The uniqueness of atomic facts in Wittgenstein’s Tractatus

D A V I D M I L L E R (University of Warwick)

I N THIS NOTE I wish to question the acceptability of Black’s explanation ([1964], pp. 46 f.; unless otherwise attributed, all quotations are from these two pages) why, in the metaphysics of the Tractatus, “the set of atomic facts must be unique”. I should warn the reader, however, that there will be no Tractatus scholar- ship displayed here: I intend merely to make one or two logical remarks that may be of interest to Tractatus enthusiasts. My quotations from Wittgenstein are based on the translation of Pears and McGuinness (Wittgenstein [ 19211) but, in some places, follow- ing Black (see pp. 38-45), I write “atomic fact” rather than “state of affairs” as a translation of Sachverhalt. However, I do not think that anything I have to say depends very much on the exactitude of the translation.

The world of Wittgenstein is a totality of facts. Facts are either atomic or molecular; they are “pictured” by propositions-in the one case, elementary propositions, in the other composite ones. I t seems to have been Wittgenstein’s view that the question of which facts are the atomic facts (and therefore the question of which propositions are the elementary ones) is determined solely by the structure of the world. That is, the world cannot be divided into atomic facts in more than one way. The atomic facts are unique. None of this, of course, is explicitly stated. Black sets out explicit- ly to state it, and to explain it.

A preliminary point concerns the independence of atomic facts. Tractatus 2.061 says: “Atomic facts are independent of one

An earlier draft of this note was most helpfully commented on by Professor Black. Sir Karl Popper, and Dr M. L. G. Redhead.

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another.” Black’s gloss is: “each might have failed to obtain while everything else remained unchanged”, an explanation of independ- .ence that is pretty much the same (apart from the vagueness of “item”) as 1.21: “Each item can be the case or not the case while everything else remains the same.” If we interpret this iteratively we obtain the idea of complete independence, originally defined by Moore [1910], p. 82, and stated quite clearly by Wittgenstein at 4.27: “For n possible (atomic) states of affairs, there are K , = = x:=o (.”)[= 2“] possibilities of existence and non-existence. Of these states of affairs any combination can exist and the remainder not exist. ” Complete independence is considerably stronger than simple independence, which is what logicians usually have in mind when they talk of the independence of the parallel postulate in Euclidean geometry, or the independence of the axiom of choice in ZermelolFraenkel set theory: n states of affairs are simply inde- pendent if any n - 1 can exist and the remaining one not exist. On the other hand, some of the other passages that Black cites (p. 72) in his note to 2.061 appear to construe independence less thorough- ly. Somewhere between simple and complete independence is 2.062: “From the existence or non-existence of one state of affairs it is impossible to infer the existence or non-existence of another.” This considers states of affairs only in pairs, but it does acknowl- edge the possibility that neither exists. Even weaker, though at the level of propositions rather than of facts, is 5.134: “One elemen- tary proposition cannot be deduced from another.”

It is natural to suppose that independence at the factual level can be routinely redeemed at the propositional level. Black actually calls 5.152, “When propositions have no truth-arguments in common with one another, we call them independent of one another”, a definition of independence (p. 72; but see the reference to Anscombe on p. 248). However, since the “truth-arguments’’ of a proposition are the elementary propositions that occur in it, all elementary propositions are vacuously independent in this sense. One must suppose that Wittgenstein meant something more by the independence of atomic facts. Moreover, if the set of atomic facts (or the set of elementary propositions) is not uniquely determined, which is the very issue we are about to investigate,

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the 5.152 idea of independence is simply not an absolute one: propositions that are independent in one language will not be so when translated into another. The same is true of any idea of statistical independence based on Wittgenstein’s primitive view of probability (in 5.15 and 5.151). For Wittgenstein’s vacillation on this issue, see Wolniewicz [ 19701.’

For reasons such as that italicized it seems best to suppose that atomic facts (or elementary propositions) are intended to be corn- pletely, not just simply, independent. To be sure, Black again hints otherwise in intoducing his example (p. 46) of two equivalent sets of independent propositions. But if “independent” here means “simply independent” then the example simply, indeed com- pletely, fails. This I shall establish in a moment.

