5.62 This remark provides the key to the resolution of the question, to what extent solipsism is a truth.

What solipsism itself [nämlich] means is completely [ganz] right, only it cannot be said, but rather shows itself.

That the world is my world shows itself by the limits of language (the only language that I understand) meaning the limits of my world.

Black (p. 309) says that meint should not be translated as ‘means’ (as P&McG, Ogden, and I have it) but as ‘intends’ or ‘wants to say.’ It means something like ‘thinks’ or ‘believes,’ and is presumably related to the English word ‘mind.’ So perhaps we could say "What solipsism has in mind..." instead.

On the translation of the parenthetical remark, see Anscombe p. 167 footnote: “Dr. C. Lewy has found a copy of the first edition of the Tractatus with a correction by Wittgenstein giving ‘the only language that I understand’.”

On understanding language, and the relation between this and the self, see Russell: “The fundamental epistemological principle in the analysis of propositions containing descriptions is this: Every proposition which we can understand must be composed wholly of constituents with which we are acquainted.”[1]

“The chief importance of knowledge by description is that it enables us to pass beyond the limits of our private experience. In spite of the fact that we can only know truths which are wholly composed of terms which we have experienced in acquaintance, we can yet have knowledge by description of things which we have never experienced.”[2]

Schopenhauer linked the riddle of existence to the connection between inner and outer, the very issue Russell seems concerned with here. Kant’s ideas on free will, on how we can conceive of ourselves as determined phenomena and as free noumena, Schopenhauer says, mark “the point at which the Kantian philosophy leads to mine, or at which mine springs out of his as its parent stem.”[3] But, what Kant says we only conceive, Schopenhauer claims to know, and his idea of the noumenal self is different from Kant’s.

How do we know that others have such a will and are not mere phenomena? It cannot be proved, but to believe in such “theoretical egoism” is madness, according to Schopenhauer (see WWR vol. I, pp. 134-5). We each feel deeply that the rest of the world shares our nature. This is not to postulate noumena as entities in any way distinct from phenomena. I am one, but can be conceived as phenomenon or as will. The same goes for everything else. Noumena do not cause phenomena. So far as noumena = will, noumena are phenomena, merely conceived under a different aspect.

“I … say that the solution of the riddle of the world must proceed from the understanding of the world itself; that … the task of metaphysics is not to pass over the experience in which the world exists, but to understand it thoroughly, because outer and inner experience is … the principal source of all knowledge; that therefore the solution of the riddle of the world is only possible through the proper connexion of outer with inner experience, effected at the right point …”[4]

See also TLP 6.5 on “the riddle.”

If what solipsism means cannot be said, can it be thought? And if not, can there be a meaning here to be correct? Presumably, if there is any truth here at all, it is what the solipsist wants to say, but without realizing it. It is not solipsism, in other words, but something that can sound like it. Here the distinction between, say, language and my language seems to be blurred or denied. So the limits of my world are the limits of language or logic or the world. Not really, I think, limits at all. So I don’t think there is any real truth being got at at all here.

Schopenhauer says: “’The world is my idea’: this is a truth which holds good for everything that lives and knows, though only man can bring it into reflected, abstract consciousness.”[5]

“[N]o truth is more certain, more independent of all others, and less in need of proof, than this: that all that is there for the knowing – that is, this whole world – is only object in relation to the subject, perception of the perceiver – in a word, idea.”[6]

“Everything that in any way belongs, or can belong, to the world is inevitably affected by this: it is conditioned by the subject, and exists only for the subject. The world is idea.”[7]

This view, Schopenhauer says, is true but one-sided. The other side which, if not frightening is at least solemn, sobering, and serious, says that each of us can and must say, “The world is my will.”[8]

“For as the world is in one aspect entirely idea, so in another it is entirely will. However, a reality which is neither of these two, but an object in itself (into which even Kant’s thing-in-itself has unfortunately degenerated in the course of his work), is the absurd product of a dream, and its credence in philosophy is a treacherous will-o’-the wisp.”[9]

Schopenhauer on solipsism: “While theoretical egoism [i.e. solipsism] can never be proved false, in philosophy it has never been used other than as a sceptical sophism, i.e. only for show. As a serious conviction, on the other hand, it could be found only in a madhouse, and as such it would need not so much a refutation as a cure. So we will concern ourselves with it no further …”[10]

Schopenhauer, then, is a self-confessed idealist (of a particular kind), but denies being a solipsist. Julian Young, though, criticizes Schopenhauer for being too egoistic. It is not that one has a triumphant sense of one’s own immortality when one experiences the sublime, as Schopenhauer suggests when he quotes the Upanishads in this connection as saying: “I am all this creation collectively, and besides me there exists no other being.” (WWR I p. 205) “It may well be that, temperamentally, Schopenhauer was a solipsist.”[11] In the experience of the sublime, it is not, Young asserts, that the world shrinks into one’s self, but rather, on the contrary, that the self expands into the world.

To get it right we need a different metaphysics, Young claims. We need “the anti-metaphysical metaphysics to be found in the work of later Heidegger.”[12] We need not idealism of any kind, let alone solipsism, but a “magic” or “poetic” realism.[13] Building on Nietzsche’s perspectivism, Heidegger sees that there are multiple ways to see things, multiple true ways to see things. Dreyfus calls this “plural” realism, apparently. “Being, that is to say “nature”, in a deep sense of the word, is multi-aspected, a “plenitude” of “facets” (PLT p. 124) nearly all of which are unknown to, indeed inconceivable by, us. So, like an iceberg, Being is, almost entirely concealed, almost entirely, as Heidegger puts it, “secret”, a “mystery”. And this makes it magical, awesome. In other words, “sublime”.”[14]

For Nietzsche on all this, see The Gay Science §374 (and 124) where, “What we need, I think Nietzsche is saying, is something like a return to the ancient Greek understanding of the holy as the “uncanny”.”[15]

For the importance of the sublime, see note on TLP 1.

LW might have discussed solipsism because of (i) Russell’s problems with knowing of other minds and with words’ being able to mean anything other than immediate, private experience (see Glock p. 444), (ii) Schopenhauer (see above and Hacker’s Insight and Illusion), or (iii) Weininger (see Haller pp. 95-96).

See TLP 6.51 on LW’s reaction to Schop’s way of dealing with solipsism, i.e. to saying that it cannot be refuted. On solipsism and LW generally, see Glock pp. 446-449, where he argues strongly that LW was a solipsist of some kind.

Rush Rhees: “Wittgenstein has never held to solipsism, either in the Tractatus or at any other time.” (Mind, 56, (1947), p. 388, quoted in Magee p. 337)

In The Problems of Philosophy p. 10 Russell writes that while solipsism “is not logically impossible, there is no reason whatever to suppose that it is true.”

[Apologies for the lack of organization in these notes.]

---

[1] Ibid., p. 206.

[2] The Problems of Philosophy (Oxford University Press, 1959), p. 59.

[3] The World as Will and Idea trans. R. B. Haldane and J. Kemp (3 volumes, Routledge and Kegan Paul, London, 1883), vol. II, p. 117.

[4] Ibid., vol. II, p. 20.

[5] First sentence of book, here quoted from Everyman edition.

