6.36 If there were a law of causality, then it could read: “There are laws of nature.”

But of course one cannot say that: it shows itself.

So is there no such law? What about 6.321? Physics unapplied is quite formal, and empty. Perhaps that is the point. For “There are laws of nature” to be a law of physics would be utterly pointless (a pointless utterance). Instead, physics tells us various laws of nature. Without such action/application the “law” would be quite empty, lacking content (what laws of nature?). With it, it loses any point it might have.

Black (p. 362): “W. should probably be read here as denying the significance of any notion of ‘causality’ (cf. his denial of the ‘causal nexus’ at 5.136). He might have agreed with other writers on the philosophy of science that the laymen’s notion of ‘cause’ comes to be superseded by a notion of ‘law’, adding a caveat about the latter being a formal notion.”

6.35 Although the spots in our picture are geometrical figures, geometry can still obviously say absolutely nothing about their actual form and position. But the net is purely geometrical, all its properties can be given a priori.

Laws, like the principle of sufficient reason, etc., deal with the net, not with what the net describes.

So geometry can be used to describe the spots, but there is no a priori knowing their shape or position. The net can be described a priori, though, so it belongs to logic. So what is the relation between a particular net and all possible nets?

Black (p. 361) says that “purely geometrical” is a reference to pure, as opposed to applied, geometry.

6.3432 We must not forget that the description of the world with mechanics is always completely general. In it there is never, e.g., talk of particular material points, but rather always only about any such points.

Pears & McGuinness have “point-masses” for “material points.” I should check this.

Black (p. 361) says that this should be read in connection with 6.35.

6.3431 Through the whole logical apparatus, throughout the physical laws still speak of the objects of the world.

Huh? Maybe I should leave “throughout” out, or else move it to the very end. Wittgenstein (Letters to Ogden p. 35) says the first “through” means the same as in “I speak through a tube.” So Pears and McGuinness seem quite wrong this time. (They have "The laws of physics, with all their logical apparatus, still speak, however indirectly, about the objects of the world.") How about: "Via the whole logical apparatus, the physical laws still speak throughout of the objects of the world."?

Black (p. 361) suggests that this comment “Can be read as a summary comment on 6.342 (2).”

6.343 Mechanics is an attempt to construct according to one plan all true propositions that we need for a description of the world.

OK, but what is logic?

6.342 And now we see the relative position of logic and mechanics. (One could also have a net consisting of different kinds of shapes, such as triangles and hexagons.) It says nothing about a picture, such as the one mentioned above, that it can be described by a net of a given form. (Since this goes for every picture of this kind.) However it does characterize the picture that it can be completely described with a specific net of a specific fineness.

Thus too it says nothing about the world that it can be described with Newtonian mechanics; but [it does say something] that it can be described in that particular way in which indeed it is described. It also says something about the world that it can be described more simply with one mechanics than with another.

So there is no absolutely right kind of mechanics, but some are more useful than others, given the way the world is. And given, presumably, what kind of thing we find easy. But what then is the relative position of logic to mechanics? Logic seems unable to choose our mechanics for us. Here I follow Wittgenstein’s comments on the translation of the penultimate sentence on p. 50 of Letters to Ogden.

6.341 Newtonian mechanics, e.g., brings the description of the world to a unified form. Let us think of a white surface with irregular black spots on it. Now we say: Whatever kind of picture these spots produce, I can always describe it as closely as you like by covering the spots with a suitably fine square netting and now say of every square that it is white or black. In this way, I will have brought the description of the spots to a unified form. This form is arbitrary, since I could have used with the same success a net with triangular or hexagonal holes. It is possible that the description would have been simpler with the help of a triangular net; meaning that we could have described the spots more closely with a bigger triangular net than with a finer square one (or vice versa), and so on. Different systems of world description correspond to different nets. Mechanics defines a form of world description by saying: All propositions of the description of the world must be obtained from a number of given propositions – the axioms of mechanics – in a given way. In this way it supplies the building stones for the construction of the scientific edifice and says: Whatever edifice you want to build, you must somehow put together with these and only these building stones.

(With the system of mechanics, one must be able to write down any arbitrary proposition of physics, as one can [write down] any arbitrary number with the number system.)

A priori axioms are normative, then, and might make our lives easier or harder, but they cannot make them possible or impossible. Nor can they tell us anything synthetic.

6.34 All these propositions, like the principle of sufficient reason, of continuity in nature, of least expenditure in nature, etc. etc., all these are a priori insights concerning the possible fashioning of propositions of science.

So a feeling that there must be a certain kind of law is the recognition that there is room in the system for such a law, a law of that (so far fairly vague) type?

Black (p. 346) says that in a letter to Russell (129, 2) Wittgenstein treats “principle of sufficient reason” and “law of causation” as synonymous.

6.33 We do not believe a priori in a law of conservation, but rather we know a priori the possibility of a logical form.

Black (p. 346) points out that “if form is itself a possibility, the phrase [i.e. “the possibility of a logical form”] shows redundancy.”

6.3211 One indeed also had a presentiment that there must be a “law of least action”, before one knew precisely how it went. (Here, as ever, the a priori certain proves to be something purely logical.)

