‘Ontology’ and Identity in the Tractatus: À Propos of Black’s Companion
‘Adequate Symbolism’ and ‘Ontology’
IF we ask what Wittgenstein means by ‘adequate symbolism’, we shall look to the relation of sign and syntax; for it depends on that. So it is pointless to say, as Black does: we must have some view of what reality is like, before we can ask if the symbolism is adequate to describe it. Black takes this as a reason for his remarks about ‘the ontology of the Tractates’, For instance: ‘Wittgenstein expects a perspicuous view of the nature of logic to have ontological implications.’ ‘Wittgenstein’s conception of the nature of language … required a standon ontological issues.’ ‘His ontology (sic) is on the whole suggested by his views about language.’1 (My italics throughout.) This is confused, and the remarks about adequate symbolism in the Tractatus do not need it.
Since there are signs, there must be a distinction of true and false propositions—a distinction to be decided finally by observation, not by logic. We could say that the truth of logical principles is tied to this. When the Tractates says in 6.1 13 that ‘What distinguishes logical propositions is that we can see by the symbol alone that they are true’, it adds: ‘So what is especially important is the fact that the truth or falsity of non-logical propositions cannot be seen in the proposition alone.’—In other words, it would have no sense at all to speak of logical propositions unless there were empirical propositions.
Mathematics is not written in tautologies; it is written in equations. But equations would be meaningless unless there were calculation: they get their reality from the general form of logical operation, and so from the internal relations of propositional forms. So we could not treat mathematics as a logical method—we could not see that mathematical proofs are logical proofs—unless empirical statements had sense and could be the bases of logical operations.
This is summed up in 4.0312, which expresses the ‘Grundge-danke’ of the book:
Die Möglichkeit des Satzes beruht auf dem Prinzip der Ver-tretung von Gegenständen durch Zeichen.
Mein Grundgedanke ist, daß die ‘logischen Konstanten’ nicht vertreten. Daß sich die Logik der Tatsachen nicht vertreten läßt.2
This is at the foundation of what Wittgenstein has to say about logical analysis—e.g. (in 4.221) ‘that in the analysis of propositions we must arrive at elementary propositions, consisting of names directly connected to one another’. He is contrasting this ‘un-mittelbare Verbindung’ of names in elementary propositions with whatever it is that logical constants express. For these appear only in the expression of the results of an operation on elementary propositions. You can always transform a proposition containing logical constants into another equivalent to it. But elementary propositions cannot be equivalent to one another.—We can carry out logical operations independently of the truth or the falsity of elementary propositions: independently of what is the case. We can do this because of the fundamental difference between elementary propositions and others: i.e., because the logical constants ‘do not stand for anything’. Otherwise we could not ‘see by the symbol alone’ that a calculation or a formal proof was correct.
The Tractatus could not begin with a discussion of logical constants and the truth of logical principles. What comes first is the truth or falsity of material propositions—in other words sense. Without this we could not even speak of possible signs.
But to call this ‘ontology’ is confusing. And to say that the discussions of logic are important because of the ontology which is built on them, is to stand the whole thing on its head. Black quotes the remark in Wittgenstein’s 1913 Notes on Logic: ‘/Philosophy/consists of logic and metaphysics, the former its basis’, and Black adds : ‘Logic as the basis of metaphysics : throughout the book Wittgenstein expects a perspicuous view of the nature of logic to have ontological implications. Logic is important because it leads to metaphysics’ (Companion, page 4). But the remark he quotes does not say that logic is the basis of metaphysics; it says it is the basis of philosophy. And Wittgenstein did not say there or anywhere else that logic has implications.
Sentences like ‘der Name bedeutet den Gegenstand’ (3.203) or ‘Der Name vertritt im Satz den Gegenstand’ (3.22) belong to the grammar of the words ‘name’ and ‘object’ and ‘proposition’. The Notes Dictated to Moore had said (Notebooks, pp. 109, no): ‘In the expression (∃y) . 
y, one is apt to say this means “There is a thing such that. .. “. But in fact we should say “There is a y, such that…”; the fact that y symbolizes expressing what we mean…. In our language names are not things : we don’t know what they are: all we know is that they are of a different type from relations, etc., etc.’ The Tractatus might not put it in just this way, but the main point holds there.
