Philosophical Investigations 29:1 January 2006 ISSN 0190-0536
The Unity of Language and Logic in Wittgenstein’s Tractatus Leo K. C. Cheung, Hong Kong Baptist University
1. Introduction Wittgenstein holds in the Tractatus that the general propositional form is the sole logical constant (or the one and only general primitive sign in logic): It is clear that whatever we can say in advance about the form of all propositions, we must be able to say all at once. An elementary proposition really contains all logical operations in itself. For ‘fa’ says the same thing as
‘ ( ∃x).fx.x = a’. Wherever there is compositeness, argument and function are present, and where these are present, we already have all the logical constants. One could say that the sole logical constant was what all propositions, by their very nature, had in common with one another. But that is the general propositional form . . . The description of the most general propositional form is the description of the one and only general primitive sign in logic. (TLP 5.47–5.472)
By the thesis, as I shall explain, he means that the general form of proposition is the general form of logical operation. The importance of the thesis consists in, first, that it brings out the unity of language and logic and, second, its crucial role in his attempt to achieve the proclaimed aim of the Tractatus (TLP, p.3) – to draw a limit to thought by drawing one to language. The purpose of this paper, besides explaining these two points, is to explain how the Tractatus employs the picture theory and the Grundgedanke in TLP 4.0312 to argue for the thesis. © 2006 The Author. Journal compilation © 2006 Blackwell Publishing Ltd.
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It is indeed not easy to see how the Tractatus can hold and prove the thesis that the general propositional form is the sole logical constant. For the thesis demands not only non-elementary propositions but also elementary propositions that they must satisfy the general form of logical operation. How can an elementary proposition, which is supposed to be an immediate, and thus non-truth-functional, combination of names (TLP 4.22–4.221), satisfy the general form of logical operation? How can an elementary proposition involve all logical constants in an intrinsic manner (TLP 5.47) if it is supposed to belong to the end products of logical analysis of propositions (TLP 4.221)? How can the application of logic be involved in an elementary proposition’s saying anything about reality? My explanation of how Wittgenstein in the Tractatus proves the thesis would also make it clear how he answers these questions. It is, however, worthy of pointing out here that while many commentators of the Tractatus are ignorant of, or have chosen to ignore, these questions,1 several prominent commentators have attempted to tackle them or to criticize the Tractatus on that. It is illuminating to see some of the latter’s views. For example, Brian McGuinness writes in his essay ‘Pictures and Form’, . . . in the first part of the Tractatus, notably in the 3’s and early 4’s, we seem to be told that the essence of a proposition is to be a picture, while in the later parts we are told that its essence is to be a truth-function, that is to say a result of applying the operation of simultaneous negation to elementary propositions. The ‘picture theory’ requires further elaboration, and the truth-function account of what it is to be a proposition seems to involve circularity by presupposing a prior understanding of what it is to be an elementary proposition. But a more serious difficulty is that the two accounts seem to be quite separate things, and, if this is 1. For example, Max Black, Robert Fogelin and James Griffin are amongst those commentators. In his A Companion to Wittgenstein’s ‘Tractatus’, Black (1964: 236–8, 270–1) discusses the general propositional form but does not explain the thesis that the general propositional form is the sole logical constant. In fact, he does not seem to have noticed the thesis, nor the fact that the Tractatus holds and attempts to argue for the unity of language and logic. Fogelin has devoted a section in his Wittgenstein, 2nd edition, (1987: 47–50) to discuss the notion of the general propositional form. But, surprisingly, he mentions neither the thesis that the general propositional form is the sole logical constant, nor the Tractatus’ proof of the unity of language and logic. In Griffin’s (1964) Wittgenstein’s Logical Atomism, the thesis and the issue of the unity of language and logic are not addressed at all. © 2006 The Author
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Here, McGuinness is criticizing the Tractatus on that the picture theory fails to explain how an elementary proposition can be a truthfunction, while the truth-function account presupposes a prior understanding of the essence of an elementary proposition. Peter Winch, in the ‘introduction’ to his edited work Studies in the Philosophy of Wittgenstein, points out that, with respect to the Tractatus, ‘it is vital to our understanding of Wittgenstein to see that the nature of logic is already being inquired into in Wittgenstein’s treatment of the puzzle about the relation between propositions and facts’.3 The application of logic in language to reality requires the structure of an elementary proposition to be logical, which in the case of the Tractatus means truth-functional.4 But it is hard to see how the structure of an elementary proposition could be truth-functional, and Winch thinks the Tractatus does not provide any account of that.5 This is a serious difficulty and ‘one of the main factors in inducing Wittgenstein to move away from the Tractatus notion of elementary propositions’.6 Rush Rhees’ view is rather different from McGuinness’ and Winch’s. He thinks that, in the Tractatus, Wittgenstein has attempted to bring out the fundamental role logic plays in a proposition’s saying something about, or picturing, reality. He writes in various places 2. McGuinness (2002: 65–6). Also, McGuinness (2002: 66) mentions in a footnote that ‘[t]he existence and importance of this problem were first, to my knowledge, pointed out and many directions for its solution (on which I have drawn freely) given by Miss G. E. M. Anscombe in lectures at Oxford (later published . . .)’. But it is not clear which passages McGuinness would be referring to in Anscombe (1971). 3. Winch (1969: 3). Winch continues to write (1969:3–4), ‘[t]his point can perhaps be expressed in the form of another problem: What is the relation between a proposition’s ability to state a fact and its ability to stand in logical relations to other propositions?’ He also thinks, for the Tractatus, ‘. . . unless propositions had logical relations with each other that they would not state fact (i.e. would not be propositions) and unless they stated facts, they would not have logical relations with other propositions’ (1969: 4). 4. Cf. Winch (1969: 2). 5. Winch (1969: 6) also says, ‘. . . an elementary proposition is also said to have a ‘structure’; and it is hard to see how this could be a truth-functional structure . . . [TLP 5] provides us with no account of what we are to understand by ‘logic’ the expression, ‘the logical structure of elementary proposition’ . . . Although Winch is referring to TLP 5 here, it is clear from his ‘introduction’ that he thinks the Tractatus does not provide any such account. 6. Winch (1969: 6). © 2006 The Author
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that the point of the picture theory is to bring out the way that logic is fundamental in connection with empirical propositions or with picturing,7 and thus shows that logic must take care of itself or its own application8 and that understanding a proposition is not anything arbitrary.9 He emphasizes in this connection the importance of a general rule – the law of projection in TLP 4.0141 – in the distinction between sense and nonsense and in showing that logic must take care of itself,10 as well as the thesis that the general form of logical operation is the general form of proposition11 and the fact that sense, as a configuration of objects, must have ‘the complexity which we express with the logical constants.’12 But Rhees does not explain in detail how the Tractatus employs the picture theory to achieve these, and, in particular, how logic is fundamental in connection with picturing or with the construction of propositions. I think Rhees is on the right track already but, unfortunately, he does not explain in detail how the picture theory can account for such intrinsic relation between logic and language.13 In the case of McGuinness and Winch, I will make it clear in this paper that, contrary to what they have thought, the Tractatus does attempt to employ the picture theory, together with the Grundgedanke, to account for the fact that logical constants, or, rather, logical operations that they 7. Rhees mentions this point in his essay ‘Miss Anscombe on the Tractatus’ (1996b: 1–15) and in Rhees (1998: 57–60). For example, he writes in Rhees (1998: 4) and (1998: 9), respectively, that ‘[w]hen Wittgenstein says that propositions are pictures of reality, one thing he wants to bring out is the way in which logic is fundamental in connexion with them’ and that ‘[w]e recognize the relation of logic to empirical propositions when we see these propositions as picturing’. 