There are remarks of Wittgenstein’s that one might try to take as tips on how to understand him – but readers may be left wondering how to understand the tips. Think, for example, of 4.0312, some – or all? – of which sets out what Wittgenstein speaks of as his fundamental idea. Or think of his remark, in a letter to C.K. Ogden, about 4.112, where he had said that the aim of philosophy was the logical clarification of thoughts. Intending to help Ogden with the translation, Wittgenstein wrote, “It cannot be the RESULT of philosophy ‘to make propositions clear’: this can only be its TASK. The result must be that the propositions now have become clear that they ARE clear” (Wittgenstein Reference Wittgenstein1973: 49, emphasis in original). This is a great remark to try to make sense of, and thinking about it can help one in one’s reading of the Tractatus – but the remark sets a task for anyone trying to understand 4.112.

My chapter is about a remark of Wittgenstein’s from his reply to the letter Russell wrote to him after first reading the Tractatus. Two sentences in Wittgenstein’s letter reply directly to Russell’s saying that he thinks Wittgenstein is correct in his main contention, that logical propositions are tautologies. Like 4.0312 and Wittgenstein’s remark to Ogden about the result of philosophy, Wittgenstein’s remark in the letter to Russell is a great stimulus to thought about the Tractatus. It sets a task, as they do – but a task more likely to leave one baffled.

I will be looking at Wittgenstein’s statement of the supposed main contention of the book, since it certainly is not obvious how it should be understood. I also want to consider what the relation is between his main contention and what he describes in the book as his fundamental idea – which I’ll put this way: In a proposition, objects have proxies, but the logic of the facts is not something capable of being represented. It has no proxy. The “logical constants” aren’t representatives. – And I also want to consider a question raised by Michael Kremer: what does Wittgenstein mean when he says that what he calls his main contention and his main point is what he takes to be the cardinal problem of philosophy?

Starting with the third issue: what is the problem? I think the sentence in the letter to Russell is one of the most compressed things Wittgenstein ever said. It’s a devil to uncompress. You could say that the question that Wittgenstein sets there is: what IS he saying is the cardinal problem of philosophy? What does it even mean to say that the theory of what can be expressed by propositions and what cannot be expressed by propositions but only shown is the cardinal problem of philosophy? – One way to approach this would be to pick out some problem that you think would be dealt with by laying out what can be expressed by propositions and what can only be shown, and you can then say that that’s what Wittgenstein really meant – that there was a main philosophical problem that you could resolve through talking about saying and showing. This is what Oskari Kuusela does: he reads Wittgenstein as meaning that there is a central philosophical problem about whether necessities can be the object of genuinely true or false sentences, and then he says that what Wittgenstein meant in the letter to Russell is that that problem can be dealt with by what he is referring to as the theory of what can be expressed by propositions and what cannot be expressed by propositions but can only be shown. So Wittgenstein is read as not actually holding that the business about saying and showing is the cardinal problem; it’s the solution (Kuusela Reference Kuusela, Kuusela and McGinn2011).

I mentioned that it was Michael Kremer who drew attention to Wittgenstein’s saying that the theory of what can be expressed by propositions and what cannot be expressed that way but only shown – that this is the cardinal problem of philosophy. Kremer himself has tried to explain this. He emphasizes that there are various ways of expressing what Wittgenstein took to be the single great problem with which he was concerned. Giving the nature of the proposition is one expression of the single great problem; giving the limits of what can be expressed in language is another; and there are still others. But Kremer argues that Wittgenstein meant to get us to see that the very idea of such a single great philosophical problem is meant to be revealed as an illusion. Kremer, that is, gives us a kind of deflationary understanding of the supposed cardinal problem of philosophy; and he also has a deflationary account of what is meant by showing. Briefly, he thinks we can take Wittgenstein’s talk of showing to be either a way of gesturing at a realm of superfacts or alternatively a bringing out of practical abilities – for example, those we exercise in making inferences (Kremer Reference Kremer and Crary2007).Footnote ¹ The trouble here is that Kremer sees those two possible understandings of showing to be the only two, and so, since he rejects the first, he accepts a deflated understanding of what Wittgenstein meant.

