Close‐Up: The Early System
Close‐Up: The Early System
Abstract and Keywords
Delineates Wittgenstein's earlier philosophy by relating its central ideas to the philosophies of his predecessors and contemporaries, most notably Russell. Concentrating mainly on Tractatus Logico‐Philosophicus and the Notebooks, Pears sets the stage for his influential realist interpretation of Wittgenstein's early philosophy.
When Wittgenstein's philosophy is put in a Kantian frame, most of its main lines stand out clearly. However, this way of looking at it does bring out its form rather better than its content. The material that he casts in that form is more variegated than has yet been indicated. In fact, one of the main difficulties of understanding him is to see how the details fit into the general pattern. Even when his treatment of a particular topic gives the impression of a strong underlying structure, it is often not at all obvious what it is.
So the description of his philosophy that has been given so far may not correspond to what people get first from his works. The discussions of particular topics stand out immediately but the general pattern only emerges gradually. That was often the way in which his ideas actually developed. For example, the work that went into the Tractatus did not begin as an attempt to fix the limits of language. That was a later development, which gave the book its final shape. The original problem was posed by logic. He was searching for a theory of meaning which would explain the necessary truth of logical formulate.
Similarly, his later work in the philosophy of mind was not undertaken in order to show what is wrong with Platonism. His interest in the mind did not even originate entirely in his investigations of language. There is also another line of development, which started from the treatment of solipsism in the Tractatus, moved on to sensations and the way we manage to communicate about them, and finally opened up a new way of looking at all mental phenomena. The traditional view had made the contents of each person's mind inaccessible to others, and, in the case of the ego, inaccessible even to the person himself. Wittgenstein's new view avoided these unacceptable consequences without toppling over into behaviourism, in something like the way in which his new view of language avoided pure realism without toppling over into arbitrary conventionalism. So here in the philosophy of mind is another example of two opposed (p.21) polarizations of a single concrete phenomenon, neither of which struck him as acceptable as a theory.
To start again at the beginning, the Tractatus, as its full title implies, is a philosophical treatise on logic. It was the foundations of logic that presented the problem, because, though we all know how to establish ordinary contingent truths, it is not clear how the familiar necessary truths on which we rely in everyday arguments are established. It is no good saying that we prove them, because anyone who has looked into Euclid knows that you cannot prove anything from nothing. Axioms and rules of inference are always needed, but then their status too can be questioned.
Much of the preliminary work for the Tractatus is in Wittgenstein's Notebooks. The text, as we now have it, opens with a memorable sentence: ‘Logic must take care of itself.’1 In other words, logic is a self‐contained system which can be validated only from within. Its formulae, therefore, must be completely different from factual sentences, which have to measure up to something outside themselves, the contingent layout of the world. This is a simple contrast, but the conclusions that he draws from it are far‐reaching. If he is right, Russell's axiomatization of logic in Principia Mathematica was mistaken in more than one way.
His main criticism of Russell's system is that it is not self‐contained. When the need arises, Russell helps himself to additional axioms which, Wittgenstein argues, are, at best, only contingently true. Now a conclusion cannot be any stronger than the weakest of the premisses from which it was drawn. Therefore, after the point at which Russell adds contingent truths to his axioms, the theorems or formulae that he proves will all belong to science rather than to logic. But that violates the fundamental distinction between logical formulae and factual sentences.
Another criticism that he makes of Russell is the one that occurs to everybody. Even if none of the axioms were contingent, they would all remain unproven, and so the task of proving the theorems from them was not worth undertaking. However strong your rope, you cannot hang anything on it unless it is attached at the other end, and then what you can hang on it will depend on the strength of that attachment. Schopenhauer makes this criticism of Euclid and goes on to argue that, if we found a method for validating the axioms—and we really need to find one—we could use it for validating the theorems (p.22) directly. So Euclid's axiomatization of goemetry was not only uncompletable, because the axioms hung in the air, but also unnecessary, because the theorems never needed any proof in the first place.2 Axioms and theorems alike stand on the same level and they are all equally in need of direct validation. This point is made against Russell by Wittgenstein,3 but, of course, with a difference: Schopenhauer's direct method of validation in geometry was pure spatial intuition, while in logic Wittgenstein's was to test for tautology.