8. Rhees (1998: 57): ‘The aim of the picture theory is to show that logic must take care of itself; that logic must look after its own application’. 9. Rhees writes in ‘Miss Anscombe on the Tractatus’ (1996b: 8), ‘. . . to say that understanding a proposition might be something arbitrary in that way, would be selfcontradictory’. He also writes in the essay ‘‘Object’ and Identity in the Tractatus’ (1996c: 27), ‘[i]n Tractatus 5: ‘A proposition is a truth function of elementary propositions.’ So the combination of signs in a proposition is not arbitrary’. 10. See Rhees (1998: 8): ‘. . . a picturing of reality is possible because there is a general rule – a rule by which we distinguish between sense and nonsense. There cannot be anything arbitrary in logic, because anything arbitrary would have to be said: and logic (the general rule) is what makes this possible . . . part of the point here is that there must be logic if there are empirical propositions – propositions which we can understand without knowing whether they are true or false’. 11. See Rhees (1998: 2–3). 12. See Rhees (1998: 13). 13. I would like to say that I was first inspired by Rhees’ views and subsequently set myself to tackle the issue. © 2006 The Author
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symbolize, are involved in an elementary proposition’s saying something about reality. Also, the attempt is not something which is plainly incoherent or trivially wrong. It is ingenious. In fact, McGuinness and Winch seem to have misconceived the issue in a similar, if not the same, way. What is crucial here is not, as they have thought, how the combination of names in an elementary proposition could be truth-functional. The Tractatus does not hold any such thing.What the Tractatus does hold, I shall argue, is that logical operations are applied in an elementary proposition’s picturing reality via naming. According to the picture theory, the operation NN (where N is the sole fundamental operation in the Tractarian system), or the existential quantifier, is present in every elementary proposition in an intrinsic manner such that it does not bind propositions together but belongs to the signifying relation between names and objects. This is the key to the Tractatus’ proof of the thesis that the general propositional form is the sole logical constant and hence the unity of language and logic.
2. Drawing a Limit to Language The thesis that the general propositional form is the sole logical constant is crucial to the Tractatus’ aim (TLP, p.3) – to draw a limit to thought by drawing one to the expression of thought, that is, language. Language is the totality of propositions (TLP 4.001). The limits of propositions, which constitute the limit of language,14 however, do not belong to language (TLP 6.43)15 but are fixed by the totality of propositions (TLP 4.51). The constructions out of all elementary propositions via logical (truth-functional) operations give rise to propositions (TLP 5), tautologies and contradictions (TLP 4.45–4.46) and, what is more, this fixes the limits of all propositions (TLP 4.51). Since the limit of language does not belong to language, it can only be constituted by tautologies, which by TLP 6.1 (or what 14. The Tractatus sometimes refers to drawing ‘a limit’ to language, for example, in TLP, p.3, and sometimes to setting ‘limits’, for example, in TLP 4.113–4.116. I take the former as referring to the limits of language collectively. 15. TLP 6.43 says that ‘[i]f the good or bad exercise of the will does alter the world, it can alter only the limits of the world, not the facts – not what can be expressed by means of language’. This implies that the limits of language do not belong to language. © 2006 The Author
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may be called ‘the truth-functionality of logical propositions’) are all logical propositions, and contradictions. In fact, the Tractatus sees tautologies and contradictions not as propositions (and thus not belonging to language) but as the limiting cases, or, simply, the limits, of propositions (TLP 4.466 and 5.143). Those lie within the limit of language are propositions, those lie on it tautologies and contradictions, and those lie outside it nonsense.16 Tautologies and contradictions constitute the limit of language and yet are fixed by the totality of propositions. It might appear that one way to draw a limit to language is to give the totality of propositions (and thus fix their limits). But the Tractatus does not, and cannot, give such totality because not even the totality of elementary propositions can be given (TLP 5.55–5.551). What it does is to give a description of the general propositional form (TLP 4.5), which is the equivalent of the general rule referred to in TLP 4.041. The general rule does not give but determines the propositions and also their limits. It is the same general rule which generates propositions, as well as tautologies and contradictions. The latter are the limiting cases of the application of the general rule. Propositions, tautologies and contradictions all satisfy the general propositional form, in a way which will be explained later. Language and logic are unified via the general rule – they are of the same nature. To draw a limit to language is to give prominence to the general rule by establishing the logical syntax of a particular language. (The fact that the Tractatus understands logical syntax in this way can be seen from TLP 3.344 and 6.124.TLP 6.124 says that ‘[i]f we know the logical syntax of any sign-language, then we have already been given all the propositions of logic’. Logical syntax gives prominence to the general rule governing the formation of logical propositions. TLP 3.344 says that ‘[w]hat signifies in a symbol is what is common to all symbols that the rules of logical syntax allow us to substitute for it’. Then logical syntax also gives prominence to the general rule of language which governs the formation of symbols capable of signifying. One may say, for the Trac-
16. Tautologies and contradictions are unsubstantial point in the centre and outer limit, respectively (TLP 5.143; also see TLP 4.466). Elementary propositions are propositions and thus cannot constitute the limits of language. Pears’ taking elementary propositions as the inner limits of language in his book Wittgenstein (1997: 67–8) is incorrect. © 2006 The Author
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tatus, language and logic are unified via logical syntax.) A limit can be drawn to language because grasping the general rule fixes not only the totality of propositions but also their limits – logical propositions (tautologies) and contradictions. Drawing a limit to language is logical, and its possibility demands the unity of language and logic. It is the crucial point of the Tractarian idea of drawing a limit to language that the general rule, or the general propositional form, cannot just determine propositions but also tautologies (logical propositions) and contradictions. Given the truth-functionality of tautologies and contradictions, such unity of language and logic can be expressed as that the general form of proposition is the general form of the combinations (applications) of logical operations, and vice versa. This is the thesis that the general propositional form is the sole logical constant. It amounts to saying that language and logic are unified via the general propositional form or the general form of logical operation. This is the unity of language and logic. (This also explains why the thesis does not only demand that a non-elementary proposition must satisfy the general form of logical operation but also that an elementary proposition must satisfy it as well. For if the general form of elementary propositions is different from the general form of logical operation, then there would be two different general rules such that one governs the formation of elementary propositions, while another the applications of logical operations. In that case, language and logic could not be unified.) Wittgenstein is true to his proclaimed aim of the Tractatus and so does attempt to argue for the thesis and hence the unity of language and logic.The Grundgedanke in TLP 4.0312 and the picture theory are crucial to his argument. The picture theory, as I shall explain, does not only account for the nature of propositions but also the unity of language and logic. An important task of the Tractatus is then the difficult one of explaining how an elementary proposition, which is an immediate combination of names, can satisfy the general form of logical operation. With this remark, I shall now explain the basic structure of the argument for the thesis in the Tractatus.
