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

Draft. Do not quote without permission.

Martin Gustafsson

Stockholm University

1. Introduction

In 1944, Wittgenstein made the following couple of remarks:

Is logical inference correct when it has been made according to rules; or when it is made

according to correct rules? Would it be wrong, for example, if it were said that p should

always be inferred from p? But why should one not rather say: such a rule would not

give the signs ‘p’ and ‘p’ their usual meaning?

We can conceive the rules of inference—I want to say—as giving the signs their

meaning, because they are rules for the use of these signs. So that the rules of inference

are involved in the determination of the meaning of the signs. In this sense rules of

inference cannot be right or wrong.1

Similar passages can be found at other places in Wittgenstein’s manuscripts from the 30’s and

onward. In such passages, he might seem to be expressing a relatively radical version of what

1 L. Wittgenstein, Remarks on the Foundations of Mathematics, 3rd ed. (Oxford: Blackwell,

1978), VII-30.

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is nowadays often called an “inferentialist” conception of the meaning of the logical

constants. In 1960, Arthur Prior famously argued that such a conception is untenable. For, if it

were true that rules of inference cannot be right or wrong because they determine the

meanings of the logical signs, then, Prior argued, nothing could stop us from introducing a

connective ‘tonk’ the meaning of which is determined by the following rules:

Introduction rule: p |- p tonk q

Elimination rule: p tonk q |- q

This, however, seems disastrous. For with this new connective, it appears possible to deduce

any proposition you like from any other proposition. No matter what propositions p and q are,

q can now be deduced from p. And this seems to do away with logic altogether. Or, as Prior

puts it, the new form ‘p tonk q’ is extremely convenient and “promises to banish falsche

Spitzfindigkeit from Logic forever.”2

Prior’s own conception of logical inference is not entirely clear. At the beginning of his

short paper, his target seems to be the very idea that the validity of inferences arises from the

meanings of logical expressions – the idea that logical inferences are, as he puts it,

“analytically valid”. But many have read him as rejecting only the “inferentialist” version of

that idea. Such readers claim that Prior’s argument has no force against the claim that logical

inferences are valid in virtue of the meanings of the logical expressions, if those meanings are

taken to be somehow determined otherwise than by the rules of inference that govern the use

of the expressions in deductions. According to such an anti-inferentialist version of the idea

that logical inferences are valid because of what the logical expressions mean, the meanings

of the connectives are somehow determined prior to the use of the connectives in deductions,

and can therefore serve to license or forbid such deductive patterns.

2 A. N. Prior, “The Runabout Inference-Ticket”, Analysis 21(1960): 38-39.

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My discussion in this paper will focus on the Tractatus, and on the question of how

Wittgenstein’s early conception of logic and the logical connectives is related to Prior’s

‘tonk’-example. At first sight, it may seem as if Prior’s attack does not concern Wittgenstein’s

early view of logic. In the Tractatus, Wittgenstein may seem to be proposing, not an

inferentialist conception of the logical connectives in terms of rules, but a semantic account of

the connectives in terms of truth-conditions. That would mean that the apparent inferentialism

that can be found in his later writings involves a sharp break with his early conception.

I will argue that this interpretation is mistaken. This is not because there are no

significant differences between early and later Wittgenstein’s conceptions of logic, but

because it locates those differences in the wrong place. As regards Wittgenstein’s early

conception, it fits none of the suggested labels: it can only misleadingly be described either as

“semantic” or as “inferentialist”. I will explain and back up this claim by contrasting

Wittgenstein’s view with two other responses to Prior that do fit one or the other of those

labels: J. T. Stevenson’s genuinely semantic conception, and Nuel Belnap’s genuinely

inferentialist one.3 From the viewpoint of early Wittgenstein, as reconstructed in the light of

the ‘tonk’-example, Stevenson’s and Belnap’s conceptions are the Schylla and Charybdis you

need to avoid in order to arrive at a truly satisfactory response to Prior.

What I provide below can also be regarded as a preamble to a study of the later

Wittgenstein’s discussions of logic. For even if passages such as the ones quoted at the

beginning of this paper may seem to suggest otherwise, I think that the classification of

Wittgenstein’s later conception as “inferentialist” is almost as misleading as the classification

of his early conception as “semantic”. In particular, this label makes it difficult to appreciate

an important continuity between Wittgenstein’s early and later conceptions of logic. In the

3 J. T. Stevenson, “Roundabout the Runabout Inference Ticket”, Analysis 21(1961): 124-128;

N. Belnap, “Tonk, Plonk and Plink”, Analysis 22(1962): 130-134.

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final section of this paper, I will say something brief about what I think this continuity

consists in.

