Wittgenstein and ‘tonk’: Inference and Representation in the Tractatus

We are now in a position to be able to clarify further why early Wittgenstein’s view should not be classified as “inferentialist”, by comparing it with another response to Prior’s ‘tonk’ example. What I have in mind is Nuel Belnap’s five-page paper, “Tonk, Plonk and Plink” from 1962. In this paper, Belnap defends the inferentialist idea that the meanings of the connectives are defined by the rules for their employment in deduction. According to Belnap, the reason why ‘tonk’ cannot be defined in terms of deducibility, whereas a connective such as ‘and’ can be so defined, is that the proposed rules for ‘tonk’, unlike those for ‘and’, are inconsistent with certain antecedent assumptions we make about deducibility. Allegedly, these assumptions are “antecedent” in the sense that they are made before or independently of the introduction of any connectives at all. To give substance and precision to this claim, Belnap employs as his characterization of this “antecedently given context of deducibility” the structural rules of Gentzen, which he states as follows:

A₁, ..., A_m, C₁, ..., C_n |- D.¹⁶

Belnap claims that it is only by failing to see the restriction provided by the antecedently given context of deducibility that Prior can argue that the illegitimacy of ‘tonk’ undermines an inferentialist account of the logical connectives. Once we make clear what the antecedently given context is, we can see that the illegitimacy of ‘tonk’ does not undercut the (correct) idea that logical connectives are defined in terms of deducibility. What it does undercut is just the (wrongheaded) idea that such definition occurs in vacuo, without any preconception of what it is to deduce one proposition from another.

Now let us consider how the Tractarian conception of inference and representation is related to Belnap’s account. In order to get clear about this, we have think harder about two intimately related issues that arise as soon as one views Belnap’s argument through Tractarian spectacles. The first issue is this: In precisely what sense is the “antecedently given context of deducibility” antecedently given? What, exactly, does it mean to say that Gentzen’s universally valid deduction statements do not involve any special connectives? In what sense can we view the connectives as “introduced” only “after” those deductive patterns are in place?

The second issue is the following. Belnap says Gentzen’s axiom and rules govern the use of sentences. This means that Gentzen’s structural rules must assume some notion of sentential identity. Indeed, this is clear already from the simple axiom, ‘A |- A’. To understand the axiom we must understand what it is to say that a certain sign written to the left of a deduction sign is an inscription of the same sentence as the sign written to the right of the deduction sign. In particular, we need to understand what identity criteria Belnap will have to presuppose if his account is going to do the sort of work he wants it to do.

The importance of these two issues becomes clear once it is noticed that if the notion of sentence that is assumed in Gentzen’s axiom and rules is the Tractarian notion of a true-or-false description of the world – the Tractarian notion of a proposition, a Satz – then it will be utterly misleading to say, as Belnap does, that the context specified by the axiom and rules is given antecedently to the subsequent introduction of specific connectives. As we have already seen, the Tractarian notion of a proposition is such that no wedge can be driven between the identification even of elementary propositions and the availability of the whole of logic. Standard logical notation makes it seem as if it is the connectives that bring with them the machinery of logic, but in fact all of this machinery is in place as soon as elementary propositions are used to assert that such-and-such is the case.

Now it is of course true that the validity of certain deductive patterns is visible even at a level of abstraction where the particular logical structure of the premises and of the conclusion is not specified. For example, no matter what particular logical structure a proposition has, you can always deduce it from itself; if one and the same proposition occurs twice among the premises in a valid inference then you can always delete one occurrence of it without making the inference invalid; and so on. Gentzen’s rules can be seen as specifying such patterns of inference at this very high level of abstraction. This is no problem for Wittgenstein.

What is doubtful, from a Tractarian viewpoint, is the further thought that Gentzen’s rules somehow specify an initially given core of deducibility relations that gets “extended” when the connectives are added. Against this sort of idea, Wittgenstein would claim that the very talk of deducing a proposition from itself, or of the double occurrence of one and the same proposition among the premises of an inference, makes clear sense only if the whole of logic is already in place. Otherwise we are no longer talking about propositions, and, hence, not of deducibility.

