Frege and Wittgenstein on Generality
It has often been noted that Frege treats the truth functions and quantifiers as particular, though quite abstract, contents, in some tension with the commonplace that logic is a purely formal discipline.11 Thus conjunction, for instance, is on his account a particular function from pairs of objects to objects—in particular ‘truth values’, though for Frege this is only to specify further its content, not its logical type—while the first-order universal quantifier is a second-level function from first-level functions (that is, functions from and into the set of objects) into the set of objects (also always having as its value a truth value).
Now, it is famously a guiding thought of Wittgenstein's Tractatus logico-philosophicus that this treatment of logical notions as particular contents cannot be right. Thus, as the Tractatus's proposition 4.0312 tells us: ‘My fundamental thought is that the “logical constants” do not represent’. The fact that the truth functions are interdefinable (which Frege already recognized12) leads Wittgenstein to conclude that a sign for a truth function in a sentence does not characterize the sense of that sentence. For if we can express a given proposition indifferently as ‘∼(A ∨ B)’, ‘∼A & ∼B’, or ‘A B’, then none of ‘∼’, ‘∨’, ‘&’ or ‘’ is essential to the expression of, nor hence to the sense of, the proposition. What is essential is the one truth function that each of these combinations of symbols expresses: that is, the truth function that is expressed in the truth table that all of these sentences share. But then, Wittgenstein argues, the task of logic must not be the study of the various truth-functional ‘logical constants’ which can (but need not) be used to express those truth functions, but rather the study of the way in which propositions can be built truth-functionally out of elementary propositions.
Something similar happens to quantification. Wittgenstein ‘separate[s] the concept all from the truth-function’ (§5.521); but as in the case of the truth-functional ‘logical constants’, he rejects the Fregean approach to generality according to which a quantifier is treated as a particular function from concepts to truth values. Let us consider Frege's approach a bit more closely, in order to appreciate better Wittgenstein's rejection of it.
A list of the symbolic apparatus of Frege's notational system might lead one to expect that Frege's explanation of his symbols would include a single, univocal treatment of the ‘concavity’ he uses to express generality. But in fact in his Grundgesetze der Arithmetik (1893) he introduces the concavity first in its application only to generality over objects (§8). Part 1.iv) of that work is titled ‘Extension of the notation for generality’, and its first section, §19, indeed makes use of what appears to be the same concavity—with, indeed, some of the same rules for its use, such as the rule for determining what the ‘corresponding function’ is to which it is applied in a given case. But his explanation of the Bedeutung of an instance of the use of the concavity to express generality over first-level functions is given entirely independently of the analogous explanation for generality over objects back at §8. And this is no coincidence. The distinction between generality over objects and generality over first-level functions, cashed out as it is in terms of the distinction between second-level and third-level functions, ‘is not made arbitrarily, but founded deep in the nature of things’ (Frege 1891: 31).13
But the following words from the Tractatus contain the material for an objection to just this approach to generality:
If logic has primitive ideas these must be independent of one another. If a primitive idea is introduced it must be introduced in all contexts in which it occurs at all. One cannot therefore introduce it for one context and then again for another. … for it would then remain doubtful whether its meaning in the two cases was the same, and there would be no reason to use the same way of symbolizing in the two cases. (§5.451)
One might see Frege's repeated use of the concavity, and some of his patter surrounding it, as indicative of a conception of a unitary notion of generality as a primitive idea of logic. But this is then betrayed by the fact that the notation needs to be reintroduced, and given a fresh explanation, for each level over which it is to be used to range. One way of understanding the treatment of generality in the Tractatus—and of putting in some context the account of ‘forms of object’ in the book's opening pages—is to recognize it as an attempt to give a genuinely unitary account of generality, in contrast to Frege's.14
There are remarks about generality scattered through the Tractatus, but it is most instructive to consider the role it is meant to play in ‘the general form of proposition’, namely ‘[, , ]’ (§6), which ‘says nothing else than that every proposition is the result of successive applications of the operation to the elementary propositions’ (§6.001). ‘ is the negation of all the values of the propositional variable ξ’ (§5.502); in other words, it is a generalization of the NOR operator in terms of which, just as in terms of the Sheffer stroke (NAND), Sheffer showed it possible to define all of the truth functions. But if N is in this way truth-functional, and represents the totality of elementary propositions, where does generality come into this purported representation of ‘that which all propositions, according to their nature, have in common with one another’ (§5.47)? In the notation of §6, it comes in through the variable ξ. About this, Wittgenstein tells us:
The values of the variable are stipulated.
