[cheung leo] the unity of language and logic in wi(bookos-z1.org)

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Philosophical Investigations 29:1 January 2006ISSN 0190-0536

The Unity of Language and Logic in

Wittgensteins Tractatus

Leo K. C. Cheung, Hong Kong Baptist University

1. Introduction

Wittgenstein holds in the Tractatus that the general propositionalform is the sole logical constant (or the one and only general prim-itive sign in logic):

It is clear that whatever we can say in advance about the form ofall 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

Wherever there is compositeness, argument and function arepresent, and where these are present, we already have all the logicalconstants.

One could say that the sole logical constant was what allpropo-sitions, 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 thedescription of the one and only general primitive sign in logic.(TLP 5.475.472)

By the thesis, as I shall explain, he means that the general form of

proposition is the general form of logical operation.The importanceof 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 theproclaimed 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 Tractatusemploys the picture theory and the Grundgedanke in TLP 4.0312

to argue for the thesis.

( x).fx.x = a.

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 provethe thesis that the general propositional form is the sole logical con-stant. For the thesis demands not only non-elementary propositions

but also elementary propositions that they must satisfy the generalform of logical operation. How can an elementary proposition,which is supposed to be an immediate, and thus non-truth-func-

tional, combination of names (TLP 4.224.221), satisfy the generalform of logical operation? How can an elementary propositioninvolve all logical constants in an intrinsic manner (TLP 5.47) if it

is supposed to belong to the end products of logical analysis ofpropositions (TLP 4.221)? How can the application of logic be

involved in an elementary propositions saying anything about reality?My explanation of how Wittgenstein in the Tractatus proves the thesiswould also make it clear how he answers these questions. It is,however, worthy of pointing out here that while many commenta-tors of the Tractatus are ignorant of, or have chosen to ignore, thesequestions,1 several prominent commentators have attempted to tacklethem or to criticize the Tractatus on that. It is illuminating to seesome of the latters views.

For example, Brian McGuinness writes in his essay Pictures andForm,

. . . in the first part of the Tractatus, notably in the 3s and early4s, we seem to be told that the essence of a proposition is to bea picture, while in the later parts we are told that its essence is tobe a truth-function, that is to say a result of applying the opera-tion of simultaneous negation to elementary propositions. Thepicture theory requires further elaboration, and the truth-func-tion account of what it is to be a proposition seems to involve

circularity by presupposing a prior understanding of what it is tobe an elementary proposition. But a more serious difficulty is thatthe two accounts seem to be quite separate things, and, if this is

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1. For example, Max Black, Robert Fogelin and James Griffin are amongst thosecommentators. In his A Companion to Wittgensteins Tractatus, Black (1964: 2368,2701) discusses the general propositional form but does not explain the thesis thatthe general propositional form is the sole logical constant. In fact, he does not seemto have noticed the thesis, nor the fact that the Tractatus holds and attempts to arguefor the unity of language and logic. Fogelin has devoted a section in his Wittgenstein,

2nd edition, (1987: 4750) to discuss the notion of the general propositional form.But, surprisingly, he mentions neither the thesis that the general propositional formis the sole logical constant, nor the Tractatus proof of the unity of language and logic.In Griffins (1964) Wittgensteins Logical Atomism, the thesis and the issue of the unityof language and logic are not addressed at all.


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so, cannot both be adequate accounts of what it is to be aproposition.2

Here, McGuinness is criticizing the Tractatus on that the picture

theory fails to explain how an elementary proposition can be a truth-function, 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 Trac-tatus, it is vital to our understanding of Wittgenstein to see that thenature of logic is already being inquired into in Wittgensteins treat-

ment of the puzzle about the relation between propositions and

facts.3

The application of logic in language to reality requires thestructure of an elementary proposition to be logical, which in the

case of the Tractatus means truth-functional.4 But it is hard to seehow the structure of an elementary proposition could be truth-func-

tional, and Winch thinks the Tractatus does not provide any accountof that.5 This is a serious difficulty and one of the main factors in

inducing Wittgenstein to move away from the Tractatus notion of ele-mentary propositions.6

Rush Rhees view is rather different from McGuinness andWinchs. He thinks that, in the Tractatus, Wittgenstein has attemptedto bring out the fundamental role logic plays in a propositions saying

something about, or picturing, reality. He writes in various places

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2. McGuinness (2002: 656). Also, McGuinness (2002: 66) mentions in a footnotethat [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 (1969:34), [t]his point can perhaps

be expressed in the form of another problem:What is the relation between a propo-sitions ability to state a fact and its ability to stand in logical relations to other propo-sitions? He also thinks, for the Tractatus, . . . unless propositions had logical relationswith each other that they would not state fact (i.e. would not be propositions) andunless they stated facts, they would not have logical relations with other proposi-tions (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 theexpression, the logical structure of elementary proposition . . . Although Winch isreferring to TLP 5 here, it is clear from his introduction that he thinks the Tracta-tus does not provide any such account.6. Winch (1969: 6).


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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 orwith picturing,7 and thus shows that logic must take care of itself or

its own application

8

and that understanding a proposition is not any-thing arbitrary.9 He emphasizes in this connection the importanceof a general rule the law of projection in TLP 4.0141 in the

distinction between sense and nonsense and in showing that logicmust take care of itself,10 as well as the thesis that the general formof logical operation is the general form of proposition11 and the fact

that sense, as a configuration of objects, must have the complexitywhich we express with the logical constants.12 But Rhees does not

explain in detail how the Tractatus employs the picture theory toachieve these, and, in particular, how logic is fundamental in con-nection with picturing or with the construction of propositions.