[6] Ibid., p. 3.

[7] Ibid., p. 4.

[8] See ibid.

[9] Ibid., p. 5.

[10] WWI book two, §19 (p. 37 in Everyman edition).

[11] Julian Young “Death and Transfiguration: Kant, Schopenhauer and Heidegger on the Sublime” Inquiry Vol. 48, No. 2, 131-144, April 2005, p. 140.

[12] Ibid., p. 139.

[13] See ibid., p. 141.

[14] Ibid., p. 141.

[15] Ibid., p. 144, note 17.

5.61 Logic fills the world; the limits of the world are also its limits.

We therefore cannot say in logic: In the world there is this and this, not that.

That would precisely [nämlich] seem to presuppose that we exclude certain possibilities and this cannot be the case, since otherwise logic would have to be beyond the limits of the world; when it could consider precisely [nämlich] these limits from the other side too.

What we cannot think, we cannot think; we therefore also cannot say what we cannot think.

Marie McGinn (p. 255) points out that there is a sense of climax here, of the limits of thought having been drawn, which in the foreword Wittgenstein said was the aim of the book.

My comment: From my world to the world. Is there a difference? Are the limits of logic the limits of language? The last sentence seems to rule out important or significant truths that cannot be said.

5.6 The limits of my language mean [bedeuten] the limits of my world.

Which is not to say that these limits cannot change. Is this a definition of "world" as much as of "language"?

5.5571 If I cannot give the elementary propositions a priori, then it must lead to obvious nonsense [offenbarem Unsinn] to try to give them.

Has this attempt been made in the Tractatus itself? Black points out (p. 306) that wollen could be rendered as ‘to want’ as well as ‘to try’ here.

5.557 The application of logic decides what elementary propositions there are.

What lies in the application, logic cannot anticipate.

This is clear: Logic must not collide with its application.

But logic must be contiguous with its application.

Therefore logic and its application may not overlap each other.

So elementary propositions do not depend on objects. Rather, what objects there are depends on elementary propositions, which depend on the application of logic, on what we arbitrarily choose to do. In this sense at least there are no elementary propositions, or objects. What it means to talk of logic and its application overlapping one another, I do not know.

White (p. 25) says that what Wittgenstein writes here and in the following remark shows that to discover what objects are would require an empirical investigation. “What we can say is that if we consider the requirements that Wittgenstein lays down for his objects, they would have to be very different from the tiny particles that we first think of when we hear the phrase ‘simple objects’.”

Black (p. 306) suggests ‘touch’ where I have ‘be contiguous with’ and Pears & McGuinness have ‘be in contact with.’

5.5563 All propositions of our ordinary language are in fact, just as they are, logically completely in order. – That simplest of things, that we should give here, is not a likeness of the truth, but rather the full truth itself.

(Our problems are not abstract, but possibly the most concrete that there are.)

What problems are these?! It sure looks abstract. See PI 97, which refers to this. Propositions are just as OK in ordinary language as they are in any concept-script, “Only it is easier for us to gather their logical form when they are expressed in an appropriate symbolism,” p. 50 Letters to Ogden. On the same page, he says that “That simplest of things” should be an expression parallel to “the highest good” or “the good and the beautiful,” so he might be alluding to a Platonic illusion of pure simplicity.

Anscombe points out (p. 91) that this contradicts Russell’s claim on p. 9 of his Introduction to the TLP that language only has meaning “in proportion as it approaches to the ideal language which we postulate.”

5.5563 All propositions of our ordinary language are in fact, just as they are, logically completely in order. – That simplest of things, that we should give here, is not a likeness of the truth, but rather the full truth itself.

(Our problems are not abstract, but possibly the most concrete that there are.)

What problems are these?! It sure looks abstract. See PI 97, which refers to this. Propositions are just as OK in ordinary language as they are in any concept-script, “Only it is easier for us to gather their logical form when they are expressed in an appropriate symbolism,” p. 50 Letters to Ogden. On the same page, he says that “That simplest of things” should be an expression parallel to “the highest good” or “the good and the beautiful,” so he might be alluding to a Platonic illusion of pure simplicity.

Anscombe points out (p. 91) that this contradicts Russell’s claim on p. 9 of his Introduction to the TLP that language only has meaning “in proportion as it approaches to the ideal language which we postulate.”

5.5562 If we know on purely logical grounds that there must be elementary propositions, then it must be known by everyone who understands the propositions in their unanalyzed form.

But such universal knowledge seems unlikely. So if this is right, it is another strike against there being elementary propositions, or at least against our knowing that there must be. But is this right? Why should it be?

5.5561 Empirical reality is limited by [to?] the totality of objects. The limit shows up again in the totality of elementary propositions.

Hierarchies are, and must be, independent of reality.

Objects in what sense? Tractarian objects, being the possibilities, limit all reality, surely. And in this sense are no limit at all. It is presumably these limits he has in mind, since the same limit appears in the totality of elementary propositions, which depend on Tractarian objects. But what is the real connection between these and empirical reality?

5.556 There cannot be a hierarchy of forms of elementary propositions. Only what we construct ourselves can we foresee.

Why must this hierarchy be foreseen? Is he saying that there is no a priori except the arbitrary, the constructed?

5.555 It is clear that we have a concept of the elementary proposition irrespective of its special logical form.

However, where one can construct symbols according to a system, there this symbol is what is important logically, and not the individual symbols.

And how would it even be possible that I should have to deal with forms in logic that I can invent; rather I must have to deal with what makes it possible for me to invent them.

Is this clear? What kind of concept is this? Logic concerns possibility again. There is arbitrariness there, but that is not what is important, or essential. So are we getting, have we got, anywhere with this investigation into logic?

5.5542 But may we then ask such a question at all? Can we erect a symbolic form and not know whether something could correspond to it?

Does the question make sense: What must be in order that something can thereby be the case?

“Could correspond” or “can correspond” as the others have? And does that question make sense? It seems to. My child must be so that I can be a parent. If existence is not a predicate, can predicates depend in any way on existence? Mustn’t possibilities of being this way or that be independent of what happens to exist or not? These possibilities of being, after all, are very close to Tractarian objects, and they are logical, not metaphysical/ontological. Wittgenstein says, Letters to Ogden pp. 33-34, that the correct answer to the initial question “would be, that we may NOT!”

Black (p. 304): “The answers to all three questions are clearly intended to be in the negative. But it is ironical to notice that the third question expresses one of the main preoccupations of the Tractatus.”

5.5541 It is supposed to be possible to determine a priori whether I can get in the position, e.g., of having to symbolize with the sign for a 27-termed relation.

Black (p. 304) says that Wittgenstein “no doubt” has Russell in mind here again, but gives no specific reference and suggests that Wittgenstein is only alleging that Russell supposes this. Ogden’s translation as “It should be possible” is just as good, as far as the German goes. It depends whether we see Wittgenstein as criticizing someone here, and then on who that person might be, etc.

5.554 The giving of any special form would be completely arbitrary.

Form of what? Why?

5.553 Russell said that there were simple relations between different numbers of things (individuals). But between which numbers? And how should this be decided? – By experience?

(There is no pre-eminent number.)

