the straight ine whose equation is 'x= y', thediversity elationby the set of all pointsnot on this straight ine. To theoperations n relations here orrespondertainoperations n pointsets. Theabsoluteoperationsrerepresentedy theusualset-theoreticperations npointsets:R+ S bytheunionofsets,RI.S by the ntersection, by thecomplement.In order o obtaintherepresentationorR.,we take thepoint et correspondingto R and rotate t (in threedimensions) hrough n angle of 1800 about theline x= y. Unfortunatelyt is moredifficult o explain the meaning of thegeometricalperationswhich orrespondo relative ddition nd relativemulti-

3This part of our axiom system is due essentially to E. V. Huntington; see his paperSets of ndependentpostulatesfor thealgebraof logic, Transactions of theAmerican Mathe-matical Society,vol. 5 (1904), p. 292 f. It is easily seen that axioms I-VII are symmetricwith respect to the absolute constants '+' and '.', as well as '1' and '0'. It would bedesirable to have a simple and independent axiom system forthe whole calculus of relationswhich would be symmetric with respect to both absolute and relative constants.4 Cf. Schr6der, op. cit., p. 52 ff.; C. Kuratowski and A. Tarski, Les operationslogiques etles ensemblesprojectifs,Fundamenta mathematicae, vol. 17 (1931), p. 240 ff.


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ON THE CALCULUS OF RELATIONS 79plication. We have to resort o three-dimensionalpace and to supplement urcoordinate ystemwitha z-axis perpendicularo thexy-plane. Then, in orderto obtain the representationfR;S (in the xy-plane),we rotate the pointsetcorrespondingo R through right ngle about the x-axis,draw through verypointof the resultinget a straightine parallelto they-axis, nd take theunionof all thesestraightines; n thisway we obtain the "cylindrical" oint set R*.Similarlywe rotatethe set correspondingo S through right ngle about they-axis,drawthe linesparallel to the x-axis,and obtainthe "cylindrical"pointset S*. Finally we take the intersection f R* and S*, and project it or-thogonally pon the xy-plane. The projectionthus obtained constitutes hegeometrical epresentationf R;S. The representationf the relative um caneasily be obtainedfromthat of the relativeproduct, n view of axiom XV.The construction ecomesmuchsimpler n the case that one of the termsofthe relative sum or the relativeproduct s the universalrelationor the nullrelation. For example, t is easilyseen that the relationR;1 holds betweenxand y if and only f there s a z suchthatxRz (so that it is, so to speak, inde-pendentof y); therefore,n orderto obtain the geometrical epresentationfR;1, we consider he set correspondingo R, drawthrough verypoint ofthisset a straightine parallelto they-axis, nd take the unionof all these straightlines. The samemethod s applied in the case of 1;R, except thatthestraightlinesare drawnparallelto the x-axis.

With the aid of thisgeometric epresentation,he content f all our axiomsfor hecalculusof relationss easilymadeintuitivelyvident. Thus,e.g.,axiomVIII correspondso theintuitivelybvious geometric act that, fwe take anypointset and rotate t throughn angle of1800 about a givenstraightine, ndthen rotate t again through 800 about the same straightine,the result s theoriginal ointset.Let us now consideraxiom XII. We have to show that either the setrepresenting;1 or the set representing;R is thewhole plane. Suppose thattheset representing;1 is not the wholeplane. As we have seen,this et is theunionof all straightineswhich reparallelto they-axis nd whichpass througha pointbelongingo the setwhich epresents . If thisuniondoesnot coincidewiththe wholeplane,theremustbe a line parallelto they-axiswhichdoes notcontainany pointof the set correspondingo R, and which thereforeonsistsexclusively f pointsof the set correspondingo R. If through verypointofthis ine we draw a straightineparallelto thex-axisand take the unionofalltheseparallel ines,we obtainthewholeplane. Hence the relation ;R is repre-sentedby the whole plane.

Continuinguraccountof thedevelopment f the calculusof relations ccord-ingto the secondmethod,wehave to describe hewayinwhich heorems f thecalculusof relations re to be derivedfrom ur axioms. Here we proceed s inthe case of the firstmethod; .e., we formulate ulesof inference,nd we calltheorems hose sentenceswhichcan be derivedfrom xiomsby applyingtherules of inference. But the situation s now simpler n that we are able torestrict urselves o two rulesof nference-the uleof substitutionnd the ruleofdetachment.