3. The Grundgedanke and the Picture Theory The thesis that the general propositional form is the sole logical constant actually consists of two parts: that there is the general propo© 2006 The Author
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sitional form and that the general propositional form is the sole logical constant. The Tractatus’ argument for the thesis employs the picture theory17 in TLP 2.1–2.225 and 4.011–4.016, the Grundgedanke (or the thesis in TLP 4.0312 that logical constants are not representatives or do not denote), the existence of the sole fundamental operation N introduced in TLP 5.5, which implies the unity of logical operation, and the analyticity thesis in TLP 5 (‘A proposition is a truthfunction of elementary propositions’). Amongst those theses, what play the crucial roles in the proof are the picture theory and the Grundgedanke. The Grundgedanke and the unity of logical operation, in a way to be explained later, imply the existence of the general propositional form. The key of the Tractatus’ proof of the major claim that the general propositional form is the sole logical constant is already contained in TLP 4.0213: The possibility of propositions is based on the principle that objects have signs as their representatives. My fundamental idea [Grundgedanke] is that the ‘logical constants’ are not representatives; that there can be no representatives of the logic of facts.
Besides the Grundgedanke, the picture theory is also referred to in TLP 4.0213 indirectly because what accounts for the dependence of the possibility of a proposition on the principle of naming is exactly the picture theory. To see that TLP 4.0213 is the key, note that the Grundgedanke implies that a logical operation is independent of the semantic content of any symbol, and this suggests that an operation is intrinsic to every elementary proposition. TLP 4.0312 also says that the possibility of a proposition is based on the possibility of naming. This further suggests that an operation is intrinsic to naming. In fact, as one will see later, the Tractatus holds that an operation is intrinsic to naming in such a manner that the general rule of logical operation is also the general rule of (the formation of ) propositions, that is, the general rule of language. The sole logical constant, or the general form of logical operation, is then the general propositional form. Of course, many details still need to be worked 17. The picture theory, as I see it, consists of an account of the notion of a picture and how a picture depicts reality, mainly in TLP 2.1–2.225 and 4.011–4.016, the thesis in TLP 4.01 that a proposition is a picture of reality and the proof of the thesis in TLP 4.02–4.021. In this paper, only those in TLP 2.1–2.225 and 4.011–4.016 will be considered, and they are taken as what constitute the picture theory. © 2006 The Author
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out, and I shall explain how the Tractatus works them out later. The Tractatus employs the picture theory, the Grundgedanke, the existence of N and the analyticity thesis to prove the thesis that the general propositional form is the sole logical constant which, together with the truth-functionality of logical necessity, implies the unity of language and logic. I already explained how the Tractatus proves the Grundgedanke, the existence of N, the analyticity thesis and the truth-functionality of logical necessity elsewhere.18 With the exception of the issue of the existence of N, I shall not repeat my explanation of the proofs here, nor shall I comment on them. One may simply regard these four theses as what are presupposed in this paper. The picture theory, however, will be explained in detail. In what follows, I shall explain the Tractatus’ proof of the thesis that the general propositional form is the sole logical constant. I shall begin with the clarification of the notions of the sole logical constant and the general propositional form in the next two sections.
4. N and the Unity of Logical Operation What is the sole logical constant? The Tractarian system of logic, as it is well known, has N as its sole fundamental operation. N is introduced in TLP 5.5 as (-----T)(ξ, . . .), where what is inside the righthand pair of brackets represents an unordered collection of propositions, and the row in the left-hand pair of brackets indicates that in the last column of the truth-table expression all but the last one are ‘F’ (TLP 4.442 and 5.5). It is also written as ‘N(ξ )’, where ‘ξ’ is a variable whose values are terms of the bracketed expression and the bar over the variable indicates that it is the representative of all its values in the brackets’ (TLP 5.501).19 The sole logical constant, however, does not symbolize N. In TLP 5.472, the sole logical constant is also called ‘the one and only general primitive sign in logic’. It is then advisable to see what a general primitive logical sign 18. In Cheung (1999), I explain and comment on the two proofs of the Grundgedanke in the Tractatus. I also explain in Cheung (2000) how N functions as the sole fundamental operation in the Tractarian system, and in Cheung (2004) how the Tractatus derives the analyticity thesis and the truth-functionality of logical necessity from the thesis that a proposition shows its sense. 19. I have dealt with the issue of the expressive capability of N elsewhere. For details, see Cheung (2000). © 2006 The Author
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is in the first place. The Tractatus holds that ‘the real general primitive signs are not ‘p ∨ q’, ‘( ∃x).fx’, etc. but the most general form of their combinations’ (TLP 5.46). A general primitive sign is then the most general form of the combinations of a logical sign. If a system has a single fundamental operation, then all operations are unified via the general form of the combinations of the sole fundamental operation – the general form of logical operation, as described in TLP 6.01, or what is in common to all operations. In that case, there is the one and only one general primitive logical sign – the sole logical constant. The sole logical constant is not really a logical constant but what symbolizes the most general form of logical operation.Thus, with respect to the Tractarian system, instead of symbolizing the sole fundamental operation N, the sole logical constant symbolizes the general form of the combinations of N. The sole logical constant, or the general form of logical operation, is given by the general term of a formal series20 [ η, ξ , N(ξ )], as in TLP 6.01. (‘[ η, ξ , N(ξ )]’ symbolizes the form of the result of a certain number of successive applications of N to a subset (ξ ) of the base ( η).) The thesis that the general propositional form is the sole logical constant can then be formulated as follows: The general propositional form is [ η, ξ , N( ξ )]. The above also shows that the Tractatus upholds the unity of logical operation. To see this, note that, according to the analyticity thesis, a proposition can be analyzed into a truth-function of elementary propositions. An elementary proposition is an immediate combination of names, and names are referential primitive symbols (TLP 3.2–3.203 and 3.206). An immediate combination of the meanings of names, or objects, is called ‘a state of affairs’ (TLP 2.01 and 2.03). The determinate way that objects are connected to one another in a state of affairs is the structure, whose possibility is the form, of the state of affairs (TLP 2.032–2.033). In general, the Tractatus seems to call a determinate way of combination ‘a structure’, and a possibility of structure, or a combinatorial possibility, ‘a form’. For instance, an object has a form which is the possibility of its occurring in states of affairs (TLP 2.0141). A possible state of affairs also has a form and, if exists, has a structure. A fact is the existence
20. For a discussion of the Tractarian notion of formal series, see Cheung (2000: 251–4). © 2006 The Author
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of states of affairs (TLP 2) and thus has a form and a structure. A propositional sign is a fact, and its form is the possibility of its constituent signs’ standing in the determinate relation to one another (TLP 3.14). A name also has a form which is its possibility of occurring in elementary propositions. The description ‘[ η, ξ , N(ξ )]’ then gives prominence to a form or a combinatorial possibility. Moreover, a form, as a combinatorial possibility, determines a rule. Thus, the sole logical constant does not only symbolize the general form of logical operation but also a rule which may be called ‘the general rule of logical operation’. This explains why, in TLP 5.4s which, amongst other things, aim to symbolize the sole logical constant,Wittgenstein writes that ‘. . . it is not a question of a number of primitive ideas that have to be signified, but rather of the expression of a rule’ (TLP 5.476). The sole logical constant symbolizes both the general form of logical operation and the general rule of an operation. Hence, the Tractatus upholds and argues for the unity of logical operation. The existence of the sole fundamental operation N enables the unification of logical operations via the general form of the combinations of N, that is, via the sole logical constant.
