Language and Logic in Wittgenstein’s Tractatus

Abstract This paper investigates Wittgenstein’s account of the relation between elementary and molecular propositions (and thus, also, the propositions of logic) in the Tractatus Logico-Philosophicus. I start by sketching a natural reading of that relation - which I call the “bipartite reading” - holding that the Tractatus gives an account of elementary propositions, based on the so-called picture theory, and a different account of molecular ones, based on the principle of truth-functionality. I then show that such a reading cannot be attributed to Wittgenstein, because he holds the view that an explanation of logical complexity is already given by a correct account of the (pictorial) nature of elementary propositions; this is implied in his claim that “an elementary proposition contains all logical constants/operations in itself”. After clarifying Wittgenstein’s notion of an operation from the Notes on Logic to the Tractatus, I finally explain why Wittgenstein claims that an elementary proposition contains all logical operations in itself, and hence why he can be said to provide a unified (and thus not bipartite) account of language and logic.


The "Bipartite Reading" andthe internalunity of language and logic
The Tractatus'a ccounto fp ropositionsi nt erms of pictures, as is wellr ecognised, directly applies only to what Wittgensteinc alls elementary propositions; an elementaryp roposition -t he "simplest kind of proposition" (TLP 4.21) 1 -i sa" nexus, ac oncatenation of names" (TLP 4.22),a nd it has ap ictorial natureb ecause its elements (names)are combined in the sameway as the elements of its corresponding state of affairs. Wittgenstein's accounto f molecular -complex -propositions (among which, importantly, propositions of logic aret ob ef ound) on the otherh and,r elieso n the idea that they are truth-functions of elementaryp ropositions, namely propositionsw hose truth-value is determined by the truthvalue of their constituent propositions;e xampleso fs uch propositions arethose obtained by theapplication of familiartruthfunctional connectivess uch as '~' (negation), '&' (conjunction), 'v' (disjunction), and so forth. As section 5o ft he Tractatus reads: "A proposition is at ruth-functiono fe lementary propositions". One theo ne hand,t herefore, we have elementaryp ropositions,w hich directly pictures tates of affairs,a nd on the other we have molecularp ropositions -truth-functions of elementaryp ropositions.
This brief summary of the Tractatus'conceptionofthe nature of the proposition encouragest he thought thatt he Tractatus'a ccount of language andlogicrests on two different principles: the principle of pictorial representation( which accountsf or elementary propositions), and the principle of truth-functionality (accounting form oleculara nd quantifiedp ropositions,a nd also, therefore,f or logical complexity);onthis view, then, it seems as if -as Elizabeth Anscombe has it -" [t]he whole theory of propositions is […] a merely externalc ombinationo ft wo theories: a 'picture theory' of elementary propositions (viz. that they have meaning by being 'logical pictures' of elementary states of affairs),a nd the theoryo f truth-functions as an accounto fn on-elementary propositions" (1959: 25-26). This is thev iew actually endorsed by Georg von Wright,a ccording to whom" Wittgenstein's Tractatus may be called as ynthesis of thet heory of truth-functions andt he idea that language is ap icture of reality" (1955: 533); K. T.F ann, similarly, holdst hat "Wittgenstein's theory of language in the Tractatus has two components:t he 'picture theory'a nd the 'truth-function theory '" (1969: 8).
This readingo fW ittgenstein'sa nalysis of language and logic in the Tractatus Ic allt he "bipartiter eading";a lthoughp erhaps a naturalw ay of conceivingo ft he relationb etween elementary and molecularp ropositions in the Tractatus,t he bipartiter eading seems also problematic in many respects. Brian McGuinnessexpresses his worriesabout it in thefollowingterms: [I]n thef irst part of the Tractatus […]w es eem to be told that the essence of ap ropositioni st ob eap icture, whilei nt he later partsw e are toldt hat itsessence is to be atruth-function […]. [A] […] serious difficultyi st hatt he twoa ccountss eem to be quite separatet hings, and, if thisisso, cannot bothbeadequateaccountsofwhatitist obe aproposition. (2002: 65-66) More recently, MichaelMorrisargued: Therei sarisk of understandingW ittgenstein's account of language […] as falling into twoc ompletelyu nconnected parts: onew hichi s appropriate to thec onceptiono fe lementarys entences as models, the otherw hich concerns the construction of others entences outo f elementary sentences. (2008: 234) Theb ipartiter eading, therefore, by holding that Wittgensteinh as an account of the nature of elementary propositions andadifferent (unrelated) oneo fm olecular propositions, seemst om aket he relationbetween elementary and molecular propositions mysterious (i.e. it does notexplain it at all); furthermore,the bipartitereading is hard to reconcile withW ittgenstein's idea that there is a general propositionalf orm,t hat is, a common essence to all propositions (elementarya nd molecular,e mpirical and logical ones,a nd so forth), which is, on the other hand,s omethingt he Tractatus is clearly committed to (cf. TLP4.5).
