Keyt Witt's Picture 20p

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Philosophical Review

Wittgenstein's Picture Theory of LanguageAuthor(s): David KeytSource: The Philosophical Review, Vol. 73, No. 4 (Oct., 1964), pp. 493-511Published by: Duke University Press on behalf of Philosophical ReviewStable URL: http://www.jstor.org/stable/2183303Accessed: 05/04/2010 14:30

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WITTGENSTEIN'S PICTURE THEORYOF LANGUAGE

"My whole task consists in explainingthe nature of the proposition."-Notebooks 2 2. I . I5

A tITTGENSTEIN S VERSION of the picture theory of language' inthe Tractatusis sharply outlined, but many of the details

are indistinct. It is hard to see, in particular, how Wittgensteinmeets the traditional objections to the picture theory. Thus it isdifficult to assess the merits of Wittgenstein's version of the picturetheory and, consequently, of the philosophy of the Tractatusas awhole.

In this paper I begin by presenting the main features ofWittgenstein's theory-the sharp outline. Then the traditionalobjections to the picture theory are set forth: these may be collect-ed together and a paradox of the picture theory formulated. Withthis paradox in mind I turn to the details of Wittgenstein'stheory. One needs here to consider the various interpretationsoffered by different commentators on the Tractatus.These inter-pretations are considered from two points of view: as interpre-tations of the Tractatus and as solutions of the paradox. Theconclusion I reach is that none of the interpretations so faroffered resolve the paradox and that probably none are correctinterpretations of the Tractatus.Finally, I present what I believeis a good interpretation of the Tractatus and also what I thinkis a satisfactory solution of the paradox, but these two turnout to be distinct.

1 The new Pears-McGuinness translation of the Tractatus London, i96i)is used throughout. The few changes of terminology that I have made areenclosed within brackets. I have used the Notebooks9i4-i9i6 (Oxford, i96i)extensively and have made whatever use I could of the backward-lookingparts of The Blue andBrownBooks(Oxford, I958) and the Philosophicalnvestiga-tions (Oxford, I953). The propositions of the Tractatusare referred to bynumber alone; the titles of the other works are given, some in abbreviatedform.493


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DAVID KEPTTHE THEORY

I shall be concerned in this paper with a single but basic facetof the picture theory of the Tractatus: the elementary propositionand how it portrays. There are other facets. One of these is theidea that the sole function of language is to picture reality (4.001together with 4.01). Another is the problem of how propositionsof ordinary language picture reality. Wittgenstein's solution verybriefly is that this is revealed by logical analysis, that the vehicleof logical analysis is a perspicuous notation along the lines ofthat of Principia Mathematica,and that upon analysis a propositionof ordinary language is resolved into a number of elementarypropositions plus some logical constants. A final evaluation ofthe picture theory of the Tractatus must take both of these otherfacets into account, but neither will be discussed in this paper.

Elementary propositions, as they are defined in the Tractatus,have the following features. They contain no logical constants,no words such as "not" or "and" or "all." They are logicallyindependent of each other (4.2II, 5.134). They are made upentirely of names (4.22), and each name stands for a simpleobject (3.203, 3.22, 2.02). Finally, an elementary proposition"asserts the existence of [an atomic] state of affairs" (4.2 I).

Many things that Wittgenstein and his commentators sayabout elementary propositions apply also to simple propositionsof ordinary language such as "Mercury is smaller than Venus."Whenever this is so, I speak simply of propositions instead ofelementary propositions and of states of affairs instead of atomicstates of affairs. But even when I speak of propositions in thisgeneral way, I still intend to refer only to propositions of the formR (xl, x2, ... x.), that is, to propositions that contain a singleverb and no logical constants.2

How does an elementary proposition portray reality? First,the negative side. Although a name stands for an object, it isnot a picture of an object: "The name is not a picture of the thingnamed!" (Notebooks, 3.10.14; cf. 3.22i). The pictorial aspect of

2 By a proposition with two verbs I mean one like "I knowthat Mercury issmaller than Venus."

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PICTURE THEORT OF LANGUAGEan elementary proposition is the arrangementof names: hownames are arranged (combined, configured, related, structured)pictures how objects are allegedly arranged (3.21). I say allegedlysince, although every elementary proposition pictures a possiblearrangement of objects, not every one (apparently) pictures anactual arrangement: every elementary proposition has a sense(cf. 4.031; Notebooks,6.io.14 par. 3) but not every one is true.I say apparentlysince it is, of course, logically possible for everyelementary proposition to be true.