Black (p. 304) gives the reference here as this: in Logic and Knowledge p. 206 Russell writes: “I see no particular reason to suppose that the simplest relations that occur in the world are (say) of order n, but there is no a priori reason against it.”

5.5521 And if this were not so, how could we apply logic? One could say: If there would be a logic, even if there were no world, then how could there be a logic, since there is a world?

Mysteriouser and mysteriouser. Logic is a priori. Perhaps that is all that 5.552 means. Now, if it were not, how could we use it? We would first have to find out how the world was. And how could we do that without logic? But he’s also saying something, or so it seems, about logic and the existence (not the nature) of the world. If logic is such that it would exist even without a world, then how can it still exist now that there is a world? If logic, perhaps that is to say, is so independent of the world as to be absolutely independent, then how can it exist or be applied here in the world, by worldly beings? I’m not sure I’m getting this.

5.552 The “experience” that we use to understand logic is not that such and such a thing is the case, but rather that something is: but that is no experience at all [eben keine Erfahrung].

Logic is before every experience – that something is thus.

It is before the How, not before the What.

The last sentence sounds almost like Heidegger. Is logic like a filter through which we experience things? Sounds almost like Kant. Mysterious.

Black (p. 303) points to 6.1222, 3.221, 6.44, 2.0271 (identifying the How with the contingent or changing), 2.024 (identifying the What with Substance), and PI 89.

5.551 Our fundamental principle is that every question that can be decided at all with logic must be decidable without anything else.

(And if we get into a state of things where we need to answer such a problem by looking at the world, then this shows that we are on a fundamentally wrong track.)

Exactly! Logic is logic, not metaphysics.

Black notes (p. 303) that Wittgenstein got onto precisely this “wrong track” in his remarks on logical form. “This must be regarded as a temporary aberration, which he later repudiated,” according to Black.

5.55 We must now answer a priori the question about all possible forms of elementary propositions.

An elementary proposition consists of names. Since we cannot give the number of names with different meanings [Bedeutung], though, we cannot give the composition of the elementary proposition either.

This last sentence is odd. What is “the elementary proposition”? Wittgenstein throughout uses the definite article where it makes more sense in English to use a plural or an indefinite article, as I have done here in the penultimate sentence. But surely being unable to number the names would not prevent us from giving the composition of some elementary propositions. He writes the last sentence as if either there is only one elementary proposition (and why would that be?) or there are more than one, but all must be given or none can be (and why would that be?). The a priori is suspect here, I feel.

5.5423 To perceive a complex means to perceive that its parts are combined in such and such a way.

Perhaps this also explains the fact that one can see the figure

as a cube in two ways; and all similar phenomena. Because we quite truly [eben wirklich] see two different facts.

(If I look first at the corners a and only cursorily at b, then a appears in front; and vice versa.)

Do we really see two different facts? Not one fact two ways? But what is a fact? Perhaps a bit of reality [taken] in a certain way. By the definition of complex (or of perceiving a complex) given here, there are two different complexes.

Would “just actually” be better than “quite truly”?

Why is this not discussed more in the literature on seeing aspects?

5.5422 The correct explanation of the form of the proposition “A judges that p” must show that it is impossible to judge a nonsense [einen Unsinn]. (Russell’s theory does not satisfy this condition.)

Black (p. 301): “Russell’s view was that a judgement has ‘several interrelated objects’ (Principia, vol. 1, p. 43). Judgement requires a relation between the mind and the various constituents of the proposition in question. ‘That is, when we judge (say) “this is red”, what occurs is a relation of three terms, the mind, and “this”, and red’ (ibid.).”

5.5421 This shows also that the soul – the subject, etc. – as it is conceived in the contemporary superficial psychology, is a nothing [Unding].

A composite soul would by definition [nämlich] be no longer a soul.

Is this psychology that of Freud, et al., or more the philosophy of mind of Russell, Moore, etc.?

5.542 It is, though, clear that “A believes that p”, “A thinks p”, “A says p” are of the form “’p’ says p”: And here it is not a question of a coordination of a fact and an object, but rather of the coordination of facts by way of the coordination of their objects.

My relevant dictionary entry on this says: “Wittgenstein’s view seems to be that one cannot judge or believe or think a piece of nonsense. One can believe that Bush is a good President or that Bush is not a good President, but one cannot believe that cockadoodledoo. This is a matter of logic, not psychology, as Wittgenstein sees it. To think (that) a proposition p (is true) is somehow (it does not matter how, except to a psychologist) to represent, express, or picture p, just as saying “p” is. So “A thinks p” means something like “Something represents/expresses/pictures/says p.” This something could as well be a sentence as anything else, such as a person. So, as Wittgenstein puts it in proposition 5.542, “A thinks p” has the same form as “‘p’ says p.””

Mounce (pp. 85-86) says that Anscombe makes a mistake here. She fails to distinguish between the contingent fact that a person A happens to have uttered p, and the non-empirical fact that p means p. Of course, the sounds or marks that make up p might have meant something else, or nothing, but given their meaning, it is not contingent that they mean p. According to Mounce (p. 86): “The point is simply that B can convey to us what A says (or thinks) simply by telling us what sounds he utters. How is this possible? Well, first, because these words possess logical form; and second because, since we ourselves have a grasp of logical form, understand a language, we do not have to be told what these say; we can tell that for ourselves.” Of course, Mounce also points out that what is believed is not an object in the ordinary sense, because what is believed must make sense.

Anscombe (p. 88) says: “It is perhaps not quite right to say that ‘A judges p’ is of the form ‘”p” says that p’; what he should have said was that the business part of ‘A judges that p’, the part that relates to something’s having as its content a potential representation of the fact that p, was of the form ‘”p” says that p’: ‘A believes p’ or ‘conceives p’ or ‘says p’ must mean ‘There occurs in A or is produced by A something which is (capable of being) a picture of p’. We should here remember the letter to Russell in which he said he did not know what the constituents of thoughts were, but he was certain that a thought must have constituents corresponding to the words of language.” The letter in question is discussed on p. 28 of Anscombe.

Friedlander (p. 113): “Only an internal connection between the act of thinking or judging and the constitution of the judgment is capable of explaining why a subject cannot judge what is not sense, what is nonsense.”

Cf. PI §358. One cannot mean a senseless string of words, and so an act of meaning is not what gives sense to (otherwise meaningless) strings of words.

Black (p. 300) says that the fact in question is the fact that A spoke the words A spoke and that the object is A.

5.541 At first glance it seems as though there is another way in which a proposition can occur in another.

Especially in certain propositional forms of psychology, like “A believes that p is the case”, or “A thinks p”, etc.

Here it seems superficially as though the proposition p stands in a kind of relation to an object A.

(And in the modern theory of knowledge (Russell, Moore, etc.) those propositions have been understood in just this way.)

