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 Do not quote without permission.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
1
L. Wittgenstein, Remarks on the Foundations of Mathematics, 3rd ed. (Oxford: Blackwell,
1978), VII-30.
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
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
2
A. N. Prior, “The Runabout Inference-Ticket”, Analysis 21(1960): 38-39.
2
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.
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
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.
3
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
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 metalinguistic 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 metalinguistic 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
4
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
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.
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.
5
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
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
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.
6
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
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’, ‘(pq)’ and ‘(pq)’ 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 truthoperation, as Wittgenstein calls it – only if the result of applying it to a couple of propositions
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.
7
(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
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
8
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,
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 Ptonk-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 truthfunctional 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
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. 252279, at p. 255.
9
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
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
10
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
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
10
P. Sullivan, “The Totality of Facts”, Proceedings of the Aristotelian Society 100(2000):
175-192.
11
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,
“[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-orfalse 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 –
11
Sullivan 2000, p. 180.
12
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 extralogically 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
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’, ‘(pq)’ and ‘(pq)’ 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
13
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.
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
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.
14
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.
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 truthoperations 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
14
Sullivan 2000, p. 189.
15
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
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
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.
16
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
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.
17
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
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
16
Belnap 1962, p. 131.
18
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
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
17
Ibid.
19
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-orfalse 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
20
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
21
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
18
Belnap 1962, p. 132.
19
Ibid.
22
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 nonTractarian 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 signdesigns 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 nonconservative 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
23
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
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
20
Belnap 1962, p. 133.
24
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’.
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
21
S. Wagner, “Tonk”, Notre Dame Journal of Formal Logic 22(1981): 289-300, at p. 291.
References omitted.
25
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
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
26
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
27
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.
28
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
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-
29
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
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
30
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
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.
31
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.
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
32
established practice. Rather, Wittgenstein thinks the very wish for such an explanation is what
makes inference seem philosophically problematic in the first place.
33