The stipulation is a description of the propositions for which the variable stands.
How the description of [these propositions] takes place is inessential.
We may distinguish three kinds of description: 1. Direct enumeration. In this case we can simply give its constant values instead of the variable. 2. Giving a function fx whose values for all values of x are the propositions to be described. 3. Giving a formal law according to which those propositions are constructed. In this case the [propositions] are all the terms of a formal series. (§5.501)
The notation of §6, in other words, has the following import: to construct any proposition whatever, we are to begin with the elementary propositions, take a selection of them (by whatever means we wish) and jointly negate that selection; then, if we wish, we may repeat the procedure, now starting with the elementary propositions together with the result of the previous application of the operation. Generality comes in through the selection: for we may, for instance, begin by selecting all the elementary propositions; or we may select all the elementary propositions containing a given expression; or again, we may select all the propositions that result from repeated application of a given operation to some beginning proposition.
What should be striking about this account of generality, especially against the background of the Fregean approach we sketched a few paragraphs back, is that it makes no essential reference to the structure of the propositions providing the basis for a given generalization. Whereas Frege, embedding his account of generality in the context of his analysis of propositional articulation in terms of objects and functions, must explain generality over objects separately from generality over functions—that is to say, must explain generality by making explicit reference to the other aspects of the structure of the propositions in which it is involved—Wittgenstein ‘locates’ generality once and for all in the selection of propositions for joint negation, however that selection is carried out. Indeed, he tells us explicitly, as we just quoted, that the procedure involved in this selection is ‘inessential’ (§5.501). He goes on to give examples of how it may go—examples which reveal that the ‘general form of proposition’ can indeed encompass (at least some of) the propositions which Frege and Russell would express using quantifiers—but he emphasizes that it is not essential to an account of generality to spell these examples out: ‘We may distinguish three kinds of description’,15 but it is not incumbent upon the logician to do so; and by the way, nothing is said to imply that there aren't other kinds besides these three.
And this helps explain the fact that the book appears to lack the kind of attention to categorial analysis that Frege's work might have led us to expect. Atomic facts, we are told, consist of ‘combination[s] of objects’ (§2.01)—rather than, say, combinations of objects with functions taking those objects as arguments.16 We are also told that these objects ‘contain the possibility of’ the states of affairs in which they can occur, the range of such possibilities being ‘the form of the object’ (§§2.014f.). This is suggestive of categorial distinctions such as Frege's: for we may think of first-level one-place functions, for example, as ‘objects’ which can occur together with (Fregean) objects to form facts; second-level functions as ‘objects’ which may combine with first-level functions; and so on.17 But very little is said definitively18 about the particular forms of object there are: examples such as those I just gave are absent. And all this is recapitulated at the level of propositions. An elementary proposition is ‘a connexion, a concatenation, of names’ (§4.22); while the ranges of possibilities of combination of simple names with one another in elementary propositions recapitulate the forms of the objects for which they go proxy, we are again given no details about these ranges. We are not given a distinction between singular term and predicate, any more than between object and function.