5. The General Propositional Form Let me now turn to the notion of the general propositional form. Amongst various passages in the Tractatus,TLP 4.5 tells us most about the general propositional form: It now seems possible to give the most general propositional form: that is, to give a description of the propositions of any signlanguage whatsoever in such a way that every possible sense can be expressed by a symbol satisfying the description, and every symbol satisfying the description can express a sense, provided that the meanings of the names are suitably chosen. It is clear that only what is essential to the most general propositional form may be included in its description – for otherwise it would not be the most general form. The existence of a general propositional form is proved by the fact that there cannot be a proposition whose form could not have been foreseen (i.e. constructed). The general propositional form of a proposition is: This is how things are [Es verhält sich so und so].
At least four important points can be gathered from this passage.The first point is that there is a proof of the existence of the general © 2006 The Author
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propositional form in the Tractatus. I shall explain the proof in the next section.The second point is that the general propositional form governs the formation of symbols capable of expressing sense. The general propositional form then completely determines the general rule which produces propositions, that is, the general rule of language. Since a form, as a combinatorial possibility, determines a rule, this is not a surprising claim. If there is the general propositional form, that is, if all propositional forms share a common general form, then there is also the general rule of language. In other words, the general propositional form characterizes the general rule of language completely. The general rule of language is the general rule mentioned in TLP 4.014–4.0141 and is, as already pointed out before, also presented by logical syntax. The third point is that the general propositional form, and thus logical syntax, is independent of the specific content of the meanings correlated with names (TLP 3.33). The fourth point, which is of immediate relevance here, is that, besides being what is shared by all propositional forms, the general propositional form can be given by a description. In the Tractatus, descriptions can be many things. A description of a complex can be right or wrong (TLP 3.24) and thus is a proposition. However, TLP 5.501 mentions three different kinds of descriptions of the terms of the bracketed expression in N(ξ ), none of which is a proposition. I suggest taking the pragmatic move in regarding a description as an expression employed to give prominence to what it is a description of. Now one can gather at least four descriptions of the general propositional form from the Tractatus. (1) The general propositional form is: Es verhält sich so und so (TLP 4.5). (2) The general propositional form is a variable (TLP 4.53). (3) The general propositional form is [ p, ξ , N(ξ )] (TLP 6). (4) The general propositional form is given by the expression ‘[ η, ξ , N(ξ )]’ or, simply, is [ η, ξ , N(ξ )] (TLP 5.47–5.472 and 6–6.01). I shall explain what (1) and (2) mean and how the Tractatus proves them later. (3) follows from the analyticity thesis directly. The focus here is, of course, on (4) because it is another formulation of the thesis that the general propositional form is the sole logical constant. Note that ‘[ p, ξ , N(ξ )]’ in (3) does not bring out the © 2006 The Author
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(formal) content of the general propositional form completely because p does not characterize the general form of elementary proposition. It is not a desirable description, especially when the concern here is the relation between language and logic, or between an elementary proposition and logical operations, that it surely cannot bring forth. However, if there is a complete description of the general form of elementary proposition, then the replacement of p in [ p, ξ , N(ξ )] by such a description would give rise to a complete description of the general propositional form. (The latter, as one will see, is actually [ η, ξ , N(ξ )].) This suggests that the first step towards explaining how the Tractatus proves the thesis that the general propositional form is the sole logical constant, or (4), is to find out from the Tractatus a complete description of the general form of elementary proposition. 6. The Grundgedanke, Logical Operation and the General Propositional Form The Tractatus attempts to prove that there is the general propositional form in TLP 4.5: . . . The existence of a general propositional form is proved by the fact that there cannot be a proposition whose form could not have been foreseen (i.e. constructed).
Presumably, the argument here is that if every proposition can be constructed according to a unified plan, then there is a general rule producing propositions and hence there is the general propositional form. Is there such a unified plan? Because of the analyticity thesis, the case of non-elementary propositions would be straightforward, once it is proven that there is a unified plan for the construction of elementary propositions. The problem is then to prove that there is such a unified plan for elementary propositions, or that there is the general form of elementary proposition. The Tractatus does not state the proof explicitly. However, as I shall explain now, in fact, the Grundgedanke and the unity of logical operation via the sole fundamental operation N entail that there is the general form of elementary proposition. The Grundgedanke implies that a logical operation, or what a logical constant symbolizes, is independent of the semantic content of any symbol. This, in turn, implies that: © 2006 The Author
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(5) An operation is completely determined by what is common to all its possible bases – the general form of its possible bases – and the specific difference between the form of its result and the form of its base. The Tractatus does not state (5). But one can see from the list of characteristics of an operation gathered from the text that Wittgenstein does hold it. Here is the list: [i] Propositions and only propositions can be the base of an operation (TLP 5.24 and 5.515). [ii] The result of an operation must share the constituent forms of its base (TLP 5.24). [iii] An operation cannot characterize any propositional form. It can only characterize a specific difference between propositional forms, that is, between the form of its result and the constituent forms of its base (TLP 5.24–5.241 and 5.254).21 [iv] Propositions can only occur in other propositions (in a nontrivial manner) as the constituents of the bases of operations (TLP 5.54). Note that [i]–[iv] imply that: (6) A proposition can be expressed as the result of an application of an operation to a finite set of other propositions if and only if, first, it shares the forms of the other propositions, and, second, there is a specific difference between its form and the forms of the other propositions. This amounts to saying that an operation is completely determined by the general form of all its possible bases and the specific difference between the form of its result and the form of its base, that is, (5). It is then reasonable to think that the Tractatus holds (5). It should be noted that (5) does not assert that there is something common to all the possible bases of an operation. What it does assert is that, first, if there is nothing common to all the possible bases of an operation, then the operation is completely determined by the specific formal difference between its result and its base. Second, if there is no specific formal difference between the result and the base of an operation, then the operation is completely determined by what is 21. Pears and McGuinness translate ‘Die Operation kennzeichnet keine Form . . .’ (TLP 5.241) as ‘An operation is not a mark of a form . . .’, while Ogden as ‘The operation does not characterize a form’. I follow Ogden here in rendering ‘kennzeichnen’ as ‘characterize’. © 2006 The Author