Whatever its merits, this test certainly occupies an interesting position in the system of the Tractatus. If logical formulae are tautologies, logic really does take care of itself, because tautologies do not depend on anything that happens in the world. They are not hostages to contingency. Factual sentences make claims and they get a grip on the world, which then verifies them or falsifies them. Tautologies make no claim and they ride loosely on the world, being neither supported nor let down by any contingency. They levitate because they say nothing. Logical formulae are radically independent when their necessary truth is explained in this way. Each of them can be validated directly without any help from the others. There is, therefore, no need to string them together in a calculus, giving some of them the role of premisses and proving others as conclusions.
If this is what logic is like, it is very unlike anything to be found in factual discourse. It is not a system of connected truths, like science: it is not even a medley of independent truths, like the ordinary record of what goes on around us. In the Tractatus Wittgenstein spends a lot of time on these differences between the formulae of logic and factual sentences, but people read this part of the book rather rapidly, because they are already converted. They ought to pause and ask themselves how he saw the point which strikes them as so obvious. He saw it as a deep difference. It is not only that logic does not cover the same ground as factual discourse: it does not cover its own ground in the same way—or, rather, it does not cover any ground. Its formulae do not express knowledge of any subject. They merely reveal connections between different forms of sentences, and so between different forms of facts. But these forms do not belong to another world, to be explored after the world of facts, as it were, on a separate (p.23) expedition. That was Russell's idea,4 and Wittgenstein was completely opposed to it. However, it did seem to him to be a natural and, therefore, a prevalent misunderstanding of logic, and that is why he spends so much time saying what logic is not like.
The system of the Tractatus is built on an idea that is the exact opposite of Russell's idea: the forms revealed by logic are embedded in the one and only world of facts and, therefore, in the language that we use to describe it. If Russell's view was Platonic, this view is approximately Aristotelian. Logic is immanent in factual discourse from the very beginning, and it emerges when we take factual sentences and combine them in various truth‐functional ways—that is, in such ways that the truth or falsehood of the combinations will depend entirely on the truth or falsehood of what went into them. Most of the combinations will make factual claims about the world and, if we want to find out exactly what claims they make, we must identify the sentences that were originally combined in them and discover what claim each of them makes on its own. When we do this, we shall find that some of the combinations make no claim about the world. These are tautologies. They always come out true and so they are the limiting case, the boundary of factual discourse, not part of it.
If tautologies reveal the connections between the forms of factual sentences, they must show us something about those forms. For if two sentences of given forms can be combined by a logical connective to produce a tautology, that must be the result of the forms themselves. It is rather like chemistry, where the properties of a compound can be entirely explained by the properties of the components that went into it. This is how Wittgenstein puts the point:
Now tautologies are the formulae of logic. So this revelation about the world is a revelation made by logic. The quoted sentence occurs in a lengthy remark which begins with these words:
It is clear that something about the world must be indicated by the fact that certain combinations of symbols—whose essence involves the possession of a determinate character—are tautologies.5
(p.24) This is not an easy passage to interpret and two questions about it need to be answered. What exactly does logic reveal about the world? And how is its revelation made?
The propositions of logic describe the scaffolding of the world, or rather they represent it. They have no ‘subject‐matter’. They presuppose that names have meaning and elementary propositions6 sense; and that is their connection with the world.
The answer to the second question is implicit in Wittgenstein's anti‐Platonic view of logic. Whatever logic conveys about the world, it does not convey it in anything like the way in which factual sentences convey information. It conveys it in a way that he calls ‘showing’, and ‘showing’ is almost a technical term in the Tractatus. Among the things that can be shown but not said7 he includes everything that can be read between the lines of factual discourse, or, to put his point in another way, everything that is expressed by logic. He calls logic ‘the great mirror’,8 and, when we read his account of it, we feel that we are standing within a limited structure which reflects the ground on which it was built. Each elementary possibility supports the sentence that reflects it, and the load‐bearing lines are carried upwards and combined in many different ways. The limiting case is tautological combination but tautologies are the outline of the structure, not part of it.
The Tractatus presents one possible view of the nature of logic. It is an anti‐Platonic view, because it denies that logic is something that we bring back from the exploration of a second world, and treats it, instead, as a peculiar extract from the results of exploring the one and only world of facts.
