1. Top of page
  2. Abstract
  3. Blank's Explication of Categorial Indeterminacy
  4. Conclusion
  5. References

Many commentators have attempted to say, more clearly than Wittgenstein did in his Tractatus logico-philosophicus, what sort of things the ‘simple objects’ spoken of in that book are. A minority approach, but in my view the correct one, is to reject all such attempts as misplaced. The Tractarian notion of an object is categorially indeterminate: in contrast with both Frege's and Russell's practice, it is not the logician's task to give a specific categorial account of the internal structure of elementary propositions or atomic facts, nor, correlatively, to give an account of the forms of simple objects. The few commentators who have hitherto maintained this view have mainly devoted themselves to establishing that this was Wittgenstein's intention, and do not much address the question why Wittgenstein held that it is not the logician's business to say what the objects are. The present paper means to fill this lacuna by placing this view in the context of the Tractatus's treatment of logic generally, and in particular by connecting it with Wittgenstein's treatment of generality and with his reaction to Russell's approach to logical form.

The question of the nature of the ‘simple objects’ of which Wittgenstein writes in the Tractatus has exercised many commentators. Among some, who took themselves to know roughly what an object was, the question took the form which things in particular Wittgenstein took to be simple: for instance, sense data or material points. Others saw that the question cuts more deeply than this: that the text appears to leave unspecified even the categorial status of objects.1 For such commentators, the interpretative question was, roughly, whether Tractarian objects include what Frege called ‘concepts’ and Russell ‘properties’ and ‘relations’,2 or are limited to what Frege too called ‘objects’ and Russell ‘particulars’.3,4

The literature already contains what I take to be the correct resolution of this set of interpretative questions.5 Wittgenstein deliberately means to leave the categorial status of objects unspecified: it is a mistake to hold either that Tractarian ‘object’ is, even more or less, synonymous with Fregean ‘object’ and Russellian ‘particular’ or that it ranges determinately over the further categories in Frege's or Russell's hierarchies. Wittgenstein holds that it is not the logician's task to give a specific categorial analysis of language or of ontology. All that is necessary for an account of logic is the very idea of categorial combination; what particular categories this combination involves is to pure logic a matter of indifference. Thus Wittgenstein is neither tacitly invoking a theory of categories along Fregean or Russellian lines nor proposing an alternative single-category theory.—And if this is so, then the particular question alluded to above, ‘Are the simple objects sense data or material points?’, is even further out of place. If we are not meant even to ascribe to Tractarian objects a determinate categorial status, we are surely not meant to assimilate them to one or another specific kind of particular thing.6

As I say, this reading of the Tractatus is already available in the literature, together with some persuasive argumentation in its favour. My main purpose in this paper is not to argue for it per se—though I will have a few things to say about other commentators' arguments—but rather to place it in context. Those who have hitherto argued for this reading have been concerned primarily to establish that it captures the Tractatus's attitude toward logical categories, and so have focused on rebutting competing interpretations such as those alluded to above. They have done less in service of explaining why the Tractatus takes this stance. (Thus, the reading's most recent, and perhaps clearest, champion, Johnston (2009), ends his paper with the remark, ‘Nothing I have said in this paper goes very far by itself towards explaining why Wittgenstein held’ the view of categorial structure in question (2009: 160).) Here, then, I mean to contribute to a discussion of this latter question. In particular, I shall show how the point of the Tractatus's categorial indeterminacy becomes clearer upon reflection on Wittgenstein's engagement with Frege's account of quantification and with Russell's treatment of logical form. After a couple of preliminary methodological remarks, I shall in Section 2 begin by considering the work of one recent commentator who does purport to give an explanation for the categorial indeterminacy of Tractarian objects, namely Blank (2000); I shall then offer my own account of the grounds for that indeterminacy in Section 3, through a discussion of the Tractarian treatment of generality in contrast with Frege's (Section 3.1) and the Tractatus's response to Russell's treatment of logical form (Section 3.2). I shall close in Section 4 with a brief synoptic account of the picture of logical categorization that emerges.