The problems toward which this theory is directed are two:the problem of the new proposition3 and the problem of the falseproposition.4

The problem of the new proposition is to explain a striking dif-ference between names and propositions. "The [references] ofsimple signs (words) must be explained to us if we are to under-stand them" (4.o26); but, on the other hand, "we understand thesense of a propositional sign without its having been explained tous" (4.02). Wittgenstein sums it up this way: "A proposition mustuse old expressions to communicate a new sense" (4.03; cf.4.025). How on the picture theory is this difference explained?Why must the reference of a new name be explained to us butnot the sense of a new proposition? The answer, quite simply, isthis. A name is not a picture of the thing named. Name and objectare connected by arbitrary convention. But a proposition is apicture of its sense. "A proposition shows its sense" (4.022; cf.Investigations, 523). Thus a proposition is essentially connectedwith the state of affairs it expresses (4.03). So the one connectionneeds to be explained, the other not. For example, if we wished,we might let "s" stand for Seattle and "k" for Spokane and layit down further that one name to the left of a second depicts onething west of another. We may adopt these as conventions or notas we please. But if we adopt them, then necessarily "sk" depictsSeattle as west of Spokane and "ks" the reverse (cf. 3.342).

The problem of the false proposition springs from a second3 "It belongs to the essence of a proposition that it should be able to com-municate a new sense to us" (4.027).4 "It must not be overlooked that a proposition has a sense that is indepen-dent of the facts" (4.o6 i).

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DAVID KEPTdifference between names and propositions. A name, if there isno object that it signifies, is meaningless: it is not a name at all(3.203; cf. Investigations, 40). But a proposition, if there is nofact that it signifies, is not meaningless but simply false. Thus whata name signifies must exist, but what a proposition signifies neednot. So the question is: how can a proposition be false withoutbeing meaningless ? (Cf. Notebooks, 15. 1.114, Blue and BrownBooks, p. 31, 11. 8-15, Investigations,5i8.) The answer given bythe picture theory is this. A proposition is composite (3.141,par. 2; 4.032). And what each part signifies exists even if what thewhole signifies does not. What a false proposition signifies issimply a nonexistent arrangementf existent objects (Blue and BrownBooks, p. 3 ', 11.I6-24). How does it do even this? Well, the namesthemselves, given the method of projection, are arranged as theobjects would be arranged if the proposition were true (4.031 1;Notebooks, 4.1 I. 14). SO it turns out that even the arrangementitself exists, though it exists as an arrangement of names, not ofwhat the names stand for.

THE PARADOX

The picture theory of language is faced with an interestingpuzzle. Consider the proposition "Seattle is west of Spokane."This is not an elementary proposition-it will be well to bearthis in mind-but let us suppose for the moment that an elemen-tary proposition is at least like this. This proposition is composedof three parts (giving a logical rather than a grammaticalanalysis): two proper names and the predicate "is west of." Nowthe first bit of the puzzle is this. The fact pictured by the proposi-tion is an arrangement of two cities but the proposition itself isan arrangement of threeparts. Thus the fact and the propositiondo not appear to have the same number of parts. But Wittgensteinholds that they must: "In a proposition there must be exactlyas many distinguishable parts as in the situation that it represents.The two must possess the same logical (mathematical) multiplic-ity" (4.04). Suppose we preserve the one-to-one correspondencebetween the fact and the proposition by dropping the predicateand writing the proposition simply as "Seattle Spokane." But if

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PICTURE THEORY OF LANGUAGEthis arrangement of names pictures the fact that Seattle is westof Spokane, how will we picture the fact that Seattle is north ofPortland? Well, we can do this by writing "Seattle" over "Port-land":

SeattlePortland

This gives us the second part of the puzzle. For this is no longera proposition but a map. Surely one important difference betweena proposition and a map is that a proposition is a linear or one-dimensional structure.5 So either the picture theory is able toexplain only a poverty-stricken language (one in which, say, therelation of being west of can be expressed, but not the relationof being north of) or else it ignores a striking difference betweenpropositions and maps.

The problem facing the picture theory is that it seems toembrace three propositions that appear to be irreconcilable:

I. There is a one-to-one correspondence between the parts ofa proposition and the objects of the state of affairs picturedby the proposition (4.04).

