Wittgenstein on Russell’s Theory of Logical Types

Journal of Philosophical Research Volume 38 2013 pp. 115–146

© 2013 Philosophy Documentation Center ISSN: 1053-8364doi: 10.5840/jpr2013387

WITTGENSTEIN ON RUSSELL’S THEORY OF LOGICAL TYPES

DAESUK HANKyung Hee university

ABSTRACT: Wittgenstein criticizes russell’s theory of logical types for involving the idea that our language must be anchored in extra-linguistic entities so that it makes a meaningful combination of signs. Calling it the “fallacy of meaning,” Wittgenstein self-consciously remains within the realm of signs. this issue of meaning vs. sign, however, has not been understood correctly, partly because of being viewed through the distorting lens of russell. siding with Wittgenstein, i will argue that our language does not go wrong because of our “transgressing the (pre-established) rules of logical syntax.” it is rather because we just happen not to use a sign in accordance with the logico-grammatical rules we arbitrarily stipulate about it. the rules of logical syntax do not, as it were, flow from some extra-linguistic entities, or anything of that nature. “the rules of logical syntax must follow of themselves, if we only know how every single sign signifies.” in general, our language is accountable to nothing but itself in order for it to make the sense that it does. When it comes to logical syntax, our language is autonomous.

I. INTRODUCTION

russell held that his theory of logical types was anchored in a principle that “has a certain consonance with common sense which makes it inherently credible” (Whitehead and russell 1910, 39). However, in the 1919 manuscript of the Tractatus Logico-Philosophicus Wittgenstein declared the theory of logical types as erroneous:

From this observation we get a further view—into russell’s Theory of Types. russell’s error is shown by the fact that in drawing up his symbolic rules he has to speak about the things his signs mean. [this is now printed as 3.331 in Wittgenstein 1922, hereafter TLP].

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if russell’s theory of types had a certain consonance with common sense, then the Wittgenstein of this remark would have to go insane in pooh-poohing it as errone-ous. the story goes on in 1919, when russell commented on the above manuscript remark as follows.

[R.] the theory of types, in my view, is a theory of correct symbolism: (a) a simple symbol must not be used to express anything complex: (b) more generally, a symbol must have the same structure as its meaning. [R is now published in Wittgenstein 1961, 129].

if the theory of types were a theory of correct symbolism, then the Wittgenstein of TLP 3.331 would have to be advocating an incorrect symbolism. But, for Wittgenstein, therein lies precisely the problem with the theory of logical types.

[W.] that [i.e., R] is exactly what one cannot say. you cannot prescribe to a symbol what it may be used to express. All that a symbol can express, it may express. this is a short answer but it is true! [W is now published in Wittgenstein 1961, 129].

russell did not incorporate W into his introduction that was printed (and still is printed) with TLP. On his side, in the introduction Wittgenstein found nothing but “superficiality and misunderstanding” (Wittgenstein 1961, 131).

Although i will try to decide who gets whom wrong, i do not imply that the present work is a mere interpretation of dead authors by any measure. What i really do is seriously challenge a prevailing view in the philosophy of logic, to say the least, by bringing alive a hitherto muffled and yet clearly important voice.

Although the topic of the present paper, i.e., russell vs. Wittgenstein on the theory of logical types, is not new, i do not think that it has been treated properly. Part of the reason comes from Wittgenstein’s notorious aphoristic writing. But there are a few exceptions: passages where he identifies target views unmistak-ably and states his own alternatives as explicit improvements. One case of this is TLP 3.333.

A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself [TLP 3.333a]. if, for example, we suppose that the function F(fx) could be its own argument, then there would be a proposition ‘F(F(fx))’, and in this the outer function F and the inner function F must have different meanings; for the inner has the form φ (fx), the outer the form ψ (φ (fx)). Common to both functions is only the letter ‘F’, which by itself signifies nothing [TLP 3.333b]. this is at once clear, if instead of ‘F(Fu)’ we write ‘(∃φ):F(φu).φu = Fu’. [TLP 3.333c] Herewith russell’s paradox vanishes [TLP 3.333d].

in spite of the aforementioned importance of TLP 3.333, i can hardly bring myself to think that the secondary literature has ever settled the score with it. At worst, some commentators in the tradition of russell have misunderstood TLP 3.333 as rehearsing the spirit while changing the letters of his solution of the paradox. At best, in my view, some were dimly aware that Wittgenstein was undermining russell’s perspective, only they eventually failed to put their finger on it. i have in

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mind particularly the situation in which the intent of TLP 3.333d is not understood properly. i hope that the present work will improve on the situation.

II. RUSSELL’S SOLUTION OF THE PARADOX

it is one thing to solve a genuine problem, but another to make an apparent prob-lem vanish. Possibly, the Wittgenstein of TLP 3.333d means that the illusion of a problem must be dissolved. to examine this possibility, it is necessary to record russell’s purported solution of the paradox as a foil.

russell determines a sign for a set—“ẑ(φz)” in his notation—as an “incomplete symbol”: a “symbol which is not supposed to have any meaning in isolation, but is only defined in certain contexts” (Whitehead and russell 1910, 69). According to him, every proposition in which ‘ẑ(φz)’ occurs can be paraphrased in the following way: “f{ẑ(φz)} = df(∃ψ)(φx.≡

x.ψ!x:f{ψ!ẑ})” (Whitehead and russell 1910, 80).

russell analyses the proposition ‘φ (ẑ(φz))’ as “(∃ψ)(φx.≡x.ψ!x:φ (ψ!x),” which

in its turn is equivalent to the proposition ‘φ (φx)’. russell regards the “common characteristic, which we may describe as self-reference or reflexiveness,” as the culprit for the paradox (Whitehead and russell 1910, 64). russell attempts to solve the paradox by ruling out the allegedly self-referential ‘φ (φx)’ as an ill- formed formula.

russell says something to the effect that once we understand the kind of entity which a propositional function φx̂ is we see why there is no value for the function φx̂ with itself as the argument.

R*. When we say that ‘φx’ ambiguously denotes φa, φb, φc, etc., we mean that ‘φx’ means one of the objects φa, φb, φc, etc., though not a definite one, but an undetermined one. it follows that ‘φx’ only has a well-defined meaning . . . if the objects φa, φb, φc, etc., are well defined. that is to say, a function is not a well-defined function unless all its values are already well defined. it follows from this that no function can have among its values anything which presupposes the function, for if it had, we could not regard the objects am-biguously denoted by the function as definite until the function was definite, while conversely, as we have just seen, the function cannot be definite until its values are definite. (Whitehead and russell 1910, 41)

the central element of russell’s explanation in R* is that the function φx̂ presup-poses its values and that it is not the other way around. in order to get clear about what exactly this explicans involves, we must pay attention to russell’s phrase “the object φa.” Although he calls ‘φa’ a proposition, it is important to note, he harbors a very peculiar conception of a proposition. russell distinguishes the “propositional function” φx̂ itself from its indeterminate value φx. the value for the “propositional function” φx̂ with the argument a, however, need not be described only in terms of φx̂ and a. this is because in construing his “propositional function” russell uses as a matrix a function in a mathematical sense that corre-lates an object with another object with the same ontological status. For example, p is the eldest son of q. the identification of p does not necessarily contain the functor the eldest son of; p can also be described as the brother of r. similarly, the

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identification of “the object φa” need not contain the functor φx̂. to the contrary, by ‘φa’ russell alludes to something that happens to be described via φx̂ and a. russell would be more appropriately thought of as using ‘φa’ as a complex name, for lack of a better expression. in this connection it bears mentioning that (circa 1913) Wittgenstein was aware of the feature of russell’s outlook we have just brought up:

Frege said “propositions are names”; russell said “propositions correspond to complexes.” Both are false; and especially false is the statement “propositions are names of complexes.” (Wittgenstein 1961, 93)

As we will see in the subsequent sections, a proposition ‘φa’ in Wittgenstein’s sense is analyzed out into its constituents φ and a. A proposition ‘φa’ (in Wittgenstein’s sense) does not pick out something that happens to be described by way of φ and a. We dwelled on the difference between a complex name and a proposition because it will become important in our subsequent discussion.

i suspect some will think that the Wittgenstein of the last quote was construct-ing a straw man and attacking him, for russell had revised his former realism about propositions understood as mind independent complexes. russell was now taking seriously the possibility that a proposition was an “incomplete symbol.” While such an “incomplete symbol” can occur as part of a meaningful sentence, it should not be regarded as representing a single entity in the corresponding proposition. this objection, however, ignores where the shoe really pinches. For Wittgenstein bypasses russell’s question whether a proposition is complete or incomplete and focuses on a certain presupposition informing that question. russell depicts our language as having a two-tier structure: a propositional func-tion and its values (or propositions as he calls them). it is this two-tier explana-tion of our language that Wittgenstein actually contests, as we will see in the subsequent sections.

Coming back to russell’s solution of the paradox, the conception of ‘φa’ as a complex name gives us a way of taking the russell of R* at his word. the complex name ‘φa’ refers to the value that happens to be described in terms of the function φx̂ and the argument a. strictly speaking, something picked out by the complex name ‘φa’ could be referred to by the simple name ‘P’ as well. similarly, we can give the complex names ‘φb’ and ‘φc’ new names: ‘Q’ and ‘R’, respectively. it is then understandable, as russell contends, why “the values of the function φx are presupposed by the function and not vice versa.” the simple values P, Q, and R do not presuppose the propositional function φx̂. Because, however, the function φx̂ “ambiguously denotes” the simple values P, Q and R, the former presupposes the latter. (We will come back to this point in the next paragraph.) that the function φx presupposes its values (and not vice versa) is central to russell’s explanation in R*. it is not difficult to notice that that central point cannot be made if ‘φa’ is a proposition, as opposed to a complex name. the proposition ‘φa’ is not “ambigu-ously denoted” by its constituent φx. instead, proposition ‘φa’ is analyzed out into its constituents, φ and a.


Once more, it is because the function φx̂ “ambiguously denotes” the simple values P, Q and R that the former presupposes the latter. We can discern a hint about why that is so in russell’s notion of a “real variable”:

in the case of euclid’s proofs, this is evident: we need (say) some one triangle ABC to reason about, though it does not matter what triangle it is. the triangle ABC is a real variable; and although it is any triangle, it remains the same triangle throughout the argument. (russell 1908, 228)

A real variable in russell’s sense plays twofold roles: it is like a fixed name in that it “remains the same triangle throughout the argument.” yet at the same time, what value we choose for it does not matter, with the result that it accommodates generality. this peculiar double office of a “real variable” is what russell has in mind in the first sentence of R*. On the one hand, the function φx̂ contains a “real variable.” strictly speaking, a “real variable” is not a fixed name (in the usual sense of the word). As a result, the function φx̂ somehow leads a parasitic existence. russell understands the parasitic existence of the function φx̂ in terms of what he takes as a fact: that the function presupposes its values. On the other hand, a real variable somehow plays the role of a name. russell packs these two apparently contradicting roles into his fancy term, “ambiguous denotation.”

