3.42 Although a proposition may determine only one place in logical space, at the same time the whole of logical space must already be given by it.

(Otherwise negation, logical sum, logical product, etc. would introduce ever new elements – in coordination.)

(The logical scaffolding around a picture reaches through the whole logical space. The proposition reaches through the whole logical space.)

A set of coordinates determines only one figure, but the coordinates themselves imply a set of axes, a space or dimension (or set of dimensions) in which the figure can be. Similarly with propositions and logical space. Since logical space applies to all things, all possibilities, as such then any possibility implies it. Remember, in case this sounds controversial, that we are pretty much still dealing with definitions here, and that there might yet prove to be no cash value whatsoever to any of this.

3.411 Geometrical and logical place agree in that both are possibilities of existence.

That is, each is the possibility of an existence, and in that they are the same. The difference is that geometrical space is more limited than logical space. Geometrical space covers geometry. Logical space covers everything.

3.41 The propositional sign and the logical coordinates: That is the logical place.

There you are.

3.4 A proposition determines a place in logical space. The existence of this logical place is guaranteed by the existence of the constituent parts alone, by the existence of the meaningful proposition.

That is, by the existence of “des sinnvollen Satzes.”

Noteworthy here is the apparent equation of the constituent parts and the meaningful proposition. But this might not be surprising if the proposition has been whittled down to an imperceptible commonality among various sentences and “the constituent parts” are simply possibility spaces. What we seem to have are possibilities, and of course each possibility guarantees the existence of a space for it in logical space. Because what follows “of course” in that sentence is a tautology.

3.3442 The sign for a complex is not arbitrarily resolved by analysis in such a way that its resolution would be different in each sentence structure.

(P&McG have ‘proposition,’ Ogden has ‘propositional structure’ here for Satzgefüge.)

Analysis must be consistent, I suppose.

3.3441 One can, e.g., express the common feature of all notations for truth-functions thus: It is common to them that they all— e.g. —can be replaced by the notation “~p” (“not p”) and “p v q” (“p or q”).

(This shows [gekennzeichnet] the way that a specific possible notation can give us general insight.)

A specific kind of logical notation or concept-script can show us something general, namely that a general range of bits of language can be translated into them. We seem to learn nothing about the world in this way though, except that different bits of it have this in common, which we could have known already, without the symbolism.

3.344 That which signifies in a symbol is the common feature of all symbols that can take its place following the rules of logical syntax.

Essentially nothing, in other words. No thing. What matters is the role within logical syntax, the place that is taken there.

3.343 Definitions are rules for translation from one language into another. Every right sign-language must allow of translation into every other by means of such rules: This is what they must all have in common.

There is no ostensive definition in this sense. To define is to determine one bit of language as equivalent to another. Any adequate language will be able to cope with all such possible translations (perhaps just by having a means for importing foreign words and phrases, so that the English for Brie is “Brie” for instance). Again, what is had in common by bits of language is just this relation between them, not anything metaphysical.

3.3421 A particular way of symbolizing may be unimportant, but it is always important that this is a possible way of symbolizing. And it is like this in philosophy generally: the particular proves unimportant time and again, but the possibility of each particular gives us an insight into the essence of the world.

What matters is not what happens to be so, but the possibilities. These constitute the essence of the world that we are concerned with. So these are what matter to the philosopher. Nothing really new here, despite the exciting reference to gaining information or clarity on the essence of the world.

3.342 In our notations there is indeed something arbitrary, but this is not arbitrary: if we have determined something arbitrarily then something else must be the case. (This stems from the essence of the notation.)

Or: This depends on the essence of notation. The best candidate for something arbitrary in a notation would seem to be the signs used, that “dog” (rather than “chien” or “Hund” say) means dog, and so on. But from this arbitrary designation, once the meaning has been defined or determined, any necessary truths about dogs (that they are animals, say) will then be necessarily true of “dogs.” It is the designation or determination or definition that produces the necessity. The essence of notation then seems to have something to do with definition, which sounds right and certainly fits the general Tractarian view of language as representing designated objects in states of affairs.

3.3411 Thus one could say: The actual name is that which all symbols that signify an object have in common. It would then follow that no composition at all is essential for a name.

P&McG are simply wrong here, as Black (p. 152) points out.

What is essential to a name is that it names the particular thing it names. If it is the name of a Tractarian object, a point in logical space, then it is as simple as a name can be. It is simply a name that simply names a simple object. Anything complex in the name would be inessential. On the other hand, the actual name referred to here would, or at least could, have no discernible features at all. No sound, or appearance, for instance. While it becomes simpler it also becomes invisible. Could it be a phantom?

3.341 The essential in a proposition is thus that which all propositions that can express the same sense have in common.

And likewise in general the essential in a symbol is that which all symbols that can fulfill the same purpose have in common.

What is common to all propositions that can express the same sense? The ability to express just that sense. Must this ability depend on some other common feature? Not that I can see. The same goes for symbols. So 3.341 seems to be quite empty, its only purpose being to confirm the suspicion about 3.34 that it was not an important metaphysical truth.

3.34 A proposition possesses essential and accidental features.

The accidental are those features that come from the particular way of producing the propositional sign. The essential are those which alone enable the proposition to express its sense.

The accidental features would then seem to be what it sounds like, if spoken, or how it looks, if written, etc. The essential would belong not to the sentence but to the proposition. What could those be? Not its sense, since they are said to be whatever it is that enables the proposition/sentence to express its sense. They are presented as non-physical sense-making powers or prerequisites. Is this metaphysics, or have we strayed into nonsense?

3.334 The rules of logical syntax must be self-evident if one only knows how each one of the signs signifies.

Are the rules of logical syntax identical with the ways of signification?

3.333 A function can therefore not be its own argument, because a functional sign already contains the prototype of its argument and it cannot contain itself.

Let us suppose that the function F(fx) could be its own argument; there would then therefore be a proposition: “F(F(fx))” and in this the outer function F and the inner function F must have different meanings, because the inner has the form ø(fx), the outer the form ψ(ø(fx)). Only the letter “F” is common to both functions, but that in itself has no significance.