Suppose that A , B, C are completely independent. Black defines K a s A * B ; L a s B - C ; a n d M a s ( B & C ) v ( i A & (BvC)).’He notes that A , B, C can be “equivalently expressed” in terms of K, L , M as follows: A is ( K & ( L - M ) ) v ( 1 K & 1 M ) ; B is (K & (L * M ) ) v ( 1 K & M ) ; C is ( 1 K & (L - M ) ) v (K & M ) . He notes further that if A , B, C are completely independent, so are K, L, M ; and conversely, if K , L , M are, so are A , B, C . (He does not say this on the first two lines of p. 47, but it is clearly what he means.) What is more, the set { A , B, C } is logically equivalent to the set { K , L, M}. Each of them, therefore, could serve as a “basic universal record”; for “in reducing a universal record we might have arrived either at the first set or the second. The view that there is a unique

Wolniewicz also undertakes to show that any two propositions independent in the sense of 5.152 are in fact completely (what he calls modally) independent. An unstated assumption is our supposition just below-the supposition that the elementary propositions are themselves completely independent. This is not a vacuous assumption in the way that the 5.152 independence of elementary propositions is. Incidentally, the proof is invalid even with the assumption. For Wolniewicz assumes (p. 163) that statistically independent propositions are compatible. This is clearly false, since a contradiction is statistically independent of every other proposition. Thus we can easily construct counterexamples to Wolniewicz’s conclusion (which is the conjunction of his (1) and (2)). For instance, if P and Q are elementary then P & i f ‘ and Q are not completely (or even simply) independent.

I have inconsequentially altered Black’s notation here and elsewhere.

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ultimate record must therefore be based upon other than purely logical grounds.”

To see that complete independence is required here, suppose that A , B, C are simply independent but cannot all be false; that is to say, { i A , i B , i C } is not consistent. Then if K and L are both true, B cannot be false-for then by the definitions of K and L both A and C would have to be false. Thus B is true, and so A and C have to be true. It follows from the definition of M that M is true, which means that the set { K , L, M } is not simply independent.

There can be no doubt that Black’s example is perfectly correct. It is, as far as I know, the first explicit recognition of the fact that two equivalent languages may be based on different sets of atomic propositions. (In algebraic terms: there is in general no unique set of generators of a free Boolean algebra.) The example is, however, rather cumbersome, though it can be simplified by noting that A is M*(K & L ) , B is l M * ( K & l L ) , and C is 1 M - ++( 1 K & 1 L ) . A more elegant one defines K and L as before, but lets M be the same as B. In this case A is simply K - M , C is simply M*L, and B is M . (This example comes from my [ 19741, p. 176. See also my [1976], p. 364 , where I acknowledge Black’s priority. The construction is not so very different from Goodman’s construction of the grue/bleen language from the greedblue language in [1954], Chapter 111, Section 4.) Black may have spurned this example because the sets { A , B , C } and { K , L , M } are not disjoint. As a further refinement I would therefore suggest the following:

K : B - A , L : B + + ( A + + C ) , M : B - C .

In terms of K , L, M we can write

A : L - M , B : L - ( M + + K ) , C : L - K .

The reader may easily check that { A , B , C } and { K , L , M } are I2 -Theorla 3: 1977

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equivalent, and that if one of them is completely independent so is the other.3

Black writes (loc. cit. ; the italics are mine) of the sets (A, B , C } and {K , L , M } (as he defines it):

If we grant that an atomic fact is composed of logical simples in immediate combination, we shall soon see that it would be impossible for both of our supposed sets to consist of elementary propositions. For the sake of illustration, suppose that the atomic facts stated by A and B , respectively, had the structures ( a , b ) and (b , c ) , respectively, where the small letters stand for objects. Since our K was equivalent to A - B , we must suppose all the objects involved in A and B to enter into K’s make-up, which must therefore be (a , 6, c , . . .). But we saw that A was also a function of K and L and M . Hence, by the same argument, the structure of A would have to be (u, b , c , . . .), contrary to our initial assumption. (I have assumed in the foregoing that atomic facts are uniquely determined by their constituents . . . .)