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common to all its possible bases. In that case, the existence of such an operation implies the existence of the general form of all its possible bases. Now the unity of logical operation via the sole fundamental operation N, together with (5), implies this: (7) Every operation is completely determined by what is common to all the possible bases of N and the specific formal difference between its result and its bases. For the Tractatus, N can only have elementary propositions and their truth-functions to be the constituents of its possible bases. It follows that: (8) Every operation is completely determined by what is common to all elementary propositions, that is, the general form of elementary proposition, and the specific formal difference between its result and its base. Similar to the case of (5), (8) does not assert the existence of the general form of elementary proposition.What it does assert, amongst other things, is that if there is no specific formal difference between the result and the base of an operation, then there is the general form of elementary proposition and the operation is completely determined by the general form of elementary proposition. It is not difficult to see that NN is such an operation. To see this, note that: (9) For any η, N[NN( η)] = N( η). The proof of (9) is very simple. Since N( η) contains no free variables, N[NN( η)] = NN[N( η)] = N( η). Hence, N[NN( η)] = N( η). It follows from (9) that (10) From the point of view of being the base of an operation, NN( η) can be seen as being equivalent to ( η). This means that there is no specific formal difference between the result and the base of NN. Two important conclusions can be drawn here. The first one is that, by (8), there is the general form of elementary proposition. Of course, there is NN because there is N. So © 2006 The Author
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the existence of the sole fundamental operation N guarantees the existence of the general form of elementary proposition. Because there is the general form of elementary proposition, given the analyticity thesis, there must be the general propositional form. So the existence of N, or, rather, that of NN, provides a unified plan according to which a proposition can be constructed and thus there cannot be a proposition whose form could not have been foreseen or constructed. With this, I have explained how a proof of the existence of a general propositional form can be constructed from the Grundgedanke and the unity of logical operation via N. The second conclusion is that, since there is no specific formal difference between the result and the base of NN, by (8), NN is completely determined by the general form of elementary proposition. This means that the operation NN is intrinsic to every elementary proposition. This is an important step towards the proof of the claim that the general propositional form is the sole logical constant. The key is to explain the intrinsic relation between the application of an operation and an elementary proposition’s saying something about reality. The Tractatus, as I shall explain later, employs the picture theory to completely characterize the nature of an elementary proposition (and thus the nature of the general propositional form) in terms of NN, and then to explain why the general propositional form is [ η, ξ , N(ξ )], or (4). The crucial idea of the picture theory is also that NN is intrinsic to every elementary proposition and, in fact, to naming.
7. The Picture Theory: A Proposition is a Picture of Reality Having explained why Wittgenstein thinks there is the general propositional form, let me now turn to the issue of a complete characterization of the general propositional form. I shall begin with the special case of elementary propositions. Given the analyticity thesis, a proposition can be analyzed into a truth-function of elementary propositions. An elementary proposition, as already mentioned, is an immediate combination of names. Names are referential primitive symbols, whose meanings (Bedeutungen) are objects. An immediate combination of objects (meanings of names) is a state of affairs. It is the major claim of the picture theory that ‘[a] proposition is a picture of reality’ (TLP 4.01). ‘A picture depicts reality by representing a © 2006 The Author
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possibility of existence and non-existence of a state of affairs’ (TLP 2.201). In particular, an elementary proposition depicts reality, or presents the existence of a state of affairs, by representing the possibility of the state of affairs (TLP 4.211). But how does the picture theory account for an elementary proposition’s depicting reality? TLP 4.0312 states that the possibility of propositions is based on the principle of naming, and this indicates where the essentials of the account are to be found. In fact, as I am going to explain, the picture theory accounts for a picture’s depicting reality via the element-objectcorrelation or, in the special case of an elementary proposition, via naming. (This of course should not be taken to be implying that naming is independent of picturing because Wittgenstein also emphasizes the dependence of naming on the relevant propositional context in TLP 3.3. What is at stake here is not whether naming is independent of picturing, which is not, but how picturing is constituted by naming.) A picture, according to the Tractatus, is a fact (TLP 2.141), and thus has a form and a structure. Indeed, the forms of its elements constitute its form. A fact is made into a picture when its form becomes a pictorial form: The fact that the elements of a picture are related to one another in a determinate way represent that things are related to one another in the same way. Let me call this connexion of its elements the structure of the picture, and let us call the possibility of this structure the pictorial form of the picture. (TLP 2.15)
In a picture, the fact that its elements are related to one another presents that objects22 are related to one another in the same determinate way. That is, the form of a fact becomes a pictorial form when the fact presents the existence of a state of affairs sharing the same form. How is this possible? By correlating objects with the constituent elements, that is, by establishing a pictorial relationship, in a certain manner: That is how a picture is attached to reality; it reaches right out to it. It is laid against reality like a measure.
22. It is clear from the content of TLP 2.151–2.1515 that ‘things’ here can be taken as objects. © 2006 The Author
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Only the end-points of the graduating lines actually touch the object that is to be measured. So a picture, conceived in this way, also includes the pictorial relationship, which makes it into a picture. The pictorial relationship consists of the correlations of the picture’s elements with things. These correlations are, as it were, the feelers of the picture’s elements, with which the picture touches reality. (TLP 2.1511–2.1515)
Here, a picture is seen as a ‘measure’ laid against reality in such a manner that ‘[o]nly the end-points of the graduating lines actually touch the object that is to be measured’. The talk of ‘measure’ and ‘the graduating lines’ here is to emphasize the fact that the form of a picture, as a measure, presents this constraint: [*] Only objects having the same form as that of a constituent element of a picture can be correlated with the element. The constraint [*] is set up by the forms of the elements of a picture. It ensures that the objects correlated to be able to produce a state of affairs of the same form as the picture. In other words, it guarantees the possibility of a picture’s representing a state of affairs sharing its form. One may also say, the form of a picture, which is constituted by the forms of its elements, acts as a constraint ensuring that only a state of affairs of the same form can be represented. The constraint [*] is actually the only constraint that the correlation of objects with elements, or the establishment of a pictorial relationship, must be subject to. The fact that the Tractatus does hold this is supported by TLP 3.315: If we turn a constituent of a proposition into a variable, there is a class of propositions all of which are values of the resulting variable proposition. In general, this class too will be dependent on the meaning that our arbitrary conventions have given to parts of the original proposition. But if all the signs in it that have arbitrarily determined meanings are turned into variables, we shall still get a class of this kind. This one, however, is not dependent on any convention, but solely on the nature of the proposition. It corresponds to a logical form – a logical prototype.
