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 Do not quote without permission.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 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.
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 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 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 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 ‘pq’, 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]
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’, ‘~(~pÚ~q)’ and ‘~(pÉ~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 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 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,