2. ‘Tonk’, Truth Tables and the Tractatus

In his comment on Prior’s article, Stevenson argues that there are two reasons why people

have been tempted by the idea that rules of inference determine the meanings of the logical

connectives. To begin with, unlike most other expressions, the connectives do not have a

denoting function: they do not purport to refer to anything. Moreover, we ordinarily validate

particular inferences by appealing to some rule of inference, such as modus ponens. Taken

together, Stevenson claims, these two points have encouraged the conclusion that rules of

inference are what gives meaning to the connectives.

However, he continues, this conclusion is premature. For a rule can validate an

inference only if the rule is sound. The rule must never permit the deduction of a false

conclusion from true premises. And whether a given rule is sound depends on the meta-

linguistic interpretation we give of the relevant connective or connectives. The interpretation

is stated by means of truth tables, and determines how the truth-value of the conclusion is

related to the truth-values of the premises.

Hence, Stevenson argues, the meanings of the connectives are provided by meta-

linguistic interpretations of the just mentioned sort. They give meanings to connectives

against which rules of inference can be tested and proven correct or incorrect, sound or

unsound. Supposedly, this suffices to handle Prior’s worry. Just try to give a truth table for

‘tonk’ that makes both the introduction rule and the elimination rule sound. You will not

succeed: an interpretation that makes one rule sound inevitably makes the other unsound.

Stevenson concludes that even if the claim that logical inferences are valid in virtue of the

meanings of the logical connectives is not defensible in its inferentialist version, it is

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defensible if we instead think of these meanings as given “in terms of truth-function

statements in a meta-language.”4

It seems fair to say that Stevenson’s conception, or closely related views, are

widespread among contemporary logicians. As Dummett notes, it is certainly a sort of view

that is encouraged by presentations found in standard textbooks in logic.5 A fruitful approach

to Wittgenstein’s thoughts on logical inference is to consider the deep-going differences

between the Stevensonian sort of conception and what the Tractatus has to say on the subject.

One thing that might immediately spring to mind is the claim, in 5.132, that “‘Laws of

inference’, which are supposed to justify inferences, as in the works of Frege and Russell,

have no sense, and would be superfluous.”6 It may be argued, however, that the tension

between this remark and a view such as Stevenson’s is not very clear. After all, Stevenson

would say that rules of inference do not justify particular inferences in any philosophically

deep sense, since their “justificatory” status is entirely parasitic on truth-function statements

in the meta-language. So, I propose that we focus elsewhere, namely, on the fact that

Stevenson’s meta-linguistic truth-function statements have no place whatsoever in the

Tractarian system.

This may seem like a surprising statement. Aren’t truth tables of crucial importance in

the Tractatus? As has been noted by many commentators, however, the truth tables in the

Tractatus are not meta-linguistic devices, and do not provide what is nowadays thought of as

4 Stevenson 1960, p. 127.

5 M. Dummett, “The Justification of Deduction”, in Truth and Other Enigmas (London:

Duckworth, 1978), pp. 290-318; see pp. 291.

6 L. Wittgenstein, Tractatus Logico-Philosophicus, transl. by D. F. Pears and B. F. McGuiness

(Routledge: London, 1961). Henceforth, references to paragraphs in this work are made

parenthetically in the text.

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semantic interpretations.7 Rather, they serve as re-articulations that make logical relations

between propositions more perspicuously visible. Tractarian truth tables are signs at the same

level as ‘p’, ‘p&q’, and so on. The difference is notational: truth tables are given in a

notation that is designed to provide an entirely clear presentation of the logical features of the

relevant propositions. Thus, the sign

‘p q ’

T T T

F T F

T F F

F F F.

expresses the same proposition as the sign ‘p&q’, though in a more perspicuous manner. It is

how ‘p&q’ gets translated into the truth table notation.

Now according to the Tractatus, it is essential to a proposition that it is determinately

true or false. And that a proposition is determinately true or false means that it constitutes “an

expression of agreement and disagreement with truth-possibilities of elementary propositions”

(4.4). The truth table notation is designed precisely with the purpose of displaying such

agreements and disagreements with truth-possibilities.8

7 See, for example, H. O. Mounce, Wittgenstein’s Tractatus. An Introduction (Oxford:

Blackwell, 1981), p. 41; T. Ricketts, “Pictures, Logic, and the Limits of Sense in

Wittgenstein’s Tractatus”, in H. Sluga and D. G. Stern (eds.), The Cambridge Companion to

Wittgenstein (Cambridge: Cambridge University Press, 1996), pp. 59-99, at p. 81; M. B.

Ostrow, Wittgenstein’s Tractatus. A Dialectical Interpretation (Cambridge: Cambridge

University Press, 2002), at p. 104.