The main point here is clearly applicable to the case of a picture, although it refers to the particular case of a proposition. Note that when talking about turning all propositional constituents into vari© 2006 The Author
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ables, the meanings of signs are said to be ‘arbitrarily determined’. The point is, of course, not that an element of a picture can correlate with any object without being subject to any constraint, as the form of the picture is also emphasized here. Rather, the point is that, given the constraint [*] that only objects having the same form as the form of an element can be correlated with the element, an element can correlate with any object. One may say, the form of an element, or the constraint [*], sorts out objects sharing its form as the candidates, and the object-element-correlation is simply arbitrarily correlating with the element an object from the candidates. Therefore, the establishment of a pictorial relationship consists in picking out objects from those sorted out by the forms of the elements of the relevant picture such that the form of the picture is instantiated. Depicting is the instantiation of the form of a picture, and the instantiation consists in arbitrarily correlating with the elements of the picture objects from the candidates determined by the constraint [*] set up by the forms of the constituent elements.
8. The Picture Theory: Naming and the Existential Quantifier Naming is conferring semantic content to a name.This, as conceived by the Tractatus, consists in correlating an object with a name as its meaning in the nexus of an elementary proposition (TLP 3.3) or, in general, in the context of depicting. Depicting is the presentation of the existence of a state of affairs by means of the instantiation of the form of a picture, and the instantiation consists in arbitrarily correlating with the constituent elements objects from those sorted out by the form of the picture. Thus, naming is the instantiation of the form of a name, and the instantiation consists in arbitrarily picking out an object as the meaning of the name from those objects sorted out by the form of the name. It follows as a corollary that naming involves the application of the existential quantifier or NN! More precisely, naming involves the application of the existential quantifier or NN to pick out an unspecified object, as the meaning of the relevant name, from those sorted out by the form of the name. Another formulation of the corollary is that the existential quantifier is intrinsic to naming. Since a propositional variable shows a form and its values signify those objects sorted out by the form (TLP 4.127), the corollary can also be put in the following way: Naming involves the application of the exis© 2006 The Author
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tential quantifier to a propositional variable such that objects are arbitrarily correlated with names as their meanings from those objects signified by the values of the propositional variable. In fact, naming involves both the application of the existential quantifier and the stipulation of a constant as the name of the object picked out by the existential quantifier. In naming, while an unspecified object is arbitrarily picked out by the existential quantifier from those objects sorted out by the relevant propositional variable, a constant is at the same time given to the object as its name.The latter is symbolized by putting the identity sign between a name and a variable name, where the latter symbolizes the pseudo-concept object (TLP 4.1272). Let me explain the above by an example. Consider, without loss of generality, the elementary proposition ‘fa’, where ‘f ’ and ‘a’ are names. ‘fa’, as a picture, asserts the existence of a state of affairs via the instantiation of its form by arbitrarily correlating with ‘f ’ and ‘a’ objects from those sorted out by the form of the picture. Employing the propositional variable φx obtained by turning the names ‘f ’ and ‘a’ in ‘fa’ into variables, the naming of ‘f ’ and ‘a’ in ‘fa’ can be seen as the application of the existential quantifier or NN to φx such that objects are arbitrarily correlated with ‘f ’ and ‘a’ as their meanings from those signified by the values of ‘φx’, respectively. (Note that a value of φx, say, ‘fa’ signifies the objects f and a.) The essentials involved here can be given prominence by ‘(∃φ, x).φx.φ = f.x = a’, which is an equivalent formulation of ‘fa’. The presence of the existential quantifier and the identity sign symbolize the arbitrary picking of the unspecified objects f and a by means of the form symbolized by φx and the stipulation of constants (names), that is, ‘f ’ and ‘a’ to the objects f and a, respectively. This explains why ‘fa’ is equivalent to ‘(∃φ, x).φx.φ = f.x = a’. Of course, if only the naming of ‘a’ in ‘fa’ is focused on, the nature of naming by ‘a’ in ‘fa’ can be given prominence in the formulation ‘(∃x).fx.x = a’, just as the Tractatus does in TLP 5.47: . . . An elementary proposition really contains all logical operations in itself. For ‘fa’ says the same thing as
‘ ( ∃x).fx.x = a’. Wherever there is compositeness, argument and function are present, and where these are present, we already have all the logical constants . . . © 2006 The Author
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Another piece of textual evidence is TLP 5.441, according to which the vanishing of apparent logical constants in propositions also occurs ‘in the case of ‘( ∃x).fx.x = a’, which says the same as ‘fa’’. This, of course, does not imply that ‘fa’ does not involve the logical operations symbolized by those logical constants. But, just as TLP 5.47 attempts to say, although the existential quantifier, as the sign of an operation, in ‘(∃x).fx.x = a’ is dispensable (as the proposition can be formulated as ‘fa’), the existential quantifier, as an operation, is still contained in ‘fa’. How can this be the case? I think the best explanation is that the application of the existential quantifier, as an operation, belongs to the naming of ‘a’ in ‘fa’ in the way described above. It is in this manner that the existential quantifier is intrinsic to ‘fa’. The existential quantifier in an elementary proposition does not bind propositions together but belongs to the signifying relation between names and objects. With this, I have explained how the picture theory accounts for the insight in TLP 4.0312 that the possibility of propositions is based on the principle of naming, and how the existential quantifier is intrinsic to naming and thus to every elementary proposition.
9. The General Form of Elementary Proposition is NN( η) I am now going to argue that, and explain how, the picture theory demonstrates that the general form of elementary proposition is given by the expression ‘NN( η)’, or that the general rule of elementary proposition is NN. The point of departure of a complete characterization of the general form of elementary proposition is, for the Tractatus, the form of reality. The Tractatus talks about the form of reality in TLP 2.18. What any picture, of whatever form, must have in common with reality, in order to be able to depict it – correctly or incorrectly – in any way at all, is logical form, i.e. the form of reality.