As the phrasing of the previous paragraph makes clear, in this paper I shall leave intact the distinction, implicit in Blank's (2000) and explicit in Johnston's (2009) remarks, between attributing a thesis to a text and explaining why the text should contain that thesis by showing how it connects with the text's other contents. Evidently these are not really two separable tasks: the more successful I am at explaining why, in the light of other aspects of the book, it makes sense that its objects should be categorially indeterminate, the more plausible the textual attribution itself becomes—and more or less contrapositively, whatever grounds are adduced for ‘finding the thesis in the text’ will not fully satisfy us if they don't show why it makes sense that the thesis should be there. But for the purposes of this paper I shall for the most part play along with the pretence that the two tasks are distinguishable and that the first has already been satisfactorily carried out elsewhere.

This is connected with another issue of textual interpretation, one especially acute for readers of the Tractatus and which has played a central role in Tractatus scholarship for the last several decades. The Tractatus, after all, claims that philosophy's result ‘is not a number of “philosophical propositions”, but to make propositions clear’ (§4.112), and indeed that its own propositions are nonsensical (§6.54).7 To avoid repetition of long-winded phrases such as ‘the reading according to which the Tractatus employs a categorially indeterminate notion of an object’, I shall sometimes simply speak as though the book straightforwardly contained a ‘categorial indeterminacy thesis’—but this will seem to stand in tension with the book's puzzling remarks about its own contents. When trying to navigate this issue I take my guidance primarily from Sullivan (2004) and Goldfarb (1997): our task is to do our best to assemble, piece by piece, a coherent account out of the words contained in the book—even if we know (because we've read the last page) that we won't in the end succeed. While we're engaged in this task, of course, we'll be thinking of the elements of the account we're trying to assemble as, precisely, elements of an account: theses, claims, arguments, and so on. As I intend this paper to be a contribution to that task, I shall allow myself to speak in such terms, on the understanding that, in one way or another, the interpretation of a stretch of words in the book as expressing or entailing a ‘thesis’ may prove to be, at best, ‘transitional’.—But this of course brings out even more clearly that the distinction between ‘finding a thesis expressed in the Tractatus’ and explaining what it's doing there can in the end be nothing more than a pretence.

Blank's Explication of Categorial Indeterminacy

  1. Top of page
  2. Abstract
  3. Blank's Explication of Categorial Indeterminacy
  4. Conclusion
  5. References

One supporter of the view that the Tractarian notion of an object is categorially indeterminate who does attempt to explain why the Tractatus employs such a notion is Andreas Blank (Blank 2000). Having argued for the thesis of categorial indeterminacy as an interpretative claim about the Tractatus, Blank goes on to offer some remarks in explanation of the book's categorial indeterminacy—but his remarks don't go much beyond Johnston's own, though as we saw, Johnston acknowledges having given little in the way of an explanation. Blank perhaps regards his remarks as illuminating partly in so far as they stand in contrast with what he sees as a competing explanation of the indeterminacy thesis: namely, McGuinness (1981) and Ishiguro (1990)'s claim that ‘object’ in the Tractatus is a concept of pure logic. It will help to make a little clearer what the indeterminacy thesis comes to, and also why we might want more explanation of it than even Blank provides, if we consider briefly his engagement with McGuinness and Ishiguro.

Now, it may come as a surprise to readers of McGuinness (1981) to find Blank invoking his work in this connection, since McGuinness nowhere articulates such a thesis as that of categorial indeterminacy, much less provides an explication of it on the grounds that the Tractarian concept of an object is a purely logical one. But Ishiguro (1990) does articulate a similar-sounding thesis: she says that ‘ “object” is a logical category for Wittgenstein’ (1990: 31), but she does not mean by this that it is a unitary category as, for instance, it is for Frege as he contrasts it with ‘function’.8 Rather, as becomes clear when later in the same paragraph she writes of ‘objects of various categories’, she means that ‘object’ is a notion arrived at through logical reflection alone: a ‘concept of pure logic’, in Blank's phrase. As she puts it, ‘it comes to us only through the logical syntax of propositions’ (ibid.: 31). And here the connection with McGuinness emerges. McGuinness, too—building, as it happens, on Ishiguro's earlier work (1969)—‘explain[s] the object as an entity definable in terms of semantic equivalence’ (1981: 66), where this in turn is explained in terms of logical properties alone: ‘whatever logical properties we give to the words we use will determine which proposition (if any) it is that we assert by means of them’ (ibid.: 70).