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80 ALFRED TARSKIIn orderto illustrate he techniqueofthisdomain,we shall outline heretheproofs f severaltheorems fthe calculusofrelations. In theseproofswe pre-supposefamiliarity ith the sentential alculusand with Boolean algebra,andwe shall apply laws of these two theorieswithout xplicitlyndicating hem.XVI. [(R;S)- T = 0*-. (S;-)-R = 01 A [(R;S). T = O*-*t;R). = 01.Proof. In XIII substituteI' for T'. This gives:

(1) (R;S); = 0 -> (S;Ti) = 0.Again in XIII substituteS', 'p', and 'R' for R', 'S', and 'T' respectively,oobtaining:(2) (S:T) .R = 0 --. = 0.ApplyingXIII forthe thirdtime, with R', 'S', and 'T' replaced by 'T', 'R',and 'S' respectively, e obtain:(3) ( ;R) * 0 --> R;S) . = O.Now (1)-(3) imply:(4) [(R;S) * = O?-* S;T) R = 0] A [(R;S) = 0 (T;R)* = 01.On the otherhand wehave byVIII:(5) T=T.From (4) and (5), theoremXVI followsdirectly.

XVII. R.S = 0 - RP. = O.Proof. By XI we have:

(6) R;1'= R A 9;1'Hence obviously:(7) (?1)* .If now in XVI we replace R', 'S', and 'T' by '1?', '1", and 'C?' respectively, eobtain,by (7):(8) (1';A)* = 0.By VIII we have:(9) R-S = 0 -R". = O.By virtue fa law of Boolean algebra, 8) and (9) imply:_ _(10) R.S = 0 -+ (1';)R = 0.ApplyingXVI again,with R', 'S', and 'T' replacedby 'Ty, 1"', and S respec-tively,we obtain:(11) (R;1') O-* (1;9).RW = O.


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ON THE CALCULUS OF RELATIONS 81From (10) and (11) follows:(12) RAM 0 --+), = 0.Fromthistogetherwith (6) we obtainXVII immediately.XVIII. R = S -R=,

Proof. We have the following ormulaewhere 13) is a well-knownaw ofBoolean algebra, 14) follows romXVII, and (15) is a particular ase of (13)):(13) R = S- (RAS 0\I R0 ).(14) (R9=0A S.R=O)-(R.?=0 A ~R=0).(15) R = B (Ray 0 A BAR = 0).From theseXVIII followsdirectly.

XIX. R+S = R+K.Proof. For simplicitywe replace R+~' by 'T'. Since obviously

(16) R.T=O A T=0,we infer y XVII that(17) R. =OA A i=0.Hence by virtueofVIII,(18) Rg= OA ST= 0,and consequently,(19) (R + S). = 0.If now in XVII we replace R' by 'R+S' and 'T' by 'p',we obtain,by (19),(20) R+S- 0=and hence,byVIII,(21) R+S- T= 0, i.e., R+S R+= 0.Since,on the otherhand,(22) R.R+S=O A S.R+S=O,we obtain,again using XVII,(23) R . R+S = 0A .R+S = 0,and hence,(24) (R+2) * R+S = 0.


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82 ALFRED TARSKIFrom (21) and (24), theoremXIX follows cf. formula 13) in the proof ofXVIII).

XX. =0 A =1.Proof. ApplyingXVII twice,the first ime with R' replaced by 'O' and 'S'by '6', and the second time with R' replacedby 1' and 'S' by 1', we obtain:(25) .=O A 1. =O.Hence by VIII:(26) 6.0= A1.1=0.Hence further:(27) 6.1 = O 1 =O.And from his,XX follows t once.

XXI. S.T=0-+(R;S) *R;T=O.Proof. We have obviously:

(28) (R;T) * = O.In XVI replace S' by 'T', and 'T' by 'R;T'. By (28) we obtain:(29) (R.T -R) *X=O.By XVII we have:(30) SoT= O =O.Formulae (29), (30) imply:(31) SIT = 0-i (R;T;R) * 0ApplyingXVI again, with R;T' put in place of T', we obtain:(32) (R;S)R;T=O(R;T;R) 0.Then XXI follows mmediately rom 31) and (32).

XXII. S = T -R;S = R;T.Proof. XXII can be derivedfromXXI in exactlythe same way as XVIIIfromXVII.XXIII. R;(S+T) = (R;S)+(R;T).Proof; For brevitywe replace (R;S)+(R;T)' by 'U'. We have obviously:

(33) (R;S)*U = 0 A (R;T).U = O.