So what was a priori certain here? The presentiment? Why must there be such a law?

6.321 “Law of causality” – that is a generic name. And as we say there are minimum laws in mechanics, – such as that of the least action – so there are in physics laws of causality, laws of the causality form.

Laws like the one discussed in 6.32, presumably.

Black (p. 346) quotes Mach (Mechanics, p. 460) saying that: “Maupertuis really had no principle, properly speaking, but only a vague formula, which was forced to do duty as the expression of different familiar phenomena not really brought under one conception.” On p. 551 Mach adds that Euler changed the principle (i.e. the law of least action) “into something new and really serviceable.”

6.32 The law of causality is not a law, but rather the form of a law.

Mounce (see pp. 75-76) says that the law of causality is the law of sufficient reason, i.e. the idea that everything has a cause. This, as he reads Wittgenstein, is not a law because it tells us nothing about the world. So far as two events can be distinguished, they must have some difference, and this difference can always be regarded as causally relevant. Saying “everything has a cause” then is not really reporting on a contingent generality but insisting a priori that every event will be interpreted as caused, as it can be.

Black (p. 345) points out that at 2.033 and 2.151 Wittgenstein links form with possibility. If he is talking about the possibility of a certain kind of empirical generalization then, Black thinks, this fits with 6.321-6.34, but not with 6.36.

My original comment: What is the law of causality? No event without a cause? But then what is an event? How are we to divide time up into events? Perhaps this is why this is not really a law. It says, in effect, no x without a y. And that is the form of a law, i.e., as Russell says in his footnote to the Ogden translation, “not the form of one particular law, but of any law of a certain sort.” I take it that “B. R.” here refers to Russell.

Black (p. 345) says that by 6.36 “the ‘law of causality’ has been emptied of any determinate meaning.”

6.31 The so-called law of induction cannot in any event be a logical law, since it is clearly a meaningful [sinnvoller] proposition. – And therefore it also cannot be an a priori law.

So it’s really no law at all. But, as Black (p. 345) says, “Which ‘law of induction’ did W. have in mind? Perhaps, with echoes of Aristotle, one allegedly permitting an inference from ‘Some A’s are B’ to ‘All A’s are B’?”

6.3 The exploration of logic means the exploration of all regularity [lawfulness]. And outside logic everything is accidental.

Logic deals with the necessary, the a priori. Outside logic everything else is a posteriori and contingent. So have we had somewhere already proof that Kant was wrong? The synthetic a priori seems to be being rejected here. On what grounds?

6.241 So the proof of the proposition 2 x 2 = 4 runs thus:

(^v)^µ'x=^v×µ'x Def.
^2×2'x = (²)²'x = (²)¹⁺¹'x = ²'²'x = ¹⁺¹'¹⁺¹'x
= (')'(')'x = ''''x = ^1+1+1+1'x = ⁴'x.

What’s going on here? I thought there was no proof in mathematics? See 6.2321.

Black (p. 343): “W.’s proposed ‘proof’ is eccentric and would not satisfy contemporary standards of mathematical rigour. It may be pointed out, for instance, that he has provided no rules for the use of variable superscripts with the lamda symbol.”

6.24 The method of mathematics, to get to its equations, is the method of substitution.

Since equations express the substitutability of two expressions and we proceed from a number of equations to new equations by substituting other expressions in accordance with the equations.

OK

6.2341 The essence of the mathematical method is to work with equations. On this method depends precisely the fact that every proposition of mathematics must go without saying.

OK. The first sentence here would be controversial, I imagine, but it isn't really a new idea at this point in the Tractatus.

6.234 Mathematics is a method of logic.

Cf. 6.2

6.2331 The process of calculation brings about precisely this intuition.

Calculation is not an experiment.

The result cannot be a surprise (even if it isn't what you expected).

6.233 The question whether one needs intuition to solve mathematical problems must be answered by the fact that language itself here supplies the necessary intuition.

No, in other words.

6.2323 An equation marks only the standpoint from which I regard the two expressions, namely the standpoint of their equality of meaning.

Black (p. 342) says “identity” would be better than “equation” here.

6.2322 The identity of the meaning [Bedeutung] of two expressions cannot be asserted [maintained, contended]. Because in order to be able to assert something about their meaning, I must be acquainted with their meaning: and in being acquainted with their meaning, I know whether the meaning is the same or different.

So in some sense I cannot say [“maintain” is another possible translation of behaupten] “The Morning Star” refers to the same thing [has the same Bedeutung] as “the Evening Star.” That is, I cannot hold or prove that this is a fact. “Facts” about expressions are tautologies, matters of definition, not contentions.

6.2321 And, that the propositions of mathematics can be proved, means indeed nothing other than that their correctness can be seen without it being necessary to compare what they express with the objects in order to determine its correctness.

Pears & McGuinness are good here. So mathematical proof is a priori, like “proof” in logic.

6.232 Frege says the two expressions have the same meaning [Bedeutung] but different senses [Sinn].

But the essence of an equation is that it is not necessary in order to show that the two expressions that the equals sign combines have the same meaning [Bedeutung], since this can be seen from the two expressions themselves.