I think Rhees is on the right track already but, unfortunately, hedoes not explain in detail how the picture theory can account for

such intrinsic relation between logic and language.13 In the case ofMcGuinness and Winch, I will make it clear in this paper that, con-

trary to what they have thought, the Tractatus does attempt to employthe picture theory, together with the Grundgedanke, to account forthe fact that logical constants, or, rather, logical operations that they

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7. Rhees mentions this point in his essay Miss Anscombe on the Tractatus (1996b:115) and in Rhees (1998: 5760). For example, he writes in Rhees (1998: 4) and(1998: 9), respectively, that [w]hen Wittgenstein says that propositions are pictures ofreality, one thing he wants to bring out is the way in which logic is fundamental inconnexion with them and that [w]e recognize the relation of logic to empiricalpropositions 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 self-contradictory. 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 propo-sitions. 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 ageneral rule a rule by which we distinguish between sense and nonsense. Therecannot be anything arbitrary in logic, because anything arbitrary would have to besaid: and logic (the general rule) is what makes this possible . . . part of the pointhere 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: 23).12. See Rhees (1998: 13).13. I would like to say that I was first inspired by Rhees views and subsequentlyset myself to tackle the issue.


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symbolize, are involved in an elementary propositions saying some-

thing about reality. Also, the attempt is not something which isplainly incoherent or trivially wrong. It is ingenious. In fact,

McGuinness and Winch seem to have misconceived the issue in asimilar, if not the same, way.What is crucial here is not, as they havethought, how the combination of names in an elementary proposi-

tion could be truth-functional.The Tractatus does not hold any suchthing.What the Tractatus does hold, I shall argue, is that logical opera-tions are applied in an elementary propositions picturing reality via

naming. According to the picture theory, the operation NN (whereN is the sole fundamental operation in the Tractarian system), or the

existential quantifier, is present in every elementary proposition inan 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 generalpropositional 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 con-stant is crucial to the Tractatus aim (TLP, p.3) to draw a limit tothought by drawing one to the expression of thought, that is, lan-guage. 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 bythe totality of propositions (TLP 4.51).The constructions out of all

elementary propositions via logical (truth-functional) operations giverise to propositions (TLP 5), tautologies and contradictions (TLP

4.454.46) 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


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14. The Tractatus sometimes refers to drawing a limit to language, for example, inTLP, p.3, and sometimes to setting limits, for example, in TLP 4.1134.116. 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 expressedby means of language. This implies that the limits of language do not belong tolanguage.


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may be called the truth-functionality of logical propositions) are all

logical propositions, and contradictions. In fact, the Tractatus sees tau-tologies and contradictions not as propositions (and thus not belong-

ing to language) but as the limiting cases, or, simply, the limits, ofpropositions (TLP 4.466 and 5.143). Those lie within the limit oflanguage are propositions, those lie on it tautologies and contradic-

tions, and those lie outside it nonsense.16 Tautologies and contradic-tions constitute the limit of language and yet are fixed by the totalityof propositions.

It might appear that one way to draw a limit to language is togive the totality of propositions (and thus fix their limits). But the

Tractatus does not, and cannot, give such totality because not eventhe totality of elementary propositions can be given (TLP

5.555.551). What it does is to give a description of the generalpropositional form (TLP 4.5), which is the equivalent of the generalrule referred to in TLP 4.041. The general rule does not give but

determines the propositions and also their limits. It is the samegeneral rule which generates propositions, as well as tautologies and

contradictions.The latter are the limiting cases of the application ofthe general rule. Propositions, tautologies and contradictions allsatisfy 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 ofa particular language. (The fact that the Tractatus understands logicalsyntax in this way can be seen from TLP 3.344 and 6.124.TLP 6.124says that [i]f we know the logical syntax of any sign-language, thenwe have already been given all the propositions of logic. Logical

syntax gives prominence to the general rule governing the forma-tion 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 logicalsyntax allow us to substitute for it. Then logical syntax also givesprominence to the general rule of language which governs the for-

mation of symbols capable of signifying. One may say, for the Trac-

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16. Tautologies and contradictions are unsubstantial point in the centre and outerlimit, respectively (TLP 5.143; also see TLP 4.466). Elementary propositions arepropositions and thus cannot constitute the limits of language. Pears taking ele-mentary propositions as the inner limits of language in his book Wittgenstein (1997:678) is incorrect.


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tatus, language and logic are unified via logical syntax.) A limit canbe drawn to language because grasping the general rule fixes notonly the totality of propositions but also their limits logical propo-

sitions (tautologies) and contradictions. Drawing a limit to languageis 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 (logicalpropositions) and contradictions. Given the truth-functionality of

tautologies and contradictions, such unity of language and logic canbe expressed as that the general form of proposition is the general

form of the combinations (applications) of logical operations, andvice versa. This is the thesis that the general propositional form is

the sole logical constant. It amounts to saying that language and logicare unified via the general propositional form or the general form of logicaloperation. This is the unity of language and logic. (This also explainswhy the thesis does not only demand that a non-elementary propo-sition must satisfy the general form of logical operation but also that

an elementary proposition must satisfy it as well. For if the generalform of elementary propositions is different from the general formof 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, lan-

guage and logic could not be unified.) Wittgenstein is true to hisproclaimed aim of the Tractatus and so does attempt to argue for thethesis and hence the unity of language and logic.The Grundgedankein 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. Animportant task of the Tractatus is then the difficult one of explain-ing how an elementary proposition, which is an immediate combi-nation of names, can satisfy the general form of logical operation.

With this remark, I shall now explain the basic structure of the argu-ment for the thesis in the Tractatus.

