Elementary Propositions and Independence

This paper is concerned with Wittgenstein’s early doctrine of the independence of elementary propositions. Using the notion of a free generator for a logical calculus—a concept we claim was anticipated by Wittgenstein— we show precisely why certain difficulties associated with his doctrine cannot be overcome. We then show that Russell’s version of logical atomism—with independent particulars instead of elementary propositions—avoids the same difficulties. We intend to discuss a basic notion of logical atomism: that of independence. Elementary propositions are clearly central to Wittgenstein’s atomism; their characterization rests on the doctrine of their independence. And a central tenet of Russell’s atomism is that the world is decomposable into independent components. Our aim is to elucidate these two uses of independence and the conceptions of elementary proposition to which they give rise. 1 Elementary propositions as free generators One of the central and most problematic concepts in the Tractatus is that of ‘elementarsatz’ or elementary proposition. By this is meant a proposition in some sense not further analyzable into simpler propositions. Of this concept it is asserted in the Tractatus: 5.3 Every proposition is the result of truth operations on elementary propositions.1 That is, elementary propositions are to be regarded as the ultimate propositional constituents or generators of propositions: in this respect elementary propositions are propositional atoms, or atomic propositions. We also find the following further assertions concerning elementary propositions: 4.211 It is a sign of a proposition’s being elementary that there can be no elementary proposition contradicting it. 5.134 One elementary proposition cannot be deduced from another. Received August 17, 1995; revised February 19, 1996 ELEMENTARY PROPOSITIONS 113 We may take these as asserting that elementary propositions are (logically) independent of one another. Now on what grounds can it be claimed that atomic propositions, in the sense of generators, satisfy this independence condition? To clarify this question, let us examine some familiar logical systems. Consider first the propositional calculus. Here propositions are built up by applying the logical operators to an initial stock of proposition letters which are in another, completely natural sense, atomic propositions, since they are of minimal length. They are also independent in the strong sense that, for any finite set {P1, . . . , Pn} of them, no conjunction of the form X1 & · · · & Xn is ever inconsistent, where each Xi is either Pi or ¬Pi. This situation has a more general algebraic description. If, following the suggestion of 5.141, we identify (provably) equivalent propositions of the propositional calculus, we obtain a Boolean algebra—the Lindenbaum-Tarski algebra associated with the propositional calculus. A subset X of a Boolean algebra B is said to be free, and its elements independent, if for any finite subset {x1, . . . , xn} of X, we have y1 ∩ · · · ∩ yn = 0 where each yi is either xi or x∗ i (= the Boolean complement of xi). X is said to generate B if B is the least subalgebra of B containing X, in other words, if every element of B can be expressed in the form y1 ∪ · · · ∪ yn , where each yi is of the form z1 ∩ · · · ∩ zm with zi ∈ X or zi ∈ X.2 Finally B is said to be freely generated if it has a free set of generators. Now the Boolean algebra PROP associated with the propositional calculus is freely generated, and the propositional letters constitute a free set of generators. Thus, the propositional calculus provides a perfectly good, if restricted, model of Wittgenstein’s scheme of elementary propositions, one that well illustrates his notion of elementarity, and the two notions of atomicity, introduced above. But Wittgenstein of course worked with a much richer language than that underlying the propositional calculus: in particular he required that universal and existential assertions be formulable. It is natural, therefore, to extend our discussion to the predicate calculus. We first consider the pure predicate calculus PC. We assume that the underlying language contains a set C of infinitely many constant symbols. Let PRED be the Boolean algebra obtained by identifying equivalent sentences in PC. PRED has a certain additional structure obtained by noting that quantified sentences are the suprema and infima of certain subsets, viz., ∃xφ(x) = sup{φ(c) : c ∈ C} ∀xφ(x) = inf{φ(c) : c ∈ C}. For this reason we shall call PRED a quantifier algebra.3 This is, of course, analogous to Wittgenstein’s discussion of quantification. Wittgenstein’s account differs from the one we have given in terms of the notion of a quantifier algebra only in the choice of truth-function with respect to which he defines the quantifiers4; Wittgenstein uses a generalization of the Sheffer stroke rather than suprema and infima. Now it can be shown that the set of atomic sentences (i.e., the set of corresponding equivalence classes) is free in PRED and that it also generates it as a quantifier algebra. (For a proof of this assertion, see Rasiowa and Sikorski [12], Chapter VIII, 24.1.) Thus, in this extended sense, PRED is freely generated by the set of atomic sen114 JOHN L. BELL and WILLIAM DEMOPOULOS tences, so that we obtain a satisfactory model of Wittgenstein’s scheme, with atomic sentences again playing the role of elementary propositions (generators). It has often been noted that Wittgenstein required of his elementary propositions that they be logically independent of one another. It has also been observed that elementary propositions constitute the atomic constituents of more complicated propositions. What appears to have been missed, so far as we can determine, is the significance of the combination of these two ideas, namely, that together they constitute an anticipation of the notion of a free generator.5 Thus, from our point of view it is not hard to see why Wittgenstein found his characterization of elementary propositions an attractive one; we can also give a complete answer to a question posed by David Pears: “It is a sign of a proposition’s being elementary that there can be no elementary proposition contradicting it.” (Tractatus, 4.211) Why did Wittgenstein require the elementary propositions of the Tractatus to pass this difficult test? ([10], p. 74) On our account, Wittgenstein had, in effect, hit upon the idea of a free generator and had correctly noted, in the applications he made of the notion to the propositional calculus and to the pure predicate calculus, its connection with independence and with the notion of atomicity which we associate with minimal size; he also perceived its central role in these two logical systems. It is often remarked that the chief mathematical contribution of the Tractatus was Wittgenstein’s presentation of the method of truth-tables. It seems to us that the articulation of the notion of a free generator together with the recognition of its centrality in the systems PROP and PRED was a contribution of no less importance. 2 Free generators and the Grundgedanke of the Tractatus Wittgenstein’s “Grundgedanke” or fundamental thought is presented at 4.0312: My fundamental thought is that the “logical constants” do not represent. That the logic of the facts cannot be represented. Negation is the logical constant with respect to which the ‘fundamental thought’ is most fully elaborated in the Tractatus. In order to better see Wittgenstein’s point in connection with negation, Ramsey ([11], pp. 146f) suggested that we should imagine the negation of an elementary proposition to be symbolically represented by inverting the sentence expressing it. His idea appears to have been that this would remove the temptation to think that in ¬p the negation sign introduces an additional representational element not already present in p, a point made explicitly—or, at least, as explicitly as any point is ever made in the Tractatus—at 4.0621 (paragraph 3): The propositions p and ¬p have opposite sense, but there corresponds to them one and the same reality. In particular this would remove any basis for supposing that there must be negative facts in anything like the sense in which Russell appears to have supposed there to be negative facts, namely, as states of affairs existing alongside atomic facts.6 It seems to us that Ramsey’s observation as well as other aspects of the fundamental thought are naturally captured by our account of elementary propositions as ELEMENTARY PROPOSITIONS 115 free generators. Let E be the set of elementary propositions and B the Boolean algebra freely generated by E, so that B is the set of all propositions. We then have an injection i: E → B such that i[E] is a free set of generators of B. We usually identify p with i(p) for p ∈ E. But this can cause problems since the map i is not uniquely determined by E. For example, suppose given any such i, call it i0 , and any subset X of E; define ix : E → B by ix(p) = i0(p) for p ∈ X, and ix(p) = i0(p)∗ for p ∈ E − X. Then ix : E → B is an injection and ix[E] is also a free set of generators of B.7 Once elementary propositions are seen to be the atomic generators of all other propositions—once it is seen, in other words, that all propositions are truth functions of the elementary propositions—it remains to be observed only that the fundamental thought holds of negation, since the latitude allowed by i extends only to the possibility of assigning i(p)∗ to an elementary proposition p. If the injection of E into B is given by iX , we may think of the elementary propositions in X as being “given positively,” and those not in X, “given negatively.” But since X is arbitrary, so is the positive/negative distinction. That is, the members of E are “without orientation” (neither positive nor negative). They acquire their orientation only after insertion into B. Wittgenstein’s pre-Tractarian notion that propositions are bipolar thus corresponds to the fact that, for p ∈ E, i0(p) as a free generator, could equally well be replaced by i0(p)∗: p itself does not determine which one we choose.8 In the Tractatus, elementary propositions both picture reality and constitute the atomic generators of all other propositions. In our scheme, the distinction between these two functions is represented by the distinction between p and i(p). By 4.0621(3) (quoted earlier), the reality corresponding to p, if p is true, is independent of its insertion into B by the injection i. We have, therefore, to distinguish between p as picture—something which does not depend on whether its image is given by i or by ix , and indeed, does not depend on p’s insertion in B at all—and p’s role as a propositional generator of B; the latter does, of course, depend on p’s insertion into B. 3 Independence lost Now it is well known that in [22] Wittgenstein came to question the doctrine of elementary propositions just outlined. What was the difficulty that caused him to do so? Consider the atomic sentences ‘Ga,’ ‘Ra.’ These are independent (free generators) in PRED. But if we now interpret ‘G’ as green, and ‘R’ as red, and ‘a’ as a speck in Ludwig’s visual field, we turn them into contraries. As we see it, the difficulty is that Wittgenstein saw his account of elementary propositions not merely as a theory of ‘empty forms,’ but as one which would encompass at least one application. But in general, any specific application will have the effect of obliterating the independence of elementary propositions. Notice that the critical point was not the incompatibility of ‘a is red’ and ‘a is green’—something already observed in the Tractatus (6.3751)—but the recognition that however the scheme of elementary propositions is applied, atomic propositions may no longer be independent, and so, a fortiori, no longer elementary. This is why, in Pears’s phrase, independence is a “difficult test” of elementarity. Wittgenstein’s reference (at 4.04, quoted below) to Hertz’s discussion of dynamical models (in [6], §§418–28), suggests that he saw his theory of elementary proposi116 JOHN L. BELL and WILLIAM DEMOPOULOS tions as a generalization of Hertz’s account of “any possible mechanical description” (a point that is noted in Griffin [5], p. 5). The central idea which Wittgenstein took from Hertz’s discussion of dynamical models was Hertz’s notion that a dynamical model must have the right “mathematical multiplicity.” A stumbling block to a correct interpretation of Wittgentein’s appropriation of Hertz’s concept of mathematical multiplicity is the juxtaposition of the two paragraphs which together make up 4.04: In a proposition there must be exactly as many distinguishable parts as in the situation that it represents. The two must possess the same logical (mathematical) multiplicity. (Compare Hertz’s Mechanics on dynamical model.) The first paragraph has misled commentators into supposing that Wittgenstein’s (and Hertz’s) use of mathematical multiplicity and dynamical models is captured by the notions of one-one correspondence and isomorphic image, with Wittgenstein (presumably following Hertz) adapting these notions to his conception of the proposition as picture. The idea, on this interpretation, is that sameness of mathematical multiplicity (which is identified with the existence of a one-one correspondence) is what is required if the names occurring in propositional signs are to be capable of “going proxy” for objects.9 4.04, however, consists of two paragraphs, and it is by no means evident that the second paragraph is merely a repetition, in other terminology, of the first. And indeed, an examination of [6] shows that the notion of mathematical multiplicity plays an altogether different and far subtler role, both in Hertz’s discussion and in Wittgenstein’s deployment of it. For Hertz, the requirement that a model have the right mathematical multiplicity meant that in an analysis of motion it is necessary to isolate the degrees of freedom characteristic of the system under study. In the simplest case—a single particle in free space—each of the three dimensions along which the particle’s position can vary constitutes a “degree of freedom,” and each component degree of freedom can be set independently of any other.10 This is the idea which is recalled when, just prior to his presentation of his general theory of propositions at §§5.5ff, Wittgenstein tells us that such a theory requires that we should construct a system of signs with a particular number of dimensions—with a particular mathematical multiplicity. (5.475) Wittgenstein’s notion of elementary proposition thus extended Hertz’s notion of mathematical multiplicity or independence to the propositional generators of pure logic—the general form of any possible description. However, Wittgenstein failed to give any indication of how to recover the particular forms of description which underlie applications of this abstract framework. Obviously, the characterization of elementary propositions was not supposed to be dependent upon the features peculiar to its physical applications; nevertheless the account was supposed to encompass any such application. In the process of characterizing elementary propositions, Wittgenstein articulated a theoretically important mathematical concept—that of a free generator; however the goal of encompassing applications of his theory of elementary propositions eluded him. If, as Dummett [3] has persuasively argued, the principal achievement of Frege’s theories of the cardinal and real numbers consisted in their ability to account for our applications of the numbers, then it is somewhat ironic that ELEMENTARY PROPOSITIONS 117 the fatal defect of Wittgenstein’s theory of propositions should consist in its inability to provide for any application of the notion of an elementary proposition. Formally, the situation may be described as follows. Let T be any (consistent) theory in PC. We obtain a new Boolean algebra PREDT by identifying sentences when they are provably equivalent in T : PREDT is a quotient of PRED. Now, not only will the atomic sentences fail to be independent, and so, not a free set of generators in PREDT , the latter is, in general, not freely generated at all. Although atomic sentences are independent relative to the ordering defined by [φ] ≤ [ψ], if φ → ψ, they may be dependent relative to the ordering