III. EXPRESSION AND VARIABLE

As we recall from the previous section, russell’s solution of the paradox con-sists in a sign for a function being unable to be its own argument because of the nature of its reference, i.e., the “ambiguous” kind of entity that a propositional function is. some commentators, especially those who have more affinity with russell than with Wittgenstein, read TLP 3.333 with pleasure, imagining that the latter’s solution of the paradox is close to that of the former. the keen reader, however, cannot ignore that the Wittg`enstein of TLP 3.333a is doing a com-pletely different thing. the Wittgenstein of TLP 3.333a repeats verbatim russell’s explicandum, i.e., the fact that a function cannot be its own argument. However, Wittgenstein’s explicans is that the functional sign (itself) already contains the “prototype” of its own argument and that it cannot contain itself. in his expli-cans Wittgenstein self-consciously remains within the realm of signs through-out. Possibly, it is not an a priori theory of any kind about the types of entities that makes a given sign construction nonsensical. that may be the backbone of Wittgenstein’s criticism.

to examine the possibility just suggested, we must get clear about the concept of the “prototype” of a sign involved in it. Let us begin with the second occurrence of the term “prototype” in TLP:

if we change a constituent part of a proposition into a variable, there is a class of propositions which are all the values of the resulting variable proposition. this class in general still depends on what, by arbitrary agreement, we mean by parts of that proposition. But if we change all those signs, whose meaning was arbitrarily determined, into variables, there always remains such a class. But this is now no longer dependent on any agreement; it depends only on

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the nature of the proposition. it corresponds to a logical form, to a logical prototype. (TLP 3.315)

Commentators of a russellian persuasion might think that few TLP-ian notions are as clearly defined as this notion “prototype.” Black is one example:

A prototype sometimes is what russell might call a fundamental irreducible form of proposition. it is a pattern for sentences, as often presented by such expressions as ‘xRy’ and the like. (1964, 126)1

Contra Black, however, the situation is not that simple. Pay attention to what Wittgenstein has to say about “a fundamental irreducible form of proposition” in the following pre-TLP-ian passage:

[W.] it is easy to suppose that ‘individual’, ‘particular’, ‘complex’, etc., are primitive ideas of logic. russell, e.g. says ‘individual’ and ‘matrix’ [such as xRy] are “primitive ideas.” this error is presumably to be explained by the fact that, by employment of variables instead of the generality sign, it comes to seem as if logic dealt with things which have been deprived of all proper-ties except complexity. We forget that the indefinables of symbols only occur under the generality sign, never outside it. (Wittgenstein 1961, 105)

W suggests we should not treat Wittgenstein’s use of “variable” in TLP 3.315 on the model of russell’s use of the same term. it should also be noted that the last sentence of W is echoed in the parenthetical remark of TLP 3.24: “the notation for generality contains a prototype.” For the reader’s information, TLP 3.24 is where the term “prototype” appears in TLP for the first time. Hence, if possible, we had better avoid reading russell’s notion of a (real) variable into Wittgenstein’s concept of a prototype.

Hence, it is possible that Black downplays the divergence between russell and Wittgenstein on the notion of a variable. the fact is, we will see, that in formulat-ing their conceptions of logic these two philosophers use the term “variable” in totally different ways. For russell, a proposition (for example, ‘fa’) is the office of a “propositional function” (fx) with a “real variable” (x) that ambiguously denotes the proposition and others of the same logical type (such as ‘fb’ and ‘fc’). in this case the logical type is “individual” in russell’s terms. According to russell, there must be the pure form of a proposition: e.g., the pure form of ‘aRb’ is attained by replacing each of the three constants with a variable. russell construes xξy as “the utmost generalization” of ‘aRb’ (russell 1984, 98).

the contrast is with what Wittgenstein says about a variable. Wittgenstein in-troduces that concept in the context of recognizing an “expression” or a “symbol” in a sign:

every part of a proposition which characterizes its sense i call an expres-sion (a symbol). (the proposition itself is an expression.) expressions are everything—essential for the sense of the proposition—that propositions can have in common with one another. An expression characterizes a form and a content. (TLP 3.31)

Also see:


An expression presupposes the forms of all the propositions in which it can occur. it is the common characteristic mark of a class of propositions. (TLP 3.311)

it is therefore represented by the general form of the propositions which it characterizes. And in this form the expression is constant and everything else variable. (TLP 3.312)

An expression is thus presented by a variable, whose values are the proposi-tions which contain the expression. (in the limiting case the variable becomes constant, the expression a proposition.) i call such a variable a “propositional variable.” (TLP 3.313)

From his use of the predicates “variable” and “constant” in TLP 3.312, it is natural to say that Wittgenstein holds the expression ‘P’ to be the common char-acteristic mark of the propositions ‘Pe’, ‘Pd’ and ‘Pf’, and the expression ‘e’ the common characteristic mark of the propositions ‘Pe’, ‘Qe’ and ‘Re’. given that he aligns the concepts of “variable” and “prototype” with each other, it is a logical conclusion from TLP 3.313 that Wittgenstein regards the “prototype” of the predicate ‘P’ as the propositions ‘Pe’, ‘Pd’ and ‘Pf’, while the prototype of the name ‘e’ as the propositions ‘Pe’, ‘Qe’ and ‘Re’. similarly, to recognize the expression ‘a’ of ‘aRb’ we examine the propositions ‘aRa’, ‘aRb’, ‘aRc’, ‘aSa’, ‘aSb’, ‘aSc’, and so on.

the contrast is with russell’s use of the term “variable.” For russell, the expression ‘a’ of ‘aRb’ is presented by ‘xRb’, where ‘x’ is a variable in his sense. According to russell, ‘xRb’ ranges over the propositions ‘aRb’, ‘bRb’, ‘cRb’, ‘dRb’, and so on. As we will see shortly, this terminological difference is very important, since it conceals a fundamental difference in outlook. nonetheless, perhaps follow-ing the lead of Black, many commentators play russell rather than Wittgenstein.

A most recent and particularly interesting example is Mcginn. On the one hand, she remarks that “variables do not have the role of expression” (Mcginn 2004, 171). Admittedly, this remark captures the whole intent of Wittgenstein’s philosophy of logic, but i doubt that Mcginn correctly executes her insightful re-mark. Particularly, i find that Mcginn still adheres to a central element of Black’s reading; she does not emphatically break away from the interpretation according to which the values of a propositional variable fx are all the sentences formed by replacing x with an appropriate name; ‘fa’, ‘fb’, ‘fc’, ‘fd’, and so on (p. 169). After all, Mcginn seems to me to be unknowingly exemplifying how difficult it is for an exegete really to play Wittgenstein, rather than other philosophers. My meaning will become clear as this paper unfolds.

in my view, the prevailing conventional interpretation of the cluster of TLP propositions displayed above run as follows:

O. An expression is either a proposition (cf. TLP 3.31) or the result of turning some (possibly all) of the constants of a proposition into variables. A prototype is a limit-case of an expression. it is the result of turning all the constants into variables (cf. TLP 3.313). turn ‘aRb’ into ‘xξy’, and the resulting ‘xξy’ is a prototype. to recognize the expression ‘a’ of ‘aRb’ we examine the propositions

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‘aRb’, ‘bRb’, ‘cRb’, ‘dRb’, and so on. A proposition is another limit-case—when all constants are left untouched.

For future reference we will refer to this as the orthodox reading. A word is in order about this dubbing. Admittedly, no commentator has ever presented the orthodox reading in its present form, but i want to say two things to counter the impression that i am setting up a straw man here to attack. First, i think that almost every commentator has embraced and even stated explicitly at least some parts of O.2 Anyway, O can be elicited in its entirety from Black (1964, 122–124). secondly and more importantly, it is a safe and minimalist description of the exist-ing secondary literature to say that no commentator has ever explicitly stated the orthodox reading as his or her foil, as i do here, and mount a criticism against it, as i will do later on.

the proponent of what i just dubbed as the orthodox reading must construe the difference between the proposition ‘aRb’ and the prototype xξy as more than a mere change of letters. Perhaps, s/he is following the lead of russell. As i said earlier, according to russell, there must be the pure form of a proposition: e.g., the pure form of aRb is xξy, in which each of the three constants has been replaced by a variable. russell calls for the “pure form” xξy to combine ‘a’, ‘b’ and ‘R’ into the meaningful proposition ‘aRb’, rather than into a meaningless sign construction such as ‘abR’. Hence, russell takes the role of a variable as presenting a rule for the construction of a class of propositions that are constructed according to a com-mon logical plan or matrix. For example, the value of the variable, F(x), will be all the propositions that contain the function as a constituent; ‘Fa’, ‘Fb’, ‘Fc’, and so on. thereby russell harbors the idea that our language has a two-tier structure: the governing (or proscribing) matrix F(x), on one side, its manifestations (‘Fa’, ‘Fb’, ‘Fc’, and so on), on the other.

Possibly, that is where Wittgenstein comes in. in his rather unusual use of “variable,” Wittgenstein determines the values of the variable to be the propositions ‘Fa’, ‘Ga’ and ‘Ha’. the identity of the expression ‘a’ is known by the proposi-tional bond into which it enters; for example, by the fact that it combines with ‘F’ to make a meaningful proposition, where meaning is not determined by anything like the matrix F(x), given in advance. Possibly, Wittgenstein contests russell by advocating, say, a one-tier view of how our language works. this possibility will become a reality, as the present paper unfolds.

recall that russell aligns a “real variable” with a “pure form.” regarding the ontological status of a “pure form,” russell equates it with the fact that “something has some relation to something” (russell 1984, 114). However, russell is not here using the word “fact” loosely to mean “possibility.” For in his detailed account of acquaintance with pure forms he says,

i do not think there is any difference between understanding and acquaintance in the case of ‘something has some relation to something’. i base this view simply on the fact that i am unable introspectively to discover any difference. (russell 1984, 130)


According to russell, one is acquainted with the logical fact that xξy in the sense that one is acquainted with a non-logical ordinary object. On the other hand, we find russell denies that xξy is a constituent of a proposition.

[if it were a constituent,] there would have to be a new way in which it and the . . . other constituents are put together, and if we take this way as again a con-stituent, we find ourselves embarked on an endless regress. (russell 1984, 98)

What is important for our present purpose is that a variable in russell’s sense is given in advance of the propositions it ranges over. this view of russell could be described as a two-tier view of language.