It is not quite clear to me where this last assumption is supposed to be used. To be sure, it guarantees that a =+c-certainly a necessary condition for anything’s being proved. But it does not seem to ensure that b + c , which also seems indispensable. And we surely do not need such a strong (even if incontestable) assumption merely to establish that there are two atomic facts with different constituent objects. Moreover, the following distribution of objects seems consistent with the assumption, but does not in any very obvious way rule out Black’s conclusion:

A =(a , b), B = ( b , c), C = ( c , d) , K = ( a , c) , L = ( d , a), M = ( b , d).

The construction is closely related to the theory of latin squares (when only two propositions are involved), or latin cubes or hypercubes. Here, of course, we are interested in the simplest case of 2 x 2 x 2 arrays, since complete independence is concerned only with propositions and their negations. (That we cannot define K , L , M perfectly symmetrically in A , B , C seems to be a direct consequence of the fact that there are no orthogonal latin squares of rank 2 . ) We would be led to general m,,x m , . . . x mn-, arrays if we were to take ascriptions of colour (for instance) as paradigms of elementary propositions, and to require of them an appropriate m- valued idea of complete independence. For brief comments on Wittgenstein’s own ideas in this direction see Black, pp. 367-369, and also Stenius [1960], pp. 42, 47. For latin squares see Denes and Keedwell[1974], in particular Chapter 5 .

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Such a distribution, however, is incompatible with the assumption that I have italicized in the above extract: i fX is a truth function of A , B , C , . . ., then all the objects involved in A , B , C , . . . are also involved in X . I shall call this the union assumption. It seems to me clear that this assumption is false.

Indeed, it must be false, in the context of Black's discussion, since the only other questionable assumption-the assumption that K , L , M can be treated as elementary propositions-plays no role at all in Black's argument. This can be seen by simply eliminating K , L , M from the argument. We would then say that the proposi- tion

( (A - B ) & ( (B-C)++ ((B & C ) v ( 1 A & (B v C ) ) ) ) ) v v(?(A-B) & l ( ( B & C ) v ( l A & ( B v C ) ) ) ) ,

which is equivalent to A , since it is merely what we get from first writing A in terms of K , L , M , and then writing K , L , M in terms of A , B, C , must contain in its make-up all the objects involved in A , B , C . On the assumption thatB and C involve some objects that A does not, this situation is an impossible one. Thus the union assumption cannot be correct.

More carefully, simply, and elegantly: suppose that A , B , C are distinct atomic facts. Then at least one of them, say A , must in- volve fewer objects than do all three together. Since A is identical with @ - ( A - C ) ) + + ( B - C ) , we have a contradiction with the union assumption. Indeed, this assumption is even more easily refuted. For, given that atomic facts are uniquely determined by their constituents, it entails that A-A, B - B , and C - C are all different propositions.

I t does not seem to me that any similar assumption is going to serve Black any better. It does not matter what functional relation- ship (if any) holds between the objects involved in A , B , C , . . . and those involved in truth functions of these propositions: in no case are we going to have materials for an argument against the possible elementary status of K , L, M , as opposed to A , B , C . Nevertheless, it is worth noting the difficulties involved in defining h(X), the set of objects involved in the proposition X . The union assumption was simply: h( ?A) = h(A) and h(A v B ) = h(A) u h(B). As we

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saw above, such an h is not single-valued when we identify logically equivalent propositions-an identification authorized by 5.141. The same goes for the definition: h( i A ) = h ( A ) and h(A vB) =h(A) A h(B). (x A y is the symmetrical difference of x and y . Using - x for the set-theoretical complement of x we can writex Ay =(x n - y ) u (-x ny).) Here we rather satisfactorily have h(A - A ) =h(A v i A ) = h(A & 1 A ) =@, a consequence in harmony with the doctrine of the equal uninformativeness of tautologies and contradiction^.^ But h(A & A ) is also empty, and so is not in general equal to h(A) . A function that at least assigns the same value to equivalent propositions is h( i A ) = - h(A) and h(A vB) = = h(A) u h(B), where - indicates complementation in the class Cl of all objects. This makes h a homomorphism from the algebra of propositions to some algebra of subsets of C l , so that h(A & B) = = h(A) n h(B). Thus if A and B have no objects in common then h(A & B ) = @ = h(A & i A ) . This conclusion certainly looks as though it ought to be an unacceptable one.