But what does this idea have to do with categorial indeterminacy? Consider how McGuinness fills out his thought:

An object in the Tractatus which is the reference of a name or simple sign can be viewed as simply the truth-value potential of a certain expression. The semantic role of the supposedly possible simple sign or name is that of being combined with other simple signs or names to produce a proposition having a truth-value. (ibid.: 65)

McGuinness's use of the phrase ‘truth-value potential’ is an allusion to Tugendhat's (1970) account of Frege's notion of Bedeutung. Tugendhat had argued that Bedeutung is best understood not at all on the model of the name-bearer relation, but merely in terms of ‘truth-value potential’ in the sense here explicated by McGuinness: and the primary motivation Tugendhat offers for this is, precisely, the transcategorial application Frege gives to his notion of Bedeutung. It is thus no surprise if, applying this conception to the Tractatus, McGuinness arrives at a transcategorial understanding of ‘object’ (even if he does not particularly flag it as such, as Ishiguro does). Somewhat sloganistically, we might sum up this conception of Tractarian objects by describing them as whatever the ultimate units of discourse go proxy for, regardless of ‘logical category’ in a Fregean or Russellian sense.

This would be a quite interesting argument for Tractarian categorial indeterminacy (though it goes a little beyond anything that McGuinness, Ishiguro or even Blank say explicitly). Indeed, it is not clear that Blank's dismissal of this argument is entirely effective. Blank points out that, if ‘object’ in the Tractatus is to be understood in terms of truth-value potential, then equally truth-value potential is to be understood in terms of objects: for by the lights of the Tractatus, what it is for a sign to have truth-value potential, in turn, is for it to be able to figure in configurations of signs standing in projective relations to configurations of objects. On the grounds that these ‘definitions’ are in this way interdependent, Blank thus disputes that ‘object’ is ‘a concept of pure logic’. Now, Blank does not make clear just what it takes for a concept to be ‘purely logical’ in his sense. But the distinction which would seem to be most salient in the context of the Tractatus, of which Blank himself makes much (as we'll see below), between logic and its application (§5.557), is one such that, I would have thought, the Tractarian notion of an object falls on the ‘logic’ side: for while—as it is the very point of the thesis of categorial indeterminacy to hold, and as we shall discuss in a little more detail below—the question what are the particular forms of elementary proposition, and correlatively what are the simple objects, is a matter for the application of logic to resolve, the point that there must be simple objects is one on which logic itself insists (cf. §§4.2211 and 5.5562: and note that the latter passage makes explicit that the point at issue there is known ‘on purely logical grounds’).9 But even if Blank were right to argue that Tractarian ‘object’ is not ‘a concept of pure logic’, it is not obvious that this would undermine the route I have just tried to reconstruct, on Blank's behalf as it were, from McGuiness's and Ishiguro's accounts of Tractarian objects to the idea of categorial indeterminacy. For even if Blank is right that ‘object’ and ‘truth-value potential’ are interdependent notions in the Tractatus rather than the former's being unilaterally dependent on the latter, it would nevertheless appear to hold that, if ‘truth-value potential’ is transcategorially applicable, then so is ‘object’—and this would seem to be so even if Blank were right (pace what §5.5562 appears to be telling us) that this interdependence suffices to show that ‘object’ is not a ‘purely logical concept’.

Now, if we were to adapt the remarks of McGuinness and Ishiguro, following Blank's hints, to yield an argument that Tractarian objects are categorially indeterminate, as we have just tried to do, the resulting argument would not, it should be granted, be knock-down. After all, a reader of the Tractatus who, for instance, insisted with the likes of Anscombe and Sellars that Tractarian objects were particulars would perhaps be unfazed by the Tugendhat-inspired appeal to ‘truth-value potential’: for such a reader might simply insist that the opening pages of the Tractatus give an account according to which, at the level of complete analysis, the only expressions with ‘truth-value potential’ are names of simple objects, which are particulars! But recall our dialectical situation: we, like Blank, are considering Ishiguro and McGuinness not as providing an argument for the indeterminist reading of the Tractatus, but as providing an explanation for Tractarian categorial indeterminacy, having granted already that the reading is correct (pretending, as flagged above, that this is an intelligible dichotomy). And if we are right to extract from their work the take-home message that an object is whatever a unit of completely analysed significant discourse goes proxy for, with emphasis on that indefinite ‘whatever’, then we have—so far as it goes—some insight into the grounds for the Tractatus's refraining from specifying the particular logical categories of objects.