Does Wittgenstein too embrace a two-tier view of language when, for example, in TLP 3.313 he introduces the term “variable” in its substantive form? Does the Wittgenstein of TLP 3.313 engage in a reification of some kind? Compare what he says about a variable in TLP 3.316:

What values the propositional variable can assume is determined. the deter-mination of the values is the variable. (TLP 3.316, his emphasis)

this emphatic use of the verb ‘to be’ here would make little sense to the proponent of the orthodox reading. if the prototype x were a group of propositions that are the value of the propositional function fx, then this group of propositions could not be the argument of that propositional function. Each proposition of the group could (at most) be the value of that propositional function fx. On the basis of this train of thought, TLP 3.316 would seem to the orthodox reader to be making a propositional function into a chimerical, impossible entity. However, it is possible that Wittgenstein self-consciously does not compose TLP 3.316 in russell’s lan-guage. it is possible that the former explicitly rejects the latter’s two-tiered view of language with “propositional functions” on one side and “real variables” on the other, by declaring that the “variable” is the determination of propositions which contain a given expression.3

there is a pressing reason for our one-tier interpretation of the “variable” of TLP 3.316. Wittgenstein says in TLP 3.313 that an expression is “presented by a variable.” now pay attention to what, for Wittgenstein, it is for something to be presented by a variable.

the general term of the formal series [aRb, (∃x):aRx. xRb, (∃x,y):aRx. xRy.yRb …] can only be expressed by a variable, for the concept symbolized by ‘term of this formal series’ is a formal concept. (this Frege and russell overlooked; the way in which they express general propositions like the above is, therefore, false; it contains a vicious circle.) (TLP 4.1273)

Wittgenstein here alludes to something that “can only be expressed by a variable” and calls it a “formal concept.” reading the modern conception of a variable into the text, one might think that Wittgenstein is endowing a formal concept with a special kind of representation. nothing could be farther from the truth, however. Wittgenstein’s intention is immediately obvious from what he says in the imme-diately preceding proposition:

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the formal concept is already given with an object, which falls under it. One cannot, therefore, introduce both, the objects which fall under a formal concept and the formal concept itself, as primitive terms. One cannot, therefore, e.g. introduce (as russell does) the concept of function and also special functions as primitive ideas; or the concept of number and definite numbers. (TLP 4.12721)

Wittgenstein here makes it clear that our symbolism does not have a sign for a formal concept on top of signs for objects that fall under it. therefore, by saying about something that it can only be expressed by a variable Wittgenstein must mean that it cannot be represented in the sense that a (non-formal) concept is rep-resented by the corresponding sign. the upshot of this consideration is that when it comes to a variable in Wittgenstein’s sense the emphasis should be on a variable not leading existence independently of its values, i.e., the propositions in which the corresponding expression can occur.

this point about the TLP-ian concept of a variable is worth following up, this time in terms of the writing of the middle Wittgenstein:

[W*.] What distinguishes a statement of number about the extension of a concept from one about the range of a variable? the first is a proposition, the second not. For the statement of number about a variable can be derived from the variable itself. (it must show itself.) But can’t i specify a variable by saying that its values are to be all objects satisfying a certain material function? in that case the variable is not a form! (Wittgenstein 1975, 133–134)

Wittgenstein here understands the ontological status of a number in terms of “form”/”variable” rather than in terms of “material function”/“concept.” this rejec-tion of “material function”/“concept” as inadequate in the context of mathemat-ics is not news. to the contrary, the Wittgenstein of W* is fleshing out a position articulated in TLP: the conception of a number as a “form.”

the same holds of the words ‘complex’, ‘fact’, ‘function’, ‘number’, etc. they all signify formal concepts and are presented in logical symbolism by variables, not by functions or classes (as Frege and russell thought). (TLP 4.1272)

the phrase “functions or classes” as in TLP 4.1272 parallels the phrase “material functions or concepts” as in W*. What is important for our present purpose is that the Wittgenstein of W* aligns the concept of a variable with the concept of a form. As is well known, a form in the sense of TLP 4.1272 can only be “shown.”

With Wittgenstein’s concept of a variable in clear sight, we submit the ortho-dox reading to criticism. the proposition ‘aRb’ in russell’s sense is the result of substituting the “constant” a for the “real variable” x of the “propositional func-tion” xRb. in Wittgenstein’s terminology, however, a “variable” is not given in advance of a “proposition.” to put it in the language of tLP 4.12721, the variable or the form is already given with the corresponding proposition. the former is shown in the latter. As a consequence, the aforementioned russellian substitution does not have an analogue in Wittgenstein’s system of thought. With the russel-lian substitution abolished, it is possible that Wittgenstein does not feel inclined to say that the prototype of ‘a’ of ‘aRb’ is presented by the propositions ‘aRb’, ‘bRb’ and ‘cRb’.


that possibility actually happens. For russell, our language has the two-tier structure: the real variable x of the propositional function Px, on one side, and the values a, b, c, and so on, on the other. russell holds the real variable to prescribe to ‘a’ what it may be used to express. But if i do not know the logical form of an “individual” then it will not help me at all to be told that the values of Px are Pa, Pb, Pc, and so on. For what i am told is just that ‘a’ has the same logical form as ‘b’. the situation is similar to the one in which we ask how many chairs there are in this room and receive the answer “As many as in that room.” that is not an answer to our question. We asked how many chairs there were, not whether there were equally many. Of course, russell will not buy this problem, who believes that he knows “by acquaintance” the logical form of an individual. What is important for our present purpose is that the orthodox reading is a very particular interpretation and that it must strike us as such. On our one-tier interpretation, the logical form of an individual is known not “by acquaintance”; it is known, say, by the combinato-rial potential of the corresponding sign. in other words, the logical form of a sign is determined when it is taken together with its logico-syntactical employment (cf. TLP 3.327). the logico-syntactical use of a sign ‘a’ is displayed by the propositions ‘Pa’, ‘Qa’ and ‘Ra’, rather than by the propositions ‘Pa’, ‘Pb’ and ‘Pc’.

the proponent of the orthodox reading will perhaps challenge me for TLP 3.31 and TLP 3.313, holding them to favor his or her reading. (1) Consider TLP 3.31 first. the orthodox reader classifies expressions into propositions and propositional functions (in russell’s sense). s/he at first sight might appear to do justice to the parenthetical sentence of TLP 3.31: “the proposition itself is an expression.” Let us refer to it as TLP 3.31P. When the orthodox reader claims that an expression is either a proposition or a propositional function, s/he must use as a matrix, for example, ‘A living thing is either an animal or a plant’. As two species of the same genus, we place an animal on an equal footing with a plant. But the ordinary “genus/species” framework is not valid here; as i said earlier, a prototype is not another thing beside a proposition.

now i will put forward my own reading of TLP 3.31P. For russell, a proposi-tion is the result of substituting a constant for the “real variable” of a “propositional function” that is given in advance. i interpret Wittgenstein as standing this whole thing on its head by saying that “the proposition itself is an expression.” thereby Wittgenstein implies that in an important sense propositions come first and “propo-sitional functions” second. Wittgenstein has no recourse to what russell calls “propositional functions.”

(2) According to the orthodox reading, a proposition is one limit-case when all constants are left untouched. the other limit-case is a “pure form” in which all the constants of a proposition are turned into variables. A “pure form” and a proposi-tion are the limit-cases of representation in the sense that the north Pole and the equator are the limit-cases of Foucault’s pendulum experiment. the proponent of the orthodox reading purports to do justice to the parenthetical remark of TLP 3.313 (hereafter TLP 3.313P). the first thing to say is that Wittgenstein makes no mention of a proposition as a limiting case. What is more, he cannot regard a proposition as a limiting case. Let me explain. it should be remembered that Wittgenstein sees no

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need for “propositional functions” or “pure forms” on top of propositions. to the contrary, Wittgenstein starts with propositions as if they are all that there is to our language and elicits “variables” from them, not the other way around. secondly, there is no evidence that Wittgenstein regards a “pure form” as a limiting case of representation, either. Consider the remark that immediately precedes TLP 3.313P. As we recall from TLP 4.1272, something “presented by a variable” does not occur as a representing component of a proposition. Hence, a “propositional variable” is not a limiting case of representation. the upshot of this consideration is that TLP 3.313P by no means supports the orthodox reading.

Below i will put forward my own interpretation of TLP 3.313P. the paral-lelisms between the expression and the variable and between the proposition and the constant are not breaking news. the difficult part of TLP 3.313P is rather the modification “in the limiting case.” For a start, an “expression” is not already laid-out there in our language as it is. to the contrary, an “expression” is by no means the easiest thing to obtain in the world. the following two examples, however, could give enough of the general way in which that is so. (1) We often have different expressions (or symbols) when the surface grammar of our language tempts us to think that we have the same expression.

On the face of it i may say “this chair is brown” and “the surface of this chair is brown.” But if i replace ‘brown’ by ‘heavy’, i can utter only the first proposition and not the second. this proves that the word ‘brown’ . . . has two different meanings. (Wittgenstein 1979, 46)

(2) Also see:

the subject-predicate form does not in itself amount to a logical form and is the way of expressing countless fundamentally different logical forms. . . . the forms of propositions: ‘the plate is round’, ‘the man is tall’, ‘the patch is red’ have nothing in common. (Wittgenstein 1975, 118)

in general, resistance to the all-leveling power of our language is the first requisite for perceiving an “expression” in a sign.

Once again, an “expression”—and by extension, a “fact”—is not ready for pickup. Nothing could be farther from the truth than that. A logician may well be remembered for the “expression” that s/he manages to reveal for what it is. Consider the once popular talk of the infinitesimal. some eighteenth-century mathematicians supposed that the differential calculus was all about producing a segment without length bounded by two distinct points; in short, about the infinitesimal. seeing it in isolation, one might believe the sign ‘dx’ refers to the infinitesimal. in so doing, one assimilates ‘dx’ with, for example, a name of a natural number. However, in order to recognize an expression in a sign, we must examine the overall logico-syntactical use of the sign concerned in the whole language to which it belongs. And in the language of the calculus, the sign ‘dx’ does not designate any quantity at all. the so-called infinitesimal is not essential for the language of the calculus. in short, it is not an “expression.”

now i propose that the Wittgenstein of TLP 3.313P intend for “the limiting case” to mark the completion of our complex and sometimes taxing endeavor to


determine the significant use of a given sign in the whole language to which it be-longs. in which case, we result in a proposition in the genuine sense of the word, i.e., something in which we have expressions and only them, where “expressions are everything essential [my emphasis] for the sense of the proposition that proposi-tions can have in common with one another.” (Cf. TLP 3.31.)

so far i have warded off the orthodox reading. it is advisable here to glean in a single passage my own positive accounts proposed sporadically throughout our discussion so far: as a logician, Wittgenstein engages in perceiving an “expression” (a “symbol”) in a sign. Wittgenstein identifies an expression with a “characteristic mark” that a whole bunch of propositions have in common. in the same vein, he says that an expression is presented by a “variable.” in his notions of “expression” and “variable,” Wittgenstein identifies the target view unmistakably and states his own alternative as an explicit improvement. the target is russell’s view: the one that propositions are the office of, more than anything else, the “forms” of propositions there in reality awaiting acquaintance. this two-tier view of russell invites one to take the realist view that language is accountable to something independent of it in order for it to make sense. By comparison, Wittgenstein has no recourse to “pure forms.” He starts with propositions as if they are all that there is to our language and elicits “pure forms” from them, not the other way around. A “variable” or a “prototype” is not an actuality in the sense that (the reference of) the corresponding expression is an actuality.