2. Propositions are linear structures.63. Every possible state of affairs can be expressed in language.7

PROPOSED SOLUTIONS

The simplest way to resolve the paradox would seem to be totake the fact pictured by "Seattle is west of Spokane" to be an

5 Strictly speaking, written propositions of English are two-dimensional.But, as Morse Code shows, this second dimension is not essential. Notice alsothat the words of a proposition are not strung out but are bunched togetherwith at most a single space between one word and the next. Thus the problemcannot be solved by reminding ourselves that the points of a plane can bemapped onto the points of a line.6 Whether or not Wittgenstein subscribes to this proposition in the Tractatusis discussed below.

7 "Man possesses the ability to construct languages capable of expressingevery sense" (4.002)."Propositions can represent the whole of reality" (4.I2; cf. 4.26)."We can indeed say: everything that is (or is not) the case can be picturedby means of a proposition" (Notebooks, 26.5.I5).

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DA VID KEPTarrangement of three parts, not two: Seattle, Spokane, and therelation of being west of. The two proper names stand for thetwo cities and the predicate for the relation, so there is a one-to-onecorrespondence between the parts of the proposition and theparts of the fact. This sort of interpretation is given by EllisEvans. Evans in fact goes one step further and counts fourelements in both fact and proposition, the fourth element of theproposition being the orderof the words and the fourth element ofthe fact being the structureof the relation and its two terms:Wittgenstein would have said, I think, that the fact that Sophiahates Amos contained four elements: the two people, the hating, andthe structure of these, i.e. that it is Sophia and not Amos that is doingthe hating and Amos and not Sophia that is receiving it. The individualwords correspond to those first three elements, and the order of thewords to the fourth.8

There are two points to be made against this interpretation.The first is that it would be better not to call order an elementof a proposition and structure an elementof a fact. For if one does,Wittgenstein's solution of the problem of the false propositionloses its point. A false proposition has sense even though thereis no fact that it pictures because each of its elements, even whenit is false, still stands for an object. (On Evans' interpretationone of these objects will be a relation.) But what does the order ofwords of a false proposition stand for? Nothing. Thus if one callsthe order of words an element of a proposition, this element mustbe treated differently from the other elements. The second pointto be made against this interpretation is more serious. Evans, incorrelating the elements of the proposition "Sophia hates Amos"with the elements of the fact that Sophia hates Amos, has over-looked one of the elements of the proposition. For if the fact hasfour elements-Sophia, Amos, the relation of hating, and thestructure of the first three-then the proposition has five-thethree words "Sophia," "hates," and "Amos," the triadic relationof one word being between two others, and the structure of therelation and its three terms. So it turns out that Evans' suggestion

8 "Tractatus 3.I432," Mind, LXIV (I955), 260.

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PICTURE THEORY OF LANGUAGEdoes not really resolve the paradox since, upon following it up,it turns out that again a proposition has one element more thanthe state of affairs that it pictures.The second solution I want to examine, given by Erik Stenius,starts at the point at which Evans' solution breaks down.9 Firstof all, Stenius does not count the structure of a state of affairs asone of its elements. Secondly, he gets rid of the extra element withwhich Evans is left by combining two of Evans' elements into one.He allows the verb in such a proposition as "Sophia hates Amos"to be absorbed by the relation of betweenness. Thus Steniustakes the predicate of the proposition to be not the word "hates,"but the relationof one word being to the left and a second to theright of "hates." This predicate is a dyadic relation that is formedfrom the triadic relation of one word being to the left and a secondto the right of a third, by filling the third place with the word"hates." Stenius calls the word "hates" a characteristicof thepredicate. The predicate, Stenius says, "is always derived froma relation between the logical subjects and one or more symbolswhich appear as 'characteristics' of the predicate" (p. 134).Finally, Stenius explicitly distinguishes propositions from dia-grams on the grounds that the former unlike the latter are one-dimensional structures (p. 147).

My objection to this solution is a metaphysical one. Supposethat Amos and Sophia hate each other. Then there are two facts:the fact that Sophia hates Amos and the fact that Amos hatesSophia. Both of these facts contain the same three elements anddiffer only in structure. Now what is structure on this analysis?It is the way the relation is linked to its terms. But this meansthat this linking relation is the only real relation, the only relationthat does any work. The relational element of each fact, the hating,is reduced to a mere term of the linking relation. Structure onStenius' analysis is, in short, the relation that relates the relationalelement of a fact to its terms. This sort of analysis is objectionablesince it interposes a second relation between a relation and itsterms. It is, nevertheless, a tempting analysis. For by covertlyreplacing the multitude of relations by a single relation, the

9 Iittgenstein's "Tractatus" (Oxford, i960), esp. ch. vii.

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DAVID KENTlinking relation, it provides neat solutions to two problemsfaced by the picture theory of language.10 One problem is howto picture in one dimension more than a single relation betweenobjects. If there is only a single relation, this problem is easilysolved. A second problem is to explain the identity of structurethat obtains between a true proposition and the fact that itpictures. Again, if there is only a single relation, then the elementsof a true proposition are related in exactly the same way as theobjects they go proxy for; and this problem also finds a neatsolution.