3. The Grundgedanke and the Picture Theory

The thesis that the general propositional form is the sole logical con-stant actually consists of two parts: that there is the general propo-


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sitional form and that the general propositional form is the sole

logical constant. The Tractatus argument for the thesis employs thepicture theory17 in TLP 2.12.225 and 4.0114.016, the Grundgedanke

(or the thesis in TLP 4.0312 that logical constants are not represen-tatives or do not denote), the existence of the sole fundamental operationN introduced in TLP 5.5, which implies the unity of logical opera-tion, and the analyticity thesis in TLP 5 (A proposition is a truth-function of elementary propositions). Amongst those theses, whatplay the crucial roles in the proof are the picture theory and the

Grundgedanke. The Grundgedanke and the unity of logical opera-tion, in a way to be explained later, imply the existence of the

general propositional form. The key of the Tractatus proof of themajor 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 thatobjects have signs as their representatives.

My fundamental idea [Grundgedanke] is that the logical con-stants are not representatives; that there can be no representativesof the logicof facts.

Besides the Grundgedanke, the picture theory is also referred to inTLP 4.0213 indirectly because what accounts for the dependence ofthe possibility of a proposition on the principle of naming is exactlythe picture theory. To see that TLP 4.0213 is the key, note that the

Grundgedanke implies that a logical operation is independent ofthe semantic content of any symbol, and this suggests that an oper-

ation is intrinsic to every elementary proposition. TLP 4.0312 alsosays that the possibility of a proposition is based on the possibility

of naming. This further suggests that an operation is intrinsic tonaming. In fact, as one will see later, the Tractatus holds that an opera-tion 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 generalpropositional form. Of course, many details still need to be worked

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17. The picture theory, as I see it, consists of an account of the notion of a pictureand how a picture depicts reality, mainly in TLP 2.12.225 and 4.0114.016, thethesis in TLP 4.01 that a proposition is a picture of reality and the proof of the thesisin TLP 4.024.021. In this paper, only those in TLP 2.12.225 and 4.0114.016 willbe considered, and they are taken as what constitute the picture theory.


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out, and I shall explain how the Tractatus works them out later. TheTractatus employs the picture theory, the Grundgedanke, the existenceof N and the analyticity thesis to prove the thesis that the general

propositional form is the sole logical constant which, together withthe truth-functionality of logical necessity, implies the unity of lan-guage and logic. I already explained how the Tractatus proves theGrundgedanke, the existence of N, the analyticity thesis and thetruth-functionality of logical necessity elsewhere.18 With the excep-tion of the issue of the existence of N, I shall not repeat my expla-

nation of the proofs here, nor shall I comment on them. One maysimply regard these four theses as what are presupposed in this paper.

The picture theory, however, will be explained in detail. In whatfollows, I shall explain the Tractatus proof of the thesis that thegeneral propositional form is the sole logical constant. I shall beginwith the clarification of the notions of the sole logical constant andthe 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 intro-duced in TLP 5.5 as (-----T)(, . . .), where what is inside the right-hand pair of brackets represents an unordered collection ofpropositions, and the row in the left-hand pair of brackets indicates

that in the last column of the truth-table expression all but the lastone are F (TLP 4.442 and 5.5). It is also written as N( ), where is a variable whose values are terms of the bracketed expressionand the bar over the variable indicates that it is the representative ofall its values in the brackets (TLP 5.501).19 The sole logical con-

stant, however, does notsymbolize N. In TLP 5.472, the sole logicalconstant is also called the one and only general primitive sign inlogic. It is then advisable to see what a general primitive logical sign


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18. In Cheung (1999), I explain and comment on the two proofs of theGrundgedanke 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 logicalnecessity 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).


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is in the first place. The Tractatus holds that the real general primi-tive signs are not p q, (x).fx, etc. but the most general formof their combinations (TLP 5.46). A general primitive sign is then

the most general form of the combinations of a logical sign. If asystem has a single fundamental operation, then all operations areunified via the general form of the combinations of the sole funda-

mental operation the general form of logical operation, asdescribed in TLP 6.01, or what is in common to all operations. Inthat case, there is the one and only one general primitive logical sign

the sole logical constant. The sole logical constant is not really alogical constant but what symbolizes the most general form of logical

operation.Thus, with respect to the Tractarian system, instead of sym-bolizing the sole fundamental operation N, the sole logical constant

symbolizes the general form of the combinations of N. The sole logicalconstant, or the general form of logical operation, is given bythe general term of a formal series20 [ , , N( )], as in TLP 6.01.

([ , , N( )] symbolizes the form of the result of a certain number

of successive applications of N to a subset ( ) of the base ( ).) The

thesis that the general propositional form is the sole logical constant

can then be formulated as follows: The general propositional form is[ , , N( )].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 ele-mentary propositions. An elementary proposition is an immediate

combination of names, and names are referential primitive symbols(TLP 3.23.203 and 3.206). An immediate combination of themeanings of names, or objects, is called a state of affairs (TLP 2.01

and 2.03). The determinate way that objects are connected to oneanother in a state of affairs is the structure, whose possibility is the

form, of the state of affairs (TLP 2.0322.033). In general, the Trac-tatus 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 itsoccurring 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

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20. For a discussion of the Tractarian notion of formal series, see Cheung (2000:2514).


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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 con-stituent signs standing in the determinate relation to one another

(TLP 3.14).A name also has a form which is its possibility of occur-ring in elementary propositions. The description [ , , N( )] thengives prominence to a form or a combinatorial possibility. Moreover,

a form, as a combinatorial possibility, determines a rule. Thus, the solelogical constant does not only symbolize the general form of logicaloperation but also a rule which may be called the general rule of

logical operation. This explains why, in TLP 5.4s which, amongstother things, aim to symbolize the sole logical constant,Wittgenstein

writes that . . . it is not a question of a number of primitive ideasthat have to be signified, but rather of the expression of a rule (TLP

5.476). The sole logical constant symbolizes both the general formof logical operation and the general rule of an operation. Hence, the

Tractatus upholds and argues for the unity of logical operation.The exis-tence of the sole fundamental operation N enables the unificationof 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 aboutthe 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 sign-language whatsoeverin such a way that every possible sense can beexpressed by a symbol satisfying the description, and every symbolsatisfying the description can express a sense, provided that themeanings of the names are suitably chosen.