My guess is that the function h(X) simply cannot be defined recursively on the structure of the truth function X . (For a slightly different stab at the problem, see Carnap [1971], pp. 61 f.) This conjecture appears all the more reasonable if we think of elemen- tary propositions in the way encouraged by most versions of the predicate calculus: they are such propositions as Pu, Qb, Rub, where P , Q, R are primitive predicates. We say that both P and u occur essentially in Pa, and both P and b in Pb. But although in Pa e P b all three letters P , a , b occur essentially, in Pa ++Pa not one does. (The union assumption sometimes works, but not always.) Thus the letters occurring essentially in a truth function

Tractutus 4.461 begins: “Propositions show what they say: tautologies and contradictions show that they say nothing.” Incidentally, this passage must surely be interpreted as meaning not that a tautology and a contradiction have equal content-for this is clearly false -but that they have equal vrrisirnilitudr. The contradiction does not “say nothing”; but it does say nothing that is particular t o this world. The interpretation is to some extent supported by 4.463 (my italics): “. . . A tautology leaves open to reality the whole-the infinite whole-of logical space: a contradiction fills the whole of logical space leaving n o point of it for reality. Thus neither of them can determine reality in any wav.”

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X are not related in any very direct way (and possibly not in any way congenial to the extreme extensionalism of the Tractatus) to those occurring in the propositions of which X is “made up”. Likewise, we might say, the constituents of a molecular fact are not entirely determined by the constituents of its atomic in- gredients. Indeed, we might go further and maintain that molecular facts are not “composed of logical simples in immediate combina- tion”, as atomic facts are, nor even, perhaps, composed of logical simples “in combination” at all, a contention that would resolve Black’s original problem in the most artless and unadventurous manner imaginable. That is, it would explain the uniqueness of the set of atomic facts simply by their being different in funda- mental structure from non-atomic facts. (This is presumably the sort of answer one would give to a question like “Why is the set of even numbers unique?”.) Atomic facts, we might put it, are intrinsically atomic. Yet although this approach is surely consistent with the main bulk of Tractatus doctrine, it cannot seriously be adopted as an answer to our problem. Too much is left unexplain- ed . For example, if atomicity is an intrinsic property, why is the set of atomic facts a completely independent one? What relation does a composite proposition bear to the elementary propositions of which it is a truth function? How, in particular, can it be that elementary propositions are truth functions of composite proposi- tions? And most important of all: why is it that all propositions are truth functions of the elementary ones (Tractatus 5)?

This last question might be rephrased by asking whether “logical space” determines, or is determined b y , the class of elementary propositions it includes. If elementary propositions are simply those that “picture” the atomic facts, it seems that their extent is determined by the extent of logical space, rather than the other way round. But then it is distinctly puzzling why all propositions are truth functions of the elementary ones. In the case of physical space, it should be noted, the analogous question is much more readily answered. What correspond to the elementary proposi- tions are the coordinate axes: they “span” the space (each point can be identified by its three coordinates; more precisely, each vector is a scalar combination of unit vectors parallel to the axes);

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they are linearly, rather than completely, independent; each corresponds to a distinct dimension (see, for example, Stenius [ 19601, Chapter IV); and so on. But though the space is determined by the axes, the reverse is certainly not true. Given physical space undissected, we cannot identify the coordinate axes. For any number of distinct trios of mutually perpendicular lines (to con- sider the simplest coordinate systems only) can serves as axes of coordinates. Thus the very question that is here our main concern (“Are the atomic facts unique?”) has an analogue in the case of physical space that is answered with a resounding No.

It seems therefore that there is no very good reason forthcoming for supposing the set of atomic facts to be uniquely determined. (I should in fairness note that Black offers a second argument for his conclusion. But I have not been able to understand it, partly, perhaps, because it seems to be attacking the thesis that A , B , K might be elementary propositions in one and the same system. But this is manifestly not the case, for they are not even simply inde- pendent.) My own view is that there are no such things as atomic facts. But there are, of course, with respect to any particular language, elementary propositions, or, rather, elementary sentences. Thus language only “pictures” the world from a particular point of view. It provides a perspective, but only one perspective amongst many. No fact is more atomic, or more composite, than any other.