Does it mean that a variable or a prototype has no reality whatsoever? if someone thinks so, s/he must assume that a value and a “variable” (a “prototype” or a “form”) are analogous with solid and gaseous. But they are analogous, if anything, with a chair and the permission to sit on a chair. it is pointless to deny existence to the permission to sit on a chair no less than a chair. similarly, to flat out reject a “logical form or logical prototype” would be beside the point. not surprisingly, the Wittgenstein of TLP 3.316 does not say that a variable is nothing but the determination of its values, or that it is nothing more than the possibility of producing propositions. to the contrary, Wittgenstein admits that the possi-bility of producing propositions is itself something, and he calls it a “form” (or a “variable”). in the same vein, an expression is more appropriately thought of as the combinatorial potential of the expression. the point is not to assimilate a variable with its values. What that means is that a prototype is not an actual item in the same way the thing we speak of in our proposition is an actual item in the world. that is why the Wittgenstein of TLP 3.316 says that “[t]he determination of the values is the variable.”

Let me draw attention to a single sentence in which Wittgenstein encapsulates the subtle and elusive nature of the subject matter under consideration: “An expres-sion characterizes a form and content” (TLP 3.31). Admittedly, a “variable” (or a “form”) and its value (or a “content”) are different. But it is important for us not to fall prey to the dualism of form and content. For Wittgenstein, form and content are different toto genere. the former cannot be placed on an equal footing with the latter. instead, an expression characterizes a form and content. By emphasizing

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‘and’, i imply that an expression is a form embodied in a content, and that that is the only way for a form to exist.

Let us conclude by reintroducing the question that led to our interpretive undertakings so far: is TLP 3.315 clear evidence of the orthodox reading O, as the proponent supposes it to be? Let me repeat the relevant part of TLP 3.315 for convenience:

if we change a constituent part of a proposition into a variable, there is a class of propositions which are all the values of the resulting variable proposition.

Change the constituent part a of the proposition ‘Pa’ into a variable. the resulting variable proposition will be ‘Px’. But it is open to debate whether the values of Px are the propositions ‘Pa’, ‘Qa’, ‘Ra’, and so on, or whether it is the propositions ‘Pa’, ‘Pb’, ‘Pc’, and so on. the proponent of O will perhaps argue that there are no good grounds for the claim that the values of Px should be ‘Pa’, ‘Qa’, ‘Ra’, and so on. But i add that there is no evidence that Wittgenstein holds the values of Px to be ‘Pa’, ‘Pb’, ‘Pc’, and so on.

there are circumstances under which no two people use the words the same way, and everybody is an expert. serious Wittgenstein scholarship is in great need of a solid method, instead of this or that impressionist interpretation. One way of settling our interpretive debate, i suggest, is to let the test be the ability to help one appreciate TLP 3.333, because, as i said earlier, TLP 3.333 is one of rare pas-sages where Wittgenstein identifies target views unmistakably and states his own alternatives as explicit improvements.

IV. TLP 3.333

A variable in russell’s sense denotes what lies beyond its values. Consonant with this, russell imagines that he is solving a problem by digging down into the condi-tion of meaningfulness, as presented by “logical types.” By contrast, Wittgenstein deploys his concept of a variable to contest russell’s attempt to interpolate beyond the realm of signs. instead Wittgenstein centers on signs on the surface and the way in which they combine with each other. that is part and parcel of the method by which Wittgenstein makes the illusion of a problem dissolve for us, as will emerge. Hence, if someone really became clear about Wittgenstein’s concept of a variable, then s/he could hardly get Wittgenstein wrong.

in order to appreciate Wittgenstein’s argument in TLP 3.333, we must first get clear about what it is not like. My foils are ishiguro and Jolley. i will take ishiguro first:

TLP 3.333b presupposes that there are such second-order predicates or propo-sitional functions which take predicate expressions, or function signs as their arguments. it describes ‘F(fx)’ as a function. it takes ‘fx’ here is a predicate variable, and the arguments that occupy its place should be particular first-order propositional signs. (1981, 52)

the point that ‘F(F(fx))’ features different logical forms of function must be proved, instead of being just “presupposed.” At least, russell would demand such proof.


For the sake of argument, i will buy into that presumption. the problem is that it does not help to reach the desired target, the vanishing of russell’s paradox. in order to reach the desired target, one might assume, as ishiguro does, that “[a] sign for a propositional function which takes names in the argument place cannot [my emphasis] take a sign for a propositional function as argument” (1981, 52). that emphasized modal term must sound familiar to the reader of russell.

it is impossible that if ψx̂ is another function such that there are arguments a with which both ‘φa’ and ‘ψa’ are significant, then ψx̂ and anything derived from it cannot significantly be argument to φx̂. this arises from the fact that a function is essentially an ambiguity. (Whitehead and russell 1910, 50)

in order not to play russell, therefore, an alleged exegete of TLP must accom-modate the modality of impossibility in question in a different way than russell does. Perhaps that is why ishiguro emphasizes that “it is not a prior theory about the types of entities which make the formula ‘φ (φ)’ ill-formed for us” (1981, 51).

What then is the sense in which “a sign for a propositional function which takes names in the argument place cannot take a sign for a propositional function as argument”? to this ishiguro’s answer is that “the result [i.e., ‘φ (φ)’] will be a non-well-formed sequence of signs and not a [genuine] proposition” (1981, 52). With no explanation offered by ishiguro, i will consider the following scenario: ishiguro just stipulates that one should use the sign ‘φ’ in a particular way and only in that particular way, for otherwise one would not have a single symbol, but two different ones. On this scenario, ishiguro will not be calling upon any prior theory of meaning in explaining why the formula ‘φ (φ)’ goes wrong. However, the problem is that the scenario under consideration makes no room for the sense in which one could say that ‘φ (φ)’ is a “non-well-formed sequence of signs.” At worst, ‘φ (φ)’ has two different symbols in the same sign, about which there is nothing worrisome by itself.

At this juncture, it might occur to the reader that s/he had better play down talk of a “non-well-formed sequence of signs” and play up talk of ‘φ (φ)’ having two different symbols. this is actually the strategy adopted by Jolley:

[in ‘F(F(fx))’], the outer function has, so to speak, two blank spots while the inner function has only one. so there is no question of the outer and the inner functions being switched in a type-transgressing way, nor is there any ques-tion of the sameness and difference of the two functions: the two functions are visibly different—our animal eyes (to use Frank ebersole’s phrase) are revelators of the difference. (2004, 285)

Also see:

in TLP 3.333[b], ‘F(F(fx))’ appears to be a case of function acting as its own argument. However, if we consider the logical syntactic application of the sign, we see—as noted above—that although the functions share a sign, ‘F’, they do not share a symbol. the outer ‘F’, considered in its significant use, has two blank spots; the inner ‘F’ has only one. Hence, the two signs are different symbols. given this, there is no danger of russell’s paradox arising. Whatever the signs may lead us (mistakenly) to believe, there will be no sentence in which a function acts as its own argument. (289–290)

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in its present form, Jolley’s reading involves a petito principii. Admittedly, accord-ing to his logical syntactic criterion of a symbol, there are two different functional symbols in ‘F(F(fx))’. However, whether the outer ‘F’ and the inner ‘F’ are dif-ferent symbols is not the issue. the issue is whether they are different symbols in such a way that one is prevented from interpreting ‘F(F(fx))’ as a case in which the same entity is represented by different symbols. nowhere in his exposition does Jolley settle that issue. Hence, russell, the interlocutor, would reject his putative conclusion that “there is no danger of russell’s paradox arising” as involving a transparent petitio principii. so much for the secondary literature.

Below i will try to offer a non-question-begging interpretation of TLP 3.333b. taking a different route (from Jolley), i work with the notion of a “prototype” as in TLP 3.333a: “the functional sign already contains the prototype of its own argument and it cannot contain itself.” it is trivially right to say that a functional sign cannot contain itself, if one conceives a sign as a mere scribble on paper. On the other hand, it is downright wrong to say that a functional sign, qua a (physical) mark on paper, can contain the prototype of its own argument. A physical mark can contain only an actual part of it, but the prototype is not an actual part of a functional sign in the same way as the argument is. Hence, it will not be much to assume that in TLP 3.333a Wittgenstein conceives a “sign” not as a mere scribble on paper but as a sign seen in connection with its prototype.

suppose that a functional “sign,” qua a union of a sign and a prototype, is given to us. “An expression is the common characteristic mark of the propositions in which it can occur” (TLP 3.311). then, according to the terminology of TLP 3.331, “the propositions in which the functional sign can occur” will be given to us. now assume that an argument and a function are essentially complementary, in the sense that the determination of a type of symbol already implies, in a cer-tain manner, the determination of the type (or types) of symbol (or symbols) that, together with the former, forms a proposition. “the determination of the values a propositional variable can assume is the variable” (TLP 3.316). Hence, according to the terminology of TLP 3.316, the preceding assumption is equivalent to saying that the prototype of a functional sign is identical with the prototype of the argument.

to say that the prototype of a functional sign is identical with the prototype of the argument is to say that their combinatorial potentials in the sense explained in the previous section are identical; in short, that their logical forms are identical. in Wittgenstein’s words,

the logical form of the proposition must already be given by the forms of its component parts. (And these have to do only with the sense of the proposi-tions, not with their truth and falsehood.) in the form of the subject and of the predicate there already lies the possibility of the subject-predicate proposition, etc.; but—fair enough—nothing about its truth or falsehood. (Wittgenstein 1961, 23)

the identity between the prototype of a functional sign and the prototype of the argument will play an important role in our exposition of TLP 3.333. so, we need a special name for it: i suggest “the principle of reciprocality.”


there seems to me to be something natural about the principle of reciprocality (or complementarity). Consider the propositional function x snores furiously. i may not know what arguments the propositional function (= F) can combine with to yield a true sentence. But it does not seem to be possible for me not to know what arguments F can combine in a meaningful way. For this reason, we could say that the functional sign ‘F’ prejudges (already contains) the sort (prototype) of its own argument. (the same holds for the argument sign, which can itself be regarded as a functional sign.) And it seems that the functional sign ‘F’ already contains the prototype of its own argument. this needs explanation.

suppose that someone said “the number one snores furiously” (= S). Knowing the prototype of the argument, i would naturally guess that s/he was talking about, for example, a prisoner in cell one. But suppose that our imaginary interlocutor informed me that s/he had meant the number one in arithmetic. What sense would i make out of the sentence S, in this case? Knowing the prototype of the number one in arithmetic, i would perhaps imagine that our imaginary interlocutor had had at the back of his or her mind some very interesting, but almost unheard of, mathematical behavior of the number one, which one could best describe by saying that the number made it absolutely impossible for all the other numbers to sleep soundly (in some mathematical sense of the phrase); perhaps in a similar way in which one can describe the number zero as the destroyer (in some mathematical sense of the word) of division. But suppose that our interlocutor confessed that s/he was not a math geek enough for anything like the aforementioned mathematical discovery, and that our interlocutor instead insisted that s/he had used F in just the same way as in ‘socrates snores furiously’. in which case, either i would simply be at a loss about what to say about the sentence S, or i should tell our imaginary interlocutor that by S s/he must be meaning something like that the prisoner in cell one snores furiously.

it is fitting here to note that we know that S is nonsense, because we know the prototypes of its constituent parts, not because we know that one constituent part by its nature cannot combine with the other. Hence, by saying “the number one cannot snore,” we mean precisely that if you understand ‘socrates snores furiously’ and ‘1 + 2 = 3’ then, for that very reason, you cannot make sense of S. We will come back to this point about nonsense.

the principle of reciprocality is also formally correct. For the sake of simplicity, let’s assume that our language has only propositions of the form φx.