According to Russell, when you have a sentence like ‘A believes p’, p cannot stand for a fact, since then you could only ever have true beliefs. Nor can p be a proposition, since propositions do not really exist. What you believe cannot be a logical fiction, but must be something real. So how can we have false beliefs? How can we really believe something that is not the case? For instance, in ‘A believes Hamlet lives in Finland’ we cannot relate A to a proposition and ‘lives in’ must be treated as a verb, even though Hamlet does not live in Finland and there is no non-existent Hamlet who does so, nor a non-existent Finland in which he lives. Russell does not know how to solve this problem, at least not in Logical Atomism. Earlier, in Theory of Knowledge, he treated propositions as functions of judgments, consisting of objects that the person making the judgment is acquainted with. These objects include relations. But then the relation becomes just another object, so how is it to be related to the other objects, and how are they to be related to it? We cannot simply assume that the objects are related correctly in the judgment, since people judge falsely sometimes. These are the problems that Wittgenstein appears to have pointed out to Russell, and that he wrestles with in Logical Atomism. Contrast Frege’s view that in the proposition ‘Copernicus thought that the planetary orbits are circular’ “the man and the thought occupy, so to speak, the same stage.”[1] I.e., Frege takes this sentence to relate two objects, a man (Copernicus) and a thought (that the planetary orbits are circular).

---

[1] Frege Philosophical and Mathematical Correspondence ed. Brian McGuinness, trans. Hans Kaal, University of Chicago Press, 1980, p. 164.

5.54 In the general propositional form propositions occur in [other] propositions only as bases of truth-operations.

OK

5.5352 Equally, one has wanted to express “There are no things” by “~ (Ex) . x = x”. But even if this were a proposition, — would it not also be true, if indeed “There were things”, but these were not identical with themselves?

Odd. The supposition that they might not be identical with themselves is nonsense (see 5.5303), at least roughly speaking. But then the symbolic notation is nonsense, and does not mean “There are no things.” Why can’t it mean that? Well, we have removed the equals sign from the notation, as unnecessary, so it doesn’t mean anything. Should we bring it back? No, for the reasons given before (redundancy). What we want here, after all, is a sign for existence, not identity.

5.5351 There are certain cases where one is led into the temptation to use expressions of the form “a = a” or “p[if…then]p” and such. And indeed this happens when one would like to speak of the prototype: Proposition, Thing, etc. Thus Russell in the “Principles of Mathematics” has rendered the nonsense [Unsinn] “p is a proposition” in symbols with “p[if…then]p” and presented it as a hypothesis in front of certain propositions whose argument places thereby could only be occupied by propositions.

(It is therefore already nonsense to put the hypothesis p[if…then]p in front of a proposition in order to ensure that its arguments are of the right form, because the hypothesis for a non-proposition as argument becomes not false but nonsensical [unsinnig], and because the proposition itself becomes nonsensical [unsinnig] with the incorrect kind of argument, therefore it saves itself from incorrect arguments just as well, or as badly, as the senseless [sinnlose] hypothesis hung on it for this purpose.)

Seems right enough.

Black (p. 297) says that ‘antecedent’ would be a better translation of Hypothese here.

5.535 Thereby also all problems that were connected with such pseudo-propositions take care of themselves.

All the problems that arise from Russell’s “Axiom of Infinity” can be solved here.

What the Axiom of Infinity is supposed to say would be expressed in language by there being infinitely many names with different meanings.

Ogden has ‘disappear’ for erledigen sich, while P&McG have ‘This also disposes of’, but my translation is more literal. It should be read as meaning ‘are dealt with by themselves’ or even ‘dispatched by themselves.’ ‘Connected’ could mean in one’s mind as well as in reality, by the way.

See also Russell’s Introduction to Mathematical Philosophy p. 142, where he says that if this axiom is not the case then “it must be theoretically possible for analysis to reach ultimate subjects, and it is these that give the meaning of “particulars” or “individuals.”” See Wittgenstein’s comment on what he means here on p. 50 of Letters to Ogden.

Black (p. 296) notes that it sounds as though Wittgenstein wants to ban certain formulas as pseudo-propositions, but that he says in 6.2 that mathematics consists entirely of pseudo-propositions. So maybe he doesn’t want to ban them after all.

5.534 And now we see that pseudo-propositions like “a = a”, “a = b . b = c . [if…then] a = c”, “(x) .x = x”, “(Ex) .x = a”, etc. cannot be written at all in a correct concept script.

I.e., one that dispenses with all that can be dispensed with.

5.533 The identity sign is thus not an essential part of the concept script.

Indeed.

5.5321 Instead of “(x): fx [if…then] x = a” we therefore write, e.g., “(Ex) .fx. [if…then]. fa: ~ (Ex, y) .fx .fy”.

And the proposition “only one x satisfies f( )” reads: “(Ex) . fx: ~(Ex, y) .fx .fy”.

All this does is eliminate (the need for) the sign “=”. OK. Clearly, it can be dispensed with.

Anscombe points out (p. 149) that he is here allowing a way of saying that only one thing has f. But in that case, “it is difficult to see how he could avoid a way of admitting formulae which say ‘There are only n things and m functions’ without using either ‘thing’ or ‘function’ as a function.” Yet at 5.535 the number of objects is supposed to be shown by the number of names with different references, not by some statement of how many objects there are. And what can be shown cannot be said. Supposedly. Looks like a problem, as Anscombe notes.

5.532 And analogously: Not “(Ex, y) . f(x, y) . x = y”, but rather “(Ex) .f(x, x)”; and not “(Ex, y) .f(x, y) .~x = y”, but rather “(Ex, y) .f(x, x)”.

(Therefore instead of the Russellian “(Ex, y) .f(x, y)”: “(Ex, y) .f(x, y) .v. (Ex) .f(x, x).)

OK

5.531 Therefore I write not “f(a, b) .a = b” but rather “f(a, a)” (or “f(b, b)”). And not “f(a, b) . ~a = b”, but rather “f(a, b)”.

No need to use the identity sign, in other words. Just the same sign for the same thing always. But what would “f(a, a)” mean? What would be the point?

5.5303 Roughly speaking: To say of two things that they are identical is a nonsense [Unsinn], and to say of one thing that it is identical with itself is to say nothing at all [gar nichts].

What’s the difference? Between speaking nonsense and saying nothing, and between saying “two objects have all their properties in common” and saying “two things are identical”? And why “roughly speaking” [beiläufig gesprochen] here?

5.5302 Russell’s definition of “=” is inadequate; because according to it one cannot say that two objects have all their properties in common. (Even if this proposition is never correct, it still has sense.)

Black (p. 292): “Russell’s definition of identity (Principia, vol. 1, definition 13.01) is based upon the principle of the identity of indiscernibles, which W. is here rejecting.”

Does the proposition in question really make sense? It depends what counts as a property.

5.5301 Identity is patently not a relation between objects. This becomes very clear if one considers, e.g., the proposition: “(x) : fx. [if…then] . x = a”. What this proposition says, is simply that only a satisfies the function f, and not that only such things satisfy the function f as have a certain relation to a.

One could of course say that in fact only a has this relation to a, but in order to express this we would need the identity sign itself.

And why is that a problem? Because we are trying to explain the meaning of the identity sign, I take it. To say that one thing is identical to another is to say something about the symbolism in use, not the objects in question. “A = A” tells you nothing about A. And a = b tells you nothing about a or b, only that these signs are used for the same thing.

5.53 I express identity of the object by identity of the sign, and not with the aid of an identity sign. [I express] difference of objects by difference of signs.

Fair enough.