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ON THE CALCULUS OF RELATIONS 83Hence by XVI we obtain:(34) (U;R)-4 = 0 A (U;R)T = O.Consequently:(35) (0U;R) _.Since, by XIX,(36) S+T=we infer rom35) that:(37) (U;R) *S+ T = 0.Now in XVI replace S' by 'S+T', and 'T' by 'U'. Using (37), we obtain:(38) [R;(S+T)].U = 0, i.e., [R;(S+T)]- (R;S)+(R;T) = O.On the otherhand we have:(39) S-S+T=0A T-S+T =O.Hence by XXI:(40) (R;S) *R;(S+T) = 0 A (R;T) * R;(S+T) = O.Hence further:(41) [(R;S)+(R;T)] R;(S+T) = O.From (38) and (41), XXIII followsdirectly cf. formula 13) in the proof ofXVIII).

XXIV. R;O = 0.Proof. By XX we have:(42) (l;R).6 =0.

Hence, by meansofXVI (with S' replacedby '0' and 'T' by '1'):(43) (R;0).1 = 0.Then XXIV is an immediate onsequenceof (43).

Proofs of the fourfollowing heorems re entirely nalogous to those ofXXI-XXIV:XXV. R.S=0-(R;T) *S;T=O.XXVI. R = S- R;T = S;T.XXVII. (R+S);T = (R;T)+(S;T).XXVIII. O;R = 0.XXIX. 1' = 1'.


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84 ALFRED TARSKIProof. By IX and XI we have:

(44) R;1'= 1';Rand(45) Pia't= A.ApplyingXVIII to (45), we get:(46) A;i' = A.Hence by (44):(47) 1';R= R.Withthe aid ofVIII and XXII we obtainfrom 47):(48) 1';R= R.Substituting ere 1" for R', we get:(49) = 1'.On theotherhand we have,byXI:(50) = 1'.And XXIX followsdirectly rom 49) and (50)..

XXX. l';R = R.Proof. ApplyingXXVI to XXIX, we obtain:

(51) 1';R = 1';R.This formula ogetherwiththe formula 48) (see the proof of XXIX) yieldsXXX directly.

XXXI. (1;S) * = 0 -+ (1;T) = 0.Proof. By XVI we have:

(52) (1;S)- T = 0 (0;1) . = 0.By applyingXXII to XX we obtain:(53) ;1= T; LOn the other hand, IX gives:(54) 1;T =Formulae (52)-(54) imply:


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ON THE CALCULUS OF RELATIONS 85In whatfollows,(1;T). S' willbe replacedby 'U'. We have obviously:(56) Us71;T = 0 A U.S= 0.Hence, applyingXVII twice,we obtain:(57) U . 1;T =0 A U[.2 = 0.In consequence f laws ofBoolean algebra, 57) leads to:(58) t 1;T.= 0.Hence:(59) 1;T . 0 aE= 0.By XVIII we have:(60) E OHence inviewof VIII and XX:(61) a = 0 -+U = 0, i.e., t = 0 -+(1;T).S = O.Formulae 55), (59), (61) together ieldXXXI.

XXXII. (R 1 = 1) *-+ (1;R);l = 1.Proof. Since

(62) R = 1 = 0,we have,byXXII,(63) R 11;R= 1;0,and hencebyXXIV,(64) R= 1 -1;R =0.From (64) we obtain,usingXXVI,(65) R = 1 -- (1;);l = 0;1,and henceby XXVIII,(66) R= 1+(1;R);l =0.From (66) resultsobviously:(67) (1;R);l = 1-+ (a. R = 1).On the otherhandwe obtain fromXII, replacingR' by '1;R":(68) (1;R);l = 1 V 1; 1;R = 1.From XXXI, substitutingR' for S' and '1;R' for T', we infer:(69) (1;1;R).R=0.


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86 ALFRED TARSKIHence:(70) 1 =.And hence:(71) 1;1;R=1-OR=1.Then (68) and (71) together mply:(72) (a R = 1) (1;R);1 = 1.AndXXXII followsmmediately rom 67) and (72).

The theoremust provedplays a very mportant olein the calculusofrela-tions, ince t allowsus to provethefollowing eneralmetalogical heorem:Every entencefthe alculusofrelations an be transformedntoan equivalentsentencef the orm R= S', and even f the ormT= 1'.5

In fact,wemayfirst, n the basisof a well-knownheorem f Boolean algebra,transformvery equationR =S

into an equationof theformT= 1,namely nto

R.S + R.S = 1.Now, on the basis of theoremXXXII, we may transformhe negation of anequationof the formT=1', i.e.,

AT= 1,into an equationof the formT= 1', namely1;T;1 = 1.Furtherwe are able to transform conjunction f two such equations, .e.,

T = 1 A U = 1,intoone equation, namely

T.U = 1.Since it is knownfrom he sententialcalculus that it is possible to eliminatefrom very sentence ll sentential onnectives xcept the signsof negation ndconjunction, t followsfromthe above transformationshat our metalogicaltheorem s valid.