P Q a Pa Qa

b Pb Qb

According to my reading of the TLP-ian notion of a prototype, the prototype of the expression ‘P’ is {Pa, Pb}, and the prototype of the expression ‘a’ is {Pa, Qa}. Contrary to my last point that the prototype of a functional sign is identical with the prototype of the argument, {Pa, Pb} is not identical with {Pa, Qa}. this apparent difficulty is not real, however. it depends on what we mean by ‘Pa’. We are not interested in ‘Pa’ as a mere sentence written on paper. instead we are concerned with

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the combination between the expression ‘P’ with its prototype and the expression ‘a’ with its prototype. Henceforth we will refer to the combination as the proposi-tion ‘Pa’ for lack of a better expression. the proposition ‘Pa’ is presented by the class {Pa, Pb, Pa, Qa}, which reduces to {Pa, Pb, Qa}. similarly, the proposition ‘Pb’ is represented by {Pa, Pb, Pb, Qb}, i.e., {Pa, Pb, Qb}. By the same token, the proposition ‘Qa’ is represented by {Qa, Qb, Pa, Qa}, i.e., {Qa, Qb, Pa}. {Pa, Pb} = {Pa, Pb, Qa, Pa, Pb, Qb} = {Pa, Pb, Qa, Qb}. And {Pa, Qa} = {Pa, Pb, Qa, Qa, Qb, Pa} = {Pa, Pb, Qa, Qb}. the upshot is that {Pa, Pb} = {Pa, Qa}, which is the desired identity.4

equipped with the notions of prototype and combinatorial potential, we now embark on cracking TLP 3.333. russell interprets ‘φ (φx)’ as a case in which a functional sign contains itself. Point i. suppose that he intends for the outer func-tion sign and the inner function sign to be logically the same. the logical identity of a sign is determined by its combinatorial potentials, i.e., by its prototype. now, according to the principle of reciprocality, the prototype of the outer functional sign ‘φ’ (= φo

p) will be identical with the prototype of the inner functional sign ‘φ’ (= φ i

p), which in its turn will be identical with the prototype of the argument (= xp).

this means, more than anything else, that you cannot retain φop while making φi

p into another thing, since what the outer functional φ is is completely determined by φ i

p. in short, φo

p is completely determined by φ ip. By the same logic, φ i

p is completely determined by x

p. From the identity between φo

p and φ ip, however, it follows that

φop is completely determined by x

p. in this way, from the logical point of view, you

cannot differentiate between φop and φ i

p. Admittedly, the letter “φ” is common to both functions. And that fact by itself signifies nothing.

now we are in a position to tackle the following part of TLP 3.333a: “[A] functional sign already contains the prototype of its argument and it cannot con-tain itself.” We first elicited from the first conjunct of TLP 3.333a what we called the principle of reciprocality. then we showed that if the principle is correct then “φ (φx)” must be simplified into “φx.” thereby, we established that if the principle of reciprocality is correct then a functional sign cannot contain itself. (the afore-mentioned simplification would be impossible insofar as a functional sign as a mere scribble on paper were concerned. But it should be remembered that Wittgenstein works with a functional sign as a sign seen in connection with its prototype.)

Point ii. Alternatively, suppose that russell intends for the outer function sign and the inner function sign to be logically different. that will mean that the prototype of the outer function sign is different from that of the inner function sign. From the logical point of view, hence, ‘φ (φx)’ will have the form of ‘ψ (φx)’, only a meaning has not yet been given to ‘ψ’.

Combining Point i and Point ii, we conclude that either ‘φ (φx)’ boils down to ‘φx’ or it can “mean” something one does not yet understand. (i use the scare quotes to indicate that ‘φ (φx)’ does not mean anything yet.) in neither case do we have a case in which a function is its own argument. if russell thinks otherwise, then that can be only because he is overly impressed by the fact that the letter ‘φ’ is common to both functions. And that fact by itself signifies nothing. thus i interpret TLP 3.333b.


russell thinks that he has discovered something, a paradox, and proposes it as an agenda to be solved. Our consideration so far enables us to tell a different story; from the outset there is no such agenda. if we correctly understand the sign ‘φx’, then we immediately see that either ‘φ (φ x)’ means φx or it “means” ψ (φx). Construed in either way, there is nothing paradoxical about ‘φ (φx)’. Contra rus-sell, the so-called paradox need only vanish, rather than being solved. And this is how i understand TLP 3.333d.

We will counterbalance the abstractness of our discussion so far by actually applying it to a version of russell’s paradox, with the hope that in the course of doing so we will be able to incorporate TLP 3.333c.

Define R as {xx ∉ x}. suppose R ∉ R. then R ∈ {xx ∉ x}. then R ∈ R. suppose that R ∈ R. then R ∈ {xx ∉ x}. then R ∉ R. therefore, R ∉ R if and only if R ∈ R.

russell here will probably explain his sign ‘∈’ by a series of examples: a ∈ {x = a}, a ∈ {x = a ∨ x = b}, {a} ∈ {x = {a}}, 1 ∈ {xx2 − 1 = 0}, etc. in fact, here i am following russell’s own definition of ‘∈’: “x ∈ (φẑ) = df φx” (1910: 188). these examples indicate what the relation ξ ∈ ζ is like. i surely follow russell up to this point. i also understand without much difficulty what he means by ‘ξ ∈ ξ’; that is a special case of the formula ξ ∈ ζ. i don’t find it hard to see what ‘ξ ∉ ξ’ means, either, and for that matter, i see that it holds for any set listed above. i also know what the sign construction ‘{xx ∉ x}’ means.

But i do not have any idea of what russell is doing when he proceeds to invite me to suppose that R ∉ R. i simply do not understand what it should be like to suppose that R ∉ R. in other words, i am at a loss about what the “question” whether {xx ∉ x} ∉ {xx ∉ x} (= Q) could be like. the only thing that i can elicit from the explanation russell has given me regarding the constituent signs is that in spelling out the truth condition of Q we can model Q on the pattern of, for example, the question whether 1 ∉ {xx2 − 1 = 0} (= Q*). i can answer Q* by seeing whether 12 − 1 = 0. But i cannot answer Q in a similar way; if i modeled Q on Q*, the former would be identical with itself. As a result, i would be at a loss about how to answer Q until i already knew the answer. the upshot is then that Q is not a question in the ordinary sense. in this way, i cannot figure out what Q should mean on the basis of the explanation russell has given to the constituent signs of ‘R’. that means that the sign construction ‘R ∉ R’ is a new addition whose meaning has not yet been explained.

Our treatment so far of russell’s paradox also helps to decode TLP 3.333c. We understand what ‘ξ ∈ ζ’ means. (i propose to take ‘ξ ∈ ζ’ to be the analogue of the ‘fx’ of TLP 3.333b.) On the basis of that knowledge we know what it is for something to belong to R =

Df {ξξ ∈ ξ}. (i regard this as the analogue of the ‘φ(fx)’

of TLP 3.333b. to be more precise, i should say “φ(gx),” where gx is a special case of fx. But this nicety does not matter much for our present purpose.) But we could not even imagine what ‘R ∉ R’ (or ‘x ∉ x

x = x ∉ x’) might mean. the issue

is not about any kind of psychological incapacity on our side. As was explained in the previous paragraph, the point is that we know that R ∉ R has not yet been

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given meaning, precisely because we know the meaning of the inner function x ∉ x [~φ (fx)]. to borrow from TLP 3.333, we know that the function x ∉ x [~φ (fx)] cannot be its own argument, because we know that the functional sign already contains the prototype of its own argument and it cannot contain itself. the outer function R (i.e., x ∉ x to the left of ‘|’) and the inner function R (i.e., x ∉ x to the right of ‘|’) are different signs. Common to both functions is only the letter ‘R’, which by itself signifies nothing. this is at once clear, if instead of ‘R(Ru)’ we write ‘(∃φ):R(φu).φu = Ru’, where Ru is u ∉ u to the right of ‘|’. in this way, we do not buy into russell’s initial supposition that R ∉ R. We cannot derive anything from a supposition whose meaning we simply do not understand. Herewith rus-sell’s paradox vanishes.