It is important to the Tractarian account that there is structure in atomic facts and in elementary propositions. That ‘the proposition is articulate’ (§3.141), and the related notions that the picture is a fact (§2.141) and that the fact is a combination of objects (§2.01), is a leitmotiv of the book. It has particular relevance to the present discussion in so far as, except in the case of ‘direct enumeration’, the collection of propositions constituting the values of a variable—to form the basis of an operation, for instance—is bound to proceed, by one or another means, on the basis of something they have in common with one another; and that two distinct symbols have something in common will be true in virtue of their being composite (§5.5261). But, again, while the fact of elementary propositional articulation is important to the account of generality in the Tractatus, the particular nature of this articulation is held not to be a matter of interest to logic. This is made explicit in the stretch of text beginning with §5.55:
[W]e cannot give the composition of the elementary proposition. …
The enumeration of any special forms would be entirely arbitrary. …
It is clear that we have a concept of the elementary proposition apart from its special logical form.
Where, however, we can build symbols according to a system, there this system is the logically important thing and not the single symbols. (§§5.55, 5.554, 5.555)
This is of course just the stretch of text on which Blank's explication of categorial indeterminacy hinges. But I hope now to have placed it in a broader context: to have brought out why Wittgenstein's more general approach to the foundations of logic leads him to say that the ‘composition of the elementary proposition’ is not of concern to logic, rather than leaving this, in turn, as an unexplained datum.
Russell and Wittgenstein on ‘Logical Forms’
We can arrive at something like the same view of the role of categorial analysis in the Tractatus if we consider its evolution from an engagement with the views of Bertrand Russell. Though it was fashionable among some Tractatus scholars in the latter half of the twentieth century to downplay the influence of Russell on Wittgenstein's Tractatus in favour of that of the ‘great works of Frege’, there is no doubt that many of the problems Wittgenstein wrestled with during his writing of the Tractatus arose from Russell's treatment of similar problems. Indeed some of what we've already discussed in connection with Frege could easily be recast as a comparison between Russell's work and the Tractatus, modulo of course Russell's differences with Frege. For instance, the ‘systematic ambiguity’ Russell is forced by his type theory to posit ‘in the meanings of “not” and “or,” by which they adapt themselves to propositions of any order’ (1910b: 43) is just as clearly anathema to Wittgenstein's approach to logic as Frege's treatment of them as particular functions, although for a different reason. I have chosen to present the foregoing discussion as a contrast with Frege, only because his treatment of the ‘logical constants’ as denoting particular functions on all fours with the functions denoted by substantive predicates, and his double introduction of generality in the Grundgesetze, make such sharp foils. A case more peculiar to Russell's work, but of no less significance for the present study, is the issue of logical forms, as it arose in Russell's struggles during the early nineteen-teens over the nature of propositions. Historians of Russell's and Wittgenstein's thought such as David Pears, Brian McGuinness, Peter Hylton and Thomas Ricketts19 have told much of this story admirably well, so I shall confine myself to a brief outline (credit for the content of which, indeed, is due in large part to them, in particular to Hylton).
At the turn of the twentieth century, G. E. Moore, and Russell following him, found themselves resisting the subjectivism they thought they perceived in idealism: in particular, for our purposes, by understanding propositions not as mind-dependent syntheses of elements, but rather as mind-independent, objective furniture of the world.20 However, the account of judgement which suited this conception—namely, as a binary relation between the judging subject and the proposition judged21—made no reference to the structure of the propositions in question; and likewise, there appeared to be no room in such a conception of propositions for an account of truth besides as a brute, inexplicable property holding of some propositions and not of others, propositions which were otherwise on an ontological par. This consequence was in fact embraced explicitly by both Moore and Russell.