There is an important and complex issue here – and I’ll mention two discussions of what showing means in the Tractatus, which illuminate in very different ways the complexity of the issue. One, that of Gilad Nir, is about Wittgenstein’s reference to “the cardinal problem of philosophy” in his reply to Russell, and the other, that of Jean-Philippe Narboux, is a far-ranging essay on showing and its significance in the Tractatus.

Gilad Nir discusses Wittgenstein on “the cardinal problem of philosophy” in an essay on philosophical riddles and their relevance to Wittgenstein’s conception of philosophy in the Tractatus (Nir Reference Nir2021). In explaining the connection that Wittgenstein makes between the “cardinal problem of philosophy” and the “theory” of what can be expressed by propositions and what cannot be expressed by propositions but can only be shown, Nir gives two examples of passages in the Tractatus where Wittgenstein speaks of something as showing itself: 5.62, where Wittgenstein says that what the solipsist means shows itself, and 6.36, where Wittgenstein speaks of there being laws of nature as something that shows itself. If you were going to look for places where Wittgenstein’s talk of what shows itself looks as if he means some kind of superfact, these two passages would be perfect. But they are not at all the kinds of cases that Wittgenstein brings up when he is explaining what he means by speaking of something as showing itself. Compare 4.1211. Here Wittgenstein is spelling out what he had been speaking of at 4.121 as what expresses itself in language. It is important that Wittgenstein chooses examples of what is supposedly seeable, and what he thinks his readers will be able to take in as seeable. The two examples that Nir cites can be connected with Wittgenstein’s favoured kind of examples, but they contrast with the kinds of case that Wittgenstein himself uses in explaining showing and what supposedly expresses itself in language (4.124, 4.125). The character of Wittgenstein’s examples has philosophical significance; and Nir’s use of examples that are quite different from Wittgenstein’s can make it harder to see what Wittgenstein means.

Narboux rejects Kremer’s interpretation of showing as merely the illustrating of practical possibility – and very centrally, he rejects Kremer’s idea that there are just two basic ways of understanding what showing means in the Tractatus (Narboux Reference Narboux2014). If there were just two ways of understanding what showing means in the Tractatus, then if you rejected the Kremer interpretation of showing as an essentially practical notion, you’d be committed to the idea of unsayable superfacts as what is supposedly shown by the propositions of the Tractatus. But it’s that conception of what the alternatives are that Narboux rejects.

I will be setting out a very different view from Kremer’s and Nir’s, though I agree with them about not just junking Wittgenstein’s talk of the cardinal problem of philosophy. I strongly disagree with Kremer’s and Nir’s deflating of the notion of showing, and I will also be disagreeing with Kremer about whether the idea of the fundamental problem of philosophy is an illusion. One further thing about which I think I’m in disagreement with Kremer and possibly Nir as well is the significance of proposition 6 of the Tractatus and proposition 6.001: “What [proposition 6] says is that every proposition is a result of successive applications to elementary propositions of the operation N$\left(\overline{\xi }\right)$”. N is the operation that negates all the values of the propositional variable written as $\overline{\xi }$. (This is explained at 5.5 and 5.501.) My disagreement with Kremer about how to read the Tractatus is basically about the importance of the idea of propositions as forming a kind of series – and this is the idea you get at 6.001, the idea of propositions as what can be given in the series constructed by the application of joint negation to elementary propositions. So that’s what I will be heading towards, but I’m nowhere near that yet. I’m still trying to see the problem situation that we are confronting.

At one point in the Notebooks Wittgenstein says that his difficulty is only ‘an – enormous – difficulty of expression’ (Wittgenstein Reference Wittgenstein, von Wright and Anscombe1979: 40). In the letter to Russell, I think that that difficulty of expression is indeed surfacing. You can try in various ways to put the problem Wittgenstein took himself to be confronting, and I will not stay with the first formulation, but to start with, we could try something like this: the cardinal problem is: “How can one make clear the logical concept of what a proposition is, and, in doing so, also make plain what is supposed to show itself in propositions?” This goes right back to Wittgenstein’s fundamental thought, his Grundgedanke: in the proposition objects have proxies, but the logic of the facts is not something capable of being represented. So what I’m suggesting now is that when you look at the expression of Wittgenstein’s fundamental thought at 4.0312 and you also read the letter to Russell, there is a connection. Both the letter to Russell and the statement of Wittgenstein’s fundamental thought involve what supposedly shows forth in propositions. This may sometimes be easy to see, but one thing that philosophy is supposed to do is to enable us to see clearly what the proposition has supposedly shown all along. (The sections of the Tractatus that I have particularly in mind at this point go from 4.01 to 4.06, and then into the 4.1s.)