This becomes clear immediately if instead of “F(F(u))” we write “(Eø) : F(øu) . øu = Fu”.

Thus Russell’s Paradox is laid to rest.

Black (p. 149) says of this reference to Russell’s paradox that Wittgenstein “presumably [means] the variant concerning functions (rather than classes) that are not themselves included in the set of their own values.”

Having rejected Russell’s solution in 3.332, here Wittgenstein rejects his problem too. (The “E” here should, by the way, be the existential quantifier.)

Ostrow (p. 67) compares this remark with 3.1432. “Wittgenstein’s aim is once more to bring out how our hold on a notion of logical form is parasitic on how we speak, on what it makes sense to say.” Russell’s belief that a theory of types is needed suggests that he confusedly thinks something like the opposite of this, Ostrow thinks.

3.332 No proposition can express something about itself, because a propositional sign cannot be contained in itself (this is the whole “Theory of types”).

Russell has made a mistake analogous to mixing up concepts and objects, in Frege’s sense of the terms.

3.331 From this remark we get a comprehensive view of Russell’s “Theory of types”: Russell’s error is shown by his having to speak of the meaning of a sign when putting together his rules for signs.

Black (p. 146) suggests ‘get a comprehensive view of’ for sehen wir in Russell’s ‘Theory of Types’ hinüber while P&McG have ‘turn to’ and Ogden has ‘get a further view – into Russell’s …’ ‘Hinüber’ means ‘over’ and ‘in’ means ‘in,’ so we are seeing into Russell’s theory, getting insight, but also getting an oversight, either looking beyond it or, as Black suggests, looking over the whole thing (but at it, not to something on the other side).

Russell’s system is impure, therefore, in the sense of TLP 3.33.

In Principles of Mathematics (1903) Russell proposed his first, simple version, and in “Mathematical Logic as Based on the Theory of Types” (1908) he proposed the ramified version. The basic idea of the theory of types is that classes are not objects. This makes for a simpler ontology, and means that it is nonsense to talk about a class being a member of a class in the way that an object (a spoon or a barber, say) can be a member of a class. But the simple version does not get rid of all paradoxes. The ramified theory of types is based on the vicious circle principle, i.e.: “Whatever involves all of a collection must not be one of that collection'; or 'If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total.” (see Principia Mathematica, 1 (1910), Introduction, ch. 2, p.1). In the simple theory, there are types of objects (real objects, classes of objects, classes of classes, and so on). In the ramified theory there are also types of properties (properties, properties of properties, etc., e.g. shyness is nice or red is a property of objects).

The theory of types is developed in order to avoid certain paradoxes, e.g. those involving infinity (how it seems possible to come up with an infinite number of objects from a finite collection, and thus prove a priori that the world contains an infinite number of objects—i.e. prove the axiom of infinity) and the class of all classes that are not members of themselves. “Now the theory of types emphatically does not belong to the finished and certain part of our subject: much of this theory is till inchoate, confused, and obscure. But the need of some doctrine of types is less doubtful than the precise form the doctrine should take.”[1]

Classes are logical fictions, and if they are treated as being real objects, whose names have real signification, then the sentences in which they are treated this way will be devoid of meaning. “The supposition that a class is, or that it is not, a member of itself is meaningless in just this way.”[2] Since we cannot know whether the axiom of infinity is true, we cannot know whether the world is infinitely or, on the contrary, finitely divisible. If the latter is the case, then logical analysis has a chance of finding the real simples or particulars that make up the world.

F. P. Ramsey, following Wittgenstein, objects to this theory. Propositional functions are symbols, while individuals are objects. So talk of functions of functions is not like talk of functions of individuals. “For the range of values of a function of individuals is definitely fixed by the range of individuals, an objective totality which there is no [getting?] away from. But the range of arguments to a function of functions is a range of symbols, all symbols which become propositions by inserting in them the name of an individual. And this range of symbols, actual or possible, is not objectively fixed, but depends on our methods of constructing them and requires more precise definition.”[3]

Wittgenstein’s criticism in the Tractatus begins (3.331) with the observation that formal logic is supposed to be purely formal, yet Russell has to refer to the Bedeutungen of the signs in the drawing up of his symbolic rules. This shows that something has gone wrong with the introduction of the theory of types. But what? The theory of types amounts to just this: “No proposition can say anything about itself, because the propositional sign cannot be contained in itself.” (3.332, Ogden).

It is, it seems, a bit like Frege’s distinction between concept and object. Talking about concepts and functions makes them sound like objects, but we need to look at their role in the system to understand them properly. The fact that ‘The class of classes’ looks and sounds like ‘The class of spoons’ does not mean that they are the same. In Frege’s terms, ‘The class of classes’ can be analyzed into the function ‘The class of ( )’ and the argument ‘classes’. Functions are not arguments. A good system of symbols will show this distinction. It does not need to be explicitly stated in a seemingly ad hoc way.

Russell’s paradox shows that there are concepts that do not determine a course of values (or value-range). It shows that Basic Law V is false. Neither Russell nor Frege really managed to rescue logicism from this problem though.

Mounce (p. 56) puts Wittgenstein’s objection to Russell’s theory of types this way: “one cannot in a correct symbolism construct a proposition which refers to itself without making it evident that the contained proposition has a different function from the proposition which contains it. But then it will be evident that one cannot construct a proposition which refers to itself. For, given such a misguided attempt, it will be evident that what one has is not one proposition, referring to itself, but different propositions. In short, a theory of types is entirely unnecessary.”

Black (p. 146): ‘It is hard to account for Wittgenstein’s evident animus in this digression. For Wittgenstein’s own programme for ‘logical syntax’ can properly be viewed as an attempt to accomplish what Russell was reaching for in his theory of types. Wittgenstein himself once said that philosophical grammar or logical syntax was ‘a theory of types’ (Phil. Bem. 3, 2). An improved version of Russell’s theory of types might well be a part of logical syntax in Wittgenstein’s conception.”

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[1] Introduction to Mathematical Philosophy p. 135.