Russell's development of his ‘multiple relation theory of judgement’ (between 1906 and 1913)22 was a result of his having come to realize the inadequacy of the accounts of judgement and truth resulting from this conception of propositions. The multiple relation theory in fact dispensed with propositions altogether as basic ontological elements, in favour of the acts of judgement on the part of judging subjects. Such an act of judgement (or, more generally, of any ‘propositional attitude’) is understood, in the first version of the theory, as the judging subject's entering into a relation to those entities which would, before the multiple relation theory, have been called the elements of the proposition. Russell next modified the theory of judgement to include, as one of the relata involved in a propositional attitude, the ‘logical form’ itself. Russell explains that mere entertainment of (what would hitherto have been called) the elements of the proposition does not suffice for its understanding; one must also know how they are to be put together.23 As he explains, ‘when all the constituents of a complex have been enumerated, there remains something which may be called the “form” of the complex, which is the way in which the constituents are combined in the complex’ (1913: 98). Such forms are ‘logical objects’ acquaintance with which, as we just saw, forms a part of an act of judgement—but ‘[i]t would seem that logical objects cannot be regarded as “entities” ’ (1913: 97): ‘the form is not a “thing”, not another constituent along with the objects that were previously related in that form’ (1913: 98). Russell suggests that when we existentially generalize on every contentful element of a proposition, we arrive at its form: thus
‘something has some relation to something’ contains no constituent at all. It is, therefore, suitable to serve as the ‘form’ of dual complexes. In a sense, it is simple, since it cannot be analyzed. At first sight, it seems to have a structure, and therefore to be not simple; but it is more correct to say that it is a structure. (1913: 114)
Nevertheless, (∃x)(∃R)(∃y)xRy is also a judgement; indeed a true one.
However, though the multiple relation theory brought back into focus the constitution of a proposition out of elements, in contrast with Moore's and Russell's earlier views, it recognized no restrictions on the range of collections of elements so unifiable into propositions—or, in Wittgenstein's words, it did not ‘show that it is impossible to judge a nonsense’ (Tractatus §5.5422). The modification of the theory to include ‘logical forms’ as relata in acts of judgement does not help it to avoid this objection,24 for no mechanism is provided to ensure that the other relata involved in a given judgement actually conform to the requirements of the ‘logical form’. That is, Russell gives no account of how the presence of a ‘logical form’ as one of the relata involved in a propositional attitude places constraints on the other relata. And it is very difficult to see how such an account could go, in view of, on the one hand, the apparently incoherent and in any case vague account of ‘logical forms’ themselves, and on the other hand the fact that, given that the point of the multiple relation theory was to account for propositions in terms of more basic propositional attitudes in order to avoid the pitfalls of the earlier anti-idealist theories of propositions as basic entities, Russell has prevented himself from making reference to the nature of propositions in an account of the possible combinations of relata available to judgement (cf. Hylton 1984: 20 and ff.).
Now, it is clear that Wittgenstein is engaging with just these puzzles when he writes these words (from which we quoted above):
The correct explanation of the form of the proposition ‘A judges p’ must show that it is impossible to judge a nonsense. (Russell's theory does not satisfy this condition.) (§5.5422.)
But when one looks for them, one can find hints of this engagement in the early part of the Tractatus as well. Indeed, a way of expressing what Wittgenstein is doing in these opening pages is to say: he is locating form in the objects themselves 25—in their possibilities of combination (cf. §2.033 ‘Form is the possibility of structure’)—rather than in mysterious further objects serving as relata involved in cognitive acts, on a plane with particulars, properties, relations and so on.26 The following Tractatus propositions, for instance, can be seen to speak directly to Russell's puzzles:
It is essential to a thing that it can be a constituent part of an atomic fact. (§2.011)
If I know an object, then I also know all the possibilities of its occurrence in atomic facts. (§2.0123)
The possibility of its occurrence in atomic facts is the form of the object. (§2.0141)
And in a passage from an early notebook, Wittgenstein gives this approach clear expression:
The logical form of the proposition must already be given by the forms of its component parts …
In the form of the subject and of the predicate there already lies the possibility of the subject-predicate proposition, etc. … (1914–16: 23)27
It is clear that, having in place this conception of objects as carrying with them the range of possibilities of combination with other objects, the picture theory is designed to place the sort of constraint on the possible objects of judgement of whose absence from Russell's account §5.5422 complains. For since (in the elementary case28) we entertain propositions by ‘mak[ing] to ourselves pictures of facts’ (§2.1), but pictures consist of elements going proxy for objects with which they share the same ranges of possibilities of combination,29 ‘[w]e cannot think anything unlogical’ (§3.03). To stand in some kind of cognitive relation to a set of elements for which combination one with another is not a possibility is, anyway, not to picture anything: and that is, not to think (§3). Indeed, it would not be too strong to say that, in such a case, one is not ‘in some kind of cognitive relation’ with meaningful elements at all. The import of the Tractatus's version of Frege's ‘context principle’ (§3.3) and its ensuing discussion, including the distinction between symbols and mere signs (see especially §§3.32ff.), secures this result.30 Such a distinction is not available in Russell's framework, since the relata involved in Russell's account of judgement correspond more closely to the Tractatus's ‘objects’ than to its symbols.