It is clear that only what is essential to the most general propo-sitional form may be included in its description for otherwiseit would not be the most general form.

The existence of a general propositional form is proved by thefact that there cannot be a proposition whose form could not have

been foreseen (i.e. constructed).The general propositional form ofa proposition is:This is how things are [Es verhlt sich so und so].

At least four important points can be gathered from this passage.Thefirst point is that there is a proof of the existence of the general


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propositional form in the Tractatus. I shall explain the proof in thenext section.The second point is that the general propositional formgoverns the formation of symbols capable of expressing sense. The

general propositional form then completely determines the generalrule which produces propositions, that is, the general rule of lan-guage. Since a form, as a combinatorial possibility, determines a rule,

this is not a surprising claim. If there is the general propositionalform, that is, if all propositional forms share a common general form,then there is also thegeneral rule of language. In other words, thegeneral propositional form characterizes the general rule of languagecompletely. The general rule of language is the general rule men-

tioned in TLP 4.0144.0141 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 thespecific 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 generalpropositional form can be given by a description. In the Tractatus,descriptions can be many things. A description of a complex can beright or wrong (TLP 3.24) and thus is a proposition. However,TLP5.501 mentions three different kinds of descriptions of the terms of

the bracketed expression in N( ), none of which is a proposition. Isuggest taking the pragmatic move in regarding a description as anexpression employed to give prominence to what it is a description of. Nowone can gather at least four descriptions of the general propositional

form from the Tractatus.

(1) The general propositional form is: Es verhlt sich so und so (TLP

4.5).(2) The general propositional form is a variable (TLP 4.53).

(3) The general propositional form is [ , , N( )] (TLP 6).(4) The general propositional form is given by the expression

[ , , N( )] or, simply, is [ , , N( )] (TLP 5.475.472

and 66.01).

I shall explain what (1) and (2) mean and how the Tractatus proves

them later. (3) follows from the analyticity thesis directly. Thefocus here is, of course, on (4) because it is another formulationof the thesis that the general propositional form is the sole logicalconstant. Note that [ , , N( )] in (3) does not bring out the

p

p

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(formal) content of the general propositional form completely

because does not characterize the general form of elementaryproposition. It is not a desirable description, especially when the

concern here is the relation between language and logic, or betweenan elementary proposition and logical operations, that it surelycannot bring forth. However, if there is a complete description of

the general form of elementary proposition, then the replacement ofin [ , , N( )] by such a description would give rise to a com-

plete description of the general propositional form. (The latter, as

one will see, is actually [ , , N( )].) This suggests that the firststep towards explaining how the Tractatus proves the thesis that thegeneral propositional form is the sole logical constant, or (4), is tofind out from the Tractatus a complete description of the generalform of elementary proposition.

6. The Grundgedanke, Logical Operation and the GeneralPropositional 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 thefact that there cannot be a proposition whose form could not havebeen foreseen (i.e. constructed).

Presumably, the argument here is that if every proposition can beconstructed according to a unified plan, then there is a general rule

producing propositions and hence there is the general propositionalform. 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 issuch a unified plan for elementary propositions, or that there is thegeneral form of elementary proposition. The Tractatus does not statethe proof explicitly. However, as I shall explain now, in fact, theGrundgedanke and the unity of logical operation via the sole fun-

damental operation N entail that there is the general form of ele-

mentary proposition.The Grundgedanke implies that a logical operation, or what a

logical constant symbolizes, is independent of the semantic contentof any symbol.This, in turn, implies that:

pp

p


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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 ofcharacteristics of an operation gathered from the text that Wittgen-stein does hold it. Here is the list: [i] Propositions and only propo-sitions 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 itsbase (TLP 5.24). [iii] An operation cannot characterize any propo-

sitional form. It can only characterize a specific difference betweenpropositional forms, that is, between the form of its result and the

constituent forms of its base (TLP 5.245.241 and 5.254).21 [iv]

Propositions can only occur in other propositions (in a nontrivialmanner) 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 ofan 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 ofthe other propositions.

This amounts to saying that an operation is completely determinedby the general form of all its possible bases and the specific differ-

ence 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 commonto 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 opera-tion, then the operation is completely determined by the specificformal difference between its result and its base. Second, if there is

no specific formal difference between the result and the base of anoperation, then the operation is completely determined by what is

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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 Theoperation does not characterize a form. I follow Ogden here in rendering kennze-ichnen as characterize.


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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 pos-sible bases.

Now the unity of logical operation via the sole fundamentaloperation N, together with (5), implies this:

(7) Every operation is completely determined by what is commonto all the possible bases of N and the specific formal differencebetween its result and its bases.

For the Tractatus, N can only have elementary propositions and theirtruth-functions to be the constituents of its possible bases. It followsthat:

(8) Every operation is completely determined by what is commonto all elementary propositions, that is, the general form of ele-

mentary proposition, and the specific formal difference betweenits result and its base.

Similar to the case of (5), (8) does not assert the existence of thegeneral form of elementary proposition.What it does assert, amongst

other things, is that if there is no specific formal difference between theresult and the base of an operation, then there is the general form of ele-mentary proposition and the operation is completely determined by the general

form of elementary proposition. It is not difficult to see that NN is suchan operation. To see this, note that:

(9) For any , N[NN( )] = N( ).