Since an elementary proposition asserts the existence of a state of affairs (TLP 4.21), elementary propositions are what establish picturing relation with reality directly. The form of reality is what is common to, shared by and intrinsic to all elementary propositions and reality. But this by itself does not imply that there is the form of reality. The Tractatus, as already explained, argues for the existence of the general propositional form and, in particular, the general form © 2006 The Author
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of elementary proposition. For the Tractatus, the existence of the general form of elementary proposition proves that there is the form of reality, but not vice versa.The form of reality, however, is the point of departure of a complete characterization of the general form of elementary proposition. For the form of reality is what is the most general concerning reality. Consider, without loss of generality, ‘fa’, or its equivalent ‘(∃φ, x).φx.φ = f.x = a’, again. The proposition ‘(∃φ, x).φx’, or ‘NN( φx)’, is in fact a description of the general form of a value of φx, that is, a description of what is common to all values of φx. This can be seen in two ways. Recall that an important point has been gathered from TLP 4.5 according to which the general propositional form is taken as something independent of the specific content of meanings correlated with names. The first way is then to ignore the specific names, or, so to speak, the particularity of the naming relations, symbolized by ‘φ = f ’ and ‘x = a’ in ‘(∃φ, x).φx.φ = f.x = a’. In other words, what ‘φ = f ’ and ‘x = a’ in ‘(∃φ, x).φx.φ = f.x = a’ symbolize are irrelevant to the general form of a value of the propositional variable φx, of which ‘fa’ is one of the values. The second way is by seeing that any value of φx entails ‘(∃φ, x).φx’. That is, if ‘fa’, ‘gb’, . . . etc., are values of φx, then fa ⊃ (∃φ, x).φx, gb ⊃ (∃φ, x).φx, . . . etc. In either way, the proposition ‘(∃φ, x).φx’ is a description of the general form of a value of φx. Since ‘NN( φx)’ is equivalent to ‘(∃φ, x).φx’, ‘NN( φx)’ is a description of a value of φx. Moreover, as explained in detail in TLP 3.314–3.315, especially in TLP 3.315 which was quoted in Section 7, a propositional variable like φx is a variable giving prominence to a combinatorial possibility, that is, to a specific logical form. Recall that, according to the Tractatus, there is the form of reality common to all specific logical forms, and that the form of reality is what is the most general concerning reality.To symbolize the general form of reality, a single variable, say, η, alone is enough. Hence, the proposition ‘NN( η)’ is a complete description of the general form of elementary proposition. One may say, the general form of elementary proposition is NN( η). NN is then the general rule of (the formation of) an elementary proposition. I have explained in Section 6 how the thesis that NN is intrinsic to every elementary proposition follows from the Grundgedanke as a corollary. The inference from the Grundgedanke, however, does not provide an account of how NN is intrinsic to every elementary © 2006 The Author
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proposition. This is probably one of the reasons why the Tractatus does not seem to take the latter as a corollary to the Grundgedanke, but, rather, as a corollary to the picture theory. In fact, as it is now proven, the Tractatus holds the following stronger thesis: (11) The operation NN is intrinsic to naming (against the background of picturing) and thus to every elementary proposition. Unlike the Grundgedanke, the picture theory also accounts for the essential role that NN plays in naming and thus in an elementary proposition’s depicting reality. The general form of elementary proposition is NN( η). NN, or the existential quantifier, is intrinsic to naming, that is, (11), and thus to every elementary proposition. The, so to speak, ‘containing’ of NN in an elementary proposition is internal, as the way that ‘internal’ is used in TLP 4.123, or as what I mean by ‘intrinsic’, or as that without which the elementary proposition could not have been an elementary proposition in the first place. This explains how, and in what way, NN is intrinsic to every elementary proposition. 10. [ η, ξ , N(ξ )], ‘Es Verhält Sich So und So’ and the General Propositional Form It is now easy to see how the Tractatus proves (1), (2) and (4), or that the general propositional form is: Es verhält sich so und so, that the general propositional form is a variable, and that the general propositional form is [ η, ξ , N(ξ )] – a formulation of the thesis that the general propositional form is the sole logical constant, respectively. Let me begin with (1), the key is the picture theory or what is exemplified in the case of the equivalents ‘fa’ and ‘(∃φ,x).φx.φ = f.x = a’. ‘(∃φ, x).φx’, or ‘NN( φx)’, as already explained, is the general form of a value of φx. To say what the Tractatus would consider ineffable, ‘(∃φ, x).φx’ describes how one thing is connected (verhälten) to another thing in a determinate way whose possibility is shown in φx. Another way of giving prominence to what ‘NN( φx)’ does is then ‘Es verhält sich so und so als φx’, which is another description of the general form of a value of φx. Moreover, ‘NN( η)’, as already explained, is a description of the general form of elementary propo© 2006 The Author
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sition. Similarly, another formulation of ‘NN( η)’ is ‘Es verhält sich so und so als η’, which is a complete description of the general form of elementary proposition. Note that the analyticity thesis entails that a proposition directs how reality is to be depicted by means of its constituent elementary propositions and the relevant logical operations. In the case of a non-elementary proposition, what directs its depiction of reality consists in two things, namely, the general form of elementary proposition and the general form of logical operation. The variable η in ‘Es verhält sich so und so als η’ cannot be left out when the general form of elementary proposition is what is to be described. No variable other than ‘so und so’ is needed, however, if the general propositional form, or the general form of all propositions, is to be described. For what is at stake here is the most general logical form and not just the logico-propositional form of an elementary proposition, nor just the logical form of reality. As a result, this explains why the Tractatus takes ‘Es verhält sich so und so’ as a description of the general propositional form. To understand the thesis that the general propositional form is a variable, that is, (2), the notions of constant and variable need to be considered first. For the Tractatus, an expression is ‘. . . presented by means of the general form of the propositions that it characterizes. In fact, in this form the expression will be constant and everything else variable. Thus an expression is presented by means of a variable whose values are the propositions that contain the expression’ (TLP 3.312–3.313). For example, the expression (name) ‘a’ can be presented by the propositional form φa whose values are propositions in which ‘a’ occurs. Here, φ is a variable and ‘a’ a constant. The general propositional form characterizes all propositions. Hence, a complete expression of the general propositional form contains no constant, as it is an expression of what is the most general. For instance, the expressions of the general proposition form in the descriptions ‘Es verhält sich so und so’ and ‘[ η, ξ , N(ξ )]’ contain no constant. In the case of the description ‘[ η, ξ , N(ξ )]’, ‘N’ is not a constant because, according to the Grundgedanke, ‘N’ does not denote. In fact, the sign ‘N’ is dispensable.23 An expression of the general propositional form is, one may say, a limiting case of expressions. It contains no constant and thus can be identified with a vari23. See Cheung (1999: 402–7).
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able. This explains why the Tractatus says that the general propositional form is a variable. It is now not difficult to understand (4), or the thesis that the general propositional form is [ η, ξ , N(ξ )], and why the Tractatus holds (4). Recall that, as pointed out in Section 5, ‘[ p, ξ , N(ξ )]’ in (3) is an incomplete description of the general propositional form because the general form of elementary proposition has not been fully characterized. It has now been proven that ‘NN( η)’ is a complete description of the general form of elementary proposition. And one should not overlook the fact that ‘NN( η)’ is a special case of ‘[ η, ξ , N(ξ )]’. Hence, the general form of proposition – elementary or non-elementary – is given by the description ‘[ η, ξ , N(ξ )]’. As a result,the general propositional form is [ η, ξ , N(ξ )], which is also the general form of logical operation. The general rule of logical operation, symbolized by [ η, ξ , N(ξ )] or [ ξ , N(ξ )]’ (TLP 6.01), is actually the general rule of language, and vice versa. This explains how the Tractatus understands [ η, ξ , N(ξ )] as a complete description of the general propositional form, as well as what the Tractatus holds by claiming that the general propositional form is the sole logical constant. 11. Language and Logic I have now explained how, according to the Tractatus, an elementary proposition satisfies the general propositional form [ η, ξ , N(ξ )], which is also the general form of logical operation. This, together with the analyticity thesis, also explains how a proposition satisfies the general propositional form. It remains to explain how the propositions of logic (and contradictions) satisfy the general propositional form. The Tractatus denies that logical propositions are propositions. Then what needs to be answered is really this: How can logical propositions satisfy the general propositional form and yet they do not belong to language (propositions)? Or, how can logical propositions be products of applying the general rule of language, which is also the general rule of logical operation, and yet fail to express sense? Let me begin by explaining how the Tractatus excludes logical propositions or tautologies from propositions or pictures of reality. Consider TLP 4.462: © 2006 The Author
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Tautologies and contradictions are not pictures of reality. They do not represent any possible situations. For the former admit all possible situations, and latter none. In a tautology the conditions of agreement with the world – the representational relations – cancel one another, so that it does not stand in any representational relation to reality.