This means that from the Tractarian viewpoint, what Belnap calls a conservative extension is not an extension at all. Rather, it is just a matter of spelling out in some further detail the logic already presupposed by the very notion of proposition that must be taken for granted if Gentzen’s rules are to be taken as rules of inference at all. And what Belnap calls a “non-conservative extension” is not an extension either. Rather, to accept such an “extension” means leaving the domain of logic and of meaningful language use altogether. It means to start doing something completely different from inferring and describing – such as, perhaps, decorating wall paper with ink-marks.

Notice here a similarity between Belnap’s and Stevenson’s seemingly antagonistic conceptions. Stevenson thinks an answer to Prior’s challenge requires a distinction between “sound” and “unsound” rules of inference, thereby suggesting that activities governed by the latter sort of rules are indeed inferences, albeit somehow illegitimate ones. As we saw, Wittgenstein thinks these notions of unsoundness and illegitimacy make no sense. Someone who is “following unsound rules of inference” is not trafficking in propositions at all, and whatever he is doing it is not a matter of inferring. Belnap, on his side, thinks an answer to Prior’s challenge requires a distinction between conservative and non-conservative extensions, thereby suggesting that there is some sort of continuity, albeit an illegitimate one, between the practice that is captured by Gentzen’s system and the practice you get if you “extend” the system non-conservatively. From Wittgenstein’s viewpoint, this “extension” just means that what you do is no longer logic.

Another way of describing the similarity between Stevenson and Belnap is to say that they both want to meet Prior’s challenge by issuing restrictions on what constitutes proper logical behavior. Wittgenstein thinks the very idea of such restrictions is based on a misunderstanding. He would claim that what Stevenson and Belnap thinks of as improper logical behavior just isn’t logical behavior at all. Nor is it anything illegitimate about it: we are perfectly free to, say, decorate wallpaper with patterns of ink-marks instead of doing inferences, if we like. There is nothing that needs to be forbidden here, and hence no restrictions to be made. Logic takes care of itself.

But now, what if the notion of a sentence involved in Gentzen’s rules, as Belnap understands them, is not the Tractarian notion of a Satz? In fact, it is quite clear that Belnap is not working with this Tractarian notion. The important point is not that Belnap says Gentzen’s characterization of deducibility “may be treated as a formal system”¹⁸ – for the question is precisely what “formal system” is supposed to mean here. What reveals that Belnap’s notion of sentence is not the Tractarian notion of Satz, is that he goes on to argue that the extension of the formal system made by a proposed definition of a connective – say, ‘plonk’ – involves an extension of the very notion of a sentence, “by introducing A-plonk-B as a sentence, whenever A and B are sentences”.¹⁹ Moreover, he argues that the extension of the formal system involves the adding of some axioms and rules governing A-plonk-B as it occurs as premise or conclusion in a deducibility statement. Obviously this is very different from the Tractarian viewpoint, according to which all possibilities of logical complexity, and hence all forms of inference, are given already with the elementary propositions. So, isn’t the Tractarian worries rehearsed above simply misplaced, since they invoke a notion of sentential (or propositional) identity that is foreign to Belnap?

In fact, there is still a pretty straightforward conflict between Belnap’s account and the Tractarian conception. The deep Wittgensteinian worry is the following. Given Belnap’s non-Tractarian conception of what a sentence is, Gentzen’s rules and axioms cannot possibly capture a conception – even a preconception – of deducibility. According to the Tractatus, to say that a sentence is deducible from others – as opposed to saying, for example, that writing down a certain concatenation of sign-designs below some other concatenations of sign-designs is to produce a permitted pattern of wallpaper decoration – means to work with a notion of a sentence as an already meaningful, true or false, entity. And to work with such a notion of a sentence is already to have introduced the whole of logic. So, from a Tractarian viewpoint, the balancing act attempted by Belnap – to claim to have access to a genuine notion of deducibility without yet having introduced the connectives – just cannot succeed.