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This has three important and interrelated consequences. First, if what is essential to a

proposition is its agreement and disagreement with truth-possibilities of elementary

propositions, then logical equivalence means propositional identity: “If p follows from q and q

from p, then they are one and the same proposition.” (5.141, 5.41) Hence, according to the

Tractatus view of what it is to be a proposition, ‘p&q’, ‘(pq)’ and ‘(pq)’ belong

to the same proposition. Consequently, their truth table translation is the same, namely, the

truth table given above. So, a translation into the truth table notation makes the notational

differences between these signs disappear.

The second consequence of how the truth table notation is supposed to work is that if, in

ordinary linguistic practice, there are two occurrences of the same propositional sign that

serve to express different propositions, then the corresponding truth table renderings will be

different. Wittgenstein thinks such cases are common. Consider a standard example: the

sentence ‘On his vacation, Max is going to Italy or Spain’. On one occasion of utterance, this

sentence may be used to say something that is true if Max is going to both Italy and Spain. On

another occasion of utterance, it may be used to say something that is false if Max is going to

both Italy and Spain. The propositions expressed by these superficially identical utterances

will then be captured by different truth tables.

The third consequence is that something qualifies as a logical connective – a truth-

operation, as Wittgenstein calls it – only if the result of applying it to a couple of propositions

(or to one proposition if the operation is negation) can itself be rendered by a truth table. If no

result that can be rendered in this sort of way is forthcoming, no determinate proposition has

been generated, and no truth-operation has been applied. Thus, suppose someone claims to

have invented a new connective, but refuses to acknowledge any translation into truth table

8 Even an elementary proposition constitutes an expression of agreement and disagreement

with truth-possibilities of elementary propositions; for it is a truth-function of itself.

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notation as a correct rendering of the construction formed by joining two propositional signs

by means of this alleged connective. Then a problem arises about the status of that

construction. It may look like some sort of logically compound proposition, but, according to

Wittgenstein, we have been given no reason whatsoever to regard it otherwise than as a

merely orthographic juxtaposition of the propositional signs and an empty scribble.

Now what does all this have to do with how the early Wittgenstein would handle the

problem about ‘tonk’? Well, remember that for Stevenson, who conceives truth tables as

providing meta-linguistic interpretations of logical connectives, it is natural to think that the

problem about capturing the envisaged use of ‘tonk’ in a truth table means that either the

introduction rule or the elimination rule must be unsound. By contrast, from the perspective of

the Tractatus, the problem about capturing the envisaged use of ‘tonk’ in one single truth

table is a problem, not about soundness, but about propositional identity. What it shows is that

insofar as the different occurrences of the sign ‘p tonk q’ belong to determinate propositions

at all, and insofar as the rules for the use of ‘tonk’ are rules of inference (and not just, say,

rules for how to decorate wall paper with ink-marks), ‘p tonk q’ must belong to one

proposition when used in accordance with the introduction rule and to another proposition

when used in accordance with the elimination rule. The impossibility of providing a joint truth

table for the two uses does not mean that the rules are unsound, but that each rule constitutes

an incomplete specification of the use of two different connectives. When ‘p tonk q’ is

inferred from ‘p’, in accordance with the introduction rule, and when the orthographically

similar ‘p tonk q’ serves as a premise from which ‘q’ is inferred in accordance with the

elimination rule, what we have are two different compound propositions that look the same on

the surface. It so happens that the connectives that occur in them are both called ‘tonk’, but

those connectives are no more similar than, say, disjunction and conjunction. As Cora

Diamond puts this Tractarian response,

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it is through logic that we can identify any proposition as the same proposition as some

proposition uttered or written earlier [...]. Logic will show us that the two rules through

which the connective “tonk” was introduced (or supposedly introduced) in fact provide

partial specifications of two logical connectives; and that a proposition P-tonk-Q

inferred from P is not in general the same proposition as the equiform proposition P-

tonk-Q used by itself as a premise from which to infer Q. We can get this out of logic if

the logic that we need in order to do the job of identifying propositions is truth-

functional logic, for that will impose the conclusion, not that the rules for “tonk” are

logically wrong, but that they go part of the way toward introducing two logically

distinct connectives.9

But what if he who introduces ‘tonk’ refuses to acknowledge that the rules for its use

specify the use of two different connectives? What if he insists that ‘tonk’ means the same in

both sorts of inferences, and that if we do not see this we have misunderstood his

explanations? Wittgenstein’s response is that such heartfelt insistence on sameness of

meaning can in no way alter the fact that it is possible to acquire unification here – to think of

the suggested rules as rules governing one and the same “piece”, as it were – only by

abandoning the very idea that the use in question is a logical use, that the pattern is a pattern

of inference, and, hence, that ‘tonk’ is a logical connective at all. This person’s constructions

may look like propositions, and if such look-alikes are written down one below another it may

look as if what is going on is logical inference. However, to the extent that those look-alikes

9 C. Diamond, “Truth Before Tarski. After Sluga, after Ricketts, after Geach, after Goldfarb,

Hylton, Floyd, and Van Heijenoort”, in E. H. Reck (ed.), From Frege to Wittgenstein.