‘Token or Type?’
In connexion with 3.203—‘A name means an object. The object is its meaning. (“A” is the same sign as “A”.)’—Black asks ‘Is the propositional sign a token or a type?’; and he goes on: ‘When we normally speak of a sentence, we use the word “sentence” in a “type-sense” rather than a “token-sense”. … That this is the way Wittgenstein himself uses the expression “propositional sign” (which takes over the rôle of “sentence” in his conception) is made quite clear by his remark at 3.203 : “ ‘A’ is the same sign as ‘A’.” If two propositional signs consist of physically similar words respectively attached to the same bearers, Wittgenstein counts the two as instances of the same propositional sign.’
But Wittgenstein himself said nothing of the sort: the text says nothing at all about physically similar signs respectively attached to the same bearers.
His remark in the Notes Dictated to Moore that ‘in our language names are not things’ says something about the grammar of ‘name’ and ‘thing’; or, as he puts it there, the difference in logical type. So, for instance, ‘identity’ has different rules of syntax when we speak of the identity of a thing and when we speak of the identity of a sign. ‘ “A” is the same sign as “A” ’ expresses—or ‘seeks to express’—the identity of a sign. It does not say anything. And it cannot be analysed in terms of ‘this scratch here resembles that scratch there’. Perhaps we’d never say ‘It’s the same sign’ unless the scratches did resemble one another, but this is not part of what we mean by ‘ “A” is the same sign as “A” ’—‘How do you know this is A?’ would be as nonsensical as ‘How do you know this is white?’.
A mark without syntax is not a sign. And this makes it hard to say what the identity of a sign is.
When Peirce3 and others write about Types and Tokens they seem to feel that we might analyse the identity of a sign in terms of the identity of a physical object or of an event. But they never manage to: this is plain when Peirce calls a Token an ‘instance’ of a Type, and when Ramsey says that ‘a proposition is a type whose instances consist of all propositional sign tokens which have in common, not a certain appearance, but a certain sense’. The ‘all’—in ‘all propositional sign tokens which …’—is bewildering, but since each of these has a sense, it ought to refer to various propositions which are equivalent to one another; and if we put it so, we are back where we started: distinguishing type and token has done nothing.—On the other hand, physical similarity between particular scratches cannot be what ‘groups tokens together into types’, in Peirce’s sense of Type; no more than grouping shells or pebbles together would be treating them as Tokens.
Black thinks that Wittgenstein uses ‘propositional sign’ for a class of token propositional signs. But what does ‘token’ add here?
When a printer estimates the number of words on a page he is speaking of physical marks, and it does not matter whether they are really words or not. Peirce calls the Token in which a Type is embodied ‘an Instance of the Type’ But then if I count the instances of various Types on a page I am not counting what the printer counts.
‘You cannot make the same scratch twice’ could have sense. ‘You cannot write the same word twice’ could not.
‘How can I be sure I am seeing the same token and not a different one? Maybe I am not reading the same copy of the book although I thought I was. Maybe someone had erased that word on that line and printed it again. Etc., etc.’ Peirce might have answered, ‘All right, you probably can’t; and in such cases it does not matter.’—I.e., ‘Tokens’ always means ‘Tokens on this page’ or ‘… in this copy of this book’—or something of this sort. It is not a term for a kind of physical object, like ‘scratch’. Ramsey and Black (and Peirce) are not clear about the grammar of it.
If nobody ever said anything, nothing would ever be said. And every time you write a word, you write a word.
But this is just as trivial as it sounds. It does not explain the meaning of ‘That’s a word’.
If I said ‘They are all instances of “word” ’, you would probably take me to mean: ‘We’d call them all words’. But of course this does not mean ‘They are all Tokens of the Type “word” ’ in Peirce’s sense.
We want to know the syntax of ‘sign’. And we think, perhaps, that then we’d know the syntax of signs—i.e., what they must have in order to be signs.