V. TLP 3.331

in the previous section, we drew exclusively on the principle of reciprocality and deduced that either ‘φ (φx)’ boils down to ‘φx’ or it is transformed into ‘ψ (φx)’. then we saw that that either-or is actually vindicated by the proper analysis of russell’s paradox. thereby we partly justified the principle of reciprocality. But this section will provide a more direct justification of the principle of reciprocality. From the aforementioned analysis of russell’s paradox we will first elicit a most natural view about how our language works. then we will show that that view is tantamount to the principle of reciprocality.

russell thinks that there is something worrisome to the sign construction ‘φ (φx)’ that should be explained away by a theory of meaning. For Wittgenstein, as we expounded him in section iv, the fact is that once one correctly understands the inner sign ‘φx’ one immediately recognizes that either ‘φ (φx)’ boils down to ‘φx’ or it “means” ψ (φx), where no meaning has been given to ‘ψ’. in which case, there is nothing to ‘φ (φx)’ for a theory of meaning to explain away. Hence, Wittgenstein’s contention is that from the outset there is no need for a theory of meaning. For Wittgenstein, the theory of logical types is superfluous. the only method of logic is to know how every individual sign signifies.

the difference between russell and Wittgenstein just brought up can be formulated in terms of the distinction between why and that. We cannot have a sign (as opposed to a physical structure on paper) prior to our understanding of it. Once, however, we understand a sign, we automatically recognize whether it is meaningful or not. And that is the be-all and end-all of the matter. there is no asking further, as russell does, why a given sign is meaningful (meaningless), as opposed to observing that it is meaningful (meaningless). For Wittgenstein, a theory of meaning is superfluous, insofar as it is intended to explain the condition under which a sign is meaningful.

the difference between russell and Wittgenstein under consideration can also be formulated in terms of the distinction between real and apparent. in all fairness to russell, we can say that he tries to explain why an apparently meaningful sign construction like ‘φ (φx)’ is really meaningless. the distinction between “appar-ently” and “really,” however, is a mistaken one in the context of logic. We already


saw that we don’t have to ask why a given sign is meaningless, rather than observe that it is meaningless. russell’s “apparent/real” framework makes it look as if logic was similar to a scientific theory, i.e., as if in logic there was such a thing as a theory of meaning. Consonant with his pursuit of a theory, russell concerns himself with sign construction in general, instead of addressing a particular sign construction. the contrast is with Wittgenstein, who raises the question of meaning only concerning a particular sign construction. in the light of the way we make sense with our signs, we see that a certain particular sign construction out of them lacks sense. We have no need for a general theory of meaningfulness, whatever it might mean. nature frequently confronts us with “black-box problems,” in which with the aid of the general laws of nature and a few observables we must see through a complex scheme or hidden mechanism. But there is no “black-box problem” in the realm of logic.

russell tries to explain why an apparently meaningful sign construction like ‘φ (φx)’ is really meaningless. For Wittgenstein, however, in logic a general theory of meaning is superfluous. Logic is all about how our individual signs are used. “A theory cannot dispel the fog” (Wittgenstein 1979, 122). in this sense, the Wittgenstein of TLP 3.333 could best be regarded as recovering pre-russellian innocence in the philosophy of logic.

so far we have proposed what the difference is between Wittgenstein and russell in their general view on how language works. We will examine whether that difference throws light on what Wittgenstein determines to be the error of russell’s theory of logical types:

in logical syntax the meaning of a sign ought never to play a role; it must admit of being established without mention being thereby made of the meaning of a sign: it ought to presuppose only the description of the expressions. (TLP 3.33)

From this observation we get a further view—into russell’s Theory of Types. russell’s error is shown by the fact that in drawing up his symbolic rules he has to speak about the things his signs mean. (TLP 3.331)

unfortunately, in TLP 3.33 Wittgenstein does not make clear what he means by the fallacy of the “meaning” of a sign. His contention cannot be that a physical entity on paper is all that there is to a sign. in that case a sign would not have a mean-ing (in the inchoate, innocent sense of the word). Wittgenstein’s intention can be gleaned from his treatment of russell’s paradox. When it comes to how language works, we saw that Wittgenstein self-consciously remains within the realm of signs throughout. By contrast, russell does not emphatically cultivate such remaining within the domain of signs. Possibly, we can discern in this contrast a hint about what the fallacy of “meaning” is. Below we will test this possibility.

As a catalyst for the identification of the fallacy of the “meaning” of a sign, let me introduce a certain formula that encapsulates the method of remaining throughout on the realm of signs. the formula is: x is a different logical type from y. it enables us to dispense with individual logical types (in the plural). All that we engage in is whether two given signs are of the same logical type or different. the aforementioned austere formula explains why Wittgenstein calls all the constituents

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of a possible proposition indiscriminatingly by one single word. As is well known, the single word is “name.” the contrast is with russell, who does not do logic in terms of the austere formula under consideration. instead he speaks of “real variables” beyond their values. russell must then have seemed to Wittgenstein to interpolate beyond the realm of signs. regarding the motivation for this russellian excess, Wittgenstein must have suspected that he engages in “real variables” (or “logical types”) with a view to regimenting our language in terms of them. the middle Wittgenstein calls this the fallacy of “meaning-bodies.”

Let us imagine cubes, prisms, and pyramids made of glass as being completely invisible in space. Only one surface of each prism, for example a square, and the base of each pyramid are supposed to be coloured. We will then, for instance, only see squares in space. However, we are unable to join together these plane figures arbitrarily because the bodies that are behind the surfaces prevent this. the invisible bodies whose surfaces are the squares determine the law according to which the surfaces can be fitted together. so a word is believed to have, as it were, a meaning-body [Bedeutungskörper] behind it, and from this meaning-body can the grammatical rules that hold true for the word be read off. Here an analogy is drawn between a cube with one painted surface, behind which is an invisible body, and the meaning of sign and something behind it; hence “meaning-body.” the position in which this surface can be placed will depend on the shape of its solid body. One naturally thinks that if we know a cube is back of the painted surface we can know the rules for arranging the surface with other surfaces. Likewise it is natural (but wrong) to think that one can deduce the rules for the use of a sign from its meaning-body. (Wittgenstein 2003, 133)

it does not seem to me to be much to assume that by the fallacy of a “meaning” of a sign in TLP 3.33 Wittgenstein has in mind what he later articulated as the fallacy of a “meaning-body.” if our assumption is right, then we can reformulate pre-russellian innocence, as we called it earlier, in the following way: we cannot have a theory about what is not there in our language. And the “meaning-body” (or “logical type”) is just not there in our language.

the identification of the fallacy of meaning as alluded to in TLP 3.33, as the fallacy of meaning-body helps to avoid a misunderstanding. Wittgenstein’s contention cannot be that a physical entity on paper is all that there is to a sign. to the contrary, a sign interests us insofar as it has a meaning (in the innocent sense of the word). For this reason, not having on hand the identification of Wittgenstein’s intention in his talk of “meaning” in TLP 3.33, we are on the verge of misinterpreting him as spreading skepticism, while he in fact is mounting a criticism.5

Wittgenstein’s criticism of russell for the fallacy of a “meaning-body” is based on the insight that the formula x is the same logical type as y is all that is necessary in logic. so, is logic really all about that austere formula? Consider the following workably simple symbolism: one arbitrarily stipulates that what symbolizes in ‘Fa’ be precisely the fact that ‘F’ is written in upper case and ‘a’ in lower case. For example, the distance between ‘F’ and ‘a’ one discounts


as irrelevant to the way for ‘Fa’ to symbolize what it symbolizes. A different type of symbolizing fact enters this symbolism along with ‘aRb’. this time one stipulates that ‘a’ (‘b’) stands to the left (right) of ‘R’ counts as the symbolizing fact. in general, our imaginary symbolism is constituted by the fact that different symbolizing facts are related to each other in a systematic way. the case ‘Fa’ serves as the first footing to make such a system of differences possible. the new comer ‘aRb’ alludes to ‘Fa’ in order to tell what different symbolizing fact is operative in it. And so on. important for our present purpose is that we do not say about the symbol ‘F’ that it is this logical type or that. We only say that in ‘Fa’ this symbolizes and not that. in short, our symbolism is a system of different symbolizing facts.

note that our symbolism as a system of different symbolizing facts is actually what Wittgenstein calls “our language”:

W*. in our language names are not things; we don’t know what they are: all we know is that they are of a different type from relations, etc. etc. the type of a symbol of a relation is partly fixed by [the] type of [a] symbol of [a] thing, since a symbol of [the] latter type must occur in it. (Wittgenstein 1961, 110)

From the context of W*, it is obvious that Wittgenstein drives at the result that “the [meta-logical] words ‘thing’, [‘relation’], etc. will disappear” (Wittgenstein 1961, 110). Meta-logic here means assertions such as “symbols like this are of such and such logical type.” Hence, we should not misinterpret him in saying “we don’t know what things are.” His real intention is rather to say that in a sense we ought to forget about what things are. instead, all that we need to know is that they are of a different type from relations. We do not need to know what relations, in their turn, are. All that we need to know is that they are a different type from things, etc. that is why, for Wittgenstein, the austere formula x is the same logical type as y is all that is necessary in logic.

the idea of our symbolism as a system of different symbolizing facts helps to recast Jolley’s reading, as displayed in section iv. there i criticized him for com-mitting a petitio principii; the difference between the outer ‘φ’ and the inner ‘φ’ must be such that one is prevented from interpreting ‘φ (φx)’ as a case in which the same entity is represented by different symbols. the same lacuna is found in the reading of ruffino:

For, if we write ‘φ (φẑ)’, even if we are using the same letter for function and argument, the constituted symbols are in fact different, since what symbolizes is not only the sign but also the fact that ‘φ’ occurs at the left of a name (or, if ‘φ’ symbolizes a function of second-order, the fact that ‘φ’ occurs at the left of a first order function symbol). As Wittgenstein said later in the Tractatus, what is common to both functions in the expression above is only the letter ‘φ’, which by itself signifies nothing. (1994, 411–412)

nonetheless, Jolley and ruffino are onto something. in order to recast their read-ings, however, we need only to put to good use what we already have established: that our symbolism is a system of different symbolizing facts. in a symbolism of a system of different symbolizing facts, a symbol is constituted by its difference

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from the symbol it combines with. Consider ‘φ (φx)’. since the symbolizing facts are not identical, it follows from the preceding claim that the outer symbol ‘φ’ is not identical with the inner symbol ‘φ’, even if we are using the same letter for function and argument. Hence, we must interpret ‘φ (φx)’ as ‘ψ (φx)’, only we do not yet understand what the latter should mean. in this way, there is no question of the outer ‘φ’ and the inner ‘φ’ being switched, and for that matter there is no question of self-containment.

the central element of this recasting of Jolley’s and ruffino’s interpretation consists in interpreting ‘φ (φx)’ as ‘ψ (φx)’, where we do not yet understand what “ψ” should mean. that decisive move, in its turn, is made possible ultimately through the principle of reciprocality, or to be more precise, through the principle that the prototype of a functional sign is identical with the prototype of the argu-ment. Assuming the identity (between the prototype of a functional sign and the prototype of the argument) and assuming that alone, it should be remembered, we deduced that ‘φ (φx)’ either boils down to ‘φx’ or “means” ψ (φx). thereby we succeeded in making russell’s paradox vanish; we demonstrated that from the very outset there is no problem to be solved.

i suspect that some will perhaps object that our success involves a petitio principii, as long as the (operative) principle of reciprocality remains an assump-tion. However, our previous discussion already offers a checkup on the truth of the principle of reciprocality. recall that we demonstrated that from the very outset there is no problem to be solved, exclusively under the assumption of the principle of reciprocality. then we proceeded to elicit from that demonstration a single formula x is of the same type as y. next we saw that formula is born out well by the general way our language works. Finally we identified the general way our language works in terms of the notion of a symbolism as a system of differences. As a reminder, in a symbolism as a system of differences, a symbol is constituted by its difference from the symbol it combines with. that delivers a sense in which a symbol essentially refers to another symbol. that is the whole force of the principle of reciprocality. in this way what we initially assumed as a hypothesis, i.e., the principle of reciprocality, has now turned out to be well grounded; it is nothing but a reflection of our actual practice. this marks the completion of our promised (non-question-begging) analysis of Wittgenstein’s criticism of russell in TLP 3.333.