Perspectives on Early Analytic Philosophy (Oxford: Oxford University Press, 2002), pp. 252-

279, at p. 255.

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resist translation into the truth table notation, they are not propositions – and there is no

inference going on either, not even an unsound one.

One might still feel unclear about the real difference between Wittgenstein’s and

Stevenson’s ways of showing why ‘tonk’ is not a viable logical connective. Stevenson uses

truth tables to distinguish between sound and unsound rules of inference. Wittgenstein seems

to be using truth tables to impose restrictions on what is to be counted as “propositions” and

“inferences”. But is this difference really deep-going? Isn’t it just a verbal issue: Wittgenstein

prefers to use a narrow conception of inference according to which only those patterns that are

governed by what Stevenson calls “sound rules of inference” are to be called inferential,

whereas what Stevenson counts as “unsound” patterns are not counted as inferences at all by

Wittgenstein? Isn’t all we have here two terminologically different ways of getting at what is

fundamentally the same point, namely, that truth-functional logic sets limits on what

constitutes logically adequate behavior?

In order to see what is mistaken about this attempt to trivialize the difference between

Wittgenstein and Stevenson, we need to understand better what it means to say that, according

to Wittgenstein, we cannot identify propositions without reference to the function those

propositions have in inferences. First of all, I want to emphasize something very important

that attentive readers may already have noted. When I introduced the problem about ‘tonk’, I

presented it in a standard sort of way, namely, as a problem about how the meaning of a

logical connective is related to the role that the connective plays in inferences. Similarly, in

presenting Stevenson’s conception of how truth tables work, I talked about them as giving

interpretations of the connectives. However, in my presentation of Wittgenstein’s view, there

was a tacit shift of focus. I started talking about how the identity of a proposition is related to

the role that the proposition plays in inferences. And I talked about truth tables, not as giving

meaning to the connectives, but as logically perspicuous re-articulations of the logical

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structure of propositions. This shift from focusing exclusively on the connectives to focusing

also on the identity of propositions is no coincidence. For it is crucial to Wittgenstein that the

functioning of the connectives is inseparable from what it is to be a proposition. Let us look in

more detail at how this idea works.

It is a central idea in the Tractatus that, unlike what standard logical notations might

tempt us to believe, the logical connectives do not contribute anything to the content of the

sentences in which they occur. The connectives are to be thought of in what Wittgenstein calls

“operational” terms, and “[a]n operation is the expression of a relation between the structures

of its result and of its bases. The operation is what has to be done to the one proposition in

order to make the other out of it.” (5.22-5.23) Peter Sullivan uses a simple example to

illustrate how the truth table notation serves to clarify this role of the connectives.10 The point

is most easily seen if we consider the condensed version of the notation, in which a

proposition is given by an expression in which the last column of the truth table is stated

within parentheses before the elementary propositions are listed. In this condensed version,

‘p&q’ is translated as (TFFF)(p,q). Now, suppose we negate this latter formula. In standard

notations, what we do is to add a negation sign: ‘(p&q)’. This construction gives the

impression that the negation sign gives some sort of genuine contribution to the content of the

sentence, and trying to understand what this contribution is leads to all sorts of puzzles. By

contrast, in order to negate ‘(TFFF)(p,q)’, we do not add a new sign. Rather, we turn the

proposition into a new one by replacing all ‘T’s with ‘F’s, and vice versa. Starting from

‘(TFFF)(p,q)’ we thus obtain ‘(FTTT)(p,q)’. This makes it clear that negating a proposition is

not a matter of adding anything to it, but of using it as a base from which a new proposition is

generated according to a determinate pattern of transformation. As Sullivan puts it,

10 P. Sullivan, “The Totality of Facts”, Proceedings of the Aristotelian Society 100(2000):

175-192.

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“[n]egation is characteristic only of the relation between two propositions, never of any

proposition itself.”11 And similarly for the other operations of propositional logic.

So what we get in the Tractatus is a conception of truth-operations according to which

their role is exhaustively displayed once a way has been found to exhibit clearly the logical

interrelations between propositions. This point is vividly manifested precisely by the fact that

in the truth table notation, where logical interconnections are made transparently visible, there

is simply no need for the connectives. Indeed, according to the Tractatus, it is precisely by

making the connectives disappear in favor of a clear exposition of logical interconnections

that the truth table notation can be said to do full justice to what truth-operations are.