Thef actt hat there is some stringent (internal) relation between truth-functional articulation (and thusalsological complexity) anda proposition's abilityt oe xpress its sense is something that Wittgensteina cknowledged very early. In 1912, when he was still studying under Russell, and when the two were mainly discussing then ature of logic and thes tatuso fl ogical propositions, WittgensteinwrotetoRussell: Ib elievet hat our problemsc an be traced downt ot he atomic propositions. This you will seei fy ou tryt oe xplainp recisely in what waythe Copula in suchaproposition has meaning.
Ic annot explain it andIt hink that as soon as an exact answer to this questioni sg ivent he problem of 'v' ando ft he apparent variablew illb eb rought very near to their solution if not solved. ( NB: 121) As olutiont ot he problem of explaining then ature of logical complexity( whatW ittgenstein refers to as thep roblemo f' v') will be achievedb yacorrect explanationo ft he role of the copulai na atomic proposition; the suggestion thus amountstothe ideathat an accounto fl ogical complexityw illb eg ivenb ya ne xplanationo f propositionalarticulationatthe elementary level.
In Wittgenstein's wartime Notebooks we findasimilar view of the relationb etween the natureo fp ropositions and of logical articulation. As regards the latter, Wittgensteinw rites: "The problemso fn egation, of disjunction,o ft ruea nd false,a re only reflections of theo ne great problemi nt he variously placedg reat and small mirrors of philosophy"( NB:4 0), wheret he one great problemi si dentified by Wittgenstein as thato fg ivinga n explanation of then ature of thep roposition: "My whole task consists in explainingthe nature of theproposition" (NB: 39). This passage suggests thatt he problems relatedt ol ogicalc omplexity ("the problemso fn egation, of disjunction" etc.)w illb es olved by an account of then atureo ft he proposition. This attests that Wittgensteinf elt there cannotb ea ny dualism between an account of the nature of the proposition and an account of logical complexity. Thesea re simply twoa spects of the samep roblem, which are to be given the same solution.
To observe thatt he bipartiter eading is ap roblematic interpretationo ft he Tractarian conception of the relationb etween elementary andcomplex propositions is by no means ap articularly novel (orv ery controversial)p roposal. PeterW inch has long ago argued: [I]t is vitaltoour understanding of Wittgenstein to see that thenature of logic is already being inquired intoi nW ittgenstein'st reatment of the puzzlea bout the relation between propositions and facts. This point canperhaps be expressedinthe form of anotherproblem:what is ther elation between ap roposition's ability to state af acta nd its ability to standi nl ogical relations to other propositions? Now Wittgenstein thought […] that therem ust be such ar elation;t hati ti s notm erelyac ontingentm atter that ap roposition can combine these twofunctions; that unless propositions hadlogical relations witheach other they would nots tatef acts (i.e. wouldn ot be propositions) and unless they statedf acts, they wouldn ot have logical relationsw ith other propositions. (1969: 3-4) In thesamevein,Marie McGinn writes: Wittgenstein is convinced that we shalls ee everything clearly -the nature and thes tatuso ft he propositions of logic, negation, disjunction, inference, trutha nd falsity -when we seet hiso ne thing clearly:the nature of a proposition. […][C]oming to seethe nature of the proposition clearly is, at thev erys amet ime,c oming to seen egation andthe status of the propositions of logicclearly: we have here, not a number of separate problems,b ut one great problem.I ft he problem is to be solved, then it must be solved alla to ncea nd in itse ntirety. The idea of thes ingle great problem is thato ncet he natureo fa propositionhas become clear, then everythingwillbeclear:the nature ands tatuso ft he propositions of logic,t he nature of negation, inference, andsoon. (2006:15-16) 2 But how does Wittgenstein,i nt he Tractatus,i mplement and develop the idea that the natureo fl ogical complexityi sg rounded in, and evene xhaustedb y, an accounto ft he nature of the (elementary) proposition? In short, Wittgenstein holds that all logical constants area lreadyp resent in an elementaryp roposition, thatis, in aproposition which, by beinganimmediate combination of names,isapicture of reality. We have statements of this position in the Notebooks: [I]f the positive fact φaisgiven then so is the possibility of (x).φx, ( x) φx, φae tc.e tc. (All logicalc onstants are already contained in the elementaryproposition). (NB:27) 3 And in the Tractatus we find: An elementaryp roposition reallyc ontains all logical operationsi n itself. For'fa'saysthe same thing as Wherevert here is compositeness,a rgument andf unction arep resent, andw here these arep resent,w ea lreadyh avea ll thel ogical constants. (TLP 5.47) Theb asic idea of these passagess eems clear (logicalc omplexityi s already givena ss oon as elementaryc ompositenessi sg iven); however, we ares till in need of af ulla ccounto ft he view thatthe nature of logic is to be clarified by ac orrecta ccounto ft he nature of elementary propositions. Although, as seen, many interpreters are aware of the shortcomingso ft he bipartiter eading, onel ooks rather in vain in the secondary literaturef or af ulla nd detailed account of the Tractatus'v iewt hat alll ogical constantsa re already presenti na ne lementaryp roposition, and thusf or an accounto f the Tractatus'conception of the unity of language and logic;this idea hasneverbeen, it seems to me,fully discussed 4 or very persuasively explained. Ourt aski nt he rest of thisp aper is to provide such an explanation.