The distinction between meaning and sense (and hence between senselessness and nonsense?) breaks down, because, roughly speaking at least, meaning is use. The notion of sense is seemingly treated here as unnecessary or irrelevant. If 1+1 = (1+1) then this should be manifest in all sorts of ways within mathematics. We do not need to be told it, as if it were an axiom or first principle. To know what (1+1) means is to know, among other things perhaps, that it means (is substitutable for) 1+1. The substitutability of these expressions for each other is not an additional fact, additional to the meaning of each.

6.231 It is a property of assertion that one can understand it as double negation.

It is a property of “1+1+1+1” that one can understand it as “(1+1) + (1+1)”.

6.23 If two expressions are combined with the identity sign, this means that they are substitutable for one another. But whether this is the case must be evident in the two expressions themselves.

It is a characteristic of the logical form of two expressions that they are substitutable for one another.

What is meant by an expression here? Is it evident in the expression “4” that it is identical with the expression “2 = 2”? Perhaps “expression” means “expression rightly and fully understood”.

6.22 The logic of the world, which the propositions of logic show in tautologies, mathematics shows in equations.

So equations are at least like tautologies. And the logic of the world is the logic of language, since their limits are the same, making them co-extensive. Looks like linguistic idealism, doesn’t it?

White (pp. 109-110): "To understand what Wittgenstein means by ‘equations’, we need to refer back to 4.241-4.242. There they are described as only ‘representational devices’ and that is what we need to understand if we are to interpret the claim that they are ‘pseudo-propositions’.”

Black (p. 341): “It is hard to see how what is shown in equations can be assimilated in this way to what is shown in tautologies.”

6.211 In life it is indeed never the mathematical proposition that we need, but rather we use mathematical propositions only in order to infer from propositions that do not belong to mathematics to others that likewise do not belong to mathematics.

(In philosophy the question “To what end do we really use this word, that proposition” leads time and again to valuable insights.)

So, for instance, “I have to give Jo this five-dollar bill” is not part of mathematics, but we might use mathematics to reach this practical conclusion from “Jo gave me a ten-dollar bill for this five-dollar soup and focaccia.” “Meaning is use” here as a guiding principle.

6.21 A proposition of mathematics expresses no thoughts.

Because it is not a science, but something of logic. An equation is a kind of tautology after all. He appears to be saying that mathematics belongs to logic, but not that it is based on or derived from logic. There is no rank in logic.

6.2 Mathematics is a logical method.

The propositions of mathematics are equations, hence pseudo-propositions.

Black (p. 341): “In discussing the proposed view of mathematics, Ramsey says: ‘this is obviously a ridiculously narrow view of mathematics, and confines it to simple arithmetic’ (Foundations, p. 17). One difficulty raised by him is the presence in mathematics of inequalities (op. cit. p. 282). But the more serious difficulty, he holds, is that of accounting, on W.’s view, for the applicability of arithmetic (op. cit. p. 19).”

6.13 Logic is not a theory [doctrine, science], but rather a mirror-image of the world.

Logic is transcendental.

Does this mean: It is not hypothetical in any way, but a direct reflection of how the world is? And transcendental in what sense?

Black (p. 340) says it means beyond experience or a priori.

6.1271 It is clear that the number of “logical basic laws” is arbitrary, since one could indeed derive logic from a single basic law, by simply, e.g., forming the logical product of Frege’s basic laws [Grundgesetzen]. (Frege would perhaps say that this basic law is now no longer immediately self-evident. But it is remarkable that so exact a thinker as Frege appealed to the degree of self-evidence as the criterion of a logical proposition.)

Yes, what is self-evident in this sense, or degree of self-evidence at any rate, is a psychological matter, not a logical one. The anti-psychologism Frege, of all people, should have seen that.

6.127 All propositions of logic have equal rights [i.e. (legal) status], there are among them no essential basic laws or derived propositions.

Every tautology shows by itself that it is a tautology.

I like the more literal translation of “equal rights”. It is all flat, all tautology.

6.1265 One can always conceive of logic in such a way that every proposition is its own proof.

Propositions of logic are rules that say that propositions of that form, i.e. their own form, are legitimate or correct. So they are self-validating. Or so we can understand them.

White (p. 108): “We may consider 6.1265 to be half truth. If it only means that a truth of logic contains within itself all the information necessary to settle its truth value, without using an idea of its derivability from other propositions as a criterion of its being a truth of logic, that can be allowed to stand. If, however, as is natural, we take ‘proof’ to be an epistemological concept, implying that we can always tell whether or not a proposition is logically true by ‘calculating the logical properties of the symbol’ (6.126), we now know that to be false.”

6.1264 A meaningful proposition states something, and its proof shows that it is so; in logic every proposition is the form of a proof.

Every proposition of logic is a modus ponens presented in signs. (And one cannot express the modus ponens with a proposition.)

Modus ponens is argument of the form: If p then q, p, therefore q. Why is all logic just this in different forms? Why can’t this be expressed in a proposition? Perhaps because it is the general form that matters. What is p, after all, but a variable? Replace it with something specific and the general form goes away. But without such replacement, we don’t have a sinnvollen proposition.