But what exactly is it that logic conveys to us about the world? That was the other question that needed to be answered. A short reply would be that logic reveals the essence of the world of facts rather than the existence of another world. This is made very clear in the continuation of the quoted passage. After saying that ‘something about the world must be indicated by the fact that certain combinations of symbols . . . are tautologies’, he continues:
That answers the question, but in doing so it raises others. Logic reveals the essence of the world, but what is that? And is there really any difference between this view of logic and Russell's Platonic view of it? If this view connects logic with the basic ontology of the Tractatus, which is uncritically realistic, it looks as if there is not much difference at this stage between Wittgenstein and Russell on the foundations of logic. Surely the real difference only begins later, when Wittgenstein's critical philosophy overran the last remaining territory of realism?
This contains the decisive point. We have said that some things are arbitrary in the symbols that we use and that some things are not.9 In logic it is only the latter that find expression: but that means that logic is not a field in which we express what we wish with the help of signs, but rather, one in which the (p.25) nature of the absolutely necessary signs speaks for itself. If we know the logical syntax of any sign‐language, then we have already been given all the propositions of logic.
A short answer to the first question about the essence of the world, revealed by logic, is that it consists of elementary possibilities which are either realized or not realized, with no third contingency. Now there is often a third contingency at the level of ordinary factual discourse. To use one of Wittgenstein's examples in the Notebooks,10 if someone says ‘My watch is on the table,’ it may or may not be on the table, but there is also a third contingency, that he has no watch, in which case the possibility, that it is on the table, is not there to be realized. Wittgenstein's point is that this third contingency is excluded at the level of completely analysed factual discourse. Or, to put it in the terms used in the previous chapter, at the ultimate level there is a grid of simple possibilities, each of which is either realized or not realized, and we cannot go below that grid to ask what things would have been like if its possibilities had not been there to be realized. The grid is ultimate and any speculation that purports to go beyond it is senseless.
A short answer to the question about the relation between Wittgenstein's view of logic in the Tractatus and Russell's contemporary view is that they really were opposed to one another. Russell believed that the logician's task is to carry out a survey of ‘logical objects’, some of which are forms while others are the real counterparts of logical connectives.11 Wittgenstein's view was that there are no logical objects,12 forms cannot be investigated and described like things,13 and logical connectives do not stand for anything in the world and are, therefore, utterly unlike names.14
(p.26) These are real differences between the two philosophers. It is one thing to say that logical connectives stand for a special kind of object and quite another thing to say that they do not stand for anything, but merely indicate different ways of producing truth‐functional combinations of the sentences that they connect. It is one thing to say that the form of a sentence is a special kind of component, and quite another thing to say that it is nothing of the sort but, rather, the possibility of a structure.15 Indeed, the latter view of form is essential to the picture theory of sentences: the form of a fact is a possibility projected into the sentence that depicts it.16 So this is not a case of two philosophers ‘agreeing to have a battle’. It may look like that, if we ask whether sentential forms exist in a transcendent world of their own or are merely immanent in this world. For how can we answer such an airy question? But we can bring it down to earth by catching it at the point of take‐off and treating it as a question about the elements of semantics, which is what it really is.
It remains true that Wittgenstein's later view of logic is even further from Russell's. However, that is a distance travelled in a different direction. In his early work Wittgenstein puts logic on a realist basis, but he does not do this in the same simple way as Russell. He does not use the link between name and object as a model to which the solution of every semantic problem presented by logic has to conform. Nevertheless, at the foundation of the system of the Tractatus there is the grid of elementary possibilities imposing certain absolute constraints on the logical structure of any language. That is uncritical realism, and later, when he abandoned it, he moved even further from Russell's Platonism, but in a different direction. In his later writings he was not merely objecting to the way in which Russell developed his semantics: he was objecting to the whole idea that the world's contribution to the necessity of logical formulae can be separated from our contribution. This is yet another example of two opposed polarizations of a single concrete phenomenon, neither of which struck him as acceptable as a theory.
There are one or two points in the short answers to the two questions about Wittgenstein's early view of logic where immediate amplification is needed. Of course, a complete account is not the aim of this chapter, but even a passable sketch requires more than has yet been done. Wittgenstein's idea that tautologies reveal the (p.27) essence of the world cannot really be appreciated without more details of his argument for the existence of the ultimate grid of elementary possibilities. There are also gaps that cannot be left unfilled in the account so far given of the connection between his early view of logic and the uncritical realism of the basic ontology of the Tractatus. The move to his later view of logic strikes most people as the most paradoxical feature of his whole philosophy. Universals may not be independently anchored in the natures of things, but can logic too slip its moorings? It would obviously be a good idea to take a closer look at the way in which its moorings are set up in the Tractatus.