Logicalconstants,operations andfunctions
At 5.2341 Wittgensteinw rites: "Negation, logical addition, logical multiplication,e tc.e tc.a re operations". Thus Wittgenstein's claim at 5.47 that an elementary propositionc ontainsa ll logical constants meanst hat it contains alll ogical operations;i no rder to understand Wittgenstein's accounto ft he unity of language and logic,w hich depends on the claim thata ll logicalo perations are present in an elementary proposition, we mustclarifyhis notionofanoperation.
Wittgensteinpresentsand discusses this notionasfollows: The In the Tractatus Wittgensteinfamously rejectsthe assimilationof operations to functions:" Operations and functions must not be confused with each other" (TLP 5.25). As Peter Hylton -following MaxB lack -s ays, however, it appears as if "everything, or almost everything, that Wittgensteins aysa bout operations could with equalc orrectnessb es aid aboutf unctions"( 2005: 140,a nd see Black 1964: 258-260). For example, thec laim that no statement is made by an operation (TLP 5.25) seems hardly helpfuli n distinguishing operations and functions,f or no statement is made by af unctione ither;t he function x 2 ,f or instance,d oes not assert anything.L ikewise, the fact that "[a] function cannot be its own argument, whereasa no peration can take oneo fi ts own resultsa s its base" (TLP 5.251) does not seem to distinguish functions from operations,f or surely operations themselves cannotb eo perated on; what can be operated on is the result of an operation (a proposition, at ruth-function in Wittgenstein'st erminology), but, likewise, the result of the saturationofafunctionwithanargument (its value) can,a tl east in some cases,b ea na rgument of that very function.
At 5.251, let us recall,W ittgensteinr emarks that "[a] function cannotbeits own argument, whereas an operation can take oneof its own results as its base".W hile an operation, "negation" for instance, can be appliedt ot he resulto fi ts own application, the "propositionalf unction",s ay, "x is wise",c annot takeo ne of its values as argument: that is, thep roposition "Socrates is wise" (one of the values of thepropositionalfunction "x is wise")cannot itself be an argument of that very propositional function: "Socrates is wise is wise" is nonsense( see Russell and Whitehead 1910: 43). A further reason forWittgenstein'sdistinction between functions and operations,a ss een,i st hat "no statement is made by an operation, buto nlyb yi ts result,a nd this dependso nt he bases of the operation" (TLP 5.25).A sH ylton (2005:1 45) notices, however, Russell and Whitehead (1910: 18,9 6-97) discuss (anda dmit) preciselyt he notiono fasserting ap ropositional function, thati s, of asserting any valueofthe propositional functioninquestion, where ap ropositionalf unctioni sd efined as "a statement containinga variable x,a nd such that it becomesap roposition when x is given any fixeddetermined meaning" (Russell and Whitehead 1910:15). 6 If Wittgensteinc ontrasted hisn otion of an operation with that of apropositionalfunction, the possibility seems open of assuming that he might have held that operations are (mathematical) 6 This,h owever, does not go very far in explaining Wittgenstein's reason forr ejecting the idea that logical constants should be understood in terms of propositionalf unctions. Hylton (2005: 144) takes Wittgenstein's claim that an operation does not characteriset he sense of the propositionw hich is the result of its application (cf. TLP 5.25) as being the relevanti ssue here; on hisv iew, ther eason depends on Russell's view that propositional functions (in particular logical constants)e nter into the determination of the sense of thosep ropositionsw hich aret heirv alues. As he writes: "[O]n Russell's account a proposition which is obtained by application of thep ropositional function disjunction to twop ropositions p and q is ad isjunctive proposition -it contains ac onstituent corresponding to disjunction. […] In short:f or Russell 'p v q'm ust representad ifferent proposition from that represented by ' ( p . q)'.B ut this is precisely the result that Wittgenstein wants to avoid" (Hylton (2005: 144)). Wittgenstein is of course adamant that since apropositionist he expression of itstruth-conditions(cf. TLP 4.431), andsince, in