6.1263 It would indeed be all too remarkable if one could prove a meaningful [sinnvollen] proposition logically from another, and a logical proposition too. It is clear in advance that the logical proof of a meaningful proposition and proof in logic must be two completely different things.

Logical propositions are not sinnvollen, of course. Proof in logic must not be proof of sinnvollen propositions. But can there be logical proof of such propositions? He doesn’t say. If there is, though, it must something other than “proof” in logic.

6.1262 Proof in logic is only a mechanical means to make perception of a tautology easier in cases where it is complicated.

Hence the quotation marks earlier (6.126) around “proof”. Proof is relative to a system. Proof of guilt in law is proof of guilt beyond a reasonable doubt. Proof in mathematics depends on the laws/rules in play. For Frege, proof means logical proof, i.e. demonstration that a sentence/formula is equivalent to an axiom or derivable from some other part of the system. Derivable, presumably, according to the rules of derivation in the system. But presumably something, whether axioms or processes of derivation, must be simply given or accepted as intuitively obvious. Experiments are particular. Manipulation of actual rods might be an experiment in physics, but there are no experiments in mathematics. If one rod breaks, this can prove nothing mathematical. Similarly if the paper tears when drawing a diagram. A diagram might prove something in geometry, but only if taken a certain way, so that physical features of the diagram are ignored, for instance.

McManus p. 85: “ontological distinctions are now [with the right mechanical means, i.e. Begriffsschrift] shown simply in that ‘they’ are shown up for what they are: namely, the confused product of word-play.”

6.1261 In logic process and result are equivalent. (Hence no surprises.)

It just is a calculus, not a machine or mechanism. Nothing is produced. There is no cause and effect. Only the application of rules (in accordance, can I say?, with themselves).

6.126 One can figure out whether a proposition belongs to logic by figuring out the logical properties of symbols.

And this is what we do when we “prove” a logical proposition. Because without concerning ourselves with a sense [Sinn] and a meaning [Bedeutung] we construct the logical proposition from others according to mere rules for signs.

The proof of a logical proposition consists in our being able to establish it from other logical propositions with successive applications of certain operations, which produce ever more tautologies from the first one. (In fact from a tautology only tautologies follow.)

Of course this way of showing that its propositions are tautologies is thoroughly inessential to logic. Because the propositions, from which the proof starts out, must show, indeed without proof, that they are tautologies.

Logic is, or belongs to, a system that cannot justify itself.

Frascolla (p. 141): “If we take the expression ‘logical proof’ in its more general meaning, and if we call ‘mechanical’ any procedure of calculation, understood as a set of effective instructions for manipulating symbols, then in the light of Church’s Theorem of Undecidability of the first-order predicative calculus, Wittgenstein’s thesis is simply false, since no mechanical procedure can exist which enables us to decide, given any arbitrary formula of the first-order predicative calculus (which is included in the calculus of Principia Mathematica), whether it is a tautology or not.”

6.1251 Hence there can also never be surprises in logic.

OK. Nothing contingent, that is. We cannot guarantee that someone won't feel surprised at something in logic.

6.125 It is possible, in fact also according to the old conception of logic, to give in advance a description of all “true” logical propositions.

It’s all a priori, in other words.

6.124 Logical propositions describe the frame of the world, or rather they present it. They “deal” with nothing. They presuppose that names have meaning and elementary propositions sense: And this is their connection with the world. It is clear that it must show [anzeigen] something about the world that certain combinations of symbols – which essentially have a specific character – are tautologies. Herein lies the decisive thing. We said that much in the symbols that we use is arbitrary, much not. In logic only this latter expresses anything: That means however that in logic we do not express what we want with the aid of signs, but rather in logic the nature of the constitutionally necessary signs exhausts itself: If we are acquainted with the logical syntax of some sign language, then all the propositions of logic are already given.

I originally had “physically necessary” for naturnotwendigen because that is the only meaning given by my dictionary for this word, but I think "constitutionally necessary" might be a better translation. Ogden has “essentially necessary” and P&McG have “absolutely necessary.” Black (p. 336) says that the sense of the expression “is that the signs needed are of a nature that is necessitated.” I think Ogden’s translation might be best, but when in doubt I prefer to be literal.

The rest I think makes sense and sounds right, apart from the slightly worrying (because it still is not wholly clear what these are) talk of names and elementary propositions, but I assume these now mean basically just “words” and “propositions” or “sentences” (like p and ~p) that are combined in logic in various ways. The really worrying thing is the claim that the existence of tautologies must show something about the world. What does it, could it, show except itself? Well, Wittgenstein does not claim that it shows anything more than this.

Black (p. 331) points out that the image of scaffolding (Gerüst) occurs also at 3.42 and 4.023.

6.1233 A world can be conceived in which the axiom of reducibility does not hold. But it is clear that logic has nothing to do with the question whether our world really is thus or not.