How, then, can the Tractatus avoid the problem about ‘tonk’? Not by a requirement of

soundness. Rather, the central idea here is the conception of what a proposition is. According

to the Tractatus, a proposition is true under some conditions and false under others. The

logical operators operate on and produce as results propositions qua such internally true-or-

false units. And the Tractarian objection against Prior’s tonk-example can now be expressed

as follows. The example presupposes that the entities on which logical operators operate, and

which figure in inferences, are not given as true-or-false units in this sense. The introduction

rule and elimination rule for ‘tonk’ can seem to determine a unified pattern of use only if it is

taken for granted that the units over which ‘p’ and ‘q’ range can be identified extra-logically,

in merely orthographic terms, as “sign-designs”, “concatenations of letters”, or whatever –

and, hence, that the so-called “use” determined by such rules is externally imposed on an

already given raw material of logically inarticulate sounds and shapes. It presupposes that the

relevant notion of inferential practice is basically a matter of manipulating such extra-

logically individuated units. By contrast, to identify those units as true-or-false propositions

means to see that the introduction rule and the elimination rule for ‘tonk’ cannot be taken to

11 Sullivan 2000, p. 180.

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govern one and the same connective. The units are propositions and the use is inferential only

if ‘tonk’ has different functions in these two different patterns of employment.

3. Inference and Representation

One way of expressing Wittgenstein’s dissatisfaction with the connectives in standard logical

notation is to say that he thinks they invite confusion between features that are essential to the

logical structure of a proposition and features that are the accidental byproducts of how the

proposition happens to have been generated from elementary propositions. For example,

Wittgenstein thinks the apparent differences between ‘p&q’, ‘(pq)’ and ‘(pq)’ are

accidental leftovers from the three different ways in which we happen to have generated one

and the same proposition by the successive applications of truth-operations to the elementary

propositions ‘p’ and ‘q’. The truth-table notation does away with this, so to speak,

“diachronic” dross, and displays only the “synchronic” essentials: what the three signs have in

common qua belonging to one and the same propositional symbol.

A misdirected striving for charity makes it easy to underestimate how radical

Wittgenstein’s view of the connectives is meant to be. For example, it is tempting to think that

Wittgenstein must still hold that connectives leave some sort of contribution to the sentences

in which they occur, if not to their content then to their “structure” or “form”. After all, the

use of a connective obviously makes some sort of difference. This, in turn, may lead to the

idea that Wittgenstein’s opposition to any account which explains the meaning of logical

connectives in terms of their being “representatives” (4.0312) means that he must be

embracing the anti-thesis to such a “representationalist” view. In other words, there is a

temptation to think that his account of the logical connectives must be what would nowadays

be described as an inferentialist conception.

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But in the case of early Wittgenstein’s view of logic, the label “inferentialism” is

dangerous. For this label suggests that Wittgenstein thinks the connectives do leave a special

contribution to the propositions of which they appear to be parts, albeit one explainable in

inferentialist rather than representationalist terms. The problem is that it seems difficult to

spell out this idea without suggesting the following division of work: what provide the

representational content of a proposition are the elementary propositions from which it has

been generated, whereas what provide the logical structure of the propositions are the

connectives. In short, the conclusion seems more or less inevitable that Wittgenstein has a

representationalist conception of the sense of elementary propositions, and an inferentialist

conception of the connectives, and that these two elements – representation and structure,

content and form – are therefore separable, one being provided by the elementary

propositions and the other by the connectives.

This, it seems, is essentially the sort of view that Anscombe is warning against early on

in her book on the Tractatus.12 According to Anscombe, Wittgenstein’s “whole theory of

propositions is [...], on this view, a merely external combination of two theories: a ‘picture

theory’ of elementary propositions [...], and the theory of truth-functions as an account of

non-elementary propositions.”13 The sort of view Anscombe criticizes here is one according to

which Wittgenstein thinks the whole domain of meaningful elementary propositions can in

principle be given before and independently of the introduction of the truth-operations. The

idea would be that Wittgenstein conceives the truth-operations – and thus the possibilities of

generating logical complexity – as add-ons to an already given set of meaningful elementary

propositions.

12 E. Anscombe, An Introduction to Wittgenstein’s Tractatus. Themes in the Philosophy of

Wittgenstein (London: Hutchinson, 1971 [1959]).

13 Anscombe 1971 [1959]: 25-6.

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It is clear that Anscombe is right to reject this sort of interpretation. Such a composite

conception is straightforwardly incompatible with Wittgenstein’s claim that “[a]n elementary

proposition really contains all logical operations in itself” (5.47), and also with the central

paragraph 3.42:

A proposition can determine only one place in logical space: nevertheless the whole of

logical space must already be given by it.