the examplea bove,' pvq 'a nd ' ( p . q)' have thes amet ruth-conditions,t heya re the sameproposition. Hylton's diagnosis of Wittgenstein's reason for rejecting Russell's view of logical constants as propositionalf unctions, therefore, seemst od epend on whether Russell takes the disjunctiveproposition 'p v q'ascontaining 'a constituent corresponding to disjunction' as Hylton claims; if so, 'p v q'w ouldi ndeed be different from' ( p. q)' which does not contain such ac onstituent. The questiont hus turns outt ob ew hether propositional functions occurintothe propositionswhich aretheirvalues. But, as Hylton himself notices (Hylton( 2005:1 44, note 12)),R ussell denies this.B esides, Russell and Whitehead (1910:7 )c laimt hat 'p . q'( Logical Product) canb ed efined in terms of the Logical Sum (disjunction) and the ContradictoryFunction(negation), and conclude that'p .q'is"merely ashortenedform of symbolism for '~(~p v~q)'", aposition which seems indeed compatible with Wittgenstein's in the Tractatus.Aswewill later see,Wittgenstein's reasons for rejecting theview that logical constants arepropositionalfunctions have to do with the fact that his conception of logicalc onstantsa soperations (rule-governed proceduresfor constructingpropositions)and this rules out thatt hey mightbesome sort of entities,such as propositional functions. functions, in roughly thew ay Frege thought. 7 Thep roposalo f identifying an operation with the mathematical notion of afunction (as opposed to thato fapropositionalfunction)h owever, makes little sense. In general, af unction takesa no bject (orm ore than one) as argument andd elivers an objecta sv alue; thati s, it maps objects onto objects. Butanoperation delivers a proposition as aresult,and a functiont hat delivers ap roposition as value just is ap ropositional function.T here are, however, deeper reasons for resistingt he identificationo fo perations and functions.O naFregeanv iew, the functional conceptionofpropositionalcomplexitycan be extended, with thei ntroduction of some particular functions (truthfunctions),t oa ccountf or logical relations between propositions; thus, on aF regean account,t he relationt hat ap ropositional constituent (a name for instance)has to the proposition of which it is ac onstituent is in principle no different from the relationa proposition has to the more complex proposition of whichi ti sa constituent. Aproposition, on thisaccount, occurs in another in no different wayfrom how asub-propositional constituent occurs in a proposition; both are arguments of some functions (see Sullivan 2000: 178). In the Notes Dictatedt oM oore,h owever, Wittgenstein remarks, contra Frege: Thereare internal relationsbetween one proposition andanother;but a proposition cannot have to another the internalr elation whichaname has to thepropositionofwhich it is aconstituent, andwhich oughtto be meant by stating it 'occurs'init. In this senseone propositioncan't 'occur' in another.(NM:116) 8 Therei saclearr ejection,h ere, of the assimilationo fl ogical operations to functions.
As we saw,the Tractatus characterises an operationas"what has to be donetoone proposition in ordertomakethe otherout of it" (TLP 5.23). This suggests thata no peration, is "less likea ne ntity, thatm ight be ac onstituent of am ore complex entity, thani ti s somethingthat we do" (Hylton2005: 152), that is to say, some kind of procedure or process.I nf act, Wittgensteind iscusses operations in terms of rules forconstructing propositions. At 5.512hewrites: [I]n '~p'itisnot '~' that negates;itisratherwhatiscommon to allthe signsofthisnotationthat negate p.
The expression of an operation should not be confusedw ith the operationitself.In'~p', it is notthe symbol '~'that negates, as well as in 'p . q', it is nott he '.'t hat conjuncts.T he negativef actor in '~p'i sr epresented, says Wittgenstein,b y' what is common to all the signsofthis notation that negate p'. Thatcommonelement (the operation' negation')i sar ule that governs thec onstructiono f propositions in which 'p'i sn egated. In 'p . q', on theo ther hand, thec onjunctivee lement( theo peration 'conjunction') is what is common to all propositions that affirm both 'p'and 'q', andwhat is common to thesep ropositions is having been constructed according to thesame rule.
