Wittgenstein and ‘tonk’: Inference and Representation in the Tractatus
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- 2. ‘Tonk’, Truth Tables and the Tractatus
- 3. Inference and Representation
| 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 Do not quote without permission.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.
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
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 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 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, 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 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 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, “[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 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. 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. 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.)
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:
(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 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. Directory: wittgenstein -> files files -> The Laws of Thought and the Power of Thinking Matthias Haase (Universität Basel) Introduction Download 87.39 Kb. Share with your friends:
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