[2] Ibid., p. 137

[3] Ramsey “Predicative Functions and the Axiom of Reducibility” in Klemke, pp. 355-368, p. 358, originally in Chapter 1 pp. 32-49 of his The Foundations of Mathematics.

3.33 In logical syntax the meaning of a sign should never play a role; it must be able to be established without anything thereby being said of the meaning of a sign, only the description of the expressions being presupposed.

McManus calls this and the following remark about Russell obscure. On p. 77 he writes: “That we should simply strive to make more apparent the difference between different symbols, rather than (per impossibile) stating what the difference between the symbols is (for instance, by discussing the different kinds of things to which they refer), helps explain” this obscure criticism of Russell.

Meaning (Bedeutung) here again seems to be used in the sense of object referred to or something like that, as we find in Frege. Logical syntax is thus independent of the world, just as algebra is independent of anyone’s bank account.

3.328 If a sign is not used then it is meaningless. That is the meaning of Occam’s razor.

(If everything behaves as if a sign had meaning, then it has meaning.)

Ogden has ‘not necessary’ and P&McG have ‘useless’ for nicht gebraucht. Black (p. 134) points out that the true meaning is ‘not used.’

Ockham’s razor is generally quoted as: Entities are not to be multiplied beyond necessity. It is a kind of metaphysical principle, a rule for generating economical theories about reality. So why link it with meaning? Perhaps this is nonsense. Or perhaps Bedeutung (“meaning”) is here used in a Fregean way, indicating something referred to or signified. When a sign in fact signifies nothing, does nothing at all, there is no point positing something as its reference. But then the fundamental question about meaning is not does a sign refer? but does it have a use? So Frege-ish questions of metaphysics become at least somewhat irrelevant. As the parenthetical comment seems to emphasize.

3.327 A sign determines a logical form only together with its logico-syntactical use.

That is, only taken together with this use does it determine a logical form.

3.326 In order to recognize the symbol in the sign one must look to the meaningful [sinnvollen] use.

To know the (meaningful, significant, or ‘sensical’) use is, it seems, to know the symbol. In Letters to Ogden, p. 59, Wittgenstein writes: “The meaning of this prop[osition] is: that in order to recognize the symbol in a sign we must look at how this sign is used significantly in propositions. I.e. we must observe how the sign is used in accordance with the laws of logical syntax. Thus “significant” here means as much as “syntactically correct”.” I have ‘meaningful’ instead of ‘significant’.

3.325 In order to avoid such errors, we must use a symbolism that excludes them by not using the same sign for different symbols and by not using signs that signify in different ways in what appears to be the same way. A symbolism then that obeys logical grammar – logical syntax.

(The concept-script of Frege and Russell is one such language, though admittedly it does not yet exclude all errors.)

Cf. 5.5563.

A thought: Wouldn’t it be ironic if right here Wittgenstein were to make just the kind of mistake he has just warned about? The first sentence seems fine. But what is “logical grammar”? Could grammar possibly be illogical? No. He means the grammar of logic, the syntactical rules of logic. But what are these? Just the rules of logic, surely. And, just as surely, grammar already obeys these. Perhaps he means a language that more clearly follows the laws of logic. But it is clear already that all grammar must, can only, do this, and the laws of logic are just generalizations (hypothetical ones, we might say) derived from natural language. So the language described might be both impossible to create satisfactorily and unnecessary anyway.

3.324 Thus the most fundamental confusions (of which the whole of philosophy is full) easily arise.

So if philosophy is full of nonsense and the point of the Tractatus is to solve this/these problem(s) then 3.323 seems to be crucial. The book presumably aims to show either how we should use words/language, or that misuse of words/language leads to confusion.

3.323 In colloquial language it is common for the same word to signify in different ways – and thus belong to different symbols --, or for two words, that signify in different ways, to be applied in a proposition in ways that are the same externally.

Thus the word “is” appears as the copula, as the sign of equality, and as the expression for existence; “to exist” as an intransitive verb like “to go”; “identical” as an adjective; we speak about something [an object], but also about something happening [an event].

(In the proposition “Green is green” – where the first word is a person’s name and the last is an adjective – these words do not simply have different meanings but they are different symbols.)

Russell in Logical Atomism: “The is of “Socrates is human” expresses the relation of subject and predicate; the is of “Socrates is a man” expresses identity. It is a disgrace to the human race that it has chosen to employ the same word “is” for these two entirely different ideas—a disgrace which a symbolic logical language of course remedies.” (p. 172)

Some clue here about Wittgenstein’s use of “external.” It refers to the extra-logical, the merely physical, the superficial. Why not say simply that “Green” and “green” have different meanings? See PI §558 and §561.

3.322 A common characteristic of two objects can never be indicated by our symbolizing them with the same signs, but by two different ways of symbolizing. Because the sign is indeed arbitrary. One could thus also choose two different signs and where would then be what was common in the symbolization?

So, for instance, that we both call something by the name “God” does not at all mean that we are talking about the same thing. The word used is arbitrary, just as it would (probably, etymological curiosities aside) be pure coincidence if a word in two historically unrelated languages looked or sounded exactly the same. What matters is the way in which the word means. What matters, one might say, is its use.

3.321 Two different symbols can thus have the same sign (written or audible etc.) in common with one another – they signify then in different ways.

Fair enough. Two words might sound and look alike but have different meanings, as the word ‘bank’ can be the side of a river or a likely target for robbers. And the same might be true of signs/symbols other than words.

3.32 A sign is what is sensibly perceptible of a symbol.

Cf. 3.11. So “2” would be the sign, but it would be a different symbol in first 23 and then 102. In “23” the sign “2” means “twenty-,” whereas in “102” it means “-and 2.”

3.318 Like Frege and Russell, I take a proposition to be a function of the expressions contained in it.

So propositions are being treated here as functions. But it does not follow, Anscombe points out (p. 103), that Wittgenstein thinks propositions just are functions. We can speak of 8 as a function of 2, she notes, without meaning that 8 just is a function and nothing else, or that thinking of it this way is the right way to think of it. So Wittgenstein is not here saying anything incompatible with Frege’s view that a proposition is not a function.

3.317 Fixing the values of a propositional variable is specifying the propositions whose common characteristic the variable is.