Importantly, thee xistence of operations,o fr ulesf or constructingpropositions,seemsfor Wittgensteintodependonthe establishment of a language.Assoonaswehavealanguage, we have rules for truth-operations;t his is, Ib elieve, whatW ittgenstein means when he says thatoperations areinplace once a notation has beenestablished: Once an otation has beene stablished,t here will be in it ar ule governing thec onstructiono fa ll propositionst hatn egate p,ar ule governingthe construction of all propositionsthataffirm p,and arule governingt he construction of all propositionst hata ffirm p or q:a nd so on […]. (TLP5.514) Operations are thusnot something over and above the workings of propositions of language, buta re rulesf or constructing propositions( ther esultso fo perations) giveno ther propositions (the bases of theo perations).I nt he next twos ections we will consider in detail Wittgenstein'sc onceptiono fa no peration from the Noteso nL ogic to the Tractatus,a nd we wills ee in what sense operations are in placew hen al anguage is established.T his will finally put us in aposition to see fully why -as Wittgenstein claims at 5.47 -an elementaryp ropositionr eallyc ontainsa ll logical operations in itself, and thusw hy the bipartiter eading cannot be a correct interpretation of Wittgenstein'sv iews on languagea nd logic.

Languageand logicinthe Notes on Logic
In the 1913 NotesonLogic Wittgensteinc onceives of propositional sense in terms of bipolarity: Everyp ropositioni se ssentially true-false: to understandi t, we must know both what must be thec asei fi ti st rue, andw hatm ustb et he case if it is false. Thusap ropositionh as two poles,c orresponding to thecaseofits truth and thecase of itsfalsehood. We call thisthe sense of aproposition.  Thep oles of ap roposition (indicatedb ythel etters 'a'a nd 'b') 9 represent itsp ossibility to be true or false. Ap roposition has two poles because it canb et ruea nd it canb ef alse. Andt hisa mounts to its sense. As Wittgenstein willl ater say in the Notes Dictated to Moore: '[T]rue' and'false'are not accidental propertieso faproposition, such that,w hen it has meaning, we cans ay it is also true or false: on the contrary, to have meaningmeans to be true or false: the being trueor falsea ctually constitutes the relation of thep roposition to reality, which we mean by saying thatithas meaning(Sinn [sense])(NM:113).
Thep roposition 'p't ogetherw ith its polesf or truth andf alsehood is rendered by Wittgenstein as 'a-p-b'.P olesa re essentiali nt he NotesonLogic's accounto fm olecular propositions, there called abfunctions (see  andobtained by the application of aboperations (see NL:9 4).' not p', fori nstance, is expressedi n Wittgenstein's notation as 'b-a-p-b-a'; what the ab-operation 'negation'does is reversing the truth-poles of the proposition a-p-b, thus turning its a-pole intoab -pole and its b-pole into an a-pole, therebyg etting as ar esultt he formula 'b-a-p-b-a', with opposite sense. Adifferent example,i nvolving twoelementary propositions, is the proposition 'p and q'. In Wittgenstein's ab-notationt his proposition is expressed by means of at ruth-diagram, 10 similart o the oneinfigure 1. In the diagrammatical representationo faconjunctive proposition the outer a-pole (the true-pole)i so nlyc onnectedw ith the pairing of the a-polesof'p'and 'q'(that is to say, with the possibility of 'p' and 'q'b eing both true),a nd the outer b-pole( the false-pole) is connected with allthe other pairings of the polesof'p'and 'q'. Abfunctions, as well as elementary propositions, have ab-poles (are essentiallytrue/false) andare thus bipolar in the sameway in which elementary propositions are. This is made clearb yt ruth-diagrams, which show howamolecular proposition gets its ab-poles (its sense) by correlation with the ab-poles of the original elementary proposition(s).O fc ourse, Wittgenstein reminds us:" As the ab functions of atomic propositions arebi-polar propositions again we canp erform ab operations on them. We shall, by doings o, correlate two newo utside poles viat he old outside poles to the poleso ft he atomic propositions" ( NL: 94). Thati s: ab-operations can be iterated.T hisi sa gain madev isible by the truth-diagram method. The proposition '(p . q) r', for example,c an be represented diagrammaticallyasshowninfigure 2. First the ab-poles of 'p'a nd 'q'a re connectedt od etermine the abpoles of 'p . q'( as we saw in thep revious diagram); thent he abpoleso f' p.q 'a re in turnc onnected to the ab-poles of 'r't o determinethe ab-poles of themolecularproposition '(p . q) r'.