In fact, Wittgenstein would not regard Belnap’s lapse into a sort of lax conventionalism as a mere accident. From a Tractarian viewpoint, the “overly formalistic viewpoint” remarked on by Wagner is precisely what makes it seem as if we can have a notion of deducibility prior to the introduction of truth-operations. More precisely, this “overly formalistic viewpoint” is precisely a viewpoint from which it looks as if entities somehow less rich than full-fledged Tractarian Sätze can work as premises and conclusions in deductions. The problem is that if we think in this sort of way, it will be very difficult to understand why the antecedently given context of “deducibility” that Belnap is talking about is not something that could have been different – just as there can be different systems for how to decorate wallpaper. Wagner wants Belnap to say that the fact that there is no such connective as ‘tonk’ has to do with the truth about deducibility, rather than with our decisions to allow some deductions and forbid others. From a Tractarian viewpoint, it is no coincidence that Belnap fails to fulfill Wagner’s wish; for his notion of “deducibility” is such that, strictly thought through, it cannot provide the truth Wagner is asking for.

My aim in this paper has been to shed light on early Wittgenstein’s conception of logic by looking at Prior’s ‘tonk’ example through Tractarian spectacles. The arguments I have rehearsed against Stevenson and Belnap are arguments that I think the author of the Tractatus would have used, had he confronted the writings of these philosophers. I have not claimed that these arguments are satisfactory. Indeed, it is clear that, in their stated form, they are not satisfactory. For they involve an untenable conception of how any meaningful proposition must be analyzable as constructed from a basis of logically independent elementary propositions. His rejection of this conception, around 1930, might be said to constitute the starting point of Wittgenstein’s tortuous journey toward his so-called “later” philosophy in the following decades.

And yet, I think a fruitful approach to Wittgenstein’s later conception of logic would be to ask if he might not still want to retain something of the fundamental orientation of his earlier view. Isn’t it possible that even later Wittgenstein would regard Stevenson’s and Belnap’s responses as the Schylla and Charybdis that must be avoided in order to reach a satisfactory response to Prior? Let me end this paper by gesturing at some features of the later conception that suggest that there is indeed a continuity with the earlier view to be found at this point.

In section 2, I raised a worry to the following effect: Isn’t early Wittgenstein simply imposing a notion of ‘description’ and ‘proposition’, and hence of ‘inference’ and ‘logic’, on our descriptive and inferential practices? And isn’t it this that makes it possible for him to claim that the logical structure of any meaningful proposition can be exhibited in the extremely simple and clear-cut framework of his truth table notation? In other words: Have early Wittgenstein really found out something essential about language and logic? Or has he merely issued a more or less arbitrary requirement? Has he merely told us what he is willing to count as ‘propositions’ and ‘inferences’?

At the time when wrote the Tractatus, I do not think Wittgenstein conceived of his account of logic as a matter of imposing anything on our linguistic and inferential practices. Rather, he seems to have thought that what he had to say about logic was a necessary development of a very obvious and very simple observation: Any meaningful description of the world purports to state that things are in a certain way. Such a description is either true or false. Either, things are as we describe them to be, or they are not. In his later manuscripts, Wittgenstein never tires of questioning the alleged obviousness and simplicity of this idea, as it is developed in the Tractatus. The Tractarian conception of the nature of the proposition is revealed as an imposed requirement, rather than some kind of innocent insight into the essence of what we do when we describe things and make inferences.