(Otherwise negation, logical sum, logical product, etc. would introduce more

and more new elements—in co-ordination.)

(The logical scaffolding surrounding a picture determines logical space. The

force of a proposition reaches through the whole of logical space. [Der Satz durchgreift

den ganzen logischen Raum.)

Sullivan agrees with Anscombe on this point, and argues that Tractarian truth-

operations must not “be understood by reference to a prior and independent conception of

their domain.”14 Rather, Sullivan notes, the role of truth-operations and the sense of

elementary propositions can only be conceived as mutually presupposing one another. As

Wittgenstein remarks already in 1914: “Just as we can see ~p has no sense, if p has none; so

we can also say p has none if ~p has none.”15

What 5.47 and 3.32 make clear is that Wittgenstein thinks that all possibilities of logical

complexity and interconnectedness are given as soon as there is picturing, as soon as there are

true-or-false descriptions of the world. It is not as if we need to add logical machinery in order

14 Sullivan 2000, p. 189.

15 Notes Dictated to G. E. Moore in Norway, in Notebooks 1914-1916, 2nd ed., ed. by E.

Anscombe and G. H. von Wright (London: Blackwell: 1979). pp. 108-119, at p. 118. Quoted

by Sullivan, in Sullivan 2000, p. 189.

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to be able to generate complex propositions out of elementary propositions. The logical

machinery is already there with the elementary propositions – with their depicting the world

truly or falsely. There is no separate carrier of the possibilities of logical complexity, aside

from the elementary propositions themselves.

More precisely, Wittgenstein’s view is this. Generating logical complexity is something

we do on a material of elementary propositions. The rules that are constitutive of such

generation is given already by the following two features of that material:

(1) Elementary proposition are true or false descriptions of the world.

(2) An elementary proposition is neither contradicted nor entailed by any other elementary

proposition (4.211) – it is a truth-function only of itself (4.53).

From the viewpoint of the Tractatus, it is misleading to separate (1) and (2), since it is one of

the central ideas of the book that the very possibility of determinately true-or-false description

requires logically independent elementary propositions. This is an idea that Wittgenstein

abandoned later on, of course. But if we want to be faithful to his early self we can simply

say: the rules that are constitutive of the generation of logical complexity are inseparable from

what it is to describe the world truly or falsely. The relation between the nature of description

and the possibility of logical complexity – between what it is for a proposition to represent a

possible state of affairs and what it is for it to be able to stand in inferential relations to other

propositions – is internal. None of these two functions – that of representing a possible state

of affairs and that of standing in determinate inferential relations to other propositions – is less

fundamental than the other. They come in a package; they are two sides of the same coin.

This is not to deny that according to the Tractatus, logic is a purely formal business.

What the formal character of logic means for the Tractatus, however, is only that logic

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disregards, or treats as arbitrary, the specific contents of propositions. Logical syntax in the

Tractarian sense does not involve the further step of abstracting from the very meaningfulness

of linguistic expressions, treating them as mere shapes or “sign designs”. Taking this further

step is characteristic of our contemporary, post-Tarskian notion of syntax; whereas in the

Tractatus, it is crucial to logic that it must “presuppose that [...] elementary propositions

[have] sense.” (6.124) Again, according to early Wittgenstein, it precisely “by representing a

possibility of existence and non-existence of states of affairs” (2.201) that elementary (and

other) propositions occupy positions in logical space. To say that a proposition represents

such a possibility is already to say that it is true-or-false, and this is already to say that it

stands in inferential relations to other propositions. It is not possible to drive a wedge between

propositions qua pictures and propositions qua nodes in an inferential network of other

propositions: there is representation only if there is inference and vice versa.

So, the reason why early Wittgenstein has neither a semantic nor an inferentialist

account of the functioning of the connectives, is that he thinks there is nothing to account for

here. According to Wittgenstein, what we need to get clear about is the nature of description –

what it is to picture the world truly or falsely. Once that has been accomplished we realize

that a theory of the connectives is a theory without subject matter.

4. Deducibility and Conservatism

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

17

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:

Axiom. A |- A

Rules. Weakening: from A1, ..., An |- C to infer A1, ..., An B |- C

Permutation: from A1, ..., Ai, Ai+1, ..., An |- B to infer A1, ..., Ai+1, Ai, ..., An |- B.