As a solution of the paradox the type of theory offered byEvans and worked out consistently by Stenius is unsatisfactory.But how does it stand as an interpretation of Wittgenstein? Itis clear from the Notebooks that Wittgenstein at one time in hislife held the sort of theory Stenius outlines. During the pre-Tractatus period Wittgenstein held the two basic propositionsupon which Stenius' theory rests: that relations and propertiesas well as individuals are objects (Notebooks,6.6. I5, par. 5) andthat the predicate of a proposition is not a word or an expressionbut a relation of one or more places. This last point is made againand again in the early "Notes on Logic" and "Notes Dictated toG.E. Moore in Norway" (Notebooks,p. 93-I i8). Wittgenstein says,for example, that "in 'aRb,' 'R' is not a symbol, but that 'R' isbetween one name and another symbolizes" (Notebooks,p. io8; cf.p. I 2 I, 11. 14-2I). There are strong reasons, however, for holding

10 One must distinguish an arrangementtself from a kind of arrangement.Beingnorth of is one kind of spatial arrangement. That Seattle is north of Portland isa particular arrangement of this kind. That Portland is north of Seattle is apossible but not an actual arrangement of the same kind. Now the word"relation" sometimes means "kind of arrangement"; at other times it meanssimply "arrangement." Thus we say that being north of is one relation,being west of another. But we also speak of the relation of Seattle to Portland,namely, that Seattle is north of Portland. This double use of the word "rela-tion" may be partially responsible for the philosophical puzzle of how theterms of a relation are related to the relation itself. In this paper, to avoidprolixity, I use "relation" in both senses; but I think it will always be clearfrom the context in which sense the word is being used. In counting relationsone will, naturally, get different answers depending upon the sense of "re-lation." In speaking in the text of a single relation I mean obviously a kind ofarrangement.

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PICTURE THEORY OF LANGUAGEthat when Wittgenstein wrote the Tractatus, he had changed hismind on the first matter.11 The key passage in the Tractatus-one, incidentally, for which there is no parallel in the Notebooks-is 2.03: "In a state of affairs objects [hang in] one another likethe links of a chain." This simile, if taken seriously, cannot bereconciled with the notion that one of the objects of a state ofaffairs is a relation. First of all, to suppose that a relation is to itsterms as one link of a chain is to other links is to violate the theoryof types, since links of a chain are all of the same type whereasa relation is one type higher than its terms. Secondly, in anatomic state of affairs, which is what Wittgenstein is discussing inthis passage, there is only one relation. Consider now a state ofaffairs in which there is a triadic relation. Such a relation is notat all like a link of a chain, for a link of a chain joins at most twoother links whereas a triadic relation ties together three terms.A triadic relation is more like a key ring that holds three keysthan a link of a chain. Thus if we suppose that relations are objects,the simile collapses completely.Is the world of the Tractatus, then, a world without relations?No. Wittgenstein simply conceives relations differently from theway they have usually been conceived. In the chain metaphorthe relation is that of hanging together, and this is what Wittgen-stein calls structure: "The determinate way in which objects[hang together] in a state of affairs is the structure of the state ofaffairs" (2.032). So relations go over into structure. Monadicrelations or qualities, however, do seem to disappear since therelation of hanging together is a polyadic relation. Thus whileStenius and the early Wittgenstein distinguish three kinds ofthing within a state of affairs-one or more individuals, a relationof one or more places, and a structure-the Wittgenstein of theTractatus distinguishes only two-two or more individuals and astructure. How does this affect Wittgenstein's analysis of predicatesin the Tractatus? One would expect the predicate of a propositionto disappear completely, its role being taken by the arrangementof names. And this is, indeed, the interpretation adopted by two

11See especially Irving M. Copi's review of Stenius' book in the PhilosophicalReview,LXXII (I963), 382.50I


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DAVID KEPTcommentators on the Tractatus: Irving M. Copi and G.E.M.Anscombe.12