What Blank goes on to say in his own voice in explication of the categorial indeterminacy of Tractarian objects, though, seems to me perfectly helpful in its own right. Blank, as I alluded to above, refers us to the §§5.55s, beginning with §5.557: ‘The application of logic [in contrast with logic itself] decides what elementary propositions there are’. He goes on to argue, correctly I think, that what is at issue here is not merely the giving of examples of particular elementary propositions, but also a catalogue of forms of elementary proposition; he supports this by adverting to other passages in the immediate vicinity as well as to some extra-textual references to Wittgenstein's other notes and writings. Now, the structure of an elementary proposition is the way in which its elements, names of simple objects, are combined, and so the possibility of its structure is inseparable from the forms of those names, that is, from their respective ranges of possible combination with other names. And of course those ranges recapitulate the ranges of possible combination of the objects of which they are names: in other words, the forms of the names recapitulate the forms of the simple objects for which they go proxy. This is, indeed, constitutive of ‘going proxy’.10 But then we should be able to give a catalogue of elementary propositional forms if only we can give the forms of simple object. So if the §§5.55s are telling us that not logic, but only its application, can yield a catalogue of elementary propositional forms, then not logic, but only its application, can give us the forms of simple object. ‘Object’, that is, is, as far as logic is concerned—and that is to say, as far as the Tractatus is concerned—categorially indeterminate.

In my view, this is all just right. But it only goes so far: for surely one who finds surprising the idea that the Tractarian notion of an object is categorially indeterminate will, in the face of the §§5.55s, simply ask why, in turn, Wittgenstein asserts that it is not a task of logic but of its application to catalogue the forms of elementary proposition. Indeed, this is presumably why Johnston (2009), who works through in greater detail just this point about the §§5.55s in support of the thesis that Tractarian objects are categorially indeterminate, ends with the remark that ‘Nothing I have said in this paper goes very far by itself towards explaining why Wittgenstein held’ the thesis (2009: 160). So in what follows I shall try to explain further how the categorial indeterminacy of the Tractarian notion of an object and the concomitant idea that logic cannot catalogue the elementary propositional forms fit in with other major themes of the work. In particular, I shall hook these ideas up with the Tractatus's treatment of generality, and with its engagement with Russell's ideas about ‘logical forms’ during the period of his work on the ‘multiple relation theory of judgement’.

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 [DOWNWARDS ARROW] B’, then none of ‘∼’, ‘∨’, ‘&’ or ‘[DOWNWARDS ARROW]’ 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 ‘[inline image, inline image, inline image]’ (§6), which ‘says nothing else than that every proposition is the result of successive applications of the operation inline image to the elementary propositions’ (§6.001). ‘inline image 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 inline image 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


  1. Top of page
  2. Abstract
  3. Blank's Explication of Categorial Indeterminacy
  4. Conclusion
  5. References

Let me attempt to synthesize.

The ‘picture theory’33 is meant to act as the foundation of an account revealing how the very idea of meaningful discourse brings with it all of logic—an account of meaningful strings of signs which reveals why they are apt for truth or falsity (and not both at once), explains the entailment relations in which they stand to one another, and shows why only they, and not other strings, are possible objects (or perhaps better, vehicles) of judgement—in one blow. For this reason, the picture theory should be completely general: it should account for what underlies all propositions' meaningfulness. The Tractarian account of propositional combination, too, should by the same token be completely general: if ‘all propositions are results of truth-operations on the elementary propositions’ (§5.3), the account of truth operations in turn should take any elementary propositions as operands, utterly indifferently to their internal structure.

This is clearest in the case of truth-functional combination: it is applicable indifferently to all propositions, including results of prior truth-functional combination. But it is also true that its results won't have any content if its inputs don't: so the idea of truth-functional combination alone can't explain content: can't provide the sought-after account described in the previous paragraph.

Now, Wittgenstein attempts to bring generality into the picture in just the same way—and indeed in the same stroke, by assimilating it to truth-functional combination—so as to show that its availability, too, emerges from the very idea of meaningful discourse. But it is natural to object here: whereas the truth-functional connectives take propositions as input in a manner palpably independent of their internal structure, this is not true of quantification. And we see this vividly when we recall how any standard logic course proceeds: we begin with the ‘propositional calculus’, with its schematic letters filling in for propositions generally, with no mechanism even for indicating internal structure; and then, when we introduce quantification, we must introduce predicate letters and individual constants together with the quantifiers and variables.

