Carnap and the Tractatus’ philosophy of logic
Oskari Kuusela
Introduction
A characteristic feature of analytic philosophy – or at least one strand of it – has been the use of symbolic or formal logic as a philosophical tool. Given the methodological importance of logic for analytic philosophy, developments in logic and the philosophy of logic have played an important part in its development. A practically unanimously accepted part of the history of logic and analytic philosophy is a particular account of the relation between Wittgenstein and Carnap, and how Carnap in the 1930s overcame the limitations of the Tractatus’ philosophy of logic, i.e. its condemnation of logic to silence through the distinction between saying and showing. Carnap’s achievement was an important step in the development of the contemporary model theoretic account of logic which, so the story goes, left the Tractatus’ philosophy of logic superseded. Arguably, however, this account of the Wittgenstein-Carnap relation, adopted rather directly from Carnap’s The Logical Syntax of Language, doesn’t do proper justice to the Tractatus. It downplays important similarities between Carnap’s and Wittgenstein’s positions whose correct understanding is a reason to regard Carnap’s position as a variant of Wittgenstein’s, rather than anything particularly novel. Described in Carnapian terms, the Tractatus’ goal is to introduce, by using only apparently metaphysical quasi-syntactical sentences, syntactical principles and concepts that constitute a calculus designed for the purpose of philosophical clarification in the formal mode. Hence, a distinction corresponding to Carnap’s distinction between the material and formal mode is already part of the Tractatus’ account of philosophy and logic. Moreover, arguably, a key point of Wittgenstein’s saying-showing distinction is to clarify the difference between logical or syntactical determinations and true/false statements, whereby logic is understood as something that is not true/false and not justifiable by reference to any facts about either language or reality. Thus construed, however, the saying-showing distinction may be taken to underlie Carnap’s principle of tolerance which is based on the very idea that logic is not true/false about anything. Similarly, the saying-showing distinction, or an equivalent distinction, is assumed by Carnap’s distinction between object-language and syntax-language. For abstractly conceived, Wittgenstein’s distinction simply concerns the difference between statements in terms of a language and logical determinations concerning that language. But this distinction is acknowledged rather than rejected or overcome when distinguishing between an object-language and a syntax- or metalanguage in logic. Hence, the key components of Carnap’s philosophy of logic are already present in the Tractatus, even if in a different form. Let us start the discussion from the generally acknowledged shared features of Carnap’s and the Tractatus’ positions.
1. Logical syntax: agreements and departures
Carnap describes the early Wittgenstein’s influence on him in his ‘Intellectual Autobiography’ as follows:
For me personally, Wittgenstein was perhaps the philosopher who, beside Russell and Frege, had the greatest influence on my thinking. The most important insight I gained from his work was the conception that the truth of logical statements is based only on their logical structure and the meaning of the terms. Logical statements are true under all conceivable circumstances; thus their truth is independent of the contingent facts of the world. On the other hand, it follows that these statements do not say anything about the world and thus have no factual content. (Carnap 1963, 25)
The Tractarian conception of logic described in this quote, which Wittgenstein put forward as a critique of Frege’s and Russell’s (early) view of logic as the most general science, constitutes the common background for both Carnap’s and Wittgenstein’s subsequent discussions of logic. Part of this conception of logic as tautological and contentless is a conception of logic as syntax, according to which, logical relations are syntactical relations determined by the rules of logical syntax. This is to regard syntactical rules as determining, not merely what kind of sentences can be constructed from subsentential constituents, but also the inferential relations between propositions or sentences. Thus, logic becomes syntax, a study of the formal or structural characteristics of language. (See, LSL, 2; Carnap 1963, 54; for relevant references to the Tractatus, see below)[1]
More specifically, for Wittgenstein logical syntax determines the logical role or the use of a sign in language, or put differently, it determines the logical form of an expression, understood as its possibilities of combination with other expressions. Given the Tractatus’ conception that the function of language is the true/false representation of reality, Wittgenstein ultimately understands the use of expressions in terms of their contribution to true/false statements about reality. This leads us to an important difference between his view and Carnap: despite their significant agreement, and a number of very similar sounding remarks they make about the notion of logical syntax, the Tractatus’ conception of logical syntax differs importantly from Carnap’s. Whereas Carnap adopts a Hilbertian formalistic conception of logical syntax as rules that apply to uninterpreted signs in the sense of, for example, ink marks on a page, or to abstract patterns of such marks interpreted as logical structures, for Wittgenstein syntax is concerned with signs with a significant or meaningful (sinnvoll) use.[2] How is this difference to be understood?