What grounds did Russell have for calling the Axiom of Reducibility a principle of logic? The axiom in question says that any higher-order property or proposition can be reduced to an equivalent first-order one. It means that the same class is determined by two propositional functions that are equivalent.[1] Every propositional function is thus logically equivalent to a predicative function. The axiom is introduced because without it the ramified theory of types makes certain mathematical proofs impossible. This kind of thing belongs to logic, in Russell’s view, because it is necessary in order to overcome logical, not just mathematical, paradoxes, e.g. about the class of classes not members of themselves as well as paradoxes about infinity. Ramsey and Wittgenstein showed that the axiom was not necessary.

---

[1] See p. 299 of the Cambridge Companion to Russell.

6.1232 One could call logical general validity essential, in contrast to accidental general validity, for instance of the proposition “All men are mortal.” Propositions like Russell’s “axiom of reducibility” are not logical propositions, and this explains our feeling that, if true, they could be true only by a propitious accident.

So Russell imports alleged facts into logic. Tut tut.

6.1231 The mark of a logical proposition is not its general validity.

To be general indeed means only: to happen to be valid for all things. An ungeneralized proposition can indeed be just as tautologous as a generalized one.

Cf. 6.031 on kinds of generality. Is there a contradiction here? Or in 6.1231 is "general" being used in a particular sense, distinct from, say, necessary or a priori?

6.123 It is clear: The logical laws must not themselves be subject to further logical laws.

(There is not, as Russell thought, a unique law of contradiction for each “type”, rather one is enough, since it is not to be applied to itself.)

Black (p. 326) quotes Russell’s Principia vol. 1, p. 127 “Negation and disjunction and their derivatives must have a different meaning when applied to elementary propositions from that which they have when applied to such propositions as (x).φx or (Ex).φx” and Logic and Knowledge p. 63 “The first difficulty that confronts us [after adopting the “vicious circle principle”] is as to the fundamental principles of logic known under the quaint name of “laws of thought”. “All propositions are either true or false”, for example, has become meaningless. If it were significant, it would be a proposition, and would come under its own scope.”

On Wittgenstein’s view, apparently, there is only logic, rules for the combination of propositions into tautologies, contradictions, and other propositions. There is no logic of logic, rules for the application of the rules.

6.1224 It becomes clear now also why logic has been called the theory of forms and of inference.

Who calls logic the theory of forms and inference?

6.1223 Now it becomes clear why it has often been felt as if “logical truths” were “postulated” by us: We can indeed [nämlich] postulate them in so far as we can postulate an adequate notation.

Why “postulated by” instead of “exacted from”? Black (p. 325) suggests ‘require’ or ‘demand.’ Anyway, the sense is clear enough. The form of logical propositions depends solely on the notation we use, which is invented by us. Which is why logic can seem to be invented, although of course it has a kind of (non-metaphysical) independence.

6.1222 This throws light on the question why logical propositions cannot be confirmed by experience, any more than [just as little as] they can be confuted by experience. Not only must a proposition of logic be capable of confutation by no possible experience, but it must also not be confirmable by any such thing.

Because a proposition of logic is, as such, a mere combination of signs.

6.1221 If, e.g., two propositions “p” and “q” in the combination “p → q” give a tautology, then it is clear that q follows from p.

That, e.g., “q” follows from “p → q . p”, we see from these two propositions themselves, but we can also see it by combining them into “p → q . p:→ : q” and now showing that this is a tautology.

OK

6.122 From which it follows that we can go on without logical propositions, since we can perceive in an appropriate notation the formal properties of propositions with a simple look at these propositions.

What would such a notation be?

Black (p. 324) suggests ‘corresponding’ for entsprechend where P&McG have ‘suitable’ and I have ‘appropriate.’ He adds that “In a sense, every notation is suitable.” On p. 325 he says that “in a sense, every notation is ‘adequate’ to express whatever meaning it expresses.”

White (p. 105): “Wittgenstein goes badly astray in his development of this train of thought [i.e. that found in the 6.1s], and subsequent developments in logic [Alonzo Church, 1936] have shown that what he says at 6.122 is demonstrably false.” On p. 107 he writes: “What Wittgenstein overlooks is that once he allows the possibility of there being infinitely many elementary propositions, he has to allow the possibility of quantification over infinite domains. If we then have propositions involving multiple quantifiers ranging over infinite domains, then even the most perspicuous notation may not be able to display the information that a given proposition is a tautology in a form that is surveyable by us. So that even if we continue to say that a tautology shows that it is such, it may not do so in a form that is recognizable by us: we may simply lack any method for extracting the fact that it is a tautology. This means that in this use of the concept of ‘showing’ at least, ‘showing’ cannot be treated as a straightforward epistemological concept.”

6.121 The propositions of logic demonstrate the logical properties of propositions, by combining them into propositions that say nothing [nichtssagenden Sätzen].

One could also call this method a null method. In a logical proposition, propositions are brought into equilibrium with one another and the state of equilibrium then shows how these propositions must be logically constituted.