Contraction: from A1, ..., An, An |- B to infer A1, ..., An |- B

Transitivity: from A1, ..., Am |- B and C1, ..., Cn, B |- D to infer

A1, ..., Am, C1, ..., Cn |- D.16

The crucial point here is that this characterization of our antecedently given conception of

deducibility is taken to be complete, in the sense that it gives us “all the universally valid

deducibility-statements not involving any special connectives.”17 What this means is that we

can demand, with respect to any subsequently introduced connective, that the rules for its

employment do not allow new valid deducibility-statements unless those statements involve

the connective in question. Thus, the idea is that if we introduce, say, ‘&’ in the ordinary way,

we extend the system by adding new deducibility-statements such as ‘A, B |- A&B’ – and all

those added deducibility-statements will themselves involve ‘&’. In Belnap’s terminology

(originally proposed by Emil Post), this extension will be conservative. By contrast, Prior’s

way of introducing ‘tonk’ allows precisely the addition of a new deducibility-statement that

16 Belnap 1962, p. 131.

17 Ibid.

18

does not contain ‘tonk’, namely, ‘A |- B’. This is not consistent with the completeness claim

above, and hence the extension is not conservative.

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.

19

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.

20

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

21

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-

18 Belnap 1962, p. 132.

19 Ibid.

22

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.

A sense that there is some problem with Belnap’s attempt to make a ban on non-

conservative extensions do the work Wittgenstein thinks is done by the very notion of what it

is to depict the world truly or falsely, may arise when one comes across the following passage

in his paper:

It is good to keep in mind that the question of the existence of a connective having such

and such properties is relative to our characterization of deducibility. If we had initially

allowed A |- B (!), there would have been no objection to tonk, since the extension

would then have been conservative. Also, there would have been no inconsistency had

we omitted from our characterization of deducibility the rule of transitivity.20

Several commentators have worried that Belnap is here spoiling his case against Prior, since

he makes it seem as if he cannot avoid precisely the sort of arbitrariness or overly lax

conventionalism that the ‘tonk’ example was meant to expose. After all, Belnap is suggesting

that we could have had a notion of deducibility such that any sentence B is deducible from

any other sentence A, and that our not having such a notion is just a matter of choice or

perhaps ingrained habit. And this was precisely the sort of idea that Prior rejected and

ridiculed. Hence, even commentators sympathetic to Belnap want to correct him at this point.

Consider Steven Wagner’s objection that

20 Belnap 1962, p. 133.

23

the fact that there is no such connective as ‘tonk’ has to do with the truth about

deducibility, not, as Belnap seems to suggest, with our beliefs. As long as ‘|-’ means

deducibility, it is nonsense to suppose that the existence of a connective satisfying

[Prior’s rules for the use of ‘tonk’] depends on our characterization of |-. As long as it is

not up to us what follows from what (except trivially due to our power to change the

meanings of words), it is similar nonsense to speak of our “allowing A |- B”. We can, of

course, falsely believe that A |- B, but that will not help the definition of ‘tonk’.

Fortunately, Belnap’s error (which we might trace to an overly formalistic viewpoint) is

easily patched up without damage to the rest of his article. The claims I have just

criticized can be dropped; what remains is the observation that [the rules for ‘tonk’] are

jointly unsatisfiable. Belnap’s general response to Prior could then be that while we can

expect contradictions to flow from an unsatisfiable definition, there is no reason to

throw out the satisfiable definitions of similar form.21

Early Wittgenstein’s reaction to this would be to say that Belnap’s error is not as easily

amended as Wagner suggests. Simply insisting that it is a truth about deducibility that it

leaves no room for ‘tonk’ is of no avail, for the problem is precisely to clarify what it is about

deducibility that makes this a truth. According to Wittgenstein, the central thing here is to

clarify what the units that figure in deductions – sentences, propositions, or whatever you

want to call them – are. And you cannot do so without thereby introducing the particular

truth-operations of propositional logic. Consequently, from a Tractarian viewpoint, Belnap’s

idea that we can specify an antecedently given context of deducibility, and conceive the

introductions of particular truth-operations as “extensions” of this context, is both incoherent

and superfluous as an explanation of why there is no room for ‘tonk’.

21 S. Wagner, “Tonk”, Notre Dame Journal of Formal Logic 22(1981): 289-300, at p. 291.

References omitted.

24

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.

5. After the Tractatus

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

25

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

26

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’

27

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, ..., fn. 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

28

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.

Wittgenstein’s reflections on color exclusion and other related difficulties lead him to

abandon the Tractarian conception of the nature of the proposition. And what finally happens

is not that he replaces it with some other general account of what it is for a proposition to state

how things are. Rather, he abandons the very idea that reflections on the general nature of

truth and falsity can provide us with a universally valid account of our concepts of

description, proposition, language and logic. Indeed, even if we lay down that a proposition is

whatever can be true or false, our concept of truth and falsity cannot thereby serve to

determine what is and is not a proposition. This seems to be the point, or one point, of the

difficult paragraph 136 in the Philosophical Investigations. In this paragraph, Wittgenstein

notes that the concepts of truth and falsity are not given antecedently to, but together with, the

concept of a proposition, and, hence, “the proposition that only a proposition can be true or

29

false can say no more than that we only predicate ‘true’ or ‘false’ of what we call a

proposition.” (PI 136)

One thing that makes this paragraph particularly difficult to understand, if one is

interested in the relation between early and later Wittgenstein, is that, as it stands, early

Wittgenstein would probably have wanted to agree with what his later self is saying there.