First, then, we need Wittgenstein's argument for the existence of the ultimate grid of elementary possibilities.17 Now an elementary possibility is one with simple objects at its nodal points. So the question is ‘What reason did Wittgenstein have to believe that ordinary factual sentences can be analysed down to factual sentences in which only simple objects are named?’ His argument for this conclusion is reductive: if there were any complex things designated in the complete analyses of ordinary factual sentences, then the analysing sentences would have senses only if certain other sentences, not included in their analyses (they have no analyses), were true. For complex things would not be there to be designated unless it were true that their components were arranged in the way required for their existence. But, Wittgenstein argued, the sense of a sentence about a complex thing cannot possibly depend on the truth of another sentence about its components. So the analysis must go on down to the next level and include the further sentence in the sense of the original one, and this process must continue until all words for complexes have been replaced by genuine names standing for simple objects.18
The assumptions on which this argument is based will be examined later. The point to be made about it now is that it pushes the level of complete analysis downwards until there are no underlying facts left, but only objects devoid of internal structure. These simple objects are the pivots on which all factual discourse turns. So logic reveals the structure imposed on all factual discourse by the ultimate structure of reality. That is its connection with the world.
(p.28) This is an audacious argument and it is easier to grasp its separate stages than to grasp it as a whole. So people often get from the Tractatus the message that the necessities of logic are not of our making, without Wittgenstein's much more remarkable explanation of their origin. He does not rest his case on the claim that the formulae of logic are in fact tautologies. He goes on to ask why we speak a language which throws up tautologies as by‐products of its structure. His answer to this further question is that the essential structure of our language is imposed on it by the ultimate structure of reality, which is a grid with simple objects at its nodal points. If this were not so, our language would not have tautologies marking the outline of its structure. For tautologies are true in all possible contingencies, and that evidently requires their component sentences to be true or false with no third possibility.
So the picture of our intellectual predicament now needs to be modified. It is true that we stand within a limited structure which is bounded by tautologies and which reflects the ground on which it was built. However, the foundations lie deeper than at first appeared. What purport to be names designating objects in ordinary factual discourse are not genuine names and those are not genuine objects. The real designation is done at a much lower level, where everything that is designated is devoid of internal structure.
The other point that needed amplification was the relation between the view taken of logic in the Tractatus and two other views of it, Russell's contemporary theory and Wittgenstein's own later account of it. Here it is necessary to be very precise about the questions that were answered in the Tractatus. The question posed at the beginning of this chapter was ‘How are the necessary truths of logic established?’ The answer given in the Tractatus was ‘By testing for tautology’. But, as has just become apparent, there is also another question asked by Wittgenstein at this point: ‘Do we have to speak a language that generates tautologies?’ Now realism answers these two questions by making two very different moves. It answers the first one by claiming that logic studies abstract objects in another world. This claim can take various forms, and Russell's version of it, in Theory of Knowledge, 1913 is very different from Frege's. It is against Russell's version that Wittgenstein's main objections are directed in the Tractatus. The second question raises a separate issue and so the realist's answer to it takes an altogether different line; the ultimate structure of reality forces us to speak a language that generates tautologies. This time the (p.29) realist is Wittgenstein. So he gives an anti‐realist answer to the first question, but a realist answer to the second one.
The strategic situation on this field of controversy is complicated. It is, therefore, useful to go back to the basic realism of the Tractatus and take a closer look at it. It was called ‘uncritical realism’ in the previous chapter, because objects are the dominant partners in their relationships with names and nothing is said about any contribution made by our minds at this point. True, we attach the names and we maintain their attachment (if the need arises), but the nature and identity of each object is fixed independently of anything done by us. This is Aristotelianism rather than Platonism, because objects exist in the one and only world, but it is uncritical realism, because the question, whether we contribute anything to the constitution of that world, is not even asked.
Standing on this platform in the Tractatus Wittgenstein criticizes the airy acrobatics of Russell's contemporary theory of logic. There is, he argues, no need for a second world to house sentential forms and logical objects. However, in order to make good his criticism, he has to give an anti‐Platonic explanation of what Russell tried to explain Platonically. Using materials drawn entirely from this world he has to explain the towering structure of logic. He did not encounter much difficulty in dealing with logical connectives once he had dealt with the problem of form. The really difficult task was to show how sentences acquire their forms from the one and only world.