5.5262 The truth or falsehood of every proposition changes something in the general structure of the world. And the range that the totality of elementary propositions would allow its structure is exactly the same as that which completely general propositions delimit.

(If an elementary proposition is true, then in any case there is thereby one more true elementary proposition.)

Is this perhaps because right now no elementary proposition is true? I don’t get it. Otherwise this seems to be saying that you don’t need elementary propositions: completely general ones are just as good.

5.5261 A completely generalized proposition is composed like every other proposition. (This is shown by the fact that in “(Ex, ø). øx” we must mention “ø” and “x” separately. Both stand independently in signifying relations to the world, as in an ungeneralized proposition.)

A characteristic of a composite symbol: It has something in common with other symbols.

The distinguishing mark of the composite symbol is also a feature of all propositions (see 5.513—they all have something in common except with their negative. And don’t they have a subject in common with their negative?) So it seems that there really are (can be) no elementary propositions, names, etc. All requires a context, some composition.

5.526 One can describe the world completely with completely generalized propositions, which means therefore without initially [or: a priori?] coordinating any name with a particular object.

In order then to get to the usual means of expression one must simply say “And this x is a” after an expression “There is one and only one x, such that…..”

Doesn’t this contradict what came earlier? How could a completely generalized proposition mean anything without being made up of elementary propositions, which must have names as their parts?

Black (p. 288): “W. does not mean that names are theoretically superfluous: as he explains in the Notebooks, ‘Names are necessary for an assertion that this thing possesses that property and so on.’ (53 (8,9)). W.’s point is that general propositions describe, without any imprecision, the general structural features—the make-up or constitution—of the actual universe. However, such a description cannot express the respects in which the actual universe differs from an isomorphic one that might have existed in its place.”

Fahrnkopf (p. 49) argues that "The second paragraph, which shows the replacement of an apparent variable by a name only for the case of 'a', a letter customarily used to represent an argument (i.e., a particular), is slightly misleading, because the point of talking about completely generalized propositions--as is made clear by 5.5621 as well as by the comparable discussion of completely generalized propositions in the Notebooks--is that in such propositions even the function is generalized.""The point, then," he continues, "of the statement in 5.562 that the description of the world can take place 'without first correlating any name with a particular object' is surely to stress that even those objects which are universals, and thus represented by function-signs, can be represented by apparent variables rather than by constants."

5.525 It is incorrect to render the proposition “(Ex). fx” in words – as Russell does – as “fx is possible”.

Certainty, possibility or impossibility of a state of things will not be expressed through a proposition, but by the fact that an expression is a tautology, a meaningful [sinnvoller] proposition, or a contradiction.

That precedent, to which one would always like to appeal, must already lie in the symbol itself.

Precedent? Black says the precedent is “the ground for an assertion of possibility, etc.” Russell implies that if something is possible then it is sometimes true. The ground for saying it is true would then be just the kind of fact that one wanted to say was possible.

Anscombe (see p. 80) sees this view of possibility as a consequence of the picture theory, an undesirable one, that is.

For Russell’s view see, e.g., Logic and Knowledge p. 231.

5.524 If the objects are given, then all objects are thereby also already given to us.

If the elementary propositions are given, then all elementary propositions are thereby also given.

Is this a pair of comments on objects and elementary propositions, or about generality? Is it a strange feature of objects that giving them means giving all objects? Or is it a fact about generality that “the objects” or “the dogs” means “all the objects”, “all the dogs”?

5.523 The symbol of generality occurs as an argument.

OK, that’s how it appears in the notation. See Letters to Ogden p. 49: “Here I want to use symbol and not symbolism because I refer to the variable x or y etc. in (Ex, y)… and not to the whole complex of symbols as before. I own this is very dark but please leave “symbol” here and don’t make it uniform with 3.24.”

5.522 What is peculiar to the symbolism of generality is first, that it points to a logical prototype, and secondly, that it emphasizes constants.

I follow Black (p. 284) on the translation of hinweist as ‘points to.’

What logical prototype? See 3.315.

What constants? Black (p. 284): “I think the ‘constants’ in question must be the names that could be substituted for the variable in the propositional function to which the quantifier is attached (not the constant formal features of the generalized proposition, i.e. those that show its form).”

Emphasized how? I'm not sure.

5.521 I separate the concept all from the truth-function.

Frege and Russell introduced generality in connection with the logical product or the logical sum. So it would be hard to understand the propositions “(Ex). fx” and “(x). fx”, in which both ideas are contained.

My translation of the last sentence is not very literal.

See Anscombe pp. 141-143 on this. She says that Frege and Russell did not at all explicitly do what Wittgenstein says here. The relevant Frege paper is “Function and Concept,” and Russell offers similar explanations of generality in his work. Frege explains his sign for generality in terms of what it means, and specifically in terms of when it means what he calls “the true.” Wittgenstein believes that the truth of a general proposition is the truth of a logical product. Hence his claim here about what he takes to be implicit in Frege and Russell. Universal propositions (For all x, …), he thinks, each say that some logical product is true, and particular propositions (For some x, …) each say that some logical sum is true.

White (p. 94) says that Wittgenstein’s target here seems mainly to be what Russell says in Principia Mathematica.

5.52 If the values of ξ are all the values of a function fx for all values of x, then N(ξ) [with a line over it] = ~(Ex). fx.

OK

5.5151 Must the sign of a negative proposition be constructed with the sign of a positive proposition? Why should one not be able to express a negative proposition by means of a negative fact. (For instance: If “a” does not stand in a specific relation to “b”, this could express the fact that aRb is not the case.)

But even here the negative proposition is given indirectly by the positive proposition.

The positive proposition must presuppose the existence of the negative proposition and vice versa.

Yes, this should be old news within the Tractatus. Cf. 5.44.

5. 515 It must be apparent in our symbols that what is connected with one another by “v”, “.”, etc. must be propositions.

And this is indeed the case, because the symbol “p” and “q” itself presupposes “v”, “~”, etc. If the sign “p” in “p v q” does not stand for a complex sign, then it cannot have sense on its own; but then also the signs “p v p”, “p. p”, etc., which have the same sense as “p”, could have no sense. If however “p v p” has no sense, then also “p v q” can have no sense.

So p must be a proposition, a complex sign.

Black (p. 279) writes: “The German text is puzzling and may have been printed incorrectly. I suggest (as an alternative to P.& McG.) that the remark might read: ‘And this is so, for the symbol p in p v q itself presupposes “v”, “~”, etc.”

5.514 If a notation has been set down, then there is in it a rule according to which all propositions negating p are to be constructed, a rule according to which all propositions affirming p are to be constructed, a rule according to which all propositions affirming p or q are to be constructed, and so on. These rules are equivalent to the symbols and in them their sense is mirrored.

More on rules. So symbols and rules are equivalent, and both depend on the notation. Although every notation, to be a notation I suppose, necessarily has these rules.

5.513 One could say: What is common to all symbols that assert p as well as q, is the proposition “p. q”. What is common to all symbols that assert either p or q, is the proposition “p v q”.

And thus one can say: Two propositions are opposed to one another if they have nothing in common with one another, and: Every proposition has only one negative, because there is only one proposition that lies completely outside of it.

It is evident also in Russell’s notation that “q: p v ~p” says the same as “q”; that “p v ~p” says nothing.