I want to try to get further into the issues here by going in a sense backwards, back from the Tractatus passages on what can be said and what cannot, to the remarks in the Notes that Wittgenstein dictated to Moore in 1914, where he begins with a whole page about logical propositions and what they show. He says that logical propositions do not say anything, but that merely by looking at them, you can see the logical properties of language that these propositions show; and then later on the page, he says that by merely looking at fa, fa ⊃ ψa, ψa, you can see that the third proposition follows from the other two. This talk of what you can see merely by looking is very striking. It’s connected with the passage in the Tractatus I mentioned earlier (4.1121), and it’s connected with 3.1432. What Wittgenstein says there is: “Instead of, ‘The complex sign “aRb” says that a stands to b in the relation R’, we ought to put, ‘That “a” stands to “b” in a certain relation says that aRb’.” This is supposed to be something you can look at. What is supposed to be doing the saying is a two-term relation between two signs, and what the fact that those two signs stand in that relation SAYS is that the two things stand in a relation. So you can just look and see dual relationality in what is doing the saying and in what is being said. This is a good example of what Wittgenstein means in saying that the logic of the facts cannot be represented: dual relationality is not represented, it is present in the saying and in what is said to be so. – Wittgenstein’s fondness for this sort of example is present also in 4.012, where Wittgenstein says that a relational proposition like “aRb” strikes us as a picture. The sign here is “obviously a likeness of what is symbolized”. – But it is essential to Wittgenstein’s understanding of what philosophy does that what we have here is only a particular case. In the particular case, we can see clearly how the logic of what we are saying is present in our signs. In the particular case, this is open to view, but this is a special case. It’s a very revealing case, but not the general case. Hence, you have room for a conception of philosophical activity as making logical showing perspicuous, because only exceptionally is this straightforwardly describable as easy to see.

Although I am talking about the Tractatus, I want to pause and mark a significant connection with Wittgenstein’s later criticism of his earlier way of thinking – the criticism at Zettel §444:

Both 4.012 and 3.1432 illustrate the particular importance of relational propositions in furnishing such intuitively convincing cases.

Getting back now to my overall argument, I want to get further into Wittgenstein’s understanding of philosophy in the Tractatus, by moving to a later example and then going back to the Tractatus. There is an idea Wittgenstein had in the Tractatus that he never gave up, but the later example is simpler and can help us to see more clearly one of the basic ideas in the Tractatus. My later example comes from his 1939 Lectures on the Foundations of Mathematics (Wittgenstein Reference Wittgenstein and Diamond1976), Lecture 9. Wittgenstein is considering an example in which we have a method for constructing polygons, using ruler and compasses. The method is very narrowly limited in that the compasses have a pre-set radius. You can use a ruler and pre-set compasses to draw a circle, then draw a diameter of the circle, and then construct a square by drawing a diameter at right angles to the original diameter. You then can go on drawing arcs and bisecting the angles created by the intersection of previously drawn lines. So you can go from drawing a square to drawing an octagon, and then you can go on and draw a polygon with sixteen sides, and so on. Wittgenstein imagines someone who is trying to draw a polygon with a hundred sides, using this method; and the person is slow to catch on that you cannot do this by going on bisecting angles and constructing polygons using the ruler and the set compass. The person keeps on trying and does not succeed. Then you give him a proof that shows him that what you can construct with this method is a series of polygons in which the number of sides is a power of 2, and then he sees that the polygon with a hundred sides is left out. It is not in the series of polygons you can construct. And this leads him to give up what he had been trying to do. He now has a clearer idea of what he is trying to do, and so he leaves off trying to construct a polygon with a hundred sides. He’s willing to accept that he now sees more clearly what it was he wanted. Juliet Floyd has written about the great importance of this sort of persuasion in philosophy as Wittgenstein understands it (Floyd Reference Floyd and Hintikka1995, Reference Floyd, Read and Crary2000).