By combining propositions into tautologies and contradictions, we show how tautologies and contradictions are made, and that such combinations do indeed produce nothing but tautologies and contradictions. Such is philosophy. Dictionary.com defines a null method thus: Zero method (Physics), a method of comparing, or measuring, forces, electric currents, etc., by so opposing them that the pointer of an indicating apparatus, or the needle of a galvanometer, remains at, or is brought to, zero, as contrasted with methods in which the deflection is observed directly; -- called also null method. The Letters to Ogden, p. 34, confirm that this is the sense Wittgenstein has in mind here.

6.1203 In order to perceive a tautology as such, one can, in cases in which no sign of generality occurs in the tautology, avail oneself of the following method: I write “TpF”, “TqF”, “TrF”, etc. in stead of “p”, “q”, “r”, etc. I express the truth-combinations with brackets, e.g.:

The coordination of the truth or falsity of the whole proposition and the truth-combinations of the truth-arguments with lines in the following way:

T (T/F)->F" shapes="_x0000_i1026" height="96" width="128">

This sign, e.g., would therefore present the proposition p → q. Now I will investigate on the strength of that whether, e.g., the proposition ~(p. ~p) (the Law of Contradiction) is a tautology. The form “~ξ” gets written in our notation as;

T, (T)->F" shapes="_x0000_i1029" height="64" width="64">T, (T)->F" shapes="_x0000_i1029" height="64" width="64">T, (T)->F" shapes="_x0000_i1029" height="64" width="64">

the form “ξ . η” thus:

F (T/T)->T" shapes="_x0000_i1032">

So the proposition ~(p. ~q) goes thus:

T, T/F->F" shapes="_x0000_i1033">T, T/F->F" shapes="_x0000_i1033">T, T/F->F" shapes="_x0000_i1033">

If we put here “p” instead of “q” and investigate the combination of the outermost T and F with the innermost, then we get the result that the truth of the whole proposition is coordinated with all the truth-combinations of its arguments, its falsity with none of the truth-combinations.

A somewhat complicated way of demonstrating an obvious truth, but I suppose it’s the demonstration that matters here. Or maybe the point is that this is all that logic can be/do.

6.1202 It is clear that one could use contradictions instead of tautologies to the same end.

Yes it is.

6.1201 That, e.g., the propositions “p” and “~p” in the combination “~(p. ~p)” give a tautology, shows that they contradict one another. That the propositions “p → q”, “p” and “q” combined with one another in the form “(p → q) . (p) :→: (q)” give a tautology, shows that q follows from p and p → q. That “(x) . fx :→: fa” is a tautology, shows that fa follows from (x) . fx, etc. etc.

I'm using the arrow for "if ... then."

6.12 That the propositions of logic are tautologies shows the formal – logical – properties of language, of the world.

The fact that its parts connecting together just so gives a tautology characterizes the logic of its parts.

For propositions, connected in a specific way, to make a tautology, they must have specific structural properties. That they [do] make a tautology when so connected shows therefore that they [do] have these structural properties.

I translate bestimmte as ‘specific,’ where P&McG have ‘certain’ and Black (p. 321) suggests ‘determinate.’ ‘Definite’ might be good too.

What properties? It’s all grammar isn’t it, arbitrary? So are the formal properties of the world arbitrary too? Maybe. They are (only) formal, after all.

6.113 It is the peculiar characteristic of logical propositions that one can perceive from the symbol alone that they are true, and this fact contains in itself the whole philosophy of logic. And thus it is also one of the most important facts, that the truth or falsehood of non-logical propositions cannot be perceived from the proposition alone.

Logic says nothing about the world, it is all (?) a matter of arbitrary conventions. Other propositions are of a wholly different kind and cannot be known a priori. Is this what he is saying? Is this the whole of the philosophy of logic, the whole message of the Tractatus?

Now, how can a tautology be true, given 4.06? Proops argues that he means ‘true’ “only in an honorary sense.” (p. 4, note 38). After all, at 6.125 he puts ‘true’ in scare quotes, and in his Notes to G. E. Moore he says that “logical propositions are neither true nor false” and refers to “what is called the truth of a logical proposition.” (Notebooks p. 109 and p. 108, quoted in Proops.)

6.112 The right explanation of a logical proposition must give it a unique position among all propositions.

Because logical propositions are not the same kind of thing as other propositions.

6.111 Theories that allow a proposition of logic to seem have content are always false. One could e.g. believe that the words “true” and “false” signify two properties among other properties, and then it would seem a remarkable fact that every proposition possesses one of these properties. This now seems [would then seem] to be anything but self-evident, just as little self-evident as the proposition “All roses are either yellow or red” would sound, even if it were true. Indeed, that proposition [of logic] now takes on completely the character of a natural scientific proposition and this is a sure sign that it has been falsely understood.

Here again we have logic versus metaphysics, clearly stated for once.

Proops (p. 1) notes that here and in 6.112, 6.1231, and 6.13, Wittgenstein rejects the universalist conception of logic, according to which “logic is a theory of the most general features of reality,” (Proops, p. 1) which he found in Frege and Russell.

6.11 The propositions of logic therefore say nothing. (They are analytical propositions.)

Black (pp. 319-320) notes that this is the only occurrence of the word analytischen in the Tractatus, and suggests that it probably does not mean “true by definition” or “true in virtue of the meanings of its component words” here, nor have its original, Kantian sense, but rather is here simply a synonym for “propositions that say nothing.”