The interpretations of Anscombe and Copi, which differslightly but significantly, rest upon the premise that in theTractatus qualities and relations are not kinds of objects. Since afully analyzed elementary proposition consists entirely of names(4.22, 5.55) and names are proxies for objects (3.203, 3.22), theyconclude that no word or expression in a fully analyzed elementaryproposition refers to a quality or a relation. A relation of objectsis expressed by a relation of their names. But how can the manydifferent relations of objects be pictured by the single relation ofnames, that of concatenation? Both solve this problem by simplydenying that this is the only relation of names. Copi says, forexample:Any relation of objects, spatial or non-spatial, can be represented bya spatial relation of the names of those objects. That a has relation Rto b can be represented by writing "a" some specified distance anddirection from "b," and that a has some different relation R' to bcan be presented by writing "a" some different distance and directionfrom '"b."'53And Anscombe even produces an illustration of what a completelyanalyzed elementary proposition might look like:

c da be 14

On this reading a fully analyzed elementary proposition doesnot differ in any respect from a diagram or a map.

Anscombe's interpretation differs in one respect from Copi's.Copi supposes that when the elementary proposition "aRb" is

12 Copi, "Objects, Properties,and Relations in the Tractatus,"Mind,LXVII(1958); Anscombe, An Introductiono Wittgenstein'sTractatus London, I959)ch. 7; "Mr. Copi on Objects, Properties and Relations in the Tractatus,"Mind, LXVIII (I959).

13 Mind, LXVII (I958), I57-I58.14 Mind, LXVIII (I 959), 404-

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PICTURE THEORY OF LANGUAGEfully analyzed and set out diagrammatically, the diagram willcontain only the two names "a" and "b." Anscombe correctlypoints out that the predicate "R" may conceal further nameswhich will emerge on analysis. Anscombe's diagram above might,for example, be the analysis of "aRb." This point is an inferencefrom two passages in the Tractatustaken in conjunction. Wittgen-stein makes the comment that every atomic state of affairs mightconsist of infinitely many objects (4.22I i), and then only a fewlines later he says that he writes elementary propositions asfunctions of names, for example, "fx," "O(xy)," and so forth(4.24). The implication is that the predicate of the elementaryproposition "fa" may conceal infinitely many names.15 This is agood point the significance of which Anscombe does not see.

The sort of solution given by Anscombe and Copi resolvesthe paradox of the picture theory by denying one of the propo-sitions composing it. According to it the only propositions thatare completely and directly pictorial are not linear. This is anunsatisfactory solution for it, in effect, concedes that the picturetheory is inconsistent with a characteristic feature of language.There may indeed be this inconsistency, but if so, this means thatthe picture theory is untenable.16

Have Anscombe and Copi interpreted the Tractatuscorrectly?There is very little direct evidence either for or against theirtheory as an interpretation. The main passage cited by bothCopi and Anscombe in support of their interpretation is 3.I43I:"The essence of a propositional sign is very clearly seen if weimagine one composed of spatial objects (such as tables, chairs,and books) instead of written signs. Then the spatial arrangementof these things will express the sense of the proposition." But thispassage is not decisive. The point Wittgenstein is making inthe entire passage 3.14-3.144 is that a propositionalsign is a factand not a name or a jumble of names.17 A propositional signformed by putting a number of spatial objects all in a line (say,a book on a chair and the chair on a table) illustrates this point

15Ibid. See also An Introductiono Wittgenstein'sTractatus,pp. 99-I02.16 Incidentally, by this solution what would a spoken proposition be like?Would pitch be essential?17 That is, not a thing.

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DAVID KENTperfectly. So this passage is consistent with the view that in theTractatus propositions are linear structures.