Indeed, my description of how early Wittgenstein would have criticized a “semantic” account

of the connectives such as Stevenson’s ascribes to him precisely the view that our “use of

‘true’ and ‘false’ “belongs to our concept ‘proposition’ but does not fit it” (PI 136). And yet,

later Wittgenstein seems to be suggesting that the author of the Tractatus has not taken the

“internal” relation between our concepts of truth and falsity and our concept of a proposition

seriously enough. According to later Wittgenstein, the Tractarian conception of truth is in fact

developed and defended in complete isolation from any careful investigation of what it

actually means to “describe the world” in real life. So, later Wittgenstein’s target is not the

Tractarian idea that the concept of truth and the concept of a proposition are internally related.

Rather, his criticism of the Tractatus is that if one really thinks through that idea one will

realize that the internal relation between the concept of truth and the concept of a proposition

cannot serve as the basis for a uniform account of the form of a proposition.

Where does this leave us with ‘tonk’? With what right can later Wittgenstein now say

that the rules for ‘tonk’ necessarily describe the use of two logically distinct connectives?

What if someone denies this? With what right can later Wittgenstein say that such a denial is

incompatible with ‘tonk’ being a logical expression at all? If there is no general, once and for

all settled determination of what a proposition is, and no general, once and for all settled

determination of what constitutes the domain of the logical, then aren’t we defenseless against

someone who insists that ‘tonk’, as defined by the introduction rule and the elimination rule

30

given by Prior, belongs to this domain of the logical? And doesn’t this show that Prior, after

all, has put his finger on a weak spot in Wittgenstein’s conception?

I think Wittgenstein answer is that we will look defenseless only on a certain conception

of what a defense here would have to look like, and that it is one of his central aims precisely

to reject that sort of conception. The sort of view against which Prior’s ‘tonk’-example has

genuine force is an inferentialism of an explanatory kind – a inferentialism meant to account

for how a logically inert raw material of mere shapes or ‘sign-designs’ can, as it were, get

beefed up so as to achieve logical potency; an inferentialism according to which rules of

inference make it the case that already identified, orthographically individuated units come to

stand in logical relations to one another. Wittgenstein never aspired to provide such an

explanation. On the contrary, he always thought that this sort of aspiration was fundamentally

misguided. Indeed, I think he would say that if we try to give rules of inference this sort of

explanatory role, any attempt to find a defense against Prior’s ‘tonk’-example has to fail.

Which means that, strictly thought through, this explanatory form of inferentialism does away

with logic; or, better, it never gets the domain of the logical into proper view at all.

Instead, Wittgenstein thinks we can get logic into proper view only if we start from that

familiarity with inferential practice that we already have as competent speakers and thinkers.

That is, we must occupy a standpoint from which the units that figure in inferences are

already given as logical units, units whose very identity is tied to the rules according to which

they are being employed. It is only through logic that we can determine whether two

occurrences of one and the same orthographic shape – say, ‘p tonk q’ – can also be

occurrences of one and the same logical unit, or whether they must have different logical

functions. An account that takes logic for granted in this sense, and therefore shuns away from

the sort of explanatory aspiration described before, will not be vulnerable to Prior’s criticism.

31

And this point seems to hold even if logic is no longer thought of in Tractarian terms –

even if the notions of logic, language, proposition, and so forth, are conceived of as multiform

and open-ended, just as the later Wittgenstein’s conception of family resemblance suggests.

The fact that there is no once and for all insight into what a proposition is and what is to be

counted as a valid inference, does not make it arbitrary what is to be so described. It will still

be true that the only way of making ‘tonk’ continuous with what we call ‘logic’ is to think of

the introduction rule and the elimination rule as rules for two different connectives. Indeed,

the very construction and patent absurdity of the ‘tonk’ example trades on precisely this

ability to perceive a discontinuity at this point. The worry about the threatening logical

breakdown is itself a manifestation of the fact that we do have a perfectly clear sense that,

whatever logic is, it is not like that. And later Wittgenstein’s claim is that this sense is enough.

It does not need support from any further explanation delivered from a standpoint outside of

established practice. Rather, Wittgenstein thinks the very wish for such an explanation is what

makes inference seem philosophically problematic in the first place.

32