OK. One could equally say the same thing the other way around; that there is only one proposition that lies completely outside of each proposition, because it has only one negative. 5.513 itself says nothing.

5.512 “~p” is true if “p” is false. Thus in the true proposition “~p”, “p” is a false proposition. How can the stroke “~” now bring it to agreement with reality [or: the truth]?

What negates in “~p” is however not the “~” but that which is common to all signs of this notation that negate p.

Thus the common rule according to which “~p”, “~~~p”, “~p v ~p”, “~p. ~p”, etc. etc. (ad infinitum) are constructed. And this commonality mirrors negation.

So “~” negates only within a system of notation. It does not stand for some thing.

5.511 How can the all-embracing, world-mirroring logic use such special hooks and manipulations? Only by all these being connected into [i.e. so as to form] one infinitely fine network, the great mirror.

What hooks and manipulations? I don’t know. But the last sentence must be ironical, surely. My parenthetical comment on its meaning follows p. 61 of Letters to Ogden. Bearn (p. 63) connects the talk of a “great mirror” here with Tolstoy’s claim in his The Gospel in Brief, that, in Bearn’s words, “The key to what Tolstoy calls peace, joy, and security is to renounce one’s personal desires and make one’s own will a great mirror of the will of God.” I don’t find this convincing. See also 6.13.

Friedlander (p. 93, note 4) notes Schopenhauer’s reference to a mirror of the world (WWR, vol. 1, pp. 287-288): “Man … is the most complete phenomenon of the will, and, as was shown in the second book, in order to exist, this phenomenon had to be illuminated by so high a degree of knowledge that even a perfectly adequate repetition of the inner nature of the world under the form of representation became possible in it. This is the apprehension of the Ideas, the pure mirror of the world.” Friedlander takes the ideas to be mirrors, but it seems more natural to read this passage as calling man’s apprehension of the Ideas the mirror of the world, doesn’t it? This is borne out by p. 206 of vol. 2, where Schopenhauer writes of “the knowing part of consciousness” becoming “the clear mirror of the world.” Cf. p. 216 and p. 380. The intellect is the mirror of the world, for Schopenhauer.

Black (p. 27) calls this image of a mirror the dominant image of the whole book.

See also Schopenhauer Fourfold Root p. 151: “What is properly called thinking in the narrower sense is the occupation of the intellect with concepts, the presence in our consciousness of that class of representations here considered. It is also expressed by the word reflection which, as a metaphor from optics, at the same time states the derived and secondary character of this kind of knowledge.” This ability, Schopenhauer says, is what places us above the animals. P. 153: “All thinking in the wider sense, and hence all inner activity of the mind generally requires either words or pictures of the imagination; without the one or the other it has no support.”

Black (p. 277) says cf. 4.121. He notes that Anscombe identifies the great mirror with language on p. 164 of her Introduction.

There are 42 entries under “mirror” in the index to Schopenhauer’s The World as Will and Representation.

5.51 If ξ has only one value then N(ξ) [with a line over it] = ~p (not p), if it has two values then N(ξ) [with a line over it] = ~p. ~q (neither p nor q).

OK

5.503 It is clear that it is easily expressed how propositions can be constructed with this operation and how propositions are not to be constructed with it, so this must also be capable of exact expression.

A bit literal as a translation, but not too bad?

5.502 Therefore I write “N(ξ) [with a line over it]” instead of “(-----T) (ξ, …..).”

N(ξ) [with a line over it] is the negation of all the values of the propositional variable ξ.

Marie McGinn (p. 232): “The operation expressed by N(ξ) [with a line over it] is not strictly equivalent to Sheffer’s stroke, which is a two-place operator. N(ξ) [with a line over it] is a multi-grade operator, which jointly negates all the propositions that are the values of the variable ξ, that is, it corresponds to the operation expressed by the sign (-----T)( ξ,…).”

5.501 An expression in brackets whose terms are propositions I indicate – if the order of terms in the brackets is indifferent – by a sign of the form “(ξ)” [but with a line over it]. “ξ” is a variable whose values are the terms of the expression in brackets, and the line over the variables indicates that it represents all its values in the brackets.

(Thus if ξ has the three values P, Q, R, then (ξ) [with a line over it] = (P, Q, R).)

The values of the variables are to be determined.

The determination is the description of the propositions that the variable represents.

How the description of the terms of the expression in brackets is done is not essential.

We can distinguish three kinds of description: 1. Direct enumeration. In this case we can put in place of the variable simply its constant values. 2. Giving a function fx, whose values for all values of x are the propositions to be described. 3. Giving a formal law, according to which those propositions are constructed. In this case the terms of the expression in brackets are all the terms of a formal series.

OK. More definitions, pretty much. Friedlander (p. 77) argues that Ogden’s ‘determination’ is preferable to P&McG’s ‘stipulation’, which sounds too arbitrary. Black (p. 276) suggests ‘prescribed’ and ‘prescription.’

The italicization of “can” suggests that there is something arbitrary about this distinction.

5.5 Every truth-function is the result of the successive application of the operation “(-----T)(ξ, …..)” to elementary propositions.

This operation negates all the propositions in the right-hand brackets and I call it the negation of these propositions.

OK. Definition.

5.476 It is clear that this is not about a number of primitive [or fundamental] concepts, that must be signified, but about the expression of a rule.

A foretaste of the later Wittgenstein's interest in rule-following?

5.475 All that matters is to build a symbol system of a definite number of dimensions – of a definite mathematical multiplicity.

And the definiteness or precision of the number is what matters, apparently. Otherwise the number is arbitrary.

Black (p. 275) points to 4.04 for more on multiplicity.

5.474 The number of necessary fundamental operations depends only on our notation.

Yes: arbitrary.

5.4733 Frege says: Every legitimately constructed proposition must have a sense; and I say: Every possible proposition is legitimately constructed, and if it has no sense, then that can only be because we have given some of its parts no meaning.

(Even if we believe that we have done so.)

So “Socrates is identical” therefore says nothing because we have given no meaning to the word “identical” as an adjective. Since if it occurs as the sign of equality then it signifies [symbolisiert] in a wholly different way – the signifying [bezeichnende] relation is different – thus the symbol too in each case is wholly different; the two symbols have only the sign [das Zeichen] in common with one another, by accident.

Marie McGinn (p. 242) says that the reference here is to §32 of Frege’s The Basic Laws of Arithmetic. She adds (same page) that “Wittgenstein’s disagreement with Frege amounts to a reassertion of the context principle, and thereby of the priority of the concept of the sense of a proposition over that of what the constituents of a proposition signify.”

Black also cites vol. 2, §92 of Frege’s book.

My comment: Frege allows propositions (and thoughts?) that are not properly constructed. Wittgenstein says these either are properly constructed or else not propositions at all. Meaning depends not on proper construction of propositions, but on propositions having parts that have been given meaning. It is arbitrary all the way down: what counts as a proposition in our system and which words have been given definitions.

5.47321 Occam’s razor is of course not an arbitrary rule, or one justified by its practical success: it says that unnecessary symbolic units mean nothing [nichts bedeuten].

Signs that fulfill a single purpose are logically equivalent, signs that fulfill no purpose are logically meaningless [bedeutungslos].