Perhaps Wittgenstein attached no importance to the linearcharacter of propositions. There is an interesting comment onthis matter in The Brown Book: "Though from certain points ofview we should call the linear character of the sentence merelyexternal and inessential, this character and similar ones play agreat role in what as logicians we are inclined to say about sen-tences and propositions" (I, 41). When Wittgenstein wrote theTractatus did he consider this linear character external and in-essential or, writing as a logician, did he allow it to play a greatrole? There is one passage in the Notebooks that suggests thatWittgenstein distinguished propositions from pictures on the basisof the linear character of the former: "It can be said that, whilewe are not certain of being able to turn all situations into pictureson paper, still we are certain that we can portray all logicalproperties of situations in a two-dimensional script" (29.9.14).Since Wittgenstein speaks of a two-dimensional script, thisappears to say the very opposite of what I say it says; but this ismere appearance. For, first, Wittgenstein is contrasting pictureson paper with portrayals in a two-dimensional script; but if aproposition were itself exactly like a diagram or map, there wouldbe no contrast. Secondly, Wittgenstein in a passage written threedays earlier asks: "What is the ground of our-certainly wellfounded-confidence that we shall be able to express any sensewe like in our two-dimensional script?" (26.9.14). Now if thetwo-dimensional script Wittgenstein is talking about is our script,he must be referring to the familiar linear script of ordinarylanguage and logical symbolism. Why, then, does he call it atwo-dimensional script? Probably because letters and words aretwo-dimensional. Finally, in support of my reading of thesepassages, we need to recall that the theory of the Notebookseasilyallows for the linear nature of propositions. Notice, incidentally,that Wittgenstein says a bit later: "It all depends on settling whatdistinguishes the proposition from the mere picture" (2.I2.I4).Jumping now from the Notebooks o the Investigations,we may noteone passage that indicates rather strongly that in the Tractatuspropositions are thought of as linear structures. Wittgenstein in

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PICTURE THEORY OF LANGUAGEthe course of his discussion of logical atomism, a discussionwhich is meant naturally to embrace the Tractatus, devises alanguage game for which the account of language given bylogical atomists is valid (48). He sets the stage by introducing twothings: first, names for colored squares and, second, the arrange-ment of squares described by an arrangement of names. Nowthe interesting point is that matters are worked out in such away that a linear arrangement of names describes a two-dimen-sional arrangement of squares. The sample proposition Wittgen-stein writes out is a linear structure consisting entirely of names.

The considerations adduced in the last paragraph do notreally count for very much. They are merely skirmishing points.The solid argument against the Anscombe-Copi interpretation isthis. Anscombe and Copi hold that fully analyzed elementarypropositions must be two-dimensional diagrams: this is an essentialand not an accidental feature of them. Now what I want to argueis that, even if it be allowed that an elementary proposition mightbe expressed by such a diagram, it cannot be an essentialfeatureof an elementary proposition that it be two-dimensional. We need,first of all, to recall Wittgenstein's distinction between the essen-tial and accidental features of a proposition:

3.34 A proposition possesses essential and accidental features.Accidental features are those that result from the particularway in which the propositional sign is produced. Essentialfeatures are those without which the proposition could notexpress its sense.

3.34I So what is essential in a proposition is what all propositionsthat can express the same sense have in common.And similarly, in general, what is essential in a symbol is whatall symbols that can serve the same purpose have in common.3.34 I So one could say that the real name of an object was what allsymbols that signified it had in common. Thus one by one, allkinds of composition would prove to be unessential to a name.

By an argument parallel to that by which it is shown that com-position is unessential to a name it can be shown that the posses-

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DAVID KEYTsion of two dimensions is unessential to an elementary proposition.Suppose that b

ais an elementary proposition written as Anscombe and Copiwould have us write it. Now there are two relations in this dia-gram that are symbolic: that "b" is at a certain distance from"a" and that "b" is in a certain direction from "a." The state ofaffairs depicted by this diagram, on the other hand, being anatomic state of affairs, contains but a single relation. That is tosay, what we have here is an atomic state of affairs pictured by amolecular state of affairs: a Sachverhaltictured by a Tatsache.But just as it must be at least possible, no matter how impractical,for one simple object to serve as the name of another, so it mustbe possible for one atomic state of affairs to serve as the propo-sitional sign that pictures a second. Consequently, the complexityendemic to a two-dimensional diagram cannot be an essentialfeature of an elementary proposition.

THE PARADOX RESOLVED

A satisfactory interpretation of the picture theory of theTractatus-if I am correct so far-must embrace six propositions:that an elementary proposition consists solely of names, that aname stands for an object, that qualities and relations are notobjects, that an elementary proposition is a linear structure,that it is also a picture in which a relation between objects isshown by a relation between names, and finally that a completelist of elementary propositions would express every possiblerelation between objects. The picture theory of the Tractatus isnot limited to these six propositions-I have not, for example,mentioned the doctrines of analysis, of the logical independenceof elementary propositions, or of the absolute simplicity of objects-but it is these six that seem especially difficult to reconcile witheach other.