The fixing is a description of these propositions.

The fixing will therefore deal only with symbols, not with their meaning.

And the only thing essential to the fixing is that it is only a description of symbols and tells nothing of the symbolized.

How the description of the propositions occurs is unessential.

Black (p. 129) says that ‘signs,’ here and in 3.33, would be better for Symbolen than ‘symbols.’

Me: Saying that X = 2 or -2 tells us that any proposition containing 2 or -2 (which the proposition “201 – 23 = 178” does not) has the variable X in common. A statement such as “X = 2” does not tell us the meaning of X. This might be easier to see with a statement of the same form in which we do not already know the meanings of the words, e.g. “A boojum is a snark.” This does not tell you what a boojum is. It simply tells you that the signs “boojum” and “snark” can be used interchangeably. (This might not be true in fact, at least in all cases. The expressions “that evil tyrant” and “your worshipful majesty” might refer to the same man, but this does not make them interchangeable exactly. Still, to whom you were referring would probably be clear enough were you to make the mistake of mixing these expressions up.) Attempts to clarify language by means of logical analysis seem likely to tell us nothing at all about the world, therefore.

3.316 What values a propositional variable may accept is fixed.

The fixing of the values is the variable.

There seems to be an interesting combination here of agreement (arbitrariness) and absolutism (non-arbitrariness). A variable may be replaced by any range of values, determined by whoever introduces the variable. But once a variable is defined, the definition fixes the range completely. X can mean anything, but if I say “X² = 4” then its meaning is settled (as either 2 or -2). “X” on its own has no meaning, and so is not a variable. In a proposition such as “X² = 4” it is a variable and has a meaning. And its meaning, its possible values, is set.

Cf. 5.501. Black (p. 128) says the procedure referred to is that described in 3.317, not 3.315.

3.315 If we convert a component of a proposition into a variable, then there is a class of propositions which are all the values of the resulting variable proposition. This class still depends in general on what we, by arbitrary agreement, mean by the parts of that proposition. But if we convert into variables all those signs whose meaning is arbitrarily determined then a class like this will still always remain. This however is now dependent on no agreement, but only on the nature of the proposition. It corresponds to a logical form – a logical prototype.

Take the proposition: Bad monkeys like good bananas. Now replace “bananas” with the variable x. We can now generate a class or set of propositions in which x is replaced by something suitable, something that fits (“apples,” say, but not “green”). This set depends on the arbitrary meanings we have given to words like “bad” (we could have used “mal” or “schlecht” instead). Now what if we replace all the arbitrary words with variables? For all m, if m is b and b is g then m likes b. Or perhaps: For all w and all y, if w is x and y is z then wLy. Something like that. Now there is still a class of propositions for which this could stand, to be generated by filling in the place of w, x, y, z, and L with grammatically appropriate words (or proposition parts of some kind). This set or class though does not depend on arbitrary agreement, the conventional meanings of words (or other symbols). Rather it depends on what I (following the later Wittgenstein) am here calling grammar, what Wittgenstein calls logic or logical form.

Mounce (p. 30): “In the Tractatus, logical form is something which, as it were, underlies the rules of language and guarantees its intelligible usage. In the Investigations, he thinks of logical form as being a kind of formalization of the rules of language and these arise out of its use; they do not underlie and guarantee its intelligibility. Common to both works, however, is the view that meaning is not some special entity or psychological process.”

3.314 An expression has meaning only in a proposition. Every variable can be taken as a propositional variable.

(Even a variable name.)

So we are really not getting away from propositions here. Perhaps just as talk of propositions does not really get us away from sentences.

3.313 An expression is thus presented by way of a variable whose values are the propositions that contain the expression.

(In the limiting case the variables become constants, the expression a proposition.)

I call such a variable a “propositional variable.”

We seem to be multiplying logical entities beyond necessity, but we’ll see where we get with all this.

3.312 It is thus presented by way of the general form of the propositions that it characterizes.

Moreover in this form the expression is constant and everything else is variable.

OK, but what is this general, unchanging form? Presumably we are dealing with logic here, not metaphysics. So does talk about constancy and variation really belong? How could a matter of logic not be constant?

3.311 An expression presupposes the forms of all propositions in which it can occur. It is the common characteristic feature of a class of propositions.

An expression (or symbol) is thus something like a meaning. In the way that a proposition can be thought of as what various sentences with the same meaning have in common, so too an expression is what various propositions with (or containing) the same meaning have in common. So do we need the concept of an expression? Do we need the concept of a proposition?

3.31 Every part of a proposition that characterizes its sense I call an expression (a symbol).

(The proposition itself is an expression.)

The expression is all that is essential for the sense of a proposition that propositions can have in common with each other.

An expression marks a form and a content.

OK. More definitions. And how do expressions differ from propositions? Well, a proposition is an expression, so there cannot be much difference. An expression is also any part of a proposition that is proposition-like too, but this might also be called a proposition surely. So we don’t seem to have gained much here.

Black (p. 123): “Wittgenstein is not defining this sense of ‘symbol’ but merely adding that an expression is a symbol.”

3.3 Only a proposition has sense; only in the context of a proposition does a name have meaning.

Wittgenstein here echoes Frege in Foundations § 62.

There is no knowing the meanings of primitive signs before one understands propositions in which they occur. So perhaps 3.263 just means that to understand an elucidation is to understand the primitive signs it contains. Perhaps. This might also throw some light on 3.1432. Propositions are primary.

Cf. Schopenhauer Fourfold Root p. 95: “It is like a word of two meanings; only from the context can we infer what is meant.”

3.263 The meanings of primitive signs can be explained through elucidations. Elucidations are propositions which contain primitive signs. They can thus only be understood if the meanings of these signs are already known.

What the…?! This sounds circular and pointless. Don’t know the meaning of a primitive sign? An “elucidation” will help. But you will only understand it if you already know the meaning of the relevant primitive sign! So either knowledge of meaning is not the same thing as understanding when it comes to signs, which seems unlikely (but who knows?). Or explanation of meaning is quite impossible (in the terms presented by the Tractatus up to now). See p. 44 and pp. 49-50 of Joan Weiner’s essay in Future Pasts.