Signs in logic, and that is where signs live, have bedeutung only so far as they do something. And what they do depends on arbitrary conventions, definitions, etc.

5.4732 We cannot give a sign the wrong sense.

We either fail to give it a sense, or we give it an OK sense.

5.4731 Self-evidence, which Russell spoke so much about, can only become superfluous in logic by language itself preventing each logical mistake. – That logic is a priori consists in the fact that nothing illogical can be thought.

Cf. 5.1363.

Black (p. 274) cites Russell’s “Philosophical importance” p. 490 and p. 492 as good sources for his views on self-evidence.

So language does prevent every mistake of a certain kind. Such mistakes cannot be thought. And this is because we have not defined some of their terms. So they might seem to be thoughts, but they are not in fact so. And it is language, not logic per se, that shows this.

5.473 Logic must take care of itself.

A possible sign must be able to signify. Everything that is possible in logic is also allowed. (“Socrates is identical” therefore denominates nothing [heisst darum nichts] because there is no property that “identical” denominates. The proposition is nonsensical [unsinnig] because there is some arbitrary definition that we have not made, but not because the symbol in and of itself would be forbidden.)

We cannot, in a certain sense, go wrong in logic.

That is, there is a certain kind of error that one cannot make. Namely, the use of absolutely (not arbitrarily, or conventionally) forbidden symbols.

The first sentence of this remark is the same as the first sentence in the Notebooks 1914-1916. This is an important idea, apparently, for Wittgenstein.

5.472 The description of the most general propositional form is the description of the one and only general primitive sign of logic.

Which is…?

5.4711 Giving the essence of the proposition means giving the essence of all description, thus the essence of the world.

This sounds very significant indeed.

5.471 The general propositional form is the essence of the proposition.

OK

5.47 It is clear that everything that can be said generally in advance about the form of all propositions must be able to be said all at once.

Indeed all logical operations are already contained in an elementary proposition. Because “fa” says the same as “(Ex). fx. x=a.”

Where there is complexity [compoundness] there is argument and function, and where these are, are already all logical constants.

One could say: the one logical constant is that which all propositions by their nature have in common with one another.

That though is the general propositional form.

So really there is no true logical constant, or this line of inquiry has misconceived the nature of logical constants.

5.4611 Logical operation signs are punctuation marks.

So none has an independent meaning, surely. Proops (p. 15): “Wittgenstein’s point is that the logical connectives share with punctuation marks the feature of lacking sense and reference while nonetheless having a meaning in their own right. The point of the comparison with punctuation is to bring out that the logical connectives make a purely structural contribution to the meanings of the sentences in which they figure.”

5.461 The apparently unimportant fact that logical pseudo-relations like v and [if…then] need brackets – in contrast to real relations – is of great importance [bedeutungsvoll].

The use of brackets with these seemingly primitive signs indicates [deutet] already indeed that these are not real primitive signs. And surely nobody is going to believe that brackets have an independent meaning.

OK, although brackets have a use, so have they as much meaning as anything else in logic? What would an “independent meaning” [eine selbständige Bedeutung] in logic be?

5.46 If one introduced logical signs correctly then one would also thereby have already introduced the sense of all their combinations; thus not only “p v q” but already also “~ (p v ~q)” etc. etc. One would thereby also already have introduced the effect of all possible combinations of brackets. And thereby it would have become clear that the proper general primitive signs are not “p v q,” “(Ex). fx,” etc., but the most general form of their combinations.

Black (p. 269) offers “real indefinable signs of logic” as an alternative to “proper general primitive signs.”

But if there are no primitive signs in logic--and haven’t we been being pushed toward this view in the last few pages?-- then there is no such thing as this most general form of the combinations of logical signs. And this general form is extremely general. Can we identify it at all? Is it a thing? Is this the form of the world? Or an idol?

5.4541 The solutions of logical problems must be simple because they set the standard of simplicity.

People have always suspected that there must be a field of questions to which the answers – a priori – are symmetrical and form a closed, regular structure.

A field in which the proposition holds: simplex sigillum veri.

If the solutions of logical problems set the standard of simplicity then whatever they are is what ‘simple’ means. So this is a definition, not information about the nature of these solutions. Is what people always suspected necessarily true? False? What is it that they have always suspected anyway (this is the hard part)?

The Latin means “simplicity is the hallmark of truth.” Black (p. 268) points out that this was a motto of Herman Boerhaave (1668-1738) of Leyden. Schopenhauer uses the phrase in his dialogue on religion (pp. 95-114 in Essays and Aphorisms edited by R. J. Hollingdale, Penguin Books, 1970, p. 106), in which Philalethes says: “Simplex sigillum veri: naked truth must be so simple and intelligible that it can be imparted to everyone in its true shape without adulterating it with myths and fables (a mass of lies) – that is, without disguising it as religion.”

Proops (p. 27, note 80, incorrectly referred to as note 79 in the text on p. 26) suggests ‘had an inkling’ where I have ‘suspected’ because Ogden’s ‘thought’ “risks making it sound as though Wittgenstein regarded the idea as some kind of delusion. Pears’ and McGuinness’s translation: “mankind has always had a presentiment,” is superior to Ogden’s, but a little grandiloquent.” But this reminds me of 6.3211, which uses the same root (here it is geahnt, a form of the verb ahnen (to suspect), there it is Ahnung, the noun (suspicion). It is not clear that Wittgenstein thinks there is no delusion here.

5.454 In logic there is no coexistence, there can be no classification.

In logic there cannot be a more general or a more specific.

No classification? No generality? That can’t be what he means (surely?). Black (p. 268) says cf. 6.127: “all logical propositions are on the same level.”)

5.453 All numbers in logic must be able to be justified.

Or rather: it must be evident that there are no numbers in logic.

There are no pre-eminent numbers.

So logic is not mathematics? Might mathematics still be derivable from logic?

5.452 The introduction of a new device in the symbolism of logic must always be an event of great consequences. No new device may be introduced into logic – with, so to speak, a wholly innocent face – in brackets or in a footnote.

(Thus in the Principia Mathematica of Russell and Whitehead there appear definitions and basic laws in words. Why suddenly words here? This would need a justification. This is missing and must be missing, since the procedure is actually forbidden.)

If, however, the introduction of a new device has proved necessary in one place, then one must ask oneself straightaway: Where must this device now always be used? Its place in logic must now be made clear.

All sounds fair enough. Black (p. 266): “An example of what Wittgenstein has in mind is the ‘primitive proposition’ *1.1: ‘Anything implied by a true elementary proposition is true’ (Principia, vol. 1, p. 94).”

5.451 If logic has primitive concepts then they must be independent of each other. If a primitive concept is introduced then it must be introduced in every combination in which it ever occurs. One cannot therefore introduce it for one combination first and then another time for another. E.g., if negation is introduced then we must now understand it in propositions of the form “~p” in just the same way as in propositions like “~(p v q,” “(Ex). ~fx” et al. We may not introduce it first for one class of cases and then for another, because it would then remain undecided whether its meaning [Bedeutung] in each case was the same, and there would be no available ground for using the same way of combining signs in both cases.