These six propositions, although difficult to reconcile, are notperhaps completely irreconcilable; for they are inconsistent, notamong themselves, but only together with a seventh proposition.The seventh proposition is that the number of possible relations

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PICTURE THEORT OF LANGUAGEbetween objects is greater than the number of possible concatena-tions of their names. This proposition is obviously true for or-dinary objects. For example: "Hitler Stalin" and "Stalin Hitler"exhaust the possible concatenations of the two names "Hitler"and "Stalin"; so if the first concatenation symbolizes, say, thatHitler hates Stalin and the second, consequently, that Stalin inturn hates Hitler, there will be no concatenation left to symbolize,say, that Hitler fears Stalin. But what is true for ordinary objectsmay or may not be true for Wittgenstein's simple objects. So thequestion needs to be raised whether this seventh proposition holdsfor Wittgenstein's simple objects. A rather satisfactory interpreta-tion of the Tractatuscan, in fact, be built upon the suppositionthat it does not hold.

Such an interpretation is not as simple-minded and as absurdas it may sound initially. For from the fact that there are manydyadic relations between two bodies such as the sun and the moonor between two cities, one cannot infer that there must also bemany dyadic relations between two simple objects. Relationalexpressions of everyday language such as "north of" occur onlyin nonelementary propositions. The elementary propositionsinto which such a proposition will be resolved upon analysis may,for all we know, be quite complex. And, further, this interpretationis consistent even with there being a multitude of elementarypropositions "aRlb," "aR2b," "aR3b," and so forth. For here weneed only recall Anscombe's point that the relational expressionsin these propositions indicate that the proposition is not yet fullyanalyzed. Upon analysis these propositions will on this inter-pretation turn out to be, say, "acb," "abc", "acdbfg," and so forth.So to suppose that the number of possible relations between Witt-genstein's simple objects is not greater than the number of possibleconcatenations of their names is at least not absurd.

Is there any support for such an interpretation? To begin with,this interpretation avoids the difficulties faced by the rival inter-pretations. This gives it some initial plausibility. Further, Wittgen-stein's metaphor of logical space strongly suggests such an inter-petation. Wittgenstein speaks of "a space of possible [atomic]states of affairs" (2.0I3). I take this to mean that a possibleatomic state of affairs is the analogue of a point in space. An

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DA VID KENTatomic state of affairs that is actual and not merely possible(an atomicfact) is the analogue of a material point (cf. Notebooks,29.IO.I4, par. 6). Thus Wittgenstein likensa world of positive andnegative facts to black spots on white paper (4.o63, 6.34I): apoint represents a possible atomic state of affairs, a black pointan existent atomic state of affairs, a white point a nonexistentatomic state of affairs. The co-ordinates of a geometrical pointare numerals; those of a point in logical space are names. Anelementary proposition asserts that a particular point in logicalspace is occupied; its denial is that it is empty. The logical productof a number of elementary propositions is like an equation ofphysics giving the exact location of all the matter in a particulararea. The logical sum, on the other hand, is like an equationgiving the size, shape, and location of the orbit of a planet: theplanet occupies one point of the orbit at any one moment, butsuch an equation does not tell us which point. Now this metaphorsupports my interpretation in this way. The number of points inspace with the co-ordinates "a," "b," and "c" is the number ofordered triples of these three numerals. There are six such triplesand six points. Thus the number of points with these co-ordinatesdoes not exceed the number of linear arrangements of the co-ordinates. But this is precisely my supposition for the points inlogical space, that the number of possible configurations of objectsdoes not exceed the number of linear arrangements of the namesof these objects. Thus this interpretation of Wittgenstein's picturetheory fits his metaphor of logical space very neatly.18

The considerations of the last two paragraphs are far fromconclusive. But they do establish, I believe, that this interpretationof the Tractatusis at least not an unreasonable one.

18 Stenius takes Wittgenstein's metaphor in a radically different way. SeeWittgenstein's " Tractatus," ch. iv. On Stenius' reading each point of logical spaceis a possible world, not a possible atomic state of affairs; and its logical co-ordinates are propositional signs, not names (see esp. pp. 54-58). The co-ordinates of each point are thus the factors of a logical product that embraceseach elementary proposition or its denial. An elementaryproposition, however,resembles the set of co-ordinates of a geometrical point much more closelythan a logical product does. A logical product is not affected by rearrangingitsfactors; but an elementary proposition, like the set of co-ordinates of a geo-metrical point, is affected by rearrangement.

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PICTURE THEORY OF LANGUAGEThe idea that the number of possible atomic states of affairs

is limited to the number that can be pictured linearly resolvesthe paradox within the framework of the Tractatus. But such asolution is not likely to rejuvenate the picture theory of language.For the paradox has only been resolved by shaping the world tofit our linguistic theory: we have cut the man to fit the bed. Thisis Wittgenstein's general strategy in the Tractatus, but it is nonethe better for that.