The proof of (9) is very simple. Since N( ) contains no free vari-

ables, N[NN( )] = NN[N( )] = N( ). Hence, N[NN( )] = N( ).It follows from (9) that

(10) From the point of view of being the base of an operation,NN( ) can be seen as being equivalent to ( ).

This means that there is no specific formal difference between theresult and the base of NN.Two important conclusions can be drawnhere. The first one is that, by (8), there is the general form of ele-mentary proposition. Of course, there is NN because there is N. So


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the existence of the sole fundamental operation N guarantees the

existence of the general form of elementary proposition. Becausethere is the general form of elementary proposition, given the ana-

lyticity thesis, there must be the general propositional form. So theexistence of N, or, rather, that of NN, provides a unified plan accord-ing to which a proposition can be constructed and thus there cannot

be a proposition whose form could not have been foreseen or con-structed.With this, I have explained how a proof of the existence ofa 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 iscompletely determined by the general form of elementary proposi-

tion. This means that the operation NN is intrinsic to every elementaryproposition. This is an important step towards the proof of the claimthat the general propositional form is the sole logical constant.The

key is to explain the intrinsic relation between the application of anoperation and an elementary propositions saying something about

reality. The Tractatus, as I shall explain later, employs the picturetheory to completely characterize the nature of an elementaryproposition (and thus the nature of the general propositional form)

in terms of NN, and then to explain why the general propositionalform is [ , , N( )], or (4). The crucial idea of the picture theory

is also that NN is intrinsic to every elementary proposition and, infact, 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 char-acterization of the general propositional form. I shall begin with thespecial case of elementary propositions. Given the analyticity thesis,

a proposition can be analyzed into a truth-function of elementarypropositions. An elementary proposition, as already mentioned, is an

immediate combination of names. Names are referential primitive

symbols, whose meanings (Bedeutungen) are objects. An immediatecombination of objects (meanings of names) is a state of affairs. It isthe major claim of the picture theory that [a] proposition is a pictureof reality (TLP 4.01). A picture depicts reality by representing a

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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 pre-sents the existence of a state of affairs, by representing the possibil-

ity of the state of affairs (TLP 4.211). But how does the picturetheory account for an elementary propositions depicting reality? TLP4.0312 states that the possibility of propositions is based on the prin-

ciple of naming, and this indicates where the essentials of the accountare to be found. In fact, as I am going to explain, the picture theoryaccounts for a pictures depicting reality via the element-object-

correlation or, in the special case of an elementary proposition, vianaming. (This of course should not be taken to be implying that

naming is independent of picturing because Wittgenstein alsoemphasizes the dependence of naming on the relevant propositional

context in TLP 3.3. What is at stake here is not whether naming isindependent of picturing, which is not, but how picturing is con-stituted by naming.)

A picture, according to the Tractatus, is a fact (TLP 2.141), andthus has a form and a structure. Indeed, the forms of its elements

constitute its form. A fact is made into a picture when its formbecomes a pictorial form:

The fact that the elements of a picture are related to one anotherin a determinate way represent that things are related to oneanother in the same way.

Let me call this connexion of its elements the structure of thepicture, and let us call the possibility of this structure the pictor-ial form of the picture. (TLP 2.15)

In a picture, the fact that its elements are related to one another pre-sents that objects22 are related to one another in the same determi-

nate way.That is, the form of a fact becomes a pictorial form whenthe fact presents the existence of a state of affairs sharing the same

form. How is this possible? By correlating objects with the con-stituent elements, that is, by establishing a pictorial relationship, in acertain manner:

That is how a picture is attached to reality; it reaches right out toit.

It is laid against reality like a measure.


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22. It is clear from the content of TLP 2.1512.1515 that things here can be takenas objects.


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Only the end-points of the graduating lines actually touch theobject that is to be measured.

So a picture, conceived in this way, also includes the pictorialrelationship, which makes it into a picture.

The pictorial relationship consists of the correlations of thepictures elements with things.

These correlations are, as it were, the feelers of the pictureselements, with which the picture touches reality. (TLP2.15112.1515)

Here, a picture is seen as a measure laid against reality in such amanner 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 ofa picture, as a measure, presents this constraint:

[*] Only objects having the same form as that of a constituentelement 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 guaran-tees the possibility of a pictures representing a state of affairs sharing itsform. One may also say, the form of a picture, which is constitutedby the forms of its elements, acts as a constraint ensuring that onlya state of affairs of the same form can be represented.

The constraint [*] is actually the only constraint that the corre-

lation of objects with elements, or the establishment of a pictorialrelationship, must be subject to.The fact that the Tractatus does holdthis is supported by TLP 3.315:

If we turn a constituent of a proposition into a variable, there isa class of propositions all of which are values of the resulting vari-able proposition. In general, this class too will be dependent onthe meaning that our arbitrary conventions have given to parts ofthe original proposition. But if all the signs in it that have arbi-trarily determined meanings are turned into variables, we shall stillget a class of this kind. This one, however, is not dependent onany convention, but solely on the nature of the proposition. It cor-

responds 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 thatwhen talking about turning all propositional constituents into vari-