But here Wittgenstein can remind us: the propositional calculus, although it didn't purport to explain this, presupposed that its schematic letters were standing in for sentences 34: contentful expressions, indeed expressions with the sort of content that is apt for truth and falsity. The picture theory shows that this requires presupposing, after all, that those expressions will be articulate: will indeed have internal structure, even if the propositional calculus abstracts from its details. But now his point about generality is that that same presupposition of internal structure is all it needs. It is not necessary to specify the particular sorts of structure exemplified in elementary propositions; all that matters for the purposes of a unified account of meaningful discourse is the bare idea of elements in structured combination. These elements of elementary propositions the Tractatus calls ‘names’, and their referents ‘objects’. Their ‘logical categories’—the particular ranges of possibility for combination with one another they admit of—can be left indeterminate.35

  1. 1

    This may be thought to be an artificial distinction, at least as applied to the Tractatus, in so far as there is material in Wittgenstein's own work for the idea that logical or grammatical categories are narrower than they are traditionally thought to be, with the implication that to specify a ‘particular kind of thing’, such as material points, may be to specify a category. This becomes especially important, I think, when considering the development of Wittgenstein's thought post-Tractatus; but I shall retain the distinction for the sake of expository simplicity.

  2. 2

    For some instances, Stenius 1960; Hintikka and Hintikka 1986; and Hacker 1986.

  3. 3

    For example, Copi 1958; Anscombe 1959; Sellars 1962; Pitcher 1964; Ishiguro 1969 and Ricketts 1996.

  4. 4

    For the present I abstract from the differences between Frege's and Russell's categorial analyses—as well as from the fact that commentators in either of these camps must after all acknowledge some differences between Tractarian objects and whatever Fregean or Russellian category or categories they compare them to. But I shall return to some of the differences between Frege and Russell, and their relation to the Tractatus, in what follows.

  5. 5

    The view I have in mind can be found expressed, with varying degrees of clarity, in Kenny 1974, Pears 1987 and Ishiguro 1990; Blank 2000 and Johnston 2009 argue for it explicitly, as does Campbell 2008, from sections of which the present paper's Sections 3.1 and 3.2 have been adapted. Blank (2000) would add McGuinness 1981 to this list. McGuinness does not seem to address directly the categorial question; when he says, ‘Wittgenstein's objects are not concrete objects … [n]or are they properties of concrete objects’ (1981: 72), we should perhaps hear the emphasis on ‘concrete’. But we shall return to McGuinness's view below.

  6. 6

    This wording suggests, in addition to the tacit assumption I flagged in note 1, that it can be taken for granted that sense data and material points are particulars. But Keyt 1965, for instance, discusses at length a ‘sense-datum interpretation’ according to which the simple objects are simple sensory qualities—i.e., universals, not particulars. I hope it is clear, though, that the point in the text applies as well to this sort of reading: this, no less than (say) a ‘material point’ interpretation, implies a determinate categorial status for objects.—Indeed, here we can discharge the tacit assumption flagged in note 1: one who holds that the Tractarian notion of an object is categorially indeterminate is bound to reject the ‘material point’ reading whether material points are understood as a species within a logical category or as constituting a logical category themselves (and mutatis mutandis for ‘sense-datum’ interpretations and all other attempts to specify the objects).

  7. 7

    Citations of the Tractatus will generally follow Ogden (and Ramsey)'s translation (Wittgenstein 1922), and will be to proposition number (§).

  8. 8

    I say this notwithstanding Ishiguro's attempt to draw an analogy between Frege's use of ‘Gegenstand’ and that in the Tractatus (1990: 26)—an analogy which rests on claims about Frege that I cannot help finding dubious.

  9. 9

    Indeed, later in his paper Blank does attempt to say in general what ‘logic’ comes to in the context of the Tractatus: ‘Logic in the sense of the Tractatus is … to be understood as the description of those formal properties of language and the world which can be given a priori’ (2000: 214; my translation). By this criterion, surely the notion ‘object’ in the technical sense of the Tractatus is a logical notion.—Indeed, in this part of Blank's paper, this is precisely his point—so it is all the more mysterious that in his discussion of Ishiguro and McGuinness he resists the idea that ‘object’ is a purely logical concept.

  10. 10

    I do not use ‘recapitulate’ here in a sense which implies the priority of the recapitulated over that which it recapitulates; the point I am making about the connection between elementary propositional forms and forms of simple object stands whether we hold that objects determine the forms of their names (cf., e.g., Pears 1987: 88), or vice versa (cf., perhaps, the works of Ishiguro and McGuinness already discussed), or indeed even if we deny both priority theses and simply hold the two notions to be interdependent, on analogy with Blank's response to Ishiguro and McGuinness.