Anscombe (p. 26) suggests that this passage, along with 3.261, provides the best evidence for thinking that the elementary propositions of the Tractatus are simple observation statements, such as “This is a red patch.” Names and only names are primitive signs. Logical signs, as he indicates elsewhere, are not primitive signs. But (see p. 27) what elucidates a name need not be an elementary proposition. And from 6.3751 it follows directly that “This is a red patch” cannot be an elementary proposition. Anscombe concludes that elementary propositions are not simple observation statements, and that this explains why Wittgenstein did not refer to observation in connection with them. What they are he cannot say, but they must exist. See, for instance, 5.5562, 3.23, 2.021, 2.0211, and 4.221.

See also 5.526.

3.262 A sign’s application shows [zeigt] whatever is not expressed in the sign itself. What signs slur over, their application speaks out.

James Conant argues that the distinction between zeigen and erläutern is important.[i] The former applies only to meaningful propositions, while the second can apply to nonsense. Without wishing to prejudge the issue, I will use ‘show’ only for zeigen. McManus also mentions this issue in footnote 8, p. 36.

The application of a sign seems almost to be working against the sign itself here. Is this slurring over a deliberate attempt to hide something? Perhaps it could be, but probably not. It might be worth asking whether signs do express anything themselves. Maybe all the meaning/signifying/saying/showing is done by the application of the sign.

The application of a sign is here linked with its meaning. In Chapter 1, §2 of Schopenhauer’s Fourfold Root (p. 2) he talks of the different applications of the principle of sufficient reason and says that the principle acquires a different meaning in each such application.

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[i] See James Conant “What ‘Ethics’ in the Tractatus is Not,” in D. Z. Phillips and Mario von der Ruhr (eds) Religion and Wittgenstein’s Legacy Ashgate, 2005, pp. 39-88, p. 82, note 49.

3.261 Every defined sign signifies via the signs through which it can be defined; and the definitions show [weisen] the way.

Two signs, one primitive and one defined by primitive signs, cannot signify in the same way. One cannot analyze names through definitions. (Nor any sign that has meaning on its own, independently.)

Working backwards through this: Signs have meaning only in the context of propositions, so there are no such signs anyway (see 3.22). Names cannot be analyzed by means of definitions or by any other means, since they are quite simple. Primitive signs (names) signify (get meaning) by some means other than definition, since they cannot be defined. Others have the meaning they are defined as having. So can primitive signs have meaning at all? It is hard to see how they could, and so hard to see how the other signs supposedly defined by means of them could be defined either.

It is hard at this point, in the terms the Tractatus gives us, to see how meaning is possible at all. Language (signs, etc.), conceived as something distinct from the world, seems to be incapable of being hooked up to it.

The “Nor any” in the last sentence is Wittgenstein’s translation. See Letters to Ogden p. 59.

Consider in this connection the fact that Frege’s goal is to get away from the ambiguities and misleading qualities of ordinary language. Hence he cannot say precisely in ordinary language what the terms of his system mean. We have to look at the system and see how they operate there. Can a sign have meaning on its own? In Frege’s view, the meaning of a word is not the mental picture associated with it, nor anything merely psychological. We should not consider the meaning of a word in isolation, but within the context of a proposition. If the proposition makes sense, then the words that make it up do. See Foundations of Arithmetic. Also see Russell’s Logical Atomism: “It is exceedingly difficult to make this point clear as long as one adheres to ordinary language, because ordinary language is rooted in a certain feeling about logic, a certain feeling that our primeval ancestors had, and as long as you keep to ordinary language you find it very difficult to get away from the bias which is imposed upon you by language.” (p. 205)

3.26 A name cannot be analyzed further by a definition: it is a primitive sign.

Propositions are complex, names are simple.

3.251 A proposition effects [expresses] what it expresses in a definite, clearly assignable way: a proposition is articulated.

Propositions are as precise as the states of affairs they describe, and as structured.

3.25 There is one and only one complete analysis of a proposition.

Obviously, since a complete analysis is an analysis into simple signs, which correspond to objects, i.e. points in possibility-space. To say an analysis is complete is to say it has been completely disambiguated or defined. If this can be done, it can be done in only one way.

3.24 A proposition that deals with a complex stands in an internal relation to a proposition that deals with a component of the complex.

A complex can only be given through its description, and this will match it or not match it. A proposition, in which there is mention of a complex, will, if this complex does not exist, be not nonsensical [unsinnig] but simply false.

One can see that a propositional element signifies a complex by a vagueness in the propositions in which it occurs. We know by this proposition that something is not yet definite [determinate]. (The notation for generality indeed contains a prototype.)

The abbreviation of the symbol of a complex in the form of a simple symbol can be expressed by a definition.

See 4.123 for the meaning of ‘internal.’ See 4.241 for the meaning of ‘definition.’ There is a logical relation between propositions about complexes and propositions about parts of complexes, presumably taking these propositions to be about complexes qua complexes and propositions about complex-parts to be about them qua complex parts (see my comment on 2.0201). In the next paragraph of 3.24 Wittgenstein seems to agree with Russell about the (non-existent) King of France. To say that the King of France is bald, when there is no King of France, is to say something that is simply false, not nonsensical. There can indeed be vagueness in propositions, something corresponding to indeterminacy. There is a sort of implication here that this vagueness might be eliminable, but there is certainly no claim that this will be so. Indeed, if the notation for generality (for all x, if x etc., I suppose) already contains a prototype for indeterminacy, then perhaps it is not dispensable from logic.

3.23 The requirement of the possibility of simple signs is the requirement of the definiteness of sense.

So if sense is to be definite and not vague then there must be, or it must be possible for there to be, such things as simple signs. See my comment on 3.2 for why we might not insist that sense be definite. But if what can be said can be said clearly, mustn’t sense be definite? I don’t think so. A vague sentence can be quite clear, as the later Wittgenstein certainly realized. If I say “Stand near the door” this is vague but, possibly, quite clear.