Let us retrace the path that we have traveled, this time with a view to seeing how Wittgenstein recasts russell’s theory of types. We first observed that a propo-sition is analyzed out into its constituent “expressions.” An “expression” already contains the prototype of what can combine with it. An “expression” for a function already contains the prototype of its argument. then we elicited from this notion of an expression and the corresponding notion of a prototype that either ‘φ (φx)’ boils down to ‘φx’ or it has the form ψ (φx). As a consequence, a propositional sign, once analyzed into “expressions,” cannot be contained in itself. thereby we made the illusion of self-containment disappear. in so doing, it should be noted, we remained throughout in the realm of expressions and depended entirely on


the way in which a given individual expression is used in the broader context of a proposition. to be clear about the logic of a given individual sign is the whole theory of types. thus i interpret TLP 3.332:

no proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the “whole theory of types”).

this minimalist recasting consists in understanding a proposition as a function of its constituent expressions. the contrast is with russell’s concept of a “complex name,” as explained in section ii. unlike a proposition, a “complex name” is not analyzed out into its constituent signs. For this reason, whether a complex name makes sense cannot be determined exclusively from the way in which its constituent signs are combined. that is why russell is prompted towards the pseudo-problem of whether ‘φ (φx)’ makes sense or not. the complex name ‘φa’ refers to the value that happens to be described in terms of the function φx̂ and the argument a. the “object φa” and other objects of the same logical type are “ambiguously denoted” by the propositional function φx̂, or to be more precise, by the “real variable” con-tained in it. A moment of reflection suffices to make evident that the complex name ‘φa’ is not a proposition in the ordinary sense of the word. the proposition ‘φa’ is not “ambiguously denoted” by its constituent φx. instead, the proposition ‘φa’ is analyzed out into its constituents, φ and a. in this way, the mistake of russell’s theory of types consists in the notion of a real variable and the correspond-ing notion of a propositional function.

to my best knowledge, however, almost all commentators have been confus-ing what Wittgenstein calls a variable with what russell calls a (real) variable. Or i should say that not a single author has expressly brought that confusion to light. (the explanation could be that russell’s language, so deeply entrenched into our thinking, does not strike us as a very particular language, still less a false one.) the price to be paid is that Wittgenstein’s intention in TLP 3.333 is completely lost on them. they implicitly assume that the only way to appreciate TLP 3.333 is to ensure that ‘φ (φx)’ has the form of ‘ψ (φx)’. in order to make that point about the (alleged) real form of ‘φ (φx)’, they point out the difference in syntax between the outer sign ‘φ’ and the inner sign ‘φ’. However, that syntactical difference does not by itself prevent one from interpreting ‘φ (φx)’ as a case in which a function is its own argument. By contrast, i emphatically do not try to prove that ‘φ (φx)’ has the form of ‘ψ (φx)’. As a matter of fact, my explication so far renders such proof as superfluous; in order to appreciate TLP 3.333, we need only derive the follow-ing either-or: either ‘φ (φx)’ boils down to ‘φx’, or it “means” ψ (φx). And it is important that the aforementioned either-or was possible precisely because of the principle of reciprocality: that the prototype of a functional sign is identical with the prototype of the argument. note that we reached the principle of reciprocality, more than anything else, by resisting the confusion between what Wittgenstein calls a variable with what russell calls a (real) variable. if i am not mistaken, in this way i might have settled the interpretive issue (of orthodox vs. unorthodox) raised in section iii.

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VI. “WE CANNOT GIVE A SIGN THE WRONG SENSE”

Our discussion so far places us in a position to tackle the question raised at the introduction of this paper, i.e., the question: who gets whom wrong. For conve-nience let me repeat what russell and Wittgenstein say about each other’s view:

[R.] [TLP 3.331 must be wrong.] the theory of types, in my view, is a theory of correct symbolism: (a) a simple symbol must not be used to express any-thing complex: (b) more generally, a symbol must have the same structure as its meaning.

[W.] that is exactly what one cannot say. you cannot prescribe to a symbol what it may be used to express. All that a symbol can express, it may express. this is a short answer but it is true!

From russell’s word “correct” in R the Wittgenstein of W rightly elicits the notion of meaning-bodies as transcendent anchor points of our language: that behind each sign there is a non-linguistic entity that prescribes the way it may be used and may not be used. But two questions remain to be answered. First, is the Wittgenstein of W right in rejecting the notion of a “meaning-body”? second, what does the Wittgenstein of W mean by saying that “that is exactly what one cannot say”?

in order to address the first question, i will introduce the contrast between anti-sense and non-sense, in the sense yet to be explained. russell holds ‘φ (φx)’ to be the result of transgressing the rules of logical syntax, as dictated by meaning- bodies. in this sense, russell regards ‘φ (φx)’ as a piece of anti-sense or as a prohibited sense. However, there is a tension involved in the notion of anti-sense. the notion of anti-sense trades on the idea that the result of violating the rules of logical syntax is an illegitimate sequence of signs. On the one hand, the result-ing illegitimate sequence of signs (= S) must be outside the logical space for any legitimate sign construction that is determined by a given prior theory of meaning (= T). On the other hand, one must be able to think S (in the broadest sense). if S were a complete gibberish as in Alice in Wonderland, then, strictly speaking, one could not even ask whether it was inside or outside the logical space determined by T. in this way, the proponent of T imagines that s/he has a thought outside of what s/he calls the logical space. in short, s/he would have a thought outside of the space of thought. the proponent of T will perhaps wish to get out of this obvious ambiguity by calling S a thought only in the psychological but not in the logical sense of the word. that would be pushing the old problem one step further back, for a psychological thought is after all a thought.

We cannot think anything unlogical, for otherwise we should have to think unlogically (TLP 3.03). to present in language anything which “contradicts logic” is as impossible as in geometry to present by its co-ordinates a figure which contradicts the laws of space; or to give the co-ordinates of a point which does not exist. (TLP 3.032)

Wittgenstein here does not say anything like that no one has ever entertained an illogical thought. Many have done so, philosophers included. Wittgenstein’s contention is rather that one does not go illogical by falling into anti-sense. if one


placed something outside logical space it would be too late, for s/he would have to somehow think. therefore, anti-sense cannot be the correct interpretation of how our language (or thought) goes astray. in other words, it cannot be the correct interpretation of how nonsense (in a sense of the word to be determined) arises.

By ‘nonsense’, we must designate something that from the outset cannot be outside logical space. What fits the bill is the notion of lack of sense (Unsinn). We must not reify a sign construction that lacks in sense into something that lacks in sense. such reification would bring us back to the same old problem of some-how thinking that which is supposed to be outside logical space. instead we rest content with saying about the sign construction that it is devoid of sense. i will call a sign construction that it is devoid of sense “non-sense,” in lieu of a better expression. the notion of non-sense, in fact, was anticipated by our exposition of TLP 3.333b in section iv. recall that once we correctly understand the inner sign ‘φx’ we immediately recognize that either ‘φ (φx)’ boils down to ‘φx’ or it “means” ψ (φx), where no meaning has been given to ‘ψ’. this indicates that our knowledge that ‘φ (φx)’ is nonsense is a function of our understanding of the meaning of an individual sign, which in our case is the inner sign ‘φx’. For another example, recall our analysis in section iv of ‘the number one snores furi-ously’. i did not try to prove—we will see shortly that there is no such proof—that it is nonsense to say of a number that it snores furiously. What i established is that the sentence in question is non-sense. i said something to the effect that “if anyone uses words with the meanings i do, [i.e., understands their prototypes as i do,] then he can connect no sense with this combination. if it makes sense to him, he must understand something different by these words from what i do” (Wittgenstein 1975, 53).

in general, our recognition of a sign construction as nonsense is a function of our understanding the senses we make with the individual signs. this account of nonsense replaces the previous notion of anti-sense, i.e., the notion that our lan-guage goes amiss by violating the rules of logical syntax. to the contrary, language goes nonsense only because it lacks sense. For example, ‘ψ (φx)’ lacks sense; no meaning has been given to ‘ψ’.

now we turn to Wittgenstein’s intention in declaring that “that is exactly what cannot be said.” Perhaps, the indexical pronoun refers to russell’s claim that a symbol must have the same structure as its meaning. At first sight Wittgenstein might appear to be saying something schizophrenic: “that a symbol must have the same structure as its meaning cannot be said.” Partly for this reason, this claim of Wittgenstein’s has baffled many commentators, as has his notion of showing, generally.

there is nothing baffling here, however. everything depends on what Witt-genstein means by ‘saying’. Let us take a detour by considering the following case: ‘a is simultaneous with b’ (= aRb) obviously represents the same fact as ‘b is simultaneous with a’. We call such a relation symmetrical. symmetry here is logical, in the sense that the relation R cannot lack it without changing into another thing. Where symmetry means logical symmetry it cannot be expressed by writing ‘(x,y)(xRy ⊃ yRx)’, or by any other proposition for that matter. the reason is simply that ‘xRy ⊃ yRx’ already presupposes that it has sense for xRy to have a different

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sense from yRx (cf. Wittgenstein 1979, 242). there is nothing baffling about it, if someone says that it cannot be said that xRy is a logically symmetrical relation.

the situation is exactly the same with the alleged proposition that a symbol must have the same structure as its meaning. it would at least have sense to say that a given symbol did not have the same structure as its meaning. if the property of having the same structure as its meaning is an essential (logical) one, then we would not have the very same symbol as before. (the situation is different when the property in question is an inessential, accidental one.) Hence, it is nonsensical to deny a symbol has the essential (logical) property in question. then it is non-sensical to assert the symbol has it as well.

the existence of an internal property of a possible situation is not expressed by means of a proposition: rather, it expresses itself in the proposition repre-senting the situation, by means of an internal property of that proposition. it would be just as nonsensical to assert that a proposition had a formal property as to deny it. (TLP 4.124)

it is impossible to distinguish forms from one another by saying that one has this property and another that property: for this presupposes that it makes sense to ascribe either property or either form. (TLP 4.1241)

in this way, that a symbol must have the same structure as its meaning cannot be said. it can only be shown. the question to be asked is: Have we lost something important here? if someone thinks so, it will be because s/he holds to russell’s notion of the two-tier structure of our language; “form” on one side and “content” on the other. For Wittgenstein, an expression is form and content. Form is embodied in content, and that is the only way for it to exist.

in conclusion, i wish to draw attention to some TLP propositions that nicely and succinctly put the logical view we have so far expounded.

if everything in the symbolism works as though a sign had meaning, then it has meaning. (TLP 3.328, my emphasis)

it cannot fail to arrest the reader’s attention that Wittgenstein delivers his intention by deliberately breaching the grammatical distinction between the subjunctive mood and the indicative mood of the verb “to have.” Also see:

now we understand our feeling that we are in possession of the right logical conception, if only all is right in our symbolism. (TLP 4.1213)

to be sure, this is not anything of a tautology. to the contrary, Wittgenstein here engages in interpreting the notion of “the right logical conception” correctly. His target is the realist interpretation in terms of “meaning-bodies,” and he re-places it with a language-oriented interpretation. We are in possession of the right logical conception, if all is right in our symbolism here, as opposed to reality out there.