And yet, one idea that is not abandoned – even if it acquires a rather different significance in the later works – is the idea that the very notions of ‘proposition’ and ‘language’ that are of importance if we want to understand what it means to describe things and perform inferences, are logical notions. As Wittgenstein puts it in 1937, “Logic, it may be said, shews us what we understand by ‘proposition’ and by ‘language’.” (RFM, I—137)

However, at this point the very notions of ‘language’ and ‘proposition’ no longer have the kind of simple unity they have in the Tractatus. They are now instead said to be family resemblance terms. According to later Wittgenstein, it is not the case that all the things we call ‘languages’ and ‘propositions’ have some one thing in common in virtue of which we use the same word for all. Rather, they are related to one another in many different ways, and it is because of these relationships that we call them all ‘language’ and ‘proposition’.

This is a fundamental change in Wittgenstein’s conception. But what follows from it, exactly? Well, at least prima facie, it does not seem to make Wittgenstein any more vulnerable to Prior’s challenge. Remember the Tractarian objection against Prior: His way of using the ‘tonk’ example presupposes that the entities on which logical operators operate, and which figure in inferences, are not individuated as expressions in a given logical employment. This Tractarian objection still stands, even if the notions of ‘logical employment’, ‘logical unit’, ‘proposition’, and so on, no longer have the kind of simple unity they have on the Tractarian conception. After all, to say that ‘logical unit’ and ‘proposition’ are family resemblance concepts is not to say that anything can adequately be counted as a ‘logical unit’ or ‘proposition’. Just think about the concept ‘game’. The fact that this is a family resemblance concept does not mean that an activity in which any behavior is just as good as another can reasonably be called a ‘game’; and, similarly, the fact that ‘logical employment’ is a family resemblance concept does not mean that the sort of employment described by the rules for the use of ‘tonk’ can properly be described as logical.

Now, I do not want to reject this way of defending later Wittgenstein against Prior. On the contrary – in what follows, I am going to argue for a version of it. However, I do think there is more to say here, and that the relation between early and later Wittgenstein on this point is considerably more complicated than the above simplistic formulation of the defense would seem to suggest.

To begin with, it is important to keep in mind that Wittgenstein does not use the notion of family resemblance to make a static point about the presently surveyable use of certain words. Rather what he does is to point at the open-endedness of this use. The notion of family resemblance is used to say something about the unforeseeable dynamics of linguistic practice. Hence, to say that our concept of a proposition is a family resemblance concept is not to suggest, say, that it is possible to capture by something like a disjunctive definition: Something is a proposition if and only if it has at least some of the properties f₁, f₂, f₃, ..., f_n. It is to make a much more radical claim, namely, that there is no determinate limit on what can or will be adequately classified as a proposition – that there is no definition in that sense. We can give examples, point out differences – but then we have to go on from there. The adequate use of the word ‘proposition’ will depend on similarities and differences we see on particular occasions between particular instances, and what those perceived similarities will be cannot be settled beforehand. The application of the concept may expand beyond anything we can control at the present, and a claim to the effect that such an expansion is illegitimate, or that it must transgress some determinate limits already set by our present concept of proposition, is nothing but philosophical prejudice.

Another key point is to realize why Wittgenstein in the Tractatus was so convinced of what he later came to recognize precisely as such a philosophical prejudice. Why, that is, was he convinced that our concepts of proposition, language, and logic, could not be family resemblance concepts in the just described sense? As I said before, he did not think of himself as issuing any restriction here, but merely as spelling out the consequences of an utterly simple and innocent observation: Propositions – descriptions of reality – are either true or false. Either, things are as we describe them to be, or they are not. What he came to realize later was that this alleged insight, as he had understood it, was not as innocent as he had thought. In fact, it involved a rather particular idea of what truth must be – an idea which gets explicated in the account of propositions as pictures, and is manifested in the requirement that any meaningful proposition must constitute a truth function of logically independent elementary propositions. Rather than a careful scrutiny of how we actually proceed when we describe things and perform inferences, it is, Wittgenstein came to think, this sweeping, all-encompassing idea that governs the Tractarian explication of logic and of the nature of the proposition.