This connects with Frege. See Basic Laws of Arithmetic 1903, v. 2, §56 and §62. Bearn discusses this on pp. 51-53. See also Notebooks p. 62.

See also Mounce, on what the point of a logical system is for Wittgenstein (this comment might be better elsewhere). (p. 48) “it is not the purpose of a logical system to provide a language more perfect logically than the ordinary. Such a project, on his view, is entirely incoherent. One thing cannot be more logical than another. A thing is either logical or it is not; it is either meaningful or it is meaningless. Thus, the purpose of a logical system is not to provide the logic that ordinary language lacks; rather it is to display the logic of ordinary language more perspicuously than ordinary language does itself. But then it follows that the cardinal sin in a logical system will be lack of perspicuity, vagueness, ambiguity.”

3.221 I can merely name objects. Signs stand for them. I can only speak them, I cannot express them. A proposition can only say how a thing is, not what it is.

The third sentence is puzzling. I can speak signs but not express them? What's the difference? Perhaps what this means is: I cannot mean things, since meaning in the relevant sense is not personal or psychological. “The cat sat on the mat” means what it means, regardless of me. It is signs that mean things, not people that do so. We can only use or utter signs. I can say “By ‘cat’ I mean dog” and then say “The cat sat on the mat” meaning “The dog sat on the mat,” but ‘my’ meaning this is done by the words themselves. I cannot mean something without words, i.e. without signs or tokens of some kind, be they (mental or physical) pictures or words or whatever. But this seems like a stretch.

What about the last sentence of 3.221? A sentence consists of signs, not of objects. It can say, in effect, “The thing designated by ‘cat’ sat on the mat.” It cannot incorporate a non-linguistic object into itself and say this is what ‘cat’ or ‘Fluffy’ stands for. Why not? Because to do so is to treat or make the object in question part of language, hence linguistic in the relevant sense. Holding up a cat and saying “This is Fluffy,” meaning “The meaning of ‘Fluffy’ is this,” is giving a definition, performing a linguistic act (an act, that is, within or belonging to language). It is, in effect, a kind of equation: Fluffy = ö. And equations tell us nothing, including what a thing is. I can of course say that Fluffy is a cat, is seriously overweight, is cowardly, and so on. But this tells you how Fluffy is, not what Fluffy is like.

What else is there to say though? What would it be to say what something is in some other sense? Nothing at all. That is why it cannot be done. The “impossibility” is logical, i.e. there is no such thing as doing that, i.e. the very idea is plain nonsense. This reminds me of Berkeley. We can say that the apple is red, mealy, soft, and so on, but we are not adding anything to say then that it is matter. Or, for that matter, idea.

3.22 In a proposition a name stands for an object.

And outside a proposition it doesn’t? I was going to say "surely not," but perhaps this is right after all. Outside a proposition is it a name at all? Does it stand for anything when not in the context of a proposition? Quite possibly not.

3.21 The configuration of the simple signs in a propositional token corresponds to the configuration of objects in a state of things.

No real surprise here, given 3.2 and so on.

3.203 A name means an object. The object is its meaning. (“A” is the same sign as “A”.)

Not the most helpful parenthetical comment, surely. Perhaps again the striking or puzzling remark is a sign of irony or something unobvious going on. See PI § 39 and 40, where the meaning of a name is distinguished from the bearer of a name.

3.202 The simple signs used in a proposition are called names.

Another definition.

3.201 I call these elements “simple signs” and the proposition “completely analyzed”.

Black (p. 108) points out that, since we have no way to know when we have reached a complete analysis, this remark does not usefully define ‘simple sign.’

3.2 In propositions thoughts can be so expressed that the objects of the thought match the elements of the propositional token.

My first attempt to comment on this was: Really? What then are the elements of the propositional token (sentence)? Not letters, presumably, or sounds. Words perhaps, or phrases. If I say “The chair was quite soft” then the words “the chair” will correspond not with one particular possibility space (or point in logical space, or object) but with many (however many such points there are in a chair, as it were, or chair-possibility). The same goes for the words “quite soft”. Do they cover a specific range of possibility-spaces? Surely not. There is vagueness here, or so it would seem. An analysis of such a propositional token as “The chair was quite soft” results in vague and infinite sets of ‘objects’. (Infinite because, by 2.0131, at least some objects exist in an infinity of objects of a similar kind.) Nothing really gets clarified or made determinate.

On a second look, I'm inclined to emphasize the word "can" in Wittgenstein's remark.

3.144 One can describe states of things, but not name them.

(Names are like points, whereas propositions, having sense, are like arrows.)

In mathematics and, I think, in German, ‘sense’ (Sinn) can mean direction, as well as meaning.

Is “name” being implicitly defined here as a word that applies to simples only? If so, is the only “impossibility” implied by the first sentence of 3.144 a definitional one, a logical one?

3.1432 Not: “The complex sign ‘aRb’ says that a stands in relation R to b” but rather: That “a” stands in a certain relation to “b” says that aRb.

Aren’t these equivalent? Perhaps that is the point. An explanation only puts the same thing another way, a way that is, in itself, neither better nor worse. On the other hand, what Wittgenstein seems to be saying is that the second of his sentences is actually preferable to the first. Why would that be? Perhaps because the first treats ‘aRb’ as needing explanation or unpacking or articulation, whereas in fact it is already fully articulate. For those who understand a sentence or proposition, its analysis is quite useless (i.e. uninformative).

Mounce says, on pp. 24-25, that it might help to substitute some actual relation for aRb. Thus, we could say “Not: ‘The complex sign ‘the painting hangs on the wall’ says that the painting stands in the relation of hanging to the wall’ but rather: That the painting stands in the relation of hanging to the wall says that the painting is hanging on the wall.” Mounce (p. 25): “In other words, the relation between a proposition and its sense is an internal one. The sense of a proposition is to be found in an arrangement of physical signs; it is not to be found in something that corresponds to that arrangement, some entity over and above it, whether in the empirical or some quasi-empirical world.” (If you understand “The painting hangs on the wall” then it does you no good at all to be told the longer version that is the alleged meaning of this complex sign.)