The crucial point here is that because a tautology admits all possible situations, the representational relations cancel one another. This means that no meaning can be suitably chosen for its constituent signs or, so to speak, ‘names’, even though before the relevant propositions are combined to yield the tautology the constituent names of the propositions have been given meanings in other propositional contexts.Therefore, a tautology cannot express a sense, that is, cannot picture any state of affair, and thus is not a proposition.24,25 Logical propositions still satisfy the general propositional form. (It is because, first, tautologies are products of applying logical operations to elementary propositions, second, elementary propositions satisfy the general proposition form and, third, the general form of logical operation is the general propositional form.) But how can tautologies satisfy the general propositional form and yet they are not propositions? It is a misunderstanding to think the Tractatus holds that whatever satisfies the general propositional form must be a proposition. Consider once again the characterization of the general propositional form in TLP 4.5: 24. I would like to thank Laurence Goldstein for keeping on reminding me that, for the Tractatus, tautologies and contradictions are not propositions. For his view and argument, see, for example, Goldstein (1999: 148–55). 25. If tautologies are not propositions, why does Wittgenstein talk about the truth of tautologies in entries like TLP 4.461 (‘unconditionally true’) and TLP 4.464 (‘[its] truth is certain’)? The answer is that the Tractatus also employs what I would call ‘a schematic way’ of talking about tautologies (and contradictions). In TLP 4.4s, he refers to Ln different groups of truth-conditions (TLP 4.45) and talks about Ln ways in which a proposition can agree and disagree with their truth-possibilities schematically, and then dismisses two of the Ln ways from propositions on the ground that one is ‘true’ and one ‘false’, respectively, for all the truth possibilities of the relevant elementary propositions, and thus do not represent any possible situation at all (TLP 4.46–4.463). In the schematic context of truth-functional logic, tautologies can be seen as groups of truth-conditions and can be said to be ‘true’ for all the truth possibilities of the relevant elementary propositions. By this, he does not mean that a tautology is true in the same way that a proposition is true, where the latter is defined via the agreement with reality and in virtue of being a picture (TLP 2.21 and 4.06). It is merely a schematic manner of speaking that a tautology is said to be true, unconditionally true (TLP 4.461) and that its truth is said to be certain (TLP 4.464). This does not contradict the claim that tautologies are not propositions. © 2006 The Author
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Philosophical Investigations . . . to give the most general propositional form [is] . . . to give a description of the propositions of any sign-language whatsoever in such a way that every possible sense can be expressed by a symbol satisfying the description, and every symbol satisfying the description can express a sense, provided that the meanings of the names are suitably chosen.
The last clause is important. A symbol satisfying the general propositional form can express a sense, and thus is a proposition, provided that meanings can be suitably chosen for its constituent ‘names’.Tautologies (and contradictions) are exactly those symbols such that meanings cannot be chosen for their constituent signs. But they still fit the characterization of the general propositional form in TLP 4.5. So, they satisfy the general propositional form, even though they are not propositions. They are still products of the application of the general rule of language.This explains why they are senseless (Sinnlos) but not nonsensical (Unsinnig) (TLP 4.461).Whatever satisfies the general propositional form, that is, all that is well-formed, cannot be nonsensical. A tautology is part of the symbolism ‘much is ‘0’ is part of the symbolism of arithmetic’ (TLP 4.4611). One may say, a tautology has no content just as 0 has no integral content, as shown by cases like ‘p v tautology ≡ p’ and ‘a + 0 = a’. A tautology still serves certain function in the symbolism just like 0 in arithmetic. In a way, however, the function is residual. Since a tautology fails to represent reality, its function confines to what its structure shows. Its form, that is, the possibility of its structure (TLP 2.033), cannot be any specific form but the general propositional form. For, otherwise, it would not be admitting all possible situations.Thus, its structure is exactly the actualization of the general propositional form. Hence, what its structure shows is already shown by any proposition which, like tautologies, satisfies the general propositional form. This explains why Wittgenstein says that ‘[t]he fact that a tautology is yielded by this particular way of connecting its constituents characterizes the logic of its constituents’ (TLP 6.2), and that ‘we can actually do without logical propositions; for in a suitable notation we can in fact recognize the formal properties of propositions by mere inspection of the propositions themselves’ (TLP 6.122). Tautologies are dispensable. Nevertheless, even though they are residual, tautologies are generated by the general rule of language. They do not say but, like any other propositions, they are products of the general rule of language. © 2006 The Author
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Both logical propositions and propositions are results of applying the general rule of language or the general rule of logical operation. For the Tractatus, saying, or picturing, is applying logical operations, though applying logical operations need not be picturing. The exceptional cases are logical propositions (and contradictions). Logical propositions are products of the general rule of language and yet they do not say anything about the world because of their having the most general form of propositions. It is in this way that one should see them as ‘the limiting cases’ (TLP 4.466) or belonging to the limits of language. Products of the general rule of language either lie within or on the limits of language.Whatever lie within the limits are propositions, while whatever on the limits logical propositions (tautologies) and contradictions. They all satisfy the general propositional form.Whatever does not satisfy the general propositional form is nonsensical. The Tractatus would not, and could not, say what the nonsensical signs are. It is enough to know what lie within and what lie on the limits of language. In fact, just knowing what lie within the limits of language is enough. For the limits of language are determined from within by the totality of propositions. The possibility of drawing limits to language depends on the unity of language and logic, which in turn is guaranteed by the general propositional form’s being the sole logical constant. In this paper, I have explained how, based on the Grundgedanke and the picture theory, the Tractatus comes to the conclusion that the general propositional form, which is satisfied by elementary and non-elementary propositions, is the sole logical constant.26
References Anscombe, G. E. M. 1971. An Introduction to Wittgenstein’s Tractatus, Philadelphia: University of Pennsylvania Press. Black M. 1964. A Companion to Wittgenstein’s ‘Tractatus’, NewYork: Cornell University Press. 26. This paper was completed while I was taking a half-year sabbatical leave from Hong Kong Baptist University and visiting both Clare Hall and the Faculty of Philosophy of Cambridge University as visiting fellow and visiting scholar, respectively, between January and July 2003. I am very grateful to their support. I would also like to thank Laurence Goldstein and Peter Hacker for their suggestions and comments on earlier drafts of this paper. © 2006 The Author
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