The difference is connected with – and easily obscured by the failure to notice – certain terminological differences between Wittgenstein and Carnap, regarding the notions of a ‘symbol’ and ‘expression’. While a symbol or an expression for Wittgenstein is a sign with a significant or meaningful use (any part of a proposition that characterizes its sense is a symbol or an expression), for Carnap a symbol or an expression is a character or a mark not assumed to possess any meaning or designate anything. (TLP 3.31; LSL, 4-6) For example, ‘a’ and ‘a’ printed on different points of a page are, in Carnap’s terminology, different symbols, even if they may be equivalent in syntactical design, insofar as their use is governed by the same syntactical rules. (LSL, 15) Wittgenstein writes in the Tractatus: ‘In order to recognize a symbol in a sign we must observe how it is used with a sense.’ (TLP 3.326) He explains this remark to Odgen, the book’s first translator as follows: ‘3.326 […] The meaning of this prop[osition] is: that in order to recognize the symbol in a sign we must look at how this sign is used significantly in propositions. I.e. we must observe how the sign is used in accordance with the laws of logical syntax. […].’ (LO, 59) And once we understand their use, i.e. how signs signify, we understand their syntax: ‘The rules of logical syntax must go without saying, once we know how each individual sign signifies.’ (TLP 3.334) Accordingly, only signs with a meaningful use have logical form: ‘A sign does not determine a logical form unless it is taken together with its logico-syntactical employment.’ (TLP 3.327; cf. NB, 53/PTLP 3.253)
The significance of this difference between Wittgenstein and Carnap is that, essentially, they are offering different accounts of the idea of the formality of logic – and therefore its nature, insofar as formality is essential to logic.[3] Both characterize the idea of formality by saying that syntax must be established without reference to the meanings of signs[4], with Carnap attributing this view to Wittgenstein, who ‘[…] made clear the formal nature of logic and emphasized the fact that the rules and proofs of syntax should have no reference to the meaning of the symbols.’ (LSL, 282, cf. 284; IS, 156, 157, cf. 232). In the Tractatus this point about meaning and syntax is presented as a criticism of Russell’s conception of logic and his theory of types as involving a reference to meaning of signs (TLP 3.331). Accordingly, it is intimately connected with Wittgenstein’s criticism of the idea of logic as a science whose statements have factual content. Wittgenstein writes: ‘In logical syntax the meaning of a sign should never play a role. It must be possible to establish logical syntax without mentioning the meaning of a sign: only the description of expressions may be presupposed.’ (TLP, 3.33) In accordance with his conception of expression as a sign with a meaningful use, by the description of an expression (qua expression) he understands a description of its use. More specifically, such descriptions are presented by means of variables. The values of the variables are possible propositions that contain the variable, whereby the determination of such a class of propositions specifies the variable, and in this way the possible uses of the expression in question. (TLP 3.313, 3.316, 3.317)
Carnap explains his conception: ‘By the logical syntax of a language, we mean the formal theory of the linguistic forms of that language – the systematic statement of the formal rules which govern it together with the development of the consequences that follow from these rules. A theory, a rule, a definition, or the like is to be called formal when no reference is made in it either to the meaning of the symbols (for example, the words) or to the sense of the expressions (e.g. the sentences), but simply and solely to the kinds and order of the symbols from which the expressions are constructed.’ (LSL, 1) The syntax of a language, in other words, concerns the structures of possible (serial) orders of its elements, or formal structures as determined by syntactical rules. (LSL, 5, 6) Or as Carnap also explains the basic idea, logical syntax as a theory of language is the ‘geometry of written pattern’ (Carnap 1963, 29) – though it is not essential here that the patterns should be written, but any patterns will do in principle (cf. LSL, 6).[5]
This explains the sense in which for Carnap logical syntax is concerned with signs irrespectively of whether they have any meaning, so that in principle uninterpreted meaningless signs too may be regarded as possessing a syntax and a logic. This is characteristic of the Hilbertian conception on which signs are treated meaningless, before an interpretation (or a model) is given to them. Wittgenstein’s view, by contrast, excludes syntactic characterizations of meaningless signs. Meaningless or nonsensical expressions whose meaning or sense has not been determined do not possess a logic or a syntax. That is, while syntax indeed is established without reference to the meanings of names or senses of sentences, only signs with a meaningful use (i.e. propositions with a sense, and by entailment their constituent expressions) have a syntax. Hence, although logic is not concerned with meanings, but rather with forms that underlie the meaningful uses of language and make the expression of meanings possible (TLP 3.34, 3.341, 3.344, 4.12), it does presuppose the meaningfulness of linguistic expressions. As Wittgenstein notes about the propositions of logic, i.e. tautologies, which he regards as bringing to view the most general and abstract formal characteristic of language, the general propositional form: ‘The propositions of logic […] presuppose that names have meaning and elementary propositions sense; […].’ (TLP 6.124; see 6.12)[6] Or as one might also explain Wittgenstein’s view, there would seem to be no reason to think of a system of marks or characters as a language and as possessing logic, unless it had a meaningful use and the signs had meaning (cf. AWL, 43).