A proposition is not a name, and the meanings of its elements are not independent of it, are not really, we might say, elements, in the sense that the proposition consists of bits that can be understood more clearly or fully when taken apart. Cf. 3.3.

Black (p. 105): “I take W. to be denying that the complex sign is a name of the situation described: a fact is needed to refer to a fact.”

Fahrnkopf discusses a nominalistic interpretation of this passage and a realistic one. Nominalist readings (e.g. Copi's and Anscombe's) take the key point to be that 'R' would have no place in an ideal symbolism. Thus relations are not real, and whatever 'aRb' tells us might just as well be expressed by, say, 'ab' or 'ba'. On p. 29 Fahrnkopf writes: "according to Wittgenstein's decimal notation, 3.1432 is a comment on 3.143, and this latter passage is concerned only to make the point that a propositional sign is a fact, not a name; this is also the context of the remark in the "Notes on Logic" which corresponds to 3.1432. On my interpretation, then, the purpose of 3.1432 is only to contrast symbolizing facts with names, and the nominalist tone of this passage--which could have been avoided altogether had Wittgenstein specified that the relation in which 'a' stands to 'b' consist in their respective relations to 'R'--is in any case minimized by the realization that the status of 'R' as a name is implied in many other contexts in the Tractatus."

Fahrnkopf also points out (p. 35) that in the Notebooks Wittgenstein wrote on 16/6/15 that relations and properties are objects.

3.1431 The essence of a propositional token becomes very clear if we think of it as made up of spatial objects (such as tables, chairs, books) instead of written signs.

The reciprocal spatial position of these things then expresses the sense of the proposition.

Ogden here has ‘mutual spatial position,’ which Black (p. 103) calls the literal translation. But ‘mutual’ can imply shared, as in “our mutual friend,” and here the spatial position is not meant to be the same.

Relative position, syntax, seems here to be presented as all that is needed for sense, although of course you need some tokens to arrange too. What about semantics?

3.143 The usual form of expression in writing or printing disguises a propositional token’s being a fact.

Because in a printed sentence, e.g., no essential difference appears between a propositional token and a word.

(This is how it was possible for Frege to call a sentence a complex name.)

‘Complex’ here is suggested by Black (p. 103). The others have ‘compounded’ (Ogden) and ‘composite’ (P&McG).

Grammar is invisible. Note also that Wittgenstein thinks it necessary to come up with an explanation for how Frege could have made a mistake.

3.142 Only facts can express a sense, a set of names cannot.

A class or set is both hypothetical and, what seems to be the point here, unstructured. Sense depends on structure, i.e. syntax or grammar. It also, as we have seen, seems to depend on what has meaning being more than merely hypothetical, i.e. on its being in some sense real.

3.141 A proposition is not a mixture of words. – (In the same way that a musical theme is not [just] a mixture of notes.)

A proposition is articulated.

Wittgenstein says (Letters to Ogden, p. 24) that the main point is that a proposition is a structure, not a mixture. Obviously the order is essential in each case. To be a proposition, to have a meaning or sense, words must be combined grammatically (at least approximately). This is the precise way in which words must be combined to have a meaning. That is to say: “The dog bit the man” means something different than “The man bit the dog.” The meaning depends on the way in which the words are combined, and which combination means what depends on grammar.

3.14 A propositional token consists in its elements, the words, relating to each other in a definite way.

A propositional token is a fact.

See 2. A sentence (propositional token) is a state of affairs that exists. True enough. It can also be a picture (see 2.16) and a thought. So the distinction between world and representation is further blurred or erased. A sentence not only expresses a thought (see 3.1), it actually is one, or at least can be. We seem to able to cut thoughts and propositions from our metaphysics, if we have one, and see that these terms are logical terms of art, no more. And, by this point, perhaps scarcely even that for Wittgenstein.

3.13 To a sentence belongs all that belongs to the projection, but not what is projected.

Thus the possibility of what is projected belongs to it, but not it itself.

Its sense is therefore not yet contained in a sentence, but [perhaps?] the possibility of expressing it is.

(“The content of a sentence” means the content of a significant [meaningful, sinnvollen] sentence.)

The form of its sense is contained in a sentence, but not its content.

I’m following Black (p. 100) here in translating Satz as ‘sentence’ rather than ‘proposition.’

You would think that only what is projected (the sense or meaning) is what belongs in common to both a proposition and the sentence that is its projection. But here Wittgenstein says that propositions are really something like potential sentences. Not only do we encounter propositions only in the form of sentences, but propositions exist only as sentences. Because until they are sentences, propositions have no content, no sense. And a proposition without sense is hardly a proposition at all, is it? If a proposition has only form then it certainly has no real existence. It is a logical fiction or hypothetical ‘entity’ used for thinking about logic. Its ‘existence’ is purely logical, not metaphysical at all.

Alternatively, the first sentence of 3.13 might be read as saying that propositions and sentences (tokens) have everything in common except the physical manifestation that is the sentence (token). That (the physical stuff) is what is projected. Thus the possibility of being communicated belongs to a proposition, but not the perceptible properties necessary for communication themselves. It therefore has no use yet (so sense is use?), but only the possibility of being used. This doesn’t sound too implausible, but what would be “the possibility of expressing the sense of a proposition” if this sense were itself the expression of a proposition/thought/sentence? The second half of 3.13 seems incompatible with the reading offered in this paragraph. So we are back with my previous paragraph. What now to make of the first sentence of 3.13, which does sound odd? I think oddness often indicates irony in the Tractatus. To a proposition belongs nothing, in other words, because what is projected exists only in sentences. (Or thoughts ‘embodied’ in some perceptible medium. I don’t see why this has to be physical and not, say, the stuff of a Cartesian mind.)

Black (p. 100) says that what belongs to the projection means “all that is internal to the representing relation, i.e. the logical form that the sentence has in common with the state of affairs it represents (2.18)” and what is projected means the sense. He also says (same page) that: “form of its sense = ‘form of the possible state of affairs presented’ = ‘the logical form’.”

3.12 I call the token through which we express a thought a propositional token. And a proposition is a propositional token in its projective relation to the world.