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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 corre-late 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 asthe form of an element can be correlated with the element, an

element can correlate with any object. One may say, the form of anelement, or the constraint [*], sorts out objects sharing its form asthe candidates, and the object-element-correlation is simply arbi-

trarily 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 ele-ments of the relevant picture such that the form of the picture is

instantiated. Depicting is the instantiation of the form of a picture, and theinstantiation consists in arbitrarily correlating with the elements of the pictureobjects 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 conceivedby the Tractatus, consists in correlating an object with a name as itsmeaning in the nexus of an elementary proposition (TLP 3.3) or, ingeneral, in the context of depicting. Depicting is the presentation of

the existence of a state of affairs by means of the instantiation of theform of a picture, and the instantiation consists in arbitrarily corre-lating with the constituent elements objects from those sorted out

by the form of the picture.Thus, naming is the instantiation of the formof a name, and the instantiation consists in arbitrarily picking out an objectas the meaning of the name from those objects sorted out by the form of thename. It follows as a corollary that naming involves the application of theexistential quantifier or NN! More precisely, naming involves the appli-cation of the existential quantifier or NN to pick out an unspeci-fied 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 propo-sitional variable shows a form and its values signify those objectssorted out by the form (TLP 4.127), the corollary can also be putin the following way: Naming involves the application of the exis-


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tential quantifier to a propositional variable such that objects are

arbitrarily correlated with names as their meanings from thoseobjects signified by the values of the propositional variable. In fact,

naming involves both the application of the existential quantifier andthe stipulation of a constant as the name of the object picked outby the existential quantifier. In naming, while an unspecified object

is arbitrarily picked out by the existential quantifier from thoseobjects sorted out by the relevant propositional variable, a constantis at the same time given to the object as its name.The latter is sym-

bolized by putting the identity sign between a name and a variablename, 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 arenames. fa, as a picture, asserts the existence of a state of affairs viathe instantiation of its form by arbitrarily correlating with f and a

objects from those sorted out by the form of the picture. Employ-ing the propositional variable x obtained by turning the names f and a in fa into variables, the naming of f and a in fa can beseen as the application of the existential quantifier or NNto x such thatobjects are arbitrarily correlated with f and a as their meanings

from those signified by the values of x, respectively. (Note that avalue of x, say, fa signifies the objects f and a.) The essentialsinvolved here can be given prominence by (, x).x. = f.x = a,which is an equivalent formulation of fa.The presence of the exis-

tential quantifier and the identity sign symbolize the arbitrarypicking of the unspecified objects f and a by means of the formsymbolized by x and the stipulation of constants (names), that is,f and a to the objects f and a, respectively. This explains why fais equivalent to (, x).x. = f.x = a. Of course, if only the namingof a in fa is focused on, the nature of naming by a in fa can begiven prominence in the formulation (x).fx.x = a, just as the Trac-tatus does in TLP 5.47:

. . . An elementary proposition really contains all logical operationsin itself. For fa says the same thing as

Wherever there is compositeness, argument and function arepresent, and where these are present, we already have all the logicalconstants . . .

( x).fx.x = a.

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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 occursin 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 opera-tions symbolized by those logical constants. But, just as TLP 5.47attempts to say, although the existential quantifier, as the sign of an

operation, in (x).fx.x = a is dispensable (as the proposition can beformulated as fa), the existential quantifier, as an operation, is stillcontained in fa. How can this be the case? I think the best expla-

nation is that the application of the existential quantifier, as an oper-ation, 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 propo-sitions together but belongs to the signifying relation between names andobjects. With this, I have explained how the picture theory accountsfor the insight in TLP 4.0312 that the possibility of propositions is

based on the principle of naming, and how the existential quanti-fier is intrinsic to naming and thus to every elementary proposition.

9. The General Form of Elementary Proposition is NN( )

I am now going to argue that, and explain how, the picture theory

demonstrates that the general form of elementary proposition isgiven by the expression NN( ), or that the general rule of ele-

mentary proposition is NN. The point of departure of a completecharacterization of the general form of elementary proposition is, for

the Tractatus, the form of reality. The Tractatus talks about the formof reality in TLP 2.18.

What any picture, of whatever form, must have in common withreality, 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 ofaffairs (TLP 4.21), elementary propositions are what establish pic-

turing relation with reality directly. The form of reality is what is

common to, shared by and intrinsic to all elementary propositionsand reality. But this by itself does not imply that there is the form

of reality.The Tractatus, as already explained, argues for the existenceof the general propositional form and, in particular, the general form


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of elementary proposition. For the Tractatus, the existence of thegeneral form of elementary proposition proves that there is the formof reality, but not vice versa.The form of reality, however, is the point

of departure of a complete characterization of the general form ofelementary proposition. For the form of reality is what is the mostgeneral concerning reality.

Consider, without loss of generality, fa, or its equivalent(, x).x. = f.x = a, again. The proposition (, x).x, orNN( ), is in fact a description of the general form of a value of

x, that is, a description of what is common to all values of x.Thiscan be seen in two ways. Recall that an important point has been

gathered from TLP 4.5 according to which the general propositionalform is taken as something independent of the specific content of

meanings correlated with names.The first way is then to ignore thespecific names, or, so to speak, the particularity of the naming rela-tions, symbolized by = f and x = a in (, x).x. = f.x = a.In other words, what = f and x = a in (, x).x. = f.x = asymbolize are irrelevant to the general form of a value of the propo-

sitional variable x, of which fa is one of the values. The secondway is by seeing that any value of x entails (, x).x.That is, iffa, gb, . . . etc., are values of x, then fa (, x).x, gb (,x).x, . . . etc. In either way, the proposition (, x).x is a descrip-tion of the general form of a value of x. Since NN( ) is equiv-alent to (, x).x, NN( ) is a description of a value of x.Moreover, as explained in detail in TLP 3.3143.315, especially in

TLP 3.315 which was quoted in Section 7, a propositional variablelike x is a variable giving prominence to a combinatorial possibil-ity, that is, to a specific logical form. Recall that, according to the

Tractatus, there is the form of reality common to all specific logicalforms, and that the form of reality is what is the most general con-

cerning reality.To symbolize the general form of reality, a single vari-able, say, , alone is enough. Hence, the proposition NN( ) is acompletedescription of the general form of elementary proposition.One may say, the general form of elementary proposition is NN( ).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 intrin-sic to every elementary proposition follows from the Grundgedankeas a corollary. The inference from the Grundgedanke, however, doesnot provide an account of how NN is intrinsic to every elementary

xx

x

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proposition. This is probably one of the reasons why the Tractatusdoes 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 back-

ground 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 elementarypropositions depicting reality. The general form of elementary

proposition is NN( ). NN, or the existential quantifier, is intrinsicto naming, that is, (11), and thus to every elementary proposition.