6.1 The propositions of logic are tautologies.

This implies, at least as Russell, e.g., uses the word ‘tautology’, that they are empty or insignificant. And that was the point of using that word, according to Burton Dreben and Juliet Floyd in “Tautology: How not to use a word,” Synthese 87 (1991), pp. 23-49.

6.031 The theory of classes is completely superfluous in mathematics.

That the generality that we need in mathematics is not the accidental [contingent] kind hangs together with this.

Russell says, “the class of all couples will be the number 2, according to our definition. At the expense of a little oddity, this definition secures definiteness and indubitableness.” (Introduction to Mathematical Philosophy, p. 18) But then the number 2 depends on the existence of a class of couples, i.e. on the existence of couples. And the number 1000 depends on the existence of 1000 things. And so on. But Russell admits that “Logical propositions are such as can be known a priori, without study of the actual world,” and it is not logically necessary that even one thing exists, he says, let alone 1000, or, even worse, infinity. (ibid., p. 204)

6.03 The general form of integers is: [0, ξ, ξ + 1].

OK, so that’s what a general form is/looks like.

6.022 The number concept is nothing other than the commonality of all numbers, the general form of the number.

The number concept is the variable number.

And the concept of numerical equality is the general form of all particular numerical equalities.

What exactly is a general form? Does this help?

6.021 A number is the exponent of an operation.

OK

6.02 And so we come to numbers. I define

x = Ω^0, x Def. and

Ω’ Ω^ν, x = Ω^ν+1, x Def.

According to these symbolic rules [or: rules about signs] we therefore write the series x, Ω’ x, Ω’ Ω’ x, Ω’ Ω’ Ω’ x, …..

thus: Ω^0, x, Ω^0+1, x, Ω^0+1+1, x, Ω^0+1+1+1, x, …..

Therefore I write instead of “[x, ξ, Ω’ ξ]”:

“[Ω^0, x, Ω^ν, x, Ω^ν+1, x]”.

And I define:

0 + 1 = 1 Def.

0 + 1 + 1 = 2 Def.

0 + 1 + 1 + 1 = 3 Def.

(and so on)

Frascolla (p. 187): “The inductive definition at the outset of 6.02 is given the task of putting the abstract notion of the application of an operation at the bottom of the construction of arithmetic. In sharp opposition to the logicist programme, number is not construed as the number of elements of a class (of the extension of a concept proper), but as the number of applications of a symbolic procedure, whose iteration gives rise to a potentially endless formal series of propositions.”

6.01 The general form of the operation Ω’ (η)is therefore: [ξ, N (ξ)]’(`η) ( = [`η, `ξ, N(`η)]).

This is the most general form of transition from one proposition to another.

OK.

6.002 If the general form is given of how a proposition is constructed, then thereby already is given also the general form of how from one proposition, by means of an operation, another one can be produced.

OK

6.001 This says nothing else than that every proposition is the result of successive applications of the operation N’ () to the elementary propositions.

OK.

6 The general form of truth-functions is: .

This is the general form of propositions.

I had had “of the truth-function” and “of the proposition,” but Wittgenstein (Letters to Ogden p. 34) says the “the” should be left out, and the words made plural if necessary to accommodate this (I think it is necessary to do so).

McManus (p. 140) renders the idea here as that “Every proposition is an elementary proposition or a (possibly very complex) complex proposition.” He sees it as problematic because no argument is really given in its favor and because it seems to bring metaphysical commitments with it. If there is a general form of propositions then there is, he suggests on p. 141, a general form of the world (see 2.04). Wittgenstein’s claim is in effect that “all logical incompatibility is a matter of contradiction,” (p. 153) but this is only a possibility, not something that Wittgenstein had proved must be the case. For instance, if a spot is red it cannot also be blue. But is “This spot is red” a contradiction of “This spot is blue”? Or could it be a kind of empirical knowledge that a spot cannot be both red and blue? Wittgenstein assumes that the meanings of the logical constants that connect elementary propositions into complex propositions are topic-neutral, the same in all contexts. He later abandoned this assumption. (See, e.g., PG 269 and RPP I 38.) At the time it probably seemed bland and inoffensive. He later saw that it brought unwanted problems and commitments.

White (p. 35) refers to 6 as “the central claim to which the book builds up.” Yet Wittgenstein has made a mistake, according to White. In note 43 (p. 152) he says that Russell silently corrects the text of 6 on p. 14 of his introduction to the Tractatus, giving “a satisfactory informal exposition of what Wittgenstein should have said.” As written “the formula at proposition 6 is radically incoherent.” (p. 103) Wittgenstein’s notation gives us a rule for moving from a propositional variable to a proposition, but what it is supposed to do is give us a rule for moving from one proposition to the next proposition in the series. 6.001 fixes the problem, or as good as does so (see pp. 103-4), according to White.

5.641 There is therefore really a sense in which there can be non-psychological talk in philosophy of the I.

The I occurs in philosophy through the fact that the “world is my world.”