We can see from the above examplest hat, on Wittgenstein's account,the link between elementary andmolecular propositions is providedbythe notion of bipolarity,which in turnexplains what it is for aproposition (elementary or not) to have sense.Aproposition's having ab-poles( truth-conditions, sense) is thus everything that is needed in order to account forl ogical relations between propositions, because ab-operations simply operate on elementary propositions' ab-poles to generate newp ropositions with ab-poles, with truth-conditions. Despitet he fact that therei sn og eneral statement, in the Notes on Logic,o ft he viewt hat logical constants are implied in an elementary proposition, Wittgenstein's accountof the sense of apropositioninthe Notes implements the general view thatl ogical complexity is containedi n( elementary)p ropositional interconnectedness. Elementary propositions alreadye nsure the possibility of molecularp ropositions( and thuso fl ogical complexity) because thel atters imply consist in ar earrangement of the ab-poles of elementary propositions, ab-poles which aregiven byand coincide with -aproposition's having sense (bipolarity). 11

Languageand logicinthe Tractatus. Why an elementary proposition really contains alllogical operations in itself
The accounto ft he relationb etween elementary and molecular propositions sketcheda bove is overall maintained in the Tractatus, Wittgenstein'sr evision of his early accounto fp ropositional sense (and of the natureofpropositional constituents)notwithstanding. 12 What therei si nt he Tractatus,w hich was notf ully present in the Noteso nL ogic,i st he conceptiono faproposition as a picture of reality, 13 as having senseb yd epicting ap articular situation( as tate of affairs). "A propositions tatess omethingo nlyi ns of ar as it is a picture"( TLP4 .03). And: "Instead of,' This proposition has such ands uchasense', we can simply say, 'This proposition represents such and such as ituation'"( TLP 4.031). But having sense still means,for the Tractatus,tobetrueorfalse.
Ap roposition can be trueo rf alse onlyi nv irtue of being apictureo f reality. (TLP 4.06) And in the Notebooks he writes: Onlyinthisway can theproposition be true or false: It canonlyagree or disagreewithrealitybybeing apicture of asituation. (NB:8) In the Notebooks entryo n3 .10.14 (repeated almost literally in the Tractatus at 4.032),W ittgenstein makest he relation between picturesa nd truth/falsityd epend on the fact thatp ictures are essentially logicallya rticulated." Thep roposition isap ictureo fa situationonlyinso far as it islogicallyarticulated". Thus a "simplenon-articulated -s ign can be neithert ruen or false"( NB:8 ). A picture's capability of being true or false thusdepends on its having a structure,bymeans of whichitrepresents (depicts) astate of affairs in reality. Ap icture is therefore a representational (or pictorial) structured fact, thati s, af actr epresenting elements in reality to be combined as itse lements arec ombined. By being ap icture of realityapropositionistherefore intrinsically true or false: if things in realityare combined as it showsthemtobethen the proposition is true,and false otherwise.Wittgenstein'sTractarian conception of a proposition also brings withitthe view that aproposition depictsa situation,astate of affairs,a nd that the latter's obtaining( or existing) makesthe proposition true: If an elementaryp roposition is true, thes tate of affairs exists:i fa n elementaryproposition is false,the state of affairs does notexist. (TLP 4.25) 14 Thus, possibilities of existence andn on-existence of states of affairsd eterminet he possibilitieso ft ruth and falsehoodo ft heir corresponding propositions: "Truth-possibilities of elementary propositions mean possibilities of existencea nd non-existenceo f states of affairs" (TLP 4.3). In thec ontexto fh is discussion of truth-possibilities Wittgensteini ntroduces thef amiliar truth-tabular (T-F) notation;i nt he truth-tabular notation, ap roposition 'p'i s assignedT -F poles, that is, truth-possibilities, where "'T'm eans 'true' [and] 'F'm eans false" (TLP 4.31). Theb ipolarityo f propositions is thusr eaffirmed: what Wittgensteinc onceived of in terms of ab-poles in the Noteso nL ogic becomes,i nt he Tractatus, truth-possibilities,ortruth-conditions. Picturing( representing, saying), therefore, generates bipolarity: by picturingi ts subject "from ap osition outside it"( TLP2 .173), a picture represents it either correctly or incorrectly,a nd thus acquirestruth-possibilities. As Thomas Ricketts observes: The conception of elementarys entencesa sp icturesm akest heir agreement or disagreement with reality -their possession of true-false poles,ofsense -intrinsic to them. (Ricketts 1996:80) Truth-possibilities, T-Fp oles, are thus embedded intot he very representationalnatureofaproposition, and are not determined by, for example, the judgemental or assertoric form with which a proposition can be expressed. As Wittgensteinh ad writteni nt he Notes on Logic:" Judgment,q uestion and command are all on the same level. What interests logic in them is only the unasserted proposition" (NL: 96). Judgementa nd assertion are, as Wittgenstein has it,m erely psychological( see NL:9 5),i .e. of no concern to logic.W hati snot psychological, becausei ntimately related to truth andt hus to logic,i saproposition's sense, namely its representationals tructure,w hichg ivesi ti ts T-Fp oles and thus makes it essentially true or false. (See the discussion in Johnston 2011.) Truth-operations, as well as ab-operations,o perateo n propositions' T-F poles to deliver propositions with T-Fp oles (bipolarp ropositions).I ti si mportant to stresst hat what an operation operateso ni saproposition's T-F poles (their truthpossibilities,orab-poles according to Wittgenstein's positionint he Noteso nL ogic); theb aseso ft he operation, say,' conjunction' are strictly speaking '(TF)(p)' and '(TF)(q)',( 'a-p-b',' a -q -b' in the Notes on Logic) and not 'p'a nd 'q'.T he resulto ft he operation -whichi n the truth-tabularnotation takesthe form '(FTTT)(p, q)' 15 -makesit evidentthat '(TF)(p)' and'(TF)(q)' do not occur in it; this is another way of puttingt he thought from the NotesD ictatedt oG .E.M oore, discussed in section 3, that "one proposition can't 'occur'i n another" (see Sullivan 2000:181).