At this point Carnap departs from Wittgenstein also in certain other respects. Wittgenstein’s conception of logic, Carnap maintains, leaves no room for talk about syntax or the logic of language, or for philosophical elucidatory statements about what is logically necessary or possible. As he says, in Wittgenstein’s view ‘syntax cannot be expressed at all’. (LSL, 53) On these grounds Carnap then regards Wittgenstein’s position as ‘certainly very unsatisfactory’ (LSL, 283; cf. Carnap 1934, 8; see below for a detailed discussion of Carnap’s disagreements with Wittgenstein). On the background of Carnap’s dissatisfaction are issues about the methodology of logic and philosophy. As he explains in the foreword to Logical Syntax with regard to the former: ‘[…] a book on logic must contain, in addition to the formulae, an expository context which, with the assistance of the words of ordinary language, explains the formulae and the relations between them; and this context often leaves much to be desired in the matter of clarity and exactitude.’ Given Carnap’s recognition that ‘[…] in this context is contained an essential part of logic […]’, ‘[…] the important thing is to develop an exact method for the construction of these sentences about sentences.’ Accordingly, the purpose of his book is to: ‘[…] give a systematic exposition of such a method, namely, of the method of “logical syntax”.’ (LSL, xiii; cf. Carnap 1963, 55.) Given the use of relevant kind of symbolic languages as tools of logical analysis, the point can also be expressed thus: ‘The aim of logical syntax is to provide a system of concepts, a language, by help of which the results of logical analysis will be exactly formulable.’ (LSL, xiii, cf. 7; cf. PLS, 58, 59)
Moreover, because Carnap regards logical analysis as the method that a scientifically respectable philosophy must adopt, questions of the nature and methodology of logic are of the greatest significance also in this sense. ‘The part of the work of philosophers which may be held to be scientific in its nature […] consists of logical analysis.’ (LSL, xiii; cf. 279) And as he intends to show: ‘[…] all philosophical questions which have any meaning belong to syntax.’ (LSL, 280) Thus, questions about the nature of logic, and logical analysis, are simultaneously questions about the nature of philosophy. Later on in his autobiography, Carnap singles as ‘the main thesis’ of his book ‘the importance of the metatheory for philosophy’. (Carnap 1963, 56; cf. LSL, xiii), where metatheory is a broader notion in the sense that it also includes semantics.[7] The point is that the adoption of the point of view of logical syntax in philosophy would make it possible to formulate philosophical questions and statements in an exact manner and enable one to sidestep the inexactitude of logical and philosophical expositions given in natural language. Consequently, Carnap believes, philosophers would be able to avoid the discussion of mere pseudo-problems, such as the questions of metaphysics. Philosophy would become more fruitful in that pointless disputes which in the garb of the traditional philosophical vocabulary misleadingly appear to concern the nature of the objects in question, while being really questions about the choice of appropriate forms of language for particular tasks, can be set aside. We can then focus on questions concerning the choice of language with a better self-understanding, and without the distraction of unnecessary disputes about who is right or wrong, given that the choice of a language is a matter of expediency, not of truth or falsity. (LSL, 277-281; see, Carus 2007, 232, 233 for discussion.)