The philosophical I is not the human being, not the human body, or the human soul that psychology deals with, but rather the metaphysical subject, the limit – not a part of the world.

What kind of talk goes on in philosophy then? Not the kind that could be included in The World as I Found It.

5.64 Here one sees that solipsism, rigorously followed through, coincides with pure Realism. The I of solipsism shrinks to an extensionless point, and the reality coordinated to it remains.

What reality coordinates to an extensionless point? Wittgenstein is not a solipsist, that much is clear. Schopenhauer on realism: “Realism aims precisely at the object without subject; but it is impossible even to envisage such an object distinctly.”[1]

“The fundamental error of all systems is the failure to recognise … that intellect and matter are correlatives, i.e., that the one exists only for the other, both stand and fall together, the one is only the reflexion of the other, and indeed, they are really one and the same thing regarded from two opposite points of view; and this one thing, I am here anticipating, is the manifestation of the will, or the thing-in-itself.”[2]

That is to say, we cannot conceive an object without a subject, and vice versa.

Weiner (p. 78) quotes a long passage from Schopenhauer WWR Vol. II, p. 193 in which Schopenhauer writes that in a sense an identity of the ideal and the real might be affirmed.

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[1] Everyman edition of Schopenhauer’s WWR, p. 19.

[2] Ibid., p. 21.

5.634 That no part of our experience is also a priori hangs together with this.

Everything that we see could also be otherwise.

Everything that we can describe at all could also be otherwise.

There is no a priori order of things.

Right. Logic, the a priori, is distinct from metaphysics (fact), the contingent.

5.6331 The field of vision has precisely [nämlich] not a form such as this:

The eye and the field do not belong to the same ‘space’, although it is easy to lose sight [ho ho] of this kind of thing when doing philosophy. We confuse logic for metaphysics, one language-game for another.

Friedlander (p. 116): “From the point of view of representation there is no limit whatsoever. This is the point of Wittgenstein’s analogy between the visual field and the field of experience as such.”

Compare Schopenhauer, The Fourfold Root of the Principle of Sufficient Reason trans. E. F. J. Payne, 1974, p. 96: “Thus since our power of vision reaches equally in all directions, we really see everything as a hollow sphere in whose centre is our eye.”

5.633 Where in the world is a metaphysical subject to be found?

You say here it is just as with the eye and the field of vision. But you do not really see the eye.

And nothing in the field of vision allows the conclusion that it is seen by an eye.

Who is “you”? Mirrors don’t justify the conclusion that we see with an eye? Not in the relevant metaphysical sense. After all, the eye in the mirror is just part of the visual field. We do not see the field of vision with the eye. Cf. PI on the visual room §§398-400. See also, perhaps, Moran’s Introduction to Phenomenology p. 43.

David Weiner (p. 60) says that “The ‘you’ he addresses in 5.633 is Schopenhauer. At issue are Schopenhauer’s two basic metaphors for the metaphysical subject, the eye and the limit. Wittgenstein’s point is that the eye metaphor is misleading, while the metaphor of the limit hits the nail on the head.” On p. 64 Weiner explains that “According to Wittgenstein, the eye suggests an empirical necessity that does not exist; the limit, on the other hand captures a logical necessity that does exist.” In a footnote referring to the p. 60 passage just quoted (note 80 on pp. 123-4), Weiner adds: “By introducing the “You” at 5.633, Wittgenstein suddenly shifts from monologue to dialogue. Up to this point, he has simply been making assertions. His voice has been like an oracle that casts out definitive truths to the world at large. Now he suddenly is arguing with an unnamed interlocutor. The reason for this change of voice is that the passage derives from a segment of Wittgenstein’s notebooks in which he is arguing against Schopenhauer. (See Notebooks 1914-16, pp. 79-80.)”

5.632 The subject does not belong to the world, rather it is a limit to the world.

But how can there be such a limit? And “world” in what sense?

5.631 There is no thinking, representing subject.

If I wrote a book The World as I Found It then I would also have to report on my body in it and say which parts are subject to my will and which not, etc., [and] this is precisely [nämlich] a means to isolate the subject, or rather to show that in an important sense there is no subject: Of it alone precisely could there not be talk in this book. –

Odd. The first sentence sounds so plainly false that you wonder what is being got at. The rest, I suppose, is the explanation. There would be reference to my body and my will and the things that I think, etc. What would be missing? In an important sense, nothing. But in some other sense, something? This sounds like a Kantian noumenal self, except that it is literally impossible to talk about it. Yet, here we are talking about it. Is the final dash a mark of irony?

Also bear in mind 6.53. And the subject here being denied seems to be the one that is identical with the world, logic, language, etc. In the relevant sense, then, they don’t exist either. So all that solipsistic talk was nonsense?

Cf. Schopenhauer Fourfold Root p. 124: “Insofar as he behaves as a purely knowing being, the movement of his body according to his will is for him merely an empirically perceived fact.” See also WWR on becoming a purely knowing being. This is the goal of the holy genius for Schopenhauer, isn’t it?

5.63 I am my world. (The microcosm.)

How seriously should/can we take this?

5.621 The world and life are one.

And so language = logic = world = life?

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.]

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[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.