This metalogical heorem uggests till anotherway of constructinghe cal-culus ofrelations. For it shows thatwe may confine urselves, n developing6 See Schroder, op. cit., p. 153.


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ON THE CALCULUS OF RELATIONS 87this calculus,to sentenceswhich have the formof equations (or whichevenhave theformT=1'), thus dispensingwith the concepts nd theorems fthesentential alculus. For thispurposewe shouldhave to put all ouraxioms ntotheform fequations, and to give ruleswhichwouldpermitus to derivenewequationsfrom ivenones. Thoughthisplanhas not beenworked utindetail,the realization f t presentsno essentialdifficulty.

In the further evelopment f the calculusofrelationswe may introduce ewconcepts definablewith the help of the fundamental onceptsof the calculus.We may,for nstance, istinguishertain specially mportant ategories frela-tions, uch as symmetric elations, ransitive elations, rdering elations, ne-many relations r functions, nd one-one relations r biunique functions. Inthis connectionthe followingpoint deserves special attention. It may benoticed hat many aws of the calculus of relations, nd in particular he axiomsVIII-XI adopted underour second method, esemble heorems f thetheory fgroups,relativemultiplication layingthe r6le of group-theoreticomposition,and the converse of a relation correspondingo the group-theoreticnverse.Let us now consider riefly he relations atisfying he condition:

R;R = 1' A R;R = 1'.This condition xpresses n ordinarymathematical erminology hat the rela-tion R maps the class of individualson itself n a one-to-onemanner. If weconfine urselves o relations atisfyinghiscondition,we can easily provethatthey atisfy ll the axiomsofthetheory fgroups. Thus it turnsout that thecalculus of relations ncludesthe elementary heoryof groupsand is, so tospeak, a unionof Boolean algebraand group heory. This fact accountsfor hedeductivepowerand mathematical ichness fthecalculus.

I should ike now to discuss ome metalogical roblems egarding he calculusofrelations.The firstproblemconcerns he relation betweenour two methodsof con-structinghe calculus. It is obviousthatevery theoremwhichcan be provedby the second method can also be proved by the firstmethod since,as men-tionedbefore, he axiomsadoptedunder the secondmethodcan be provedastheorems nder hefirstmethod, nd all therulesof nference sed inthesecondmethodwerealso assumedin the firstmethod). It is, however, y no meansobvious that the converse s also true, nd that consequently ur two methodsare entirely quivalent. Since it follows rom resultofK. Godelrthat everysentencewhich s truein every domainof individuals s provable by our firstmethod, heproblemcan also be put in thefollowingway: Is it the case thateverysentenceof the calculus of relationswhich s true in every domain ofindividuals s derivablefrom he axioms adopted under the second method?This problempresents omedifficultiesnd stillremains pen. I can only saythat I am practically urethatI can prove with the help of the second method

6 Die VollstandigkeitderAxiome des logischenFunktionenkalkils,Monatshefte fUrMathe-matik und Physik, vol. 37 (1930), pp. 349-360.


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88 ALFRED TARSKIall of the hundreds ftheoremso be found n Schroder'sAlgebra ndLogic derRelative.The nextproblem s the so-calledrepresentationroblem. Is everymodelofthe axiom system fthe calculusofrelationssomorphic ith class ofbinaryrelationswhich contains the relations 1, 0, 1', O' and is closed under all theoperationsconsidered n this calculus? As is known,the analogous problemforBoolean algebrawas raised, nd solved n theaffirmative,y M. H. Stone.7For the calculus ofrelations heproblem emains pen; a particular ase of t-for n atomisticystem fthetheory frelations-wasrecentlyolvedbyJ.C. C.McKinsey.8 The twoproblems-thatoftheequivalence four twoconstructionmethods, nd therepresentationroblem initsapplication o the axiomsystemadopted underthe second method)-are relatedto each other:from n affirma-tive solutionofthe secondproblemwe couldobtainan affirmativeolutionofthe first.In discussing he remainingmetalogicalproblemswe shall have in mind ourfirstmethodofconstructinghe calculusofrelations.The first f these problems s knownas the decisionproblem. It can beformulatedn thefollowingway: Is therea methodwhich would enable us ineveryparticular ase to decidewhether givensentence xpressedn the termsof the calculus of relations s a theorem f this calculus? We knowfromresultof A. Church9hatno such method xistswith regard o what we calledtheelementaryheory frelations-i.e.,withregard o sentenceswhichcontainnot only relationvariables but also individualvariables. With the aid of thisresult t can be shown hat the solution fthe decisionproblems likewisenega-tivefor hecalculusofrelations roper.10The next problem oncerns he connection etween heelementary heory frelations-or, what is practically he same, the restricted unctional alculus-and the calculusof relations. Since the calculus of relations s only a properpartof thetheory frelations,heproblem riseswhethervery entence ormu-lated in theelementary heory f relations nd concerned ssentially nly withthe properties f relations i.e., containing nlyrelationvariablesas freevari-ables) canbe transformedntoan equivalent entence fthecalculusofrelations.In a less exact form hisproblem an be put as follows: s it true that everyproperty f relations,relation between relations,operation on relations, tc.which can be definedn the restricted unctional alculus can be expressed lsoin thecalculusofrelations? It was shownby A. Korseltthat the answer. o thisquestionis negative."1His proof,however,dependsessentially n admitting