  11. 11

    MacFarlane 2000 constitutes an admirable study of this constellation of notions. Somewhat oddly to my ear, he understands Frege's idea that logic ‘provides constitutive norms for thought as such’ as a perfectly intelligible thing to mean by ‘formal’, indeed making it one of the three competing conceptions of the formality of logic among which he adjudicates. It strikes me as odd, though, as I say, to call a conception one of formality if (as MacFarlane recognizes explicitly) it does not involve a contrast between form and content or matter.

  12. 12

    See, e.g., Frege 1880-1: 37.

  13. 13

    The words I have quoted are actually used by Frege of the distinction between first- and second-level concepts; but surely the slight adaptation to the present context is perfectly legitimate.

  14. 14

    Anscombe 1959 quotes §5.451, and in her chapter on generality at that, but without applying it to the case of quantification itself as I do: rather, she focuses on the way the Russellian treatment of quantification requires us to violate the point of §5.451 as applied to the truth-functional connectives (as Wittgenstein's own example there of the role of negation in ‘∼p’ and ‘(∃x)∼fx’ illustrates).

  15. 15

    Ogden's translation overlooks Wittgenstein's emphasis on ‘können’.

  16. 16

    I don't mean to suggest that the italicized alternative is a gloss of Frege's view; on the contrary, Frege's extensionalism leads him, to hold, famously, that the reference of a thought is a truth value. The realm of reference, though it does contain objects and functions, does not contain ‘facts’ composed of them. Compare also his remark in Begriffsschrift that the function-argument distinction ‘has nothing to do with the conceptual content, but only with our way of grasping it’ unless that content involves generality (1879: §9).

  17. 17

    It is of course no less suggestive of Russell's categorial distinctions than of Frege's—indeed in certain respects more suggestive, not least because of the point made in the previous footnote. I discuss an aspect of Wittgenstein's engagement with Russell in Section 3.2. On the other hand, see Linsky 1992 for a useful discussion of how Tractarian ‘forms of object’ can be understood as a generalization of Frege's notion of ‘unsaturatedness’ or ‘incompleteness’. (Linsky presents this precisely as a contrast with Russell's treatment of all ‘terms’ as in a sense self-standing, but of course Russell himself developed the categorial distinctions alluded to above in response to problems which arise for, among other things, that very notion of a ‘term’, as Linsky recognizes.)—Note in this connection that the ‘unitary treatment of generality’ I describe Wittgenstein as seeking should not be confused with the ‘absolute and unrestricted notion of all’ Linsky (1992: 262) associates with Russell's use of ‘term’, contemplation of which leads one to the problems just mentioned. That Wittgenstein treats generality in a unified manner does not at all imply that he employs or envisions quantifiers ranging unrestrictedly over all Tractarian objects.

  18. 18

    I phrase this a little weakly because Wittgenstein does give us a handful of examples, but they are problematic. It is difficult, for instance, to see how to reconcile them with the axiom that atomic facts are logically independent: hence the notorious ‘colour incompatibility problem’, widely thought to be partly responsible for Wittgenstein's later repudiation of the book.

  19. 19

    Pears 1981, 1987; McGuinness 1974; Hylton 1984, 1990; and Ricketts 1996.

  20. 20

    For Moore, see ‘The Nature of Judgment’ (1899). For Russell see, e.g., Principles of Mathematics (1903); he also makes some delightfully quotable remarks in his 1904, as Hylton 1984 brings out.

  21. 21

    This may not seem quite to capture the structure of Moore's theory, in which ‘concepts’ are more fundamental than ‘propositions’. But ‘a proposition is nothing other than a complex concept’ (p. 180), and though the account of propositions' truth and falsity makes some reference to their composition out of concepts, it is entirely schematic: ‘A proposition is constituted by any number of concepts, together with a specific relation between them; and according to the nature of this relation the proposition may be either true or false. What kind of relation makes a proposition true, what false, cannot be further defined, but must be immediately recognised’ (ibid.). So I think the simplification in the text is harmless. As for Russell, on the one hand Principles of Mathematics Ch. IV is an enquiry into the ‘philosophical grammar’ of the proposition; but on the other hand, Russell there makes it explicit that propositions are no less terms than their constituents. It is Russell himself who later describes his view from this period as a dyadic theory of judgement.