So a Denken is a Gedanke, it seems, and we must take 3.11 logically, not metaphysically, just as I suggested. And what I said about sentences and their relation to propositions seems to be confirmed here too.

3.11 We use the physically perceptible token (audible or written, etc.) of the proposition as a projection of a possible state of things.

The method of projection is the thinking of the proposition’s sense.

This is tricky and important. How do we use sentences (sensibly perceptible representatives of propositions) as projections (representations?) of possible states of things? A possible state of things is a state of affairs. A proposition is a picture of a state of affairs. A sentence is a physical version of a proposition, i.e. (I suppose) a proposition with additional, physical characteristics. Since this is the form in which (it seems) we must deal with propositions, we could identify the two, but the proposition is the sentence conceived under the aspect of logic. Its external features (font, language, etc.) are irrelevant. Perhaps this is where the earlier talk of “external properties” of objects comes in. If objects are quasi-fictional ‘entities’ that we get by analyzing propositions, and we only ever encounter propositions in a form where they have external properties, then perhaps one might talk of the external properties of objects. To do so would surely be at best misleading though, since to think of a sentence as a proposition is precisely to think of it without its external properties, or not to think of those properties, or to think of it as if they were irrelevant.

Anyway, the first sentence of 3.11 seems to say no more than that we use sentences in place of (or as) propositions. The second equates doing this with thinking (of) the sense of the proposition. If thinking means doing something with (or having) a thought (as this term has been used and understood up to now) then this cannot be a psychological matter. But Wittgenstein could be using das Denken (“the thinking”) to mean something other than the thought (der Gedanke). So let’s not jump the gun. If thinking in a psychological (metaphysical, not logical) sense is a method for using sentences to get at or convey propositions, then 3.11 tells us that sentences get hooked up to propositions by means of a psychological act. There are multiple problems with this idea. Some have to do with Wittgenstein’s Fregean opposition to psychologism. But perhaps saying so begs the question. More obviously, the theory under consideration is massively implausible. If I use a sentence to convey a thought to another, how does that person know what I’m thinking? Language use (i.e. successful language use) would seem to not only require but be a kind of telepathy. Surely Wittgenstein cannot have meant this.

The other way to take the second sentence of 3.11 is as an equation or definition: thinking of the proposition’s sense is (means, equals) using the sentence that expresses or corresponds to it. What I say or write, I also (thereby) think. This seems to make insincerity impossible, but it does not do so if “thinking” is simply being (re-)defined in this way.

Anscombe says (p. 69, note 1) that “Wittgenstein’s use of ‘projection’ is a metaphorical extension of the mathematical use, which may be explained thus: ‘The drawing of straight lines through every point of a given figure, so as to produce a new figure each point of which corresponds to a point of the original figure.’”

3.1 In a sentence a thought is expressed perceptibly.

Ogden’s “through the senses” is surely wrong, unless it means “through the senses and into the mind.” P&McG have “an expression that can be perceived by the senses.” Black (p. 99) notes that this is “somewhat laboured,” and suggests “In a sentence the thought expresses itself perceptibly.”

3.05 We could only know a priori that a thought was true if its truth could be known from the thought itself (without any object of comparison).

Because we are talking about a priori truth here and because, it seems to be implied, no other thoughts can be involved, either because they would not help (because the truth of each is independent of the truth of others) or because other thoughts themselves would count as objects of comparison.

3.04 An a priori correct thought would be one such that its possibility implied its truth.

Cf. 2.202.

3.0321 We could well represent spatially a state of affairs that went against the laws of physics, but not one that went against the laws of geometry.

Because the idea is quite empty: a geometric representation of anti-geometry.

3.032 One can no more in language present “the logically contradictory” than one can in geometry present through its coordinates a figure that contradicts the laws of space; or give the coordinates of a point that does not exist.

The “impossibility” is logical, not metaphysical. There is not something that one cannot do, just as there is no such thing as a geometric figure that contradicts geometry (not geometry as we know it, but geometry as such). Talk of such things is gibberish.

3.031 It used to be said that God could create everything, only nothing that would be contrary to the laws of logic. – That is [?], we could not say of an “illogical” world how it would look.

I wonder whether Wittgenstein commented on the translation of this. Both Ogden and P&McG have “The truth is…” for the start of the second sentence, but I see nothing corresponding in the German. It looks literally to be: “We could of course not say of an “illogical” world how it would look.”

We cannot picture it, we cannot conceive of it, even God could not create it. These are all ways of saying (in a misleadingly metaphysical-sounding way) that the idea has no sense. I could add “for us”, but then all pictures are made by us and for us, or so at least it seems so far for Wittgenstein.

McManus (p. 59, note 24) cites the first sentence of this remark as an example (others are in 3.323, 4.002, 4.003, and 5.02) of straightforwardly empirical claims that could not possibly be interpreted as nonsensical, even if they are false. That is, it is evidence that not every sentence in the book is meant to be simply nonsensical.

3.03 We cannot think anything illogical, because we would then have to think illogically.

Cf. 5.4731. "Thinking illogically" is a contradiction in terms if "thinking" is understood as Wittgenstein means it. That is why we cannot do it. There is no such thing to do. And so what is illogical is utterly inconceivable. There cannot, as a matter of logic, of sense, be an illogical state of things. The word "cannot" here sounds metaphysical, but it can't be. It would be absurd to say that that which is unthinkable or inconceivable cannot occur in the world. Why on earth not, after all? And what are we talking about? What is the reference of "the unthinkable"? Asking this starts to sound metaphysical and mystical, but here that is plainly a mistake. “Thinking illogically” is self-contradictory, neither possible nor impossible, that is why we ‘cannot’ “think anything illogical,” because that too is, or implies, a contradiction. What could it mean to have a thought of “an illogical state of things”? It would mean to have a logical picture, a kind of copy or representation of, the logic of an illogical thing, the internal structure of something with no internal structure. This is, again, neither possible nor impossible but sheer nonsense.

3.02 A thought contains the possibility of the state of things that it thinks. What is thinkable is also possible.

So the limits of the thinkable are determined by possibility, i.e. by logic, not by any psychological limits of thinkers.