It might be misleading to say, as Idid in the previous paragraph, that operations operateo np ropositions'T -F polesa nd deliver propositions with T-Fp oles, if this is takenl iterally. Once elementary propositions areg iven, in fact, by means of truthoperations all propositions can be obtained; this procedure can of course also produce propositions which aren ot bipolar, in that theyd on ot have eitheraT -pole( contradictions) or aF -pole (tautologies). These propositions, the propositions of logic, are peculiar truth-functions because,u nlike other truth-functions, are either true fora ll truth-possibilitieso fe lementaryp ropositions (tautologies) or false fora ll truth-possibilities of them (contradictions). In this sense, although they aret he "extreme cases" (TLP 4.46)-or the "limiting cases"( TLP 4.466)-of the process of thet ruth-functionalc onstruction of propositions,t hey are integral part of thatv eryp rocess,i ntegral "part of the symbolism" ( TLP 4.4611). It is in this context, Ib elieve,t hat Wittgenstein'sc laim that" [i]f we knowt he logical syntax of any sign-language, then we have alreadybeen given allthe propositions of logic"( TLP6 .124)s hould be read. Whenw ek nowt he logical syntax of language, thus when we knowhow signs signify andthus how theyc an be combined in meaningful symbols -propositions that are true or false -we alreadyhaveall thepropositionsoflogic, thati s, we have the resources to operate on propositions' truthvaluest og enerate propositionsw hich are true or falsen om atter what.
On Wittgenstein's conception, thus, operations are rules fort he constructionofmolecular propositions:they are not representatives, names of entities (cf. TLP4 .0312), such as Fregeanf unctions or Russellian propositional functions;t he proposition 'p.q ', whose formulation in Russelliann otation might encouraget he idea that thereissomething corresponding to the dot, can -as seen above -be rewritten as '(FTTT)(p,q)', where nothing corresponds to the logical constant.A gain,W ittgenstein's notation makesi tc lear that the molecularp roposition in question does not contain anything over and abovew hat is contained in thee lementary propositionso f which it is at ruth-function, because therei sn onew material element, in it, whichw as otherwise missing from the elementary propositions 'p'and 'q'.
This, however, might seem problematic on thel ightso f Wittgenstein's claim that an elementary proposition really contains all logical operations -constants -in itself. Sincea ne lementary proposition is an immediate concatenationo fn ames, if logical operations were present in an elementaryp roposition, they would have to be names,a nd thus they wouldb er epresentatives of objects, contrary to whatW ittgensteins ays at 4.0312. Clearly, therefore, Wittgensteind oes not mean that, literally,l ogical operations are present in an elementaryp roposition,t hat is, containedi ni ts sense. 16 Ourd iscussion of then otion of an operation, however, gives us the resources to understandw hat Wittgensteinm eans.E lementaryp ropositions, by depictingr eality trulyo rf alsely, arec orrect or incorrect, and thush avet ruthpossibilities. This, as seen, is all thati sn eededt op roduce, by means of the application of operations, molecularp ropositions. Thus what elementary propositions really containisthe possibility of alll ogical operations,t he possibility of generatinga ll molecular propositions; in this sense theyc ontain all logical operationsi n themselves. In the Notebooks entryo n5 .11.14, quoted towards the endo fs ection 1, Wittgensteine xplicitlyd iscusses the relation between an elementary proposition and itst ruth-functions by sayingthat the possibility of thel atterisgiven as soona sthe former is given. 17 Elementary propositions'T-F articulation (givenb ytheir being pictures of reality) makes them suitable for beingl ogically 16 If we take the claim that an elementary proposition contains all logicalo perations literally, this would be in striking contrast to what Wittgenstein says at 5.233:"Operations cannotmaketheirappearanceb eforethe point at which oneproposition is generated out of anotherinalogicallymeaningful way; i.e.,the point at which the logical constructionof propositions begins." 17 In the Tractatus,l ikewise, Wittgenstein says that" [t]he possibility of negation is already written into affirmation" (TLP 5.44, my emphasis). operated on, and thus for potentiallygenerating all truth-functional molecularpropositions. 18

Conclusion
We aren ow in ap osition to seef ully why the bipartiter eading, discussed in section 1, is wrong.A ccording to it, therei sn o intrinsic connectionb etween elementary propositions and the truth-functions obtainedb yt he applicationo ft ruth-operations on them, because to be an elementaryp roposition and to be at ruthfunction areq uite different things. But Wittgensteinr epeatedly remarkst hat nothing moret han the senseo fe lementary propositionsi sn eeded in order to account fort ruth-functional articulation,and hencefor logical complexity as awhole.O nce the pictorial (andb ipolar) charactero faproposition is understood, then everythingi sg iveni no rder to understand how propositions cane nter into logical relations witho ne another. Thoser elations are already implied by propositions' own nature (their T-F articulation),bypropositions' determining their sense, hence by the conditions on whicht heya re true or false. As Wittgenstein summarisest his in the Notebooks:" Thel ogical constantso ft h e proposition aret he conditions of itst ruth"( NB: 36). This is, ultimately, the reason why Wittgensteinc laims,a ss een at the end of section 2, thatl ogical operations area lreadyi np laceo ncea language has been established;t heir possibilityi si mpliedo nce we have anotation in which we express thoughts thatare trueorfalse.