For my purposes here, the point of rehearsing this history of Wittgenstein's engagement with Russell's work of the nineteen-teens has been to bring out, from another direction, the importance of the following aspect of the Tractarian approach. Where Russell takes himself to be obliged to provide an account of the logical categories, as well as an account of the logical forms in which elements of those categories are unified into propositions, the Tractatus purports to provide an account of propositions sufficient for the purposes of logical theory without having to enter into the details of the types of object or of combination at all. This is another instance of that recurring theme of the Tractatus, the importance of that which is most general which makes the more specific possible (cf. §3.3421), which we have already seen in connection with its treatment of the truth-functional ‘logical constants’.
In fact, the parallel between ‘logical forms’ and the truth-functional ‘logical constants’ is close indeed. Russell in the Theory of knowledge manuscript treats his ‘logical forms’ as of a piece with the truth functions and quantifiers, under the heading of ‘logical objects’; he supposes that analogous problems are associated with the epistemological status of all of these. Wittgenstein's claim that logic needn't and hence oughtn't treat the ‘logical constants’ as separate special contents is more explicit in the case of the truth functions; but we can see in the light of the story I've just rehearsed that he holds the same view of Russell's ‘logical forms’.31 Wittgenstein's view is that, as soon as we have available an account of elementary propositions that explains their fitness to be truth-bearers, we thereby also secure, at once, the whole of the truth-functional and quantificational apparatus for them. (Compare §5.442, though similar ideas are expressed frequently in the vicinity.) Necessary for this, Wittgenstein takes it, is the picture theory. The picture theory explains why truth and falsity are, unlike on Moore's and Russell's early view, not brute, inexplicable properties of propositions.
(Here compare §6.111:
One could e.g. believe that the words ‘true’ and ‘false’ signify two properties among other properties, and then it would appear as a remarkable fact that every proposition possesses one of these properties. This now by no means appears self-evident, no more so than the proposition ‘All roses are either yellow or red’ would sound even if it were true.
The example of the roses echoes explicitly a figure Russell uses in a paper of 1904, in which he embraces just this consequence of his own view. In this connection it is worth remarking that Moore's and Russell's account of truth from that period made the possibility of an account of the inferential relations in which propositions stand to one another also utterly obscure; from this angle, too, we can see how Wittgenstein takes it that a correct account of truth will bring an account of logic with it.)
The picture theory also provides an account of the impossibility of nonsensical thought, as we have seen. But the picture theory can be articulated without entering into details about the nature of the categories of object—without, that is, a classification of the various ranges of possibility of combination into which objects can enter one with another—and likewise without giving a specific account of the forms of elementary proposition. In this way, again, the Tractatus presents a conception of logic according to which, though the very fact of categorial structure is essential—for essential to the picture theory is the idea that facts, and pictures, and propositions, are articulate—it is beyond the purview of logic to give the details of that categorial structure. Compare §5.5571: ‘If I cannot give elementary propositions a priori then it must lead to obvious nonsense to try to give them’.32