The picture theory of sentences was his solution to this problem. Form is the possibility of structure and sentences pick up the forms of facts through the links between their names and the objects embedded in reality. The details will be given later, and the immediate point is only that this theory gave Wittgenstein a platform from which to criticize Russell's Platonic theory of logic. The platform is Aristotelian in this sense:19 it brings down sentential forms from Russell's transcendent world and treats them as immanent in this world. This, as already remarked, is a real difference between the two philosophers.
At this stage the common ground between them was the mandatory character of logic. Of course, both of them believed that some features of our language are optional. There are even necessary truths which we ourselves generate by choosing to define certain non‐logical words in one way rather than another. Those are connections between (p.30) concepts which do not have to be marked in any way in our language. But there are other features of our language which are mandatory. These are the connections which hold between sentences in virtue of their forms, and they have to be marked in our language in one way or another. Wittgenstein and Russell agreed that these logical features are mandatory. They only disagreed about their source, which Russell placed in a transcendent world, but Wittgenstein placed in this world. So Wittgenstein's problem, like Aristotle's, was to explain how we get the abstract out of the concrete.
Wittgenstein deserted this common ground in his later writings. In so doing he abandoned both kinds of realist theory of logic, Platonic and Aristotelian. This strikes almost everybody as an extremely paradoxical move. Can logic really slip its moorings in this way? The question is beyond the scope of this book. However, a few brief points, which may take some of the shock out of the paradox, can be made here.
First, it really is necessary to identify the position from which Wittgenstein started. According to the Tractatus, the formulae of logic are tautologies and we are forced by the one and only world to use a language that generates tautologies. It is surely not too difficult to understand how he came to abandon the second of these two theses. The move away from it would start with the reflection that it may not be the ultimate grid of elementary possibilities that forces this logic on us. It may be the other way round: we have this logic already, and looking at the world through its formulae, we think that we can see an ultimate grid‐like structure supporting it. The apparent independence of the grid may be an illusion, because it may just be the shadow cast on the world by the logic that we use. And how can anything be supported by its own shadow?20
Second, this interpretation of Wittgenstein's move fits in very well with the anti‐realist move that he made simultaneously in the theory of descriptive language. That move was introduced in the previous chapter. Instead of holding that the criteria of identity of objects and their types are simply imposed on us, he came to think that a considerable contribution is made at this basic point by our minds. However, that was not the beginning of a stampede into arbitrary conventionalism. On the contrary, he was redrawing the map without the traditional boundary between arbitrary conventionalism and pure realism. May that not be what is happening in the case of logic too?
(p.31) We have to be careful at this point, because the line between the immanent, Aristotelian version of realism and the transcendent, Platonic version is not always easy to draw. Of course, Russell's theory, that we understand logic when we achieve acquaintance with objects of a special kind, was clearly Platonic. But was the rival theory put forward by Wittgenstein in the Tractatus Platonic or Aristotelian? The case for calling it ‘Aristotelian’ is that it represents logic as something that is forced on us by the structure of the grid that we find underlying the phenomenal world. This looks like the immanent version of realism, and it needs to be labelled in a way that will distinguish it from Russell's transcendent version.
However, a moment's reflection will show that the difference between the two theories is not really so great. According to the Tractatus, the nature of reality is there to be discovered in objects. If we could encounter an object, it would imprint its form on our minds. This form would give us two things, one specific, and the other more general. Specifically, it would give us an unambiguous representation of the correct use of the word in the future. More generally, it would be part of the grid‐like structure of reality which forces us to hold our sentences true or false, with no third possibility and so throws up tautologies on the outer edge of language. Is it really so easy to decide whether these two results of an encounter with an object are Platonic or Aristotelian?
The point can be made in another way. Wittgenstein's move against Russell's theory of logic gave his own theory an inner momentum which would carry it to self‐destruction later. After 1929, the specific aspect of the realism of the Tractatus would be destroyed by the ‘antiPlatonic’ argument of Philosophical Investigations,21 which could equally well be called ‘anti‐Aristotelian’; and its general aspect would be destroyed by the development of logic without realism of either type.