Three solutions of the paradox have been considered. TheEvans-Stenius solution reinterprets the first proposition of thetriad, the one about one-to-one correspondence. The Anscombe-Copi solution denies the second, that propositions are necessarilylinear structures. And my interpretation of the Tractatus inds away out through the third by imposing an unexpected limitationon the number of atomic states of affairs of any one form. Thefirst two are unsatisfactory both as solutions of the paradox andas interpretations of the Tractatus. The third seems to me to bea good interpretation of the Tractatus but a bad solution of theparadox. In conclusion I want to present what I think is a goodsolution of the paradox. My solution, like the Evans-Steniussolution, looks for a way out through the first proposition of thetriad; and also, like theirs, mine applies to any proposition of theform R(x1, x2, . . . xj) and not simply to the very specialelementary propositions of the Tractatus. My solution, unliketheirs, however, works only for n greater than one.

The dilemma facing the picture theory is this. Either a proposi-tion contains a predicate or else it does not. (It is like "Seattleis west of Spokane" or like "Seattle Spokane.") If it does, thereis no one-to-one correspondence between the parts of the proposi-tion and the parts of the state of affairs expressed by the proposi-tion. If it does not, then of the many possible relations between thethings named by the names in the proposition, only a single one canbe symbolized. This might be called the problem of the predicate.If the proposition "Seattle is west of Spokane" contains onepart too many for there to be a one-to-one correspondencebetween proposition and state of affairs, why not simply notcount the predicate? Would this be cheating? This is the questionI want to consider.

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DA VID KEYTTake maps. Here, surely, is the ideal case of a one-to-one

correspondence between symbol and state of affairs symbolized.Is the case, however, so ideal? If we think so, we are ignoringcertain symbols customarily found on maps: the arrow and thescale. But both of these, we want to reply, are written in themargin: this is why they can be ignored. Why, however, do wefeel that they are, so to speak, written in the margin? The reason,I think, is that the arrow does not enter into a triadic relationwith the symbols for, say, Seattle and Spokane. The arrow,rather, indicates how the dyadic relation between these twosymbols is to be taken: if the one symbol is left of the other, thearrow indicates (perhaps) that this relation pictures the one cityas west of the other.

The predicate of a proposition, I want to argue, functions verymuch like the arrow on a map. The following discussion makesthe best sense if we think of propositions written in logical notation.Let us take as our sample proposition "sWk" ("Seattle is westof Spokane"). One point that has been often repeated in thispaper is that the order of symbols in a proposition is itself sym-bolic: "sWk" and "kWs" are different propositions. A secondpoint, which has not been made, is that we get different proposi-tions only if we rearrange names; if we change the position of thepredicate, we get either a different notation (" Wsk")or nonsense("skW"). The position of the names in a proposition symbolizessomething; the position of the predicate does not. This suggeststhat the predicate of the proposition "sWk," like the arrow ofa map, does not enter into a triadic relation with the two names.It would make sense to write the predicate first followed by acolon: "W: sk." The predicate indicates, as this notation suggests,how the relation between the two names is to be taken: that"s" is left of "k" shows Seattle as west of Spokane, and that itshows this is indicated by "W:." Every proposition of the formR(xl, x2,. . . x.), where n is greater than one, is literally a mini-mum map. Thus there is as much of a one-to-one correspondencebetween a proposition and the corresponding state of affairs asthere is between a map and its corresponding state of affairs.Surely, this is as much as one can demand of the picture theoryof language.

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PICTURE THEORY OF LANGUAGEI once thought that this sort of theory could be found in the

Tractatus, but it does not seem possible to reconcile the notionthat a fully analyzed elementary proposition contains a predicatewith Wittgenstein's statement that "an elementary propositionconsists of names. It is a nexus, a concatenation, of names" (4.22).This passage strongly implies that elementary propositions consistof names alone. Wittgenstein at one place does discuss two symbolsthat are a bit like the arrow on a map, the sharp and flat symbolsin musical notation (4.oI3).19 But he does not appear to haveany special theory regarding them.

DAVID KEYTUniversity of Washington

19 Cf. G. E. Moore, "Wittgenstein's Lectures in 1930-33," Mind, LXIII(1954), i2; reprinted in Philosophical apers(London, 1959), p. 2 64.

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