Is it not the syntax of words that determines the syntax of ‘word’? Unless you understood words in their syntax, you would not know what a word is, and you could not use the word ‘word’.—If we say this, we do not mean—we deny—that the syntax of ‘word’ is given (or determined) by instances of the Type ‘word’.
Black lands in this confusion because he thinks there is something by which our grammar is determined—or that this is how the Tractatus must be read. This goes with what he says about ontology in his discussion of ‘adequate symbolism’.
‘The Naming Relation’
We speak in a different sense (1) of a proposition corresponding with reality (this is Abbildung or ‘projection’) and (2) of a name corresponding to what it means or to what is called by it.
When the Tractatus says, in 3.3 : ‘it is only in the connexion of a proposition that a name has meaning’, it means that without the picturing or projection in a proposition there would be no correspondence at all. A proposition can describe a state of affairs in a language. Apart from a language it would not be a proposition. In Tractatus 5: ‘A proposition is a truth-function of elementary propositions.’ So the combination of signs in a proposition is not arbitrary.4 I am committed to the signs I use and the ways I combine them—by the general rule, the syntax of the language. It is through this that the marks and sounds become symbols.
‘There might have been a different corrrelation (of signs and things).’ Alternatives are possible in a language. But a jumble of sounds or scratches would not be an alternative; it would mean nothing to call it one.
‘But assigning names is arbitrary—definitions are arbitrary.’—What makes it a definition? If I give a name to a colour or a shape I must have distinguished these as I distinguish expressions of a language. And within the language my definition commits me in certain ways, not in others. What the definition establishes—the relation of the name to what it stands for—is not an external relation.
We could say that the rules of multiplication are fixed by definition; in certain algebras these rules mean nothing. Or we might say that 4 is the result of 2 × 2 by definition; and this would not make the relation of result and multiplication a contingent one.— Words are related to what they say as a result to its calculation.
The Tractatus hardly distinguishes naming and calling something by its name. And 3.3 shows that this is not an oversight. ‘Nur im Zusammenhange des Satzes hat ein Name Bedeutung.’5 So we may think that what the word ‘red’ means is expressed by the sentence ‘a is red’.
Someone might say: ‘The name must correspond to some reality. It cannot describe anything if there is nothing which it signifies.’ Or suppose I told you: ‘I call each of these roses red because each of them is red. The word I use corresponds to the colour of the flower.’—But what corresponds is the sentence. The Tractatus supposed that ‘red’ determines how I use it.
Wittgenstein rejected this later. It confuses giving a sample and using a sample. I may give a sample—a piece of coloured paper—to explain what I mean by ‘vermilion’. Or I may use the sample in place of the word and tell you ‘the flowers in that bed are this colour’. But I cannot use the sample to explain what colour this sample is.
The idea had been that the sample can serve as a ‘primary sign’—one which explains itself and cannot be misunderstood. Other signs may be explained by the primary signs; but without the primary signs we’d never know what we were saying. Wittgenstein brought out the confusions in all this. But it showed that the distinction between what a name means and what is called by it is not always simple or easy.
Black knows that the meaning and the bearer of a name are different. But in his remarks on ‘difficulties about the naming relation’ he seems to think that arbitrary means contingent and that this means empirical. On page 116 he says: ‘It is only contingently the case that the elements of [the propositional sign] F have the bearers that are attached to them, since it is perfectly conceivable that F might have had a different sense’ I have put the last clause in italics, for this does not show that: ‘F says (so and so)’ is an empirical proposition.
If the meanings of names are arbitrarily fixed, this does not mean that the sense of a sentence is arbitrarily fixed. What fixes the meaning of a name is a rule. But if someone says ‘an arbitrary rule is a contingent proposition’, he confuses a rule with a generalization.
Identity
What shows that a name now means what it meant then? What shows that this statement speaks of the same thing as that?