The, so to speak, containing of NN in an elementary propositionis internal, as the way that internal is used in TLP 4.123, or as whatI mean by intrinsic, or as that without which the elementary propo-

sition could not have been an elementary proposition in the firstplace.This explains how, and in what way, NN is intrinsic to every

elementary proposition.

10. [ , , N( )], Es Verhlt Sich So und So and the GeneralPropositional Form

It is now easy to see how the Tractatus proves (1), (2) and (4), or thatthe general propositional form is: Es verhlt sich so und so, that thegeneral propositional form is a variable, and that the general propo-sitional form is [ , , N( )] 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 exem-

plified in the case of the equivalents fa and (,x).x. = f.x = a.(, x).x, or NN( ), as already explained, is the general formof a value of x. To say what the Tractatus would consider ineffable,(, x).x describes how one thing is connected (verhlten) toanother thing in a determinate way whose possibility is shown inx. Another way of giving prominence to what NN( ) does is

then Es verhlt sich so und so als , which is another descriptionof the general form of a value of x. Moreover, NN( ), as alreadyexplained, is a description of the general form of elementary propo-

x

x

x


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sition. Similarly, another formulation of NN( ) is Es verhlt sich so

und so als , which is a complete description of the general formof elementary proposition. Note that the analyticity thesis entails that

a proposition directs how reality is to be depicted by means of itsconstituent elementary propositions and the relevant logical opera-

tions. In the case of a non-elementary proposition, what directs itsdepiction of reality consists in two things, namely, the general formof elementary proposition and the general form of logical operation.

The variable in Es verhlt sich so und so als cannot be left outwhen the general form of elementary proposition is what is to be

described. No variable other than so und so is needed, however, ifthe general propositional form, or the general form of all proposi-

tions, is to be described. For what is at stake here is the most generallogical form and not just the logico-propositional form of an ele-mentary proposition, nor just the logical form of reality. As a result,

this explains why the Tractatus takes Es verhlt sich so und so as adescription of the general propositional form.

To understand the thesis that the general propositional form is avariable, that is, (2), the notions of constant and variable need to beconsidered first. For the Tractatus, an expression is . . . presented bymeans of the general form of the propositions that it characterizes.In fact, in this form the expression will be constant and everythingelse variable. Thus an expression is presented by means of a variablewhose values are the propositions that contain the expression (TLP

3.3123.313). For example, the expression (name) a can be pre-sented by the propositional form a whose values are propositionsin which a occurs. Here, is a variable and a a constant. Thegeneral propositional form characterizes all propositions. Hence, a

complete expression of the general propositional form contains noconstant, as it is an expression of what is the most general. Forinstance, the expressions of the general proposition form in the

descriptions Es verhlt sich so und so and [ , , N( )] contain no

constant. In the case of the description [ , , N( )], N is not a

constant because, according to the Grundgedanke, N does notdenote. In fact, the sign N is dispensable.23 An expression of the

general propositional form is, one may say, a limiting case of expres-

sions. It contains no constant and thus can be identified with a vari-

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23. See Cheung (1999: 4027).


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able. This explains why the Tractatus says that the general proposi-tional form is a variable.

It is now not difficult to understand (4), or the thesis that the

general propositional form is [ , , N( )], and why the Tractatusholds (4). Recall that, as pointed out in Section 5, [ , , N( )] in(3) is an incomplete description of the general propositional formbecause the general form of elementary proposition has not been

fully characterized. It has now been proven that NN( ) is a com-plete description of the general form of elementary proposition.Andone should not overlook the fact that NN( ) is a special case of

[ , , N( )]. Hence, the general form of proposition elementary

or non-elementary is given by the description [ , , N( )]. As aresult,the general propositional form is [ , , N( )], which is also

the general form of logical operation. The general rule of logical

operation, symbolized by [ , , N( )] or [ , N( )] (TLP 6.01), isactually the general rule of language, and vice versa. This explainshow the Tractatus understands [ , , N( )] as a complete descriptionof the general propositional form, as well as what the Tractatus holdsby 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 elementaryproposition satisfies the general propositional form [ , , N( )],which is also the general form of logical operation. This, togetherwith the analyticity thesis, also explains how a proposition satisfies

the general propositional form. It remains to explain how the propo-sitions 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 logicalpropositions satisfy the general propositional form and yet they do

not belong to language (propositions)? Or, how can logical propo-sitions 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:

p


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Tautologies and contradictions are not pictures of reality. They donot represent any possible situations. For the former admit all pos-sible situations, and latter none.

In a tautology the conditions of agreement with the world

the representational relations cancel one another, so that it doesnot stand in any representational relation to reality.