To conclude, I wish to ask, half seriously, half frivolously, whether logical atomism could adjust itself to a world where there is no unique set of atomic facts or elementary propositions. That even what is fundamental should turn out to be a relative matter may seem to some the very antithesis of atomism as a significant doctrine. (Professor Popper has pointed out to me that Wittgen- stein himself moved in this direction. See Wittgenstein [ 19531, section 47.) Yet physical atomism has recently been prepared to entertain, if not to embrace, a similar relativity in the theories of Chew, and others, that are variously referred to as nuclear democracy or the hadron b0otstrap.j According to Polkinghorne

Hadrons are strongly interacting particles, like protons, neutrons. and other baryons, and mesons. My attention was drawn to these ideas by Dr Redhead, and

THE U N I Q U E N ~ S S OF ATOMIC‘ FACTS IN WITTGFNSI EIN s TRAC TATUS 183

[1971], p. 568: “such a theory cuts the embarrassing gordian knot of what particles are elementary and what composite. From the bootstrap point of view everything is made of everything; all particles are composites. This pleasing equalitarianism has some- times been given the name of nuclear democracy.” Capra [1975], p. 313, writes:

The picture of hadrons which emerges from these bootstrap models is often summed up in the provocative phrase, “every particle consists of all other particles”. . . . all hadrons are composite structures whose com- ponents are again hadrons, and none of them is any more elementary than the others. The binding forces holding the structures together manifest themselves through the exchange of particles, and these exchanged particles are again hadrons. Each hadron, therefore, plays three roles: it is a composite structure, it may be a constituent of another hadron, and it may be exchanged between constituents and thus constitute part of the forces holding a structure together. . . . Thus, “each particle helps to generate other particles, which in turn generate it.” . . . The whole set of hadrons generates itself in this way or pulls itself up, so to say, by its “bootstraps”. . . .

The similarity between the structure of hadrons, so imagined,fi and the structure of the propositions A , B , C , K , L , M , displayed on p. 177 above, is striking. But there are important differences, clearly revealed by what Capra says at the end of the paragraph quoted above. He writes:

The idea, then, is that this extremely complex bootstrap mechanism is self-determining, that is, that there is only one way in which it can be achieved. In other words, there is only one possible self-consistent set of hadrons-the one found in nature.

What constitutes the bootstrap element in the bootstrap philosophy is therefore not so much that the hadrons are composed of one another, but that they are determined by one another-or rather that the physical parameters characterizing them are determined by the mathematical equations. Nothing like this holds in the propositional case: the propositions A , B , C , K , L , M may be

I have benefited a good deal from correspondence with him. For bibliographical assistance 1 am indebted to Mr M. J. Davies. ti For further references see Chew [1968]. [1970]. and Capra [1975].

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composed of one another, but whether they, or perhaps their negations, are true is something quite independent of their logical interconnections. Nevertheless, the similarity is significant. But since Capra sees in the bootstrap also a striking likeness to the doctrines of Mahayana Buddhism, and in them a likeness to Leibniz’s Monadology, and since there are-in view of the last quotation-obvious connections with the coherence theory of truth, it is perhaps unwise to take any of these parallels too serious- ly. Nothing could show more clearly the non-transitivity of the relation of similarity than the fact that the Avatamsaka Sutra and the Tractatus logico-philosophicus are similar to the same sophisticated theory of mathematical physics.

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CHEW, G . F. [1%8] “‘Bootstrap’: A scientific idea?” Science, vol. 161, pp. 762- 765.

CHEW, G . F. [ 19701 “Hadron bootstrap: Triumph or frustration’?’’, Physics today , vol. 23, pp. 23-28.

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THE UNIQUENESS OF ATOMIC FACTS IN WITTGENSTEIN’S TRACTATUS 185

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Received on June 18, 1976. Revised version received o n June 8, 1977.