4.243: ‘Can we understand two names without understanding whether they signify the same thing or two different things?— Can we understand a proposition in which two names occur without knowing whether their meaning is the same or different? …’
5.53: ‘That what is meant is the same I express by using the same sign, and not by using a sign for identity. [und nicht mit Hilfe eines Gleichheitszeichens.] …’
Wittgenstein does not ask ‘What shows that this is the same sign?’—nor can this be asked. Yet Black seems to think the Tractatus tries to answer it. He sees that in 5.53–5.534 Wittgenstein wants ‘to show that identities are not truth functions of elementary propositions, as genuine propositions are’ (Companion, page 290), but then he says: ‘The basic idea is to show identity of objects, whether identified by names or included in the ranges of given variables by means of physical similarities in the signs for such names and such variables.’ I have put that phrase in italics.
Wittgenstein’s point is that identity is not a function, not a tautology and not a logical principle; also that there is no ‘general concept of logical identity’ (compare e.g. Tarski’s Introduction, page 61).
Wittgenstein could call the laws of logic—the modus ponens, say, or double negation—propositions about propositions. The Tractatus brings out this relation of tautologies to genuine propositions by writing them both as truth-functions. But x = y, or x = x, is not a truth-function. This is the main point. The details of Principia Mathematical definition—‘x and y are called identical when every predicative function satisfied by x is also satisfied by y’—are less important.
Principia Mathematica distinguished the ‘=’ of definition from the ‘=’ in x = y but assumed that here (in x = y) it is the same as the sign of equality in mathematics. Wittgenstein called this a confusion.
To arrive at cardinal numbers Principia Mathematica treats of Unit Classes and of Cardinal Couples; and so (e.g. *51.232) of ‘the class whose only members are x and y’ To express this: if any third term, 2, be assumed to belong to the given class, then z = x.v.z = y; and PM treats this formula as a function. A little later it says, ‘the class of all couples of the form ι‘x ∪ ι‘y (where x ≠ y) is the cardinal number 2’—where the sign in the parenthesis does not express mathematical inequality. Couples, apparently, are entities correlated in the form x = a.y = b.v. x = c.y = d.v. … etc. Here the sign of identity would be used to express logical correlation but not mathematical equality.
For ‘only x and y have a given property’ the Tractatus gives a notation in 5.5321:
The Tractatus does not introduce numbers in this way. But it shows that what Principia Mathematica wants to express can be written without the ambiguous z = x.v.z = y. Here it seems to be using the P M symbolism. But the apparent variables are different signs from those which Principia Mathematica writes the same way; for there the apparent variables seem to have the generality of a concept. The criticism of identity is also a criticism of the Principia Mathematica use of quantifiers—which the Tractatus has just been discussing.
It seems as if Principia Mathematica explains what it is we say about x and y when we call them identical. Just as it seems to say of those things which form a couple that they stand in this (which?) relation to one another. Perhaps Russell thought that unless he did treat x = y as a function he could not write the propositions of arithmetic in logical notation.
Ramsey seemed to accept the criticism of Russell’s definition of identity. But he wanted to keep x = y as a function ; so it looks at first as though he had kept the substance of Russell’s theory. ‘x = y is a function in extension of two variables. Its value is tautology when x and y have the same value, contradiction when x, y have different values.’6 But these are not functions in Russell’s or Frege’s sense; so that when Ramsey speaks of ‘an apparent variable e’, for instance, we do not know what he is saying. What he did share with Russell was a confusion about the application of mathematics and the reference to things.
He said (page 49) that
Wittgenstein’s convention (regarding identity) … puts us in a hopeless position as regards classes, because … we can no longer use x = y as a propositional function in defining classes. So the only classes with which we are now able to deal are those defined by predicative functions. … Mathematics then becomes hopeless because we cannot be sure that there is any class defined by a predicative function whose number is two; for things may all fall into triads which agree in every respect, in which case there would be in our system no unit classes and no two-member classes.
Apparently Ramsey rejects the formulations of Tractatus 5.321 on the ground that any such proposition might be false—we cannot be sure that the facts would justify it: ‘things may all fall into triads’. And the advantage of (Ramsey’s) functions in extension would be that the correlation here is arbitrary; the assertion of such functions does not depend on whether individuals agree or disagree in their properties.
Wittgenstein would say then as he said later: ‘The application of mathematics in our language does not say what is true and what is false, but what is sense and what is nonsense.’