Back to the Tractatus. I said Wittgenstein had an idea that he used in the Tractatus that also comes up later, the idea of someone being able to see that something is not going to turn up in a certain series, and the person can thereby get clearer about something that she was previously unclear about. The way Wittgenstein sets up the Tractatus, we are meant to get clear about what a proposition is partly by seeing propositions as having a place in a series of propositions constructed from elementary propositions. Heaven knows it is not easy to see how exactly this is supposed to work, and people can differ about what Wittgenstein’s idea of the construction of propositions as members of a formal series is supposed to be. But, if you have a way of constructing propositions using the operation of negating all the values of a propositional variable, you are going to be able to see what you are going to be able to get that way, and you’ll also be able to see what that construction will not give you. This is something Wittgenstein says about this general method of constructing propositions at 5.503. Logical methods of construction anticipate what you can construct. Just as the method of bisection of angles anticipates, you could say, all the polygons with a number of sides equal to a power of 2, a logical method of construction of propositions anticipates a series of propositions. And there are things the series passes by, just as the series of constructible polygons with the ruler and fixed compasses bypasses the construction of a polygon of a hundred sides. But we need to note here that the construction of propositions as a formal series depends on what the elementary propositions are, and Wittgenstein did not think we knew what they were. (I also want to add here that the idea of a method of construction of signs as anticipating what you are going to get by the method is significant later in Wittgenstein’s discussion of rule-following.)

I am going to get back to this, but I want first to look further at the ideas reflected in Wittgenstein’s letter to Russell. I suggested earlier that the letter itself and the statement of Wittgenstein’s fundamental idea are connected with the understanding of philosophy as making perspicuous what is internal to our propositions. When Wittgenstein writes about philosophy as logical clarification of thoughts, you get the rather puzzling idea that the thoughts are in some sense clear already, but their clarity is clouded over in our view of them, and philosophizing should turn our thoughts into ones whose clarity is open to view. This idea is in a letter of Wittgenstein’s to Ogden, about how to translate the Tractatus remark at 4.112 about philosophy as an activity of clarification (Wittgenstein Reference Wittgenstein1973: 49) . This comes right before passages on drawing the limits on what can be thought by presenting clearly what can be said, and before the passages on what propositions show. I want to look briefly at a couple of examples of this business of making perspicuous, making open to view, what is internal to our thoughts.

My first case is based on remarks by Brian McGuinness, which bear on 4.44 to 4.461 (McGuinness Reference McGuinness2005: 308–309). This part of the Tractatus is where Wittgenstein introduces truth-tables as a way of writing a propositional sign. We often in philosophy talk about the truth-table “for” a proposition, but this is not what Wittgenstein is doing. The truth-table gives a notation in which we can construct propositional signs. Thus, the sign that Wittgenstein has at 4.442 (with the Ts and Fs) is a propositional sign for the same proposition we might write instead with the material implication sign. In the TF notation what shows up clearly is the range that is left open to the facts by the proposition. This is what Wittgenstein says at 4.463. The truth-table for a tautology shows that reality is in no way determined by the proposition; it leaves everything open. If we go back to 4.05 and 4.06, where Wittgenstein connects a proposition’s being a picture and its being something with which reality gets compared, you can see in the truth-table representation of the tautology that there is no such comparison for the tautology. So what shows up in the truth-table notation for propositions is the logical contrast between propositions, which do determine how things are if the proposition is true, and tautology and contradiction, which do not do that.

The second case I want to mention is that of seeing inferential relations. I will not go into details here, but I did want to point out the image that Wittgenstein uses at 5.1311. He suggests a way of writing the premises and conclusion of a familiar form of argument, and then says that, before the suggested rewriting of the premises, the internal relation between the propositional forms is masked (verhüllt). The new way of writing is a kind of unmasking of the internal relation between propositional forms.