Andt his, in turn, perfectlyc oheres with Wittgenstein's view that, at bottom, thereisonly one logical constant,which is "what all propositions,b yt heirv eryn ature, had in common with one another";t his -Wittgensteins ays -" is the general propositional form" (TLP 5.47), the "essence of thep roposition" (TLP 5.471). The general form of thep ropositioni s" ad escription of the 18 My discussion of Wittgenstein's conception of logical complexity does not address quantification. In the Tractatus,h owever, as is well known, Wittgenstein gives at ruthfunctionala ccount of quantifiers,a nd thus quantified propositions are not a counterexample to the thesis that logical complexityi sg iven by the application of truthoperations. For recentdiscussions of Wittgenstein's treatment of generalpropositions see McGinn (2006: 234-240)and Morris(2008: 215-225). propositions of any sign-language whatsoever"s ucht hat" every possible sensec an be expressedb yasymbols atisfying the description.
[…] Thegeneral form of aproposition is: This is how things stand" ( TLP 4.5). 19 The general propositionalf ormi st hus a proposition's capability of expressing sense-representing realityand thusbeing trueorfalse. As MarieMcGinn writes: The general form of apropositionexpresses […]whatall propositions thatrepresent states of affairs have in common. It is given as soonasa languageinwhich we express judgementsabout theworldisgiven. In acquiring language we have already grasped theg eneralf ormo fa proposition, that is, we have already grasped the wholeo fl ogic,t he essence of representation as such. (2006:240) The bipartite reading however,p resents ap icture of the Tractarian conception of languaget hat can hardlyb er econciled witht he idea thatt here is ag eneral propositional form, an essence of representation, becauseo nt hat viewt he essence of some propositions (elementaryo nes) is to be pictures, the essence of others (molecularo nes) is to be truth-functions.B ut on Wittgenstein's viewe veryp roposition is the valueo ft he general propositional form, becausep ropositions haveac ommone ssence (pictorials tructure), by which they sayt hat "sucha nd suchi st he case",and which makes them true or false.
We haves een, therefore, that on Wittgenstein's viewl ogical operations (the rulesf or thef ormationo fm olecular propositions) areg iven by -and impliedi n-ther epresentationalc haracter of (elementary)propositions; their possibility is already givenwhen we havee lementary propositions depicting realityt ruly or falsely. This is the reason why logic as awhole,for Wittgenstein, is givenatthe level of elementary propositions,t hat is to say, is given as soona s propositions saying something about realitya re given; a proposition'sh aving sense (depicting reality) is thus all that is needed to carry outlogicaloperations,and thus to obtain molecular (and logical) propositions out of elementaryo nes; this is the core motivation for Wittgenstein's idea that logical complexity is contained in -forms part of -linguistic complexity,a nd thus for his ideat hat an elementary proposition alreadyc ontains all logical constants (operations)i nitself. Thus," [l]ogic is given as soon as a language in whichweexpress judgements aboutthe world is given; it is […]a lready complete or entirew henw eh ave al anguaget hat we use to say how things are" (McGinn 2006: 69).
If the oned evelopeda bove is ac orrect reading of Wittgenstein's view of logical complexity, we have thus reacheda n explanation of Wittgenstein'sc onceptiono ft he unity of language andl ogic,o fw hy the problems related to individual logical constants (the problem of logical complexity),a sW ittgenstein repeatedlys tates, areo nlyr eflections of the" oneg reatp roblem" (NB: 40),the problemofg iving the essence of ap roposition(TLP 5.471),a nd thus thee ssence (of ther epresentationalc haracter) of language. 20