(Briefly, what Frege (Grundgesetze der Arithmetik) has said about the introduction of signs through definitions goes, mutatis mutandis, for the introduction of primitive signs.)

Agreeing with Frege and therefore disagreeing with Frege, then.

White (p. 89): “We understand the point of this paragraph best if we see its target as Russell and the way the primitive logical constants were introduced in Principia Mathematica.” Russell and Whitehead first introduced signs for ‘or’ and ‘it is not the case that’, using ‘~’ only where there were no quantifiers used. When they later introduced quantifiers they had to explain how these worked together with the negation sign. Then they define what these combinations of signs mean. White says that Wittgenstein objects to this piecemeal approach. “Either the negation sign means the same as it did when it was first introduced, in which case the significance of its combination with the quantifiers ought to follow from the way it was initially explained, or it means something different, in which case to use the same sign leads to confusion. Wittgenstein is claiming that this situation can only be avoided if we introduce all the primitive signs of logic, not in serial order, but all at one go.”

5.45 If there are primitive signs of logic then a correct logic must make clear their position with regard to one another and justify their being. The construction of logic out of its primitive signs must be made clear.

Fair enough.

5.442 If we are given a proposition then with it we are already given the results of all truth-operations that have it as their basis.

OK

5.441 This vanishing of the apparent logical constants also occurs if “~(Ex) . fx. X=a” says the same as “(x). fx,” or “(Ex). Fx. X=a” the same as “fa.”

OK

5.44 Truth-functions are not material functions.

If one can produce, e.g., an affirmation by double negation, is then negation – in any sense – contained in the affirmation? Does “~~p” negate ~p, or affirm p; or both?

The proposition “~~p” does not deal with negation as with an object; but the possibility of negation is already presupposed in affirmation.

And were there an object called “~” then “~~p” would have to say something other than “p.” Since the one proposition would then deal with ~, the other not.

Yes. Clearly, logic is not metaphysics.

5.43 It is scarcely credible that from a fact p infinitely more others should follow, namely ~~p, ~~~~p, etc.. And it is no less remarkable that the infinite number of propositions of logic (of mathematics) follow from half a dozen “primitive propositions.”

All propositions of logic say the same thing however. Namely, nothing.

Wittgenstein suggests leaving out any translation of “von vornherein” (“from the very beginning”) in the first sentence here. See Letters to Ogden, p. 31. Those propositions follow from p, of course, but the idea that they are different facts is not credible (i.e. false?). Logic and mathematics cannot be derived from primitive propositions of logic. Now what about Wittgenstein’s own elementary propositions?

5.42 It is obvious that v [i.e. or], [if…then], etc. are not relations in the sense that right and left etc. are.

The possibility of the crosswise definition of the logical “primitive signs” of Frege and Russell shows already that these are not primitive signs, and much less still signs for relations.

And it is obvious that the “[if…then]” that we define by means of “~” and “v” is identical with that by which we define “v” with the help of “~”, and that this “v” is the same as the first, and so on.

We have inter-definable signs, in other words, which are functions or operations. None is primitive, since they are inter-definable, and none is a relation. Translation of the last sentence based on Wittgenstein’s suggestion, p. 61 of Letters to Ogden.

5.41 Because all results of truth-operations on truth-functions are identical when they are one and the same truth-function of elementary propositions.

OK?

5.4 Here it becomes apparent that there are no “logical objects” or “logical constants” (in Frege’s and Russell’s sense).

What about objects in Wittgenstein’s sense?

5.32 All truth-functions are results of the successive application of a finite number of truth-operations to elementary propositions.

OK

5.31 The schemata in no. 4.31 also then have a meaning [Bedeutung], if “p,” “q,” “r,” etc. are not elementary propositions.

And it is easy to see that the propositional sign in no. 4.42, even if “p” and “q” are truth-functions of elementary propositions, expresses a single truth-function of elementary propositions.

Is this easy to see? And doesn’t this all still depend on the idea of elementary propositions?

Fahrnkopf (p. 41) says that this is the only place in the Tractatus where 'Bedeutung' is used not in connection with objects (but rather in "a wide, non-technical sense").

5.3 All propositions are results of truth-operations on elementary propositions.

Truth-operations are the way that truth-functions are produced from elementary propositions.

In accordance with the essence of truth-operations, a new one arises from truth-functions in the same way that their truth-functions arise from elementary propositions. Every truth-operation begets from truth-functions of elementary propositions another truth-function of elementary propositions, a proposition. The result of every truth-operation on the results of truth-operations on elementary propositions is again the result of a single truth-operation on elementary propositions.

Every proposition is the result of truth-operations on elementary propositions.

Clear? Or should I re-translate?

So the whole machinery here rests on elementary propositions. And wasn’t that concept called into doubt above? See 4.221.

5.254 An operation can vanish (e.g. the negation in “~~p”: ~~p = p).

OK

5.253 One operation can cancel the work of another. Operations can neutralize one another.

OK

5.2523 The concept of successive application of an operation is equivalent to the concept “and so on.”

Ironic banality?

5.2522 The general term of a formal series a, O’ a, O’ O’ a, … I write thus: “[a, x, O’ x].” This bracketed expression is a variable. The first term of the bracketed expression is the beginning of the formal series, the second the form of an arbitrary term x of the series, and the third the form of the term of the series that follows immediately after x.

OK, more definitions.

5.2521 I call the repeated application of an operation to its own result its successive application (“O’ O’ O’ a” is the result of three successive applications of “O’ ξ” to “a”).

In a similar sense I speak of the successive application of multiple operations to a number of propositions.

OK, definition.

5.252 Only thus is the advance from term to term possible in a formal series (from type to type in the hierarchy of Russell and Whitehead). (Russell and Whitehead did not admit the possibility of this advancing, but they made use of it again and again.)

Black (p. 260): “W.’s criticism can properly be leveled against the use of ‘typical ambiguity’ in Principia (for which see vol. 1, p. 65, and the ‘prefatory statement’ to vol. 2). It has often been pointed out that Russell needs, for example, not a single axiom of reducibility, but an indefinite number of such axioms, applicable to entities of different types. For a specific criticism of this sort, see Anscombe, Introduction, p. 130.”

5.251 A function cannot be its own argument, but the result of an operation can be a basis for that operation.

Cf. 3.333.

5.25 The occurrence of an operation does not characterize the sense of a proposition.

An operation indeed asserts nothing, only its result does, and this depends on the bases of the operation.

(Operation and function must not be confused with one another.)

Fair enough.

5.242 The same operation that makes “q” from “p,” makes “r” from “q” and so on. This can only be expressed by the fact that “p,” “q,” “r,” etc. are variables that give general expression to certain formal relations.

The first sentence seems banal, the second a little odd. Why can the first sentence’s point be expressed only by the way the second sentence puts it?

5.241 An operation characterizes no form but only the difference between forms.

OK

5.24 An operation shows itself in a variable: it shows how from one form of proposition one can arrive at another.

It gives expression to the difference between the forms.

(And what is common to the bases and the result of an operation is just the bases.)

OK

5.2341 The sense of a truth-function of p is a function of the sense of p.

Negation, logical addition, logical multiplication, etc., etc. are operations.

(Negation reverses the sense of a proposition.)

OK