The last thing I want to get to is a pair of very familiar remarks from the Preface to the Tractatus:

Those remarks from the Preface are also connected with what I said about Wittgenstein’s claim that his theory of what can be said and what cannot be said but can only be shown is the cardinal problem of philosophy. I said that what is meant by the cardinal problem is the problem of how to make clear the logical concept of a proposition – to make clear, that is, what it is for something to be included in what we can think, what we can say – and through that to make plain the showing forth of what shows itself. This is what is at stake in what Wittgenstein speaks of in the Preface as drawing the limits of the expression of thoughts from within language. To get clear about this we need to keep in mind the idea I mentioned earlier: that producing a series of constructions, or a series of propositions, can help someone see what is not going to be in the series. A very crucial proposition in the Tractatus, for the issues here, is 6.001, about the general form of a proposition. This is about a recipe for constructing propositions, a recipe for constructing a series of propositions as truth-functions of elementary propositions. Even if we do not know what elementary propositions there are, this recipe for constructing propositions was supposed by Wittgenstein to make clear the general kind of way a series of propositions can be produced, and this would make clear what it is for signs to be included in the series we will get – and also what it is for signs not to be included. Anscombe described in An Introduction to Wittgenstein’s Tractatus how this was supposed to work. She also explained why it would work only if the number of elementary propositions was finite; and she also argued that Wittgenstein had not been aware of the problems (Anscombe Reference Anscombe1971: 132–137). Here I shall discuss how it was supposed to work, ignoring the issues about the infinite number of elementary propositions. I shall be assuming that TLP 6 and 6.001 do give us a series of propositions generated from elementary propositions; and that we are thereby given a clarification of “the expression of thoughts”. What then comes out in this clarification is that “the expression of thoughts” has a limit. We can construct propositions from elementary propositions, and there are some signs that will not come up. If we say they aren’t within the limits of the sayable, what does that mean? That’s what I’m now trying to get to.

I’ll give two conflicting answers, using two examples.

One example of something we will not get to is “(∃ x).x = a”, “There is an object a”. This is the kind of proposition that Russell thought would go into a catalogue of what there is in the world. But the identity sign in that would-be propositional sign has not been given any meaning, and the sign therefore is nonsense. (See remarks in the Tractatus beginning 5.53, on the identity sign.) You can say that “(∃ x).x = a” is on the “far side” of the limits of what can be said, but this means only that it’s a sign-construction, which could be given a use as a propositional sign, but which contains a sign to which no meaning has been given. Putting this another way: you are given clarity about what it is to say something by proposition 6.001, which is a recipe for constructing propositions. And thereby what is made clear is what is not going to be got to by constructing propositions using that recipe, that is, what are merely signs without meaning. “There is an object a” is then one example of such a sign, a sign with no meaning.

The other example I want to consider is one discussed by Roger White in his book on the Tractatus. I want to get to his answer to the question about what lies beyond the limits, because he does not take it that what lies beyond the limits is merely signs with no meaning. He is considering the propositions of the Tractatus itself, like “A proposition is a truth-function of elementary propositions”. He says that propositions like that one “appear to be presented as necessary a priori truths, and therefore to fall outside the scope of the general form of proposition, and hence to be nonsense” (White Reference White2006: 116–117). He also says that the propositions that Wittgenstein used in seeking to establish the limits of language “constantly transgressed the limits they were establishing, and thus fell on the wrong side of the limits, and hence were nonsense” (White Reference White2006: 125). What comes to the surface here is a question about what Wittgenstein means by talking about “limits”. This question is important for us now, but it is also at the centre of how we can understand Wittgenstein’s letter to Russell, and his remarks in that letter about what cannot be said and what shows itself. And it is also absolutely essential to how we think about the propositions in the Tractatus itself, like the one I mentioned, “A proposition is a truth-function of elementary propositions”.

Roger White’s account is very helpful to us, I think, in enabling us to see that there are two alternative conceptions of being outside the limits. These are two alternative understandings of what Wittgenstein means in his talk of limits. I’m going to draw on how Peter Sullivan has explained the two conceptions: he speaks of limits in the one case and limitations in the other. Unfortunately, the easier conception to grasp is the one that is not Wittgenstein’s. Here now is Peter Sullivan on this.