The theory frepresentationsorBoolean algebras, Transactions oftheAmericanMathe-matical Society, vol. 40 (1936), pp. 37-111; see in particular p. 106.8 Postulates for thecalculus of binaryrelations, this JOURNAL, Vol. 5 (1940), pp. 85-97;see in particular p. 94.9 A note on theEntscheidungsproblem,his JOURNAL, Vol. 1 (1936), pp. 40-41 (and Correc-tion, bid., pp. 101-102).0The proofwill be given later in a separate paper.11Korselt's result was published in the paper of L. LMwenheim,OberMoglichkeiten mRelativkalkWl,Mathematische Annalen, vol. 76 (1915), pp. 447-470.


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ON THE CALCULUS OF RELATIONS 89finite omains of individualswith various numbers f elements. I have shownthat the resultremainsvalid if we confine urselves o infinite omains of ndi-viduals. It followsfrommy reasoningthat even the properties f relationsexpressedby such simpleformulae f the restricted unctional alculus as11I >2xRu A yRuA zRu)X y 2 tSor

, Z(xRy AxRz AxRu AyRzA yRuA zRu)X V 2 Ucannotbe expressedn the terms fthecalculusofrelations; .e., no sentence fthecalculusofrelationss satisfied y exactly he same binary elations betweenelementsof an infinite et) whichsatisfyeither one of the above formulae.Moreover, have succeeded n considerably xtending his result. For I haveshownthat the answerremainsnegativeeven if we enrich he calculus of rela-tions by the additionof any finite umber f new constantsdenoting ixed ela-tions,properties frelations, perations n relations, nd so on, providedthat.all concepts ntroducednthis way are invariant nderany one-to-onemappingofthe class of individualson itself.10 It followsfrom n earlierresultby A.Lindenbaum nd myself'2hat this nvarianceproperty elongs to all conceptsdefinablewithin he restricted unctional alculus,or even withinmuchmorecomprehensiveogical systems, uchas thatofPrincipiamathematicalOur last metalogicalproblem s closely relatedto the preceding ne. Sincethere re sentences ftheelementary heory frelationswhichcannotbe trans-formed ntoequivalent entences fthe calculusofrelations,we maylook foracriterion hichwouldenableus to decide neveryparticular ase whetheruchatransformationspossible. We are here onfronted ith new decisionproblem.This problem s so far unsolved, but it seems plausible that its solution snegative.

The aim of thispaper has been, not so much to presentnew results, s toawaken nterestn a certainneglected ogical theory,nd to formulateomenewproblems oncerninghis theory. I do believe that the calculusofrelations e-servesmuchmoreattention han it receives. For, aside from hefact that theconceptsoccurringn thiscalculuspossessan objective mportance nd are inthesetimes lmost ndispensablen anyscientificiscussion, he calculusof rela-tionshas an intrinsic harm nd beautywhichmakes t a sourceof ntellectualdelight o all whobecomeacquaintedwith t."3HARVARD UNIVERSITY

12 Uber die Beschrdnktheitder Ausdrucksmittel deduktiver Theorien, Ergebnisse einesmathematischen Kolloquiums, no. 7 (1936), pp. 15-22. (There are several misprints n thatpaper. In particular, read 'metamathematisch' and 'Metamathematik' instead of mathe-matisch' and 'Mathematik.')18 I wish to take this opportunityto express my gratitude to Dr. J. C. C. McKinsey forhis assistance in preparing this paper.

calculus of relations

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