For the Wittgenstein of TLP 3.328 and TLP 4.1213, logic does not con-sist in our having a “meaning” beyond the sign concerned that permits us to use it in the way we do. instead, “a possible sign must also be able to signify. everything which is possible in logic is also permitted.” A correct logical point


of view is incorporated by signs seen together with their whole logical space. A sign seen together with the whole logical space is a world on its own. it is not to be regarded as the reflection of something behind it. A sign seen together with the whole logical space is not a symptom of something else, but the thing itself. the logic of the world is known to us, whole and undivided, from signs in logical space.

Once again, a correct logical point of view is not a function of “meanings,” qua transcendent anchor points of our language. the logic of the world is pres-ent at the surface of our language. to put this point in terms of “anti-sense,” we cannot give a sign the anti-sense. And it is this impossibility that makes logic a priori. “What makes logic a priori is the impossibility of illogical thought” (TLP 5.4731). instead, it is only possible for us to fail to give meaning to a sign, even if we believe that we have done so. thus i interpret the Wittgenstein of TLP 5.4732: “We cannot give a sign the wrong sense.”

A proposition is senseless because we have not made some arbitrary deter-mination about the sign concerned, not because the sign is in itself impermissible. Otherwise, it would be possible for us to make mistakes in logic by giving a sign the wrong sense. in which case, it would not be the case that logic is accountable to nothing but itself.

Logic must take care of itself. A possible sign must also be able to signify. everything which is possible in logic is also permitted. (“socrates is identical” means nothing because there is no property which is called “identical.” the proposition is senseless because we have not made some arbitrary determina-tion, not because the symbol is in itself impermissible.) in a certain sense we cannot make mistakes in logic. (TLP 5.473)

the sense in which we cannot err in logic is known from the fact that we cannot give a sign the wrong sense, i.e., that our language cannot go wrong because of “meanings” hidden from us. given that we cannot give a sign the wrong sense, logical errors should not be compared with those we make because our present knowledge is limited in this way or that.

Let us turn our attention to yet another formulation of the view about the nature of logic we are concerned with.

the rules of logical syntax must follow of themselves, if we only know how every single sign signifies. (TLP 3.334)

Before proceeding, a terminological point is in order; i define logic as that which our language is accountable to in order for it to make sense. this need not be a kind of definition that comes at the end of investigation, but it is a kind of definition that at least tells us what we are talking about, what is the target of our investigation. A conception of logic depends on how to interpret the italicized “that.” in one conception, the reference of “that” is reality independent of language, while in the other conception language itself. russell embraces the first conception; behind each sign there is a non-linguistic entity or “meaning-body” which prescribes the way it may be used and may not be used. Once you know what the word signifies, you understand it, you know its whole application.

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According to Wittgenstein, there is no explaining or justifying the rules for signs in terms of their meaning-bodies. For him, therefore, the rules in accordance with which our signs are used, as they stand now, are all that there is to our signs. Because logic is defined as that which language is accountable to for it to make sense, we can say, the rules in accordance with which signs are used, as they stand now, are the rules of logical syntax. that is one interpretation of the phrase “rules of logical syntax” (= I

W); the subscript stands for Wittgenstein.

russell embraces a different interpretation (= Ir). (Of course, russell does not

make use of the phrase “rules of logical syntax” anywhere. i am concerned with the idea underlying what he actually says.) By ‘rules of logical syntax’, russell means the rules in accordance with which signs should be used in order for us to make sense with them. the rules of logical syntax in russell’s sense are supposed to make language logical. Wittgenstein’s notion of rules of logical syntax stands this view on its head. Language is logical, say, from the very beginning. if anything, logic follows language, not vice versa. Language is accountable to nothing but itself in order for it to make sense.

With these two interpretations of “rules of logical syntax”—IW

and Ir—in

mind, we turn to TLP 3.334. suppose that in TLP 3.334 Wittgenstein has in mind the interpretation I

W of “rules of logical syntax.” On the interpretation I

W, the

rules in accordance with which signs are used, as they stand now, are the rules of logical syntax. We know how each individual sign signifies by knowing the rules in accordance with which they are used. As a result, on the interpretation I

W, TLP 3.334 turns into a vacuous remark. that is sufficient to exclude I

W as

a possible reading of TLP 3.334. suppose, on the other hand, that Wittgenstein harbors the interpretation I

r of “rules of logical syntax.” On the interpretation I

r,

“rules of logical syntax” are things to be excavated. As such, they do not “follow of themselves” in the ordinary sense; they are not understood of themselves. On the interpretation I

r, therefore, TLP 3.334 must be simply false. the interpreta-

tion Ir of “rules of logical syntax” is attractive only to those who suppose rules

of logical syntax, thus interpreted, to explain our knowledge of how each indi-vidual sign should signify. But obviously in TLP 3.334 Wittgenstein intends for our knowledge of how each individual sign should signify to be an explanans. therefore, it is obvious that Wittgenstein does not work with the interpretation I

r of “rules of logical syntax.”

in this way, Wittgenstein’s intention in TLP 3.334 might appear to be lost be-tween I

W and I

r. now that we have ruled out I

W as a possible reading of TLP 3.334,

i suggest rehabilitating TLP 3.334 by modifying Ir. We need not look far, though.

We must realize that in TLP 3.334 Wittgenstein does not use the phrase ‘rules of logical syntax’ in the same way that russell does. to the contrary, Wittgenstein is putting his finger on his target. TLP 3.334 can be then paraphrased into TLP 3.334*: “the so-called ‘rules of logical syntax’ [as russell understands them] must be superfluous, for they must be understood of themselves once we know how each individual sign signifies.” TLP 3.334* is yet another unmistakable formulation of the rejection of the notion of a “meaning-body.”6


ENDNOTES

1. these sentences are from the dissertation thesis of D. s. shwayder, which Black approvingly quotes in his comment on TLP 3.315.

2. One exception is Brockhaus 1991, 167.

3. At this juncture, the orthodox reader will perhaps complain that i am playing fast and loose with the verb ‘to be’ of TLP 3.316. Wittgenstein may intend just that the values of a propositional variable are settled in advance and a new value cannot be discovered later. that putative intention of Wittgenstein, i should say immediately, is contradicted by the two-tiered view of language. this is because the two-tiered view involves the notion of a real variable and because the real variable of φx “ambiguously denotes” ‘φa’, ‘φb’, ‘φc’, etc.

4. it is fitting in this connection to compare the orthodox reading, as introduced and criti-cized in the previous section. According to it, the proposition ‘Pa’ is presented by the set {Pa, Pb, Qa, Qb}. that is because, according to the orthodox reading, the prototype of the expression ‘P’ is {Pa, Qa, Pb, Qb}, and the prototype of the expression ‘a’ {Pa, Pb, Qa, Qb}. And every other proposition is presented by the same set {Pa, Pb, Qa, Qb}. the analogue within the framework of the orthodox reading of the question (= Q1) whether {Pa, Pb} = {Pa, Qa} would be the question (= Q2) whether {Pa, Qa, Pb, Qb} = {Pa, Qa, Pb, Qb}. By Q1, we address the problem of identity between the prototype of the expression ‘a’ and the prototype of the expression ‘P’. And that problem is what Wittgenstein TLP 3.333a tries to answer when he says something to the effect that the prototype of a functional sign is identical with the prototype of the argument. to be sure, Q2 is not a problem of identity in the sense that Q1 is a problem of identity. What is more, Q2 does not even appear to be a problem. therefore, TLP 3.333a must be completely lost on the orthodox reading.

5. some authors, without scruples, have imputed the notion of a “meaning-body” to TLP. A case in point is glock 1996, 239. russell did not recognize that the Wittgenstein of TLP 3.33 caught him embracing the notion of a meaning-body. Decades later, glock played russell in his reading of TLP.

6. i offer my emphatic thanks to Michael Kremer for his helpfully harsh and forthright comments on the original draft. About them was there always something playful and good-natured. nonetheless, i lay claim to any of the shortcomings that remain.

BiBliOgrApHy

Black, Max. 1964. A Companion to Wittgenstein’s ‘Tractatus.’ ithaca: Cornell university Press.

Brockhaus, r. r. 1991. Pulling Up the Ladder: The Metaphysical Roots of Wittgenstein’s tractatus Logico-Philosophicus. illinois: Open Court.

glock, Hans-Johann. 1996. A Wittgenstein Dictionary. Oxford: Blackwell.ishiguro, Hide. 1981. “Wittgenstein and the theory of types.” in Perspectives on the Phi-

losophy of Wittgenstein, edited by ian Block, 43–59. Oxford: Basil Blackwell.Jolley, K. D. 2004. “Logic’s Caretaker—Wittgenstein, Logic and the vanishment of russell’s

Paradox.” The Philosophical Forum 35: 281–309.doi: http://dx.doi.org/10.1111/j.1467-9191.2004.00175.x

Mcginn, Mary. 2004. Elucidating the tractatus. Oxford: Clarendon Press.

146 Daesuk Han

ruffino, M. A. 1994. “the Context Principle and Wittgenstein’s Criticism of russell’s theory of types.” Synthese 98: 401–414. doi: http://dx.doi.org/10.2307/2369948

russell, Bertrand. 1908. “Mathematical Logic as Based On the theory of types.” American Journal of Mathematics 30: 222–262. doi: http://dx.doi.org/10.2307/2369948

———. 1984. Theory of Knowledge: The 1913 Manuscript. London: george Allen & unwin.Whitehead, A. n., and russell, Bertrand. 1910. Principia Mathematica, volume 1. London:

Cambridge university Press.Wittgenstein, Ludwig. 1922. Logico-Philosophicus Tractatus, translated by C. K. Ogden.

London: routledge & Kegan Paul.———. 1961. Notebooks 1914–1916. Oxford: Basil Blackwell.———. 1975. Philosophical Remarks. Oxford: Blackwell.———. 1979. Ludwig Wittgenstein and the Vienna Circle. Oxford: Blackwell.———. 2003. The Voices of Wittgenstein: The Vienna Circle. London: routledge.

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Wittgenstein on Russell’s Theory of Logical Types