The crucial point here is that because a tautology admits all possi-ble situations, the representational relations cancel one another. This

means that no meaning can be suitably chosen for its constituentsigns or, so to speak, names, even though before the relevant propo-sitions 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, cannotpicture any state of affair, and thus is not a proposition.24,25

Logical propositions still satisfy the general propositional form. (Itis because, first, tautologies are products of applying logical opera-

tions to elementary propositions, second, elementary propositionssatisfy the general proposition form and, third, the general form of

logical operation is the general propositional form.) But how cantautologies satisfy the general propositional form and yet they are

not propositions? It is a misunderstanding to think the Tractatus holdsthat whatever satisfies the general propositional form must be aproposition. Consider once again the characterization of the general

propositional form in TLP 4.5:

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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 andargument, see, for example, Goldstein (1999: 14855).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 aschematic way of talking about tautologies (and contradictions). In TLP 4.4s, herefers to Ln different groups of truth-conditions (TLP 4.45) and talks about Ln waysin which a proposition can agree and disagree with their truth-possibilities schemat-ically, and then dismisses two of the Ln ways from propositions on the ground thatone is true and one false, respectively, for all the truth possibilities of the relevantelementary propositions, and thus do not represent any possible situation at all (TLP4.464.463). In the schematic context of truth-functional logic, tautologies can beseen as groups of truth-conditions and can be said to be true for all the truth pos-sibilities 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 definedvia 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.


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. . . to give the most general propositional form [is] . . . to give adescription of the propositions of any sign-language whatsoever insuch a way that every possible sense can be expressed by a symbolsatisfying the description, and every symbol satisfying the descrip-

tion can express a sense, provided that the meanings of the namesare suitably chosen.

The last clause is important.A symbol satisfying the general proposi-

tional form can express a sense, and thus is a proposition,provided thatmeanings can be suitably chosen for its constituent names.Tautologies (andcontradictions) are exactly those symbols such that meanings cannotbe chosen for their constituent signs. But they still fit the characteri-

zation of the general propositional form in TLP 4.5. So, they satisfythe general propositional form, even though they are not proposi-tions.They are still products of the application of the general rule of

language.This explains why they are senseless (Sinnlos) but not non-sensical (Unsinnig) (TLP 4.461).Whatever satisfies the general propo-

sitional form, that is, all that is well-formed, cannot be nonsensical.Atautology is part of the symbolism much is 0 is part of the symbol-

ism of arithmetic (TLP 4.4611). One may say, a tautology has nocontent just as 0 has no integral content, as shown by cases like p vtautology p and a + 0 = a.A tautology still serves certain functionin 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), cannotbe any specific form but the general propositional form. For, other-

wise, it would not be admitting all possible situations.Thus, its struc-ture is exactly the actualization of the general propositional form.

Hence, what its structure shows is already shown by any propositionwhich, like tautologies, satisfies the general propositional form. This

explains why Wittgenstein says that [t]he fact that a tautology isyielded by this particular way of connecting its constituents charac-terizes the logic of its constituents (TLP 6.2), and that we can actu-

ally do without logical propositions; for in a suitable notation wecan 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, tautolo-gies are generated by the general rule of language.They do not say

but, like any other propositions, they are products of the general ruleof language.


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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. Theexceptional 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 havingthe most general form of propositions. It is in this way that oneshould see them as the limiting cases (TLP 4.466) or belonging to

the limits of language. Products of the general rule of language eitherlie 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 proposi-

tional form.Whatever does not satisfy the general propositional formis nonsensical. The Tractatus would not, and could not, say what thenonsensical signs are. It is enough to know what lie within and what

lie on the limits of language. In fact, just knowing what lie withinthe limits of language is enough. For the limits of language are deter-

mined 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 forms being the sole logical constant. In this paper, Ihave explained how, based on the Grundgedanke and the picture

theory, the Tractatus comes to the conclusion that the general propo-sitional 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 Wittgensteins Tractatus,Philadelphia: University of Pennsylvania Press.

Black M. 1964.A Companion to Wittgensteins Tractatus, NewYork:Cornell University Press.

Leo K. C. Cheung 49

2006 The Author

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 Phi-losophy of Cambridge University as visiting fellow and visiting scholar, respectively,between January and July 2003. I am very grateful to their support. I would alsolike to thank Laurence Goldstein and Peter Hacker for their suggestions and com-ments on earlier drafts of this paper.


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Cheung, L. K. C. 2004. Showing, Analysis and the Truth-

Functionality of Logical Necessity in Wittgensteins Tractatus,Synthese, 189: 81105.

Cheung, L. K. C. 2000. The Tractarian Operation N and ExpressiveCompleteness, Synthese, 123: 24761.Cheung, L. K. C. 1999. The Proofs of the Grundgedanke in

Wittgensteins Tractatus, Synthese, 120: 395410.Fogelin R. 1987. Wittgenstein, 2nd edition, London: RKP.Goldstein, L. 1999. Clear and Queer Thinking, London: Duckworth.Griffin J. 1964. Wittgensteins Logical Atomism, Oxford: OUP.McGuinness, B. 2002.Approaches to Wittgenstein. London: Routledge.Pears, D. 1997. Wittgenstein, 2nd edition, London: Fontana.Rhees R. 1998. Wittgenstein and the Possibility of Discourse, D. Z.

Philips, ed., Cambridge: CUP.Rhees R. 1996a. Discussions of Wittgenstein, Bristol:Thoemmes Press.Rhees R. 1996b. Miss Anscombe on the Tractatus, in R. Rhees

(1996a: 115).Rhees R. 1996c. Object and Identity in the Tractatus, in R. Rhees

(1996a: 2336).Winch P., ed. 1969. Studies in the Philosophy of Wittgenstein, London:

RKP.

Wittgenstein, L. 1974. Tractatus Logico-Philosophicus [TLP], D. F. Pearsand B. F. McGuinness, trans., London: RKP.

Wittgenstein L. 1981. Tractatus Logico-Philosophicus, C. K. Ogden,trans., London: RKP.

Department of Religion and PhilosophyHong Kong Baptist University

Kowloon TongHong [email protected]


2006 The Author

[cheung leo] the unity of language and logic in wi(bookos-z1.org)

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