The most important point in Sullivan’s attempt to explain Wittgenstein’s understanding of limits is his remark that “there is nothing thought-like excluded by the limits of thought for lacking thought’s essential nature.” There is nothing excluded except what is simply nonsense. I think we can be helped to see what is at stake here by seeing Roger White’s account of Wittgenstein on limits because it involves exactly the understanding of limits that Sullivan argues we tend to fall into when we do not grasp what Wittgenstein means by limits.

Moving on now to White’s account. White argues that the propositions of the Tractatus fall outside of a boundary, which has on one side of it propositions that are constructed truth-functionally from elementary propositions and thus have the general form of propositions, while on the other side of the boundary, there are propositions like those of the Tractatus, about which White says that they appear to be presented as necessary a priori truths, and for that reason, then, they are on the far side of the limit that they help to establish, and hence are nonsense. It is then part of White’s reading of the Tractatus that what there is on the far side of the limits are thought-like constructions, which are excluded because they lack what counts as the essential nature of thought. On White’s reading, those proposition-like signs on the far side are not mere constructions that can be anticipated not to turn up when we construct propositions by the recipe given at 6.001. On the alternative view to White’s, the reason these proposition-like signs do not turn up on the right side is not that they lack the essential nature of thoughts. It’s rather that they can be shown to contain one or more signs with no meaning in the particular context. In the case of many of these propositions, the meaningless signs are signs that can in some contexts be used for formal concepts, but in their Tractarian contexts they do not have that use, and there is no other kind of meaning that they have been given.

This then connects directly back to Wittgenstein’s letter to Russell and what he says is his main contention. A reading like Roger White’s has at its heart the idea of what cannot be said as something that is thought-like. So on that sort of reading, the idea would be that “there is an object a” is not merely something with no sense. The Roger White view involves our continuing to think of “There is an object a” as quasi-propositional, as something thought-like, which lacks what Wittgenstein counts as the essential nature of thought. And the idea then is that that’s why “There is an object a” counts as nonsense. But Wittgenstein’s view is that what shows in the uses of “a” as a name is not a matter of some quasi-propositional content that cannot be put into a genuine proposition. When we think of it as a sort of quasi-propositional content, it appears to be something out of reach of saying, as opposed to being something that is already expressed on this side of the limit, in the ways we use all the propositions that say something about a.

I want to end by getting back to Wittgenstein’s letter telling Russell that he had misread the Tractatus and had missed Wittgenstein’s main contention. I have a couple of points.

One is that Wittgenstein does not in that letter speak of the “distinction between saying and showing”. Jean-Philippe Narboux has discussed the way that the expression, “the distinction between saying and showing” can be misleading. Here I will just note that talking of the distinction between saying and showing encourages the idea that what shows itself is something that is thought-like but which lacks the essential nature of thought, and that that is why it is unsayable. It encourages the idea that we are drawing a distinction between genuine expressions of thought and what are thought-like but not genuinely thoughts. This involves the idea of limits as limitations, not as limits. Anyway, I think it is worthwhile to note that Wittgenstein himself does not speak in terms of such a distinction in his letter to Russell.

The last thing I wanted to get to was the business about what the cardinal problem of philosophy is, how to formulate that. My idea was that we could see better how to formulate the problem after reading through some of the paragraphs I’ve focused on. So here is my reformulation. What Wittgenstein means by the cardinal problem is the problem of making completely clear the limit of the expression of thoughts, so that it is clear that everything that can be said lies within the limit thus drawn, and so that it is also clear how what shows itself shows itself in what lies within the limit. The problem will be thus to delimit at the same time what cannot be thought, by making clear what can be thought. The problem, that is, is that of making clear how what can be thought can be thought clearly, and to determine in that way what is not included. – If this is a formulation of the cardinal problem of philosophy, the Preface statement of the “whole sense of the book” can be read as announcing that the problem has been solved: what can be said can be said clearly; and, as for what is not included in the sayable, darüber muss man schweigen.