  22. 22

    The earlier version of the theory discussed in the text is expounded in Russell 1910a; the later version is given in the 1913 manuscript Theory of Knowledge.

  23. 23

    I am omitting from this summary the way Russell, in earlier versions of the theory, tried variously to appeal to what he called the ‘sense’ of the relating relation in the judgement and to the order of the relata in order to settle the question how the elements are to be put together in the judgement. Ricketts 1996 is helpful on this.

  24. 24

    Indeed, as several commentators note, Wittgenstein apparently made his objection upon reading the 1913 manuscript, which contains the version of the multiple relation theory including ‘logical forms’.

  25. 25

    This slogan, though irresistibly catchy, risks misleading: it might suggest a preference of Pears's view to McGuinness's and Ishiguro's on the issue of realism flagged at note 10. It might also suggest an order of explanation running from part to whole rather than from whole to part—in conflict with, for instance, the reading of the role of the ‘context principle’ (§3.3) in the Tractatus in Diamond 1981 or Linsky 1992. Allow me to disavow both suggestions. All I mean by these words is that, when I analyse a proposition (or understand a fact as dividing) into constituents, whatever that involves, I don't need a further account (and certainly not one which appeals to a further constituent) of how these constituents cohere: rather, my very understanding of the constituents is as things which fit together so as to yield a proposition, or a fact, like the one with which I began. (See note 29 below for more on the easy switching back and forth between ‘ontological’ and ‘linguistic’ modes I'm engaging in here, in the face of the univocally ‘ontological’ phrasing of the remark occasioning the present note.) This point stands whatever our view of the relative priority of object and name, or of constituent and whole.

  26. 26

    Some of Russell's remarks on logical forms quoted earlier show that he, too, wanted to avoid treating them as on all fours with the (other) constituents involved in judgement. But it is hard not to hear this as mere lip service, given the role they actually play in the multiple relation theory of 1913.

  27. 27

    Readers ought not be distracted by Wittgenstein's reference to subject and predicate as putative forms of object, worrying that it gives the lie to the claim of categorial indeterminacy. Wittgenstein plays with a range of examples of forms of object and of atomic fact, of name and of elementary proposition, in the Notebooks, which examples are not obviously consistent even one with another. One starting point for the inquiry that ends with the ascription of categorial indeterminacy to the Tractatus might be precisely the observation that such examples generally don't survive in the final manuscript. (For the qualification ‘generally’, though, see note 18 above.)

  28. 28

    The case of non-elementary propositions is more complicated to describe; but since my goal here is only to show how Wittgenstein is engaging with Russell's problems about ‘logical forms’, and those problems are patent at the elementary level, it suffices for us to confine our attention to this level.

  29. 29

    And so I could have said above that Wittgenstein is locating form ‘in the names themselves’ just as well as ‘in the objects themselves’: the idea that form is in, rather than additional to, the objects has an exact analogue at the linguistic level, in that the ‘forms of elementary propositions’ under discussion e.g. in the §§5.55s are nothing over and above ways for names to hang together—which ways, far from being components of propositions separable from the names themselves, are such that their possibility is constitutive of those names' being the symbols they are.

  30. 30

    Cf. Diamond 1981.

  31. 31

    McGuinness makes this clear in his 1974.

  32. 32

    Linsky 1992's illuminating discussion of the Tractatus's treatment of the unity of the proposition is clearly closely allied with that of the present section (for, after all, Russell's ‘logical forms’ were devised precisely to account for the unity of the propositional attitudes). However, Linsky does not draw explicitly the conclusion most central to the present paper: namely, that Wittgenstein holds that it is not the logician's task to draw up a system of categories, and that, instead, his term ‘object’ is left deliberately categorially indeterminate.

  33. 33

    In this paragraph in particular, I indulge quite shamelessly in the sort of talk for which I preemptively apologized in the methodological remarks closing Section I.

  34. 34

    I do not mean to imply, what would be false, that Wittgenstein is working with a ‘schematic’ conception of logic; I am merely envisioning (anachronistically) the sort of reply he might give if confronted with the objection of the previous paragraph.

  35. 35

    I'm grateful to David Berger, Ian Blecher, John McDowell, Henry Pickford, Tom Ricketts and especially Ben Laurence for conversations about the Tractatus, and to Ian Hacking, Joe Gonda and Doris Olin for encouragement.


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  2. Abstract
  3. Blank's Explication of Categorial Indeterminacy
  4. Conclusion
  5. References
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