There is, however, a difference between the liberation of logic and the liberation of descriptive language. In the case of descriptive words it is fairly easy to understand both the realism that is rejected by Wittgenstein and the effect of rejecting it. It is the kind of realism that claims that there are standards for the correct use of our descriptive vocabulary fixed independently of us, and perhaps fixed even before we arrived on the scene. When this kind of realism is rejected, the result is not that we are left wildly improvising. A contribution to stability is still made by the natures of the things to which we apply our (p.32) words. It is just that another contribution is made by the ways in which we find it natural to apply them, and the two contributions cannot be disentangled from one another.
So it ought to be the same with logic. However, though it is easy enough to say this, it is far more difficult to understand in the case of logic than it is in the case of descriptive words. This is not the place to deal with these difficulties. Perhaps they can all be explained as consequences of the differences between logic and factual discourse without relinquishing Wittgenstein's view, that in the end realism and conventionalism about logical necessity are two aspects of a single concrete phenomenon which can be polarized but not treated as rival theories. Or perhaps he is mistaken in this case. The only way to settle the question would be to work through the details, but that will not be done in this book.
There are, however, some general observations about the problem which can be made quickly. Suppose that we really did encounter people who seemed to follow eccentric patterns of thought instead of our logic. We ought not to protest immediately, ‘The idiots! Why don't they make the inferences that we make?’ That would be too like criticizing the colour vocabulary of the ancient Greeks, on the ground that they must have seen that they were really grouping together colours that were quite different from one another and separating colours that were really the same. We would have to give our eccentrics the benefit of the hypothesis that they really were thinking in a regular way.
So we would need to observe them carefully, in order to see what use they made of their thought processes. Did anything in their environment enter the strange patterns on one side, pass through them, and come out on the other side as intelligible actions? To ask the same question less obscurely, was there any real reason to call their idiosyncrasies ‘a logic’? Here we must remember that Wittgenstein always regarded logic as a handy guide for moving from one ordinary truth to another, rather than as a vehicle with which to explore a transcendent world for truths of a special kind.
Now many commentators have pointed out that his dramatizations of encounters with radically eccentric thinkers are extremely thin. Is this merely because it is difficult to present such cases from our standpoint? If so, we ought to reflect that, if we were in their shoes, we would have just as much difficulty in appreciating the logic that is now ours, and we ought to show tolerance. After all, in the case of (p.33) descriptive words, though we can report what eccentric speakers do, it is hard for us to accept that it is really the same kind of thing that we do. So hasty ridicule is out of place.
But this way of putting it masks a real difference between the two cases. An eccentric user of a descriptive word has his own way of applying it, and we cannot fault him on the ground that it does not serve the general purpose of descriptive language. But if the general purpose of logic is to get from true premisses to true conclusions, then, provided that we know the criteria of truth used by the eccentric thinker for his premisses and conclusions, we shall be in a position to fault him. He may, of course, reject our criticism, but, if he does so, it will look as if he is not engaged in a new kind of logical inference, but in improvisations with no attachment to the world. Even if he also uses descriptive words in an eccentric way, so that we have to pick up his criteria of truth as he goes along, there is still a resource available to us. We can give him blind tests, by asking him his reaction to each actual situation after we have made him forget the inference that he had made about it. That would put him on the spot. For if his patterns of inference would often have forced him off course, they would not count as an alternative logic, and this verdict will not be affected by the fact that the course that he steers with each descriptive word is, from our point of view, eccentric.
(3) TLP 6.126–6.127.
(5) TLP 6. 124.
(6) i.e. fully analysed propositions.
(7) TLP 4.12–4. 1212.
(8) Ibid., 5.511.
(9) This refers back to 3.342, which draws the distinction between the optional and the mandatory, discussed in ch. 1 (p. 10 and pp. 15–16). ‘Although there is something arbitrary about our notations, this much is not arbitrary—that when we have determined one thing arbitrarily, something else is necessarily the case. (This derives from the essence of notation.)’
(10) NB 15 and 16 June 1915.
(12) TLP 4.441.
(13) Ibid., 2.172
(14) Ibid., 4.0312.
(15) TLP 2.033.
(17) i.e. for the thesis which Russell called ‘logical atomism’.
(19) But, of course, Wittgenstein's forms, unlike Aristotle's, are sentential. It is only his view of their source that is Aristotelian.
(20) See PI I § 104.
(21) PI I §§ 137–242. See above, pp. 10–11.