Arithmetic
Ramsey and Russell wanted to express arithmetic—mathematics—in logical terms: in terms of relations between functions. The Tractatus holds that the propositions of mathematics are equations and that these show the logic of the world as tautologies do: but they are not tautologies. (‘The logic of the world’—roughly, we speak in the same way of necessity and of impossibility here as in logic.)
Russell’s notation for numerical expressions does not show their interconnexion with operations such as addition and multiplication. The Tractatus holds that we understand numbers when we see them as features of a formal system or a calculation. ‘Die Zahlen treten mit dem Kalkül in die Logik ein.’7 A correlation between signs on one side and the other of an implication will not provide this; no more than it gives a conception of formal series.—Suppose we showed that it is the expression of an identity. What is there mathematical about this? How does the conception of ‘… and so on’ come into it? How is it the expression of a rule?
This is a criticism of Russell’s view of the generality of mathematics and of logic.
Suppose we said that the result of a calculation holds universally. This is of the same form as saying that the development of a given decimal is periodic. It is what Wittgenstein in the Tractatus expressed by ‘the general term of a formal series’ or ‘the general form of number’. And, as he remarked later, the ‘generality’ of [1, x, x + 1] cannot be expressed by ‘(x).[1, x, x + 1]’. (He was speaking of induction and the idea of ‘holding for all numbers’.)
We might feel like saying that the general form of operation was the same as the general concept of a formal series—except that there is not really any such concept; it is a form. And we need to keep this distinction especially when we are speaking of generality. The sign for an operation is the general sign for a member of a formal series. Take that given in 5, 2522: [a, x, O’x]. This is also the general form of successive application of an operation. For the general form of operation is the general form of its successive application. But it would be misleading to say that this shows ‘the kind of generality which an operation has’. We might distinguish more general and more special operations, but this would be something else. It would not be the generality of the form (as opposed, say, to the generality of a concept).
‘Numbers are exponents of operations.’ They are not properties of aggregates, nor properties of the defining properties of aggregates. To say ‘the successive applications of an operation form an aggregate’ would be nonsense. It would be treating ‘repetitions of the operation’ as physical events. It would confuse the form, or the possibility, with my carrying out the operation. This would be like a confusion of counting in mathematics—counting the roots of an equation or the inner and the outer vertices of a pentagram (Wittgenstein’s examples)—and counting outside mathematics : counting the jars on a shelf or the white corpuscles in a blood sample.
We may want to say ‘The order of successive application is a temporal order: one after another.’ This is all right if we remember that it is an order of possibilities—the order in a construction. ‘We cannot construct the polygon before the triangle’ (Simone Weil).8 This does not refer to the times of actual happenings.
One reason for speaking of numbers as exponents of an operation was to show that the expressions of arithmetic belong to a system. Otherwise equations would be arbitrary rules of substitution. We should not know where they belonged; i.e., we should not know what to do with them. Black seems confused about this when he speaks of ‘the applications of arithmetic to counting’ (page 314).
He is more seriously confused in what he says of 6.02. Here (page 314) Black seems to take the successive application of an operation (his phrase ‘the self-application of an Ω-operation’ is not in the Tractatus and is misleading) to be something like the logical addition of a truth-function to itself, p v p = ρ; so if the operation were ‘v’, then O’O’O’a would be the same as O’a. (cf. 5.2521.) Black may have been led to this by the fact that when Wittgenstein begins his definitions or rather constructions of numbers he decides that the sign for the repetition of the same operation shall be an exponent written first as a succession of ‘+1’s’. But this is the ‘+’ of arithmetical addition. And whatever difficulties it may carry here, it is not logical addition
As the Tractatus uses ‘operation’, it would be nonsense to speak of a logical sum of operations. And although throughout its successive application the operation is the same, this does not mean that the successive application is no different from a single application of it.
Black concludes: ‘It would seem that for all m and n greater than zero we must have Ωm’x = Ωn’x. Does it follow that m = n for all positive integers?’—But what does ‘=’ mean in the first of these sentences? If it means ‘is the same operation as’, then nothing follows about arithmetical equality. If it is the sign of numerical equality, then I have no idea what the whole expression means. (Black’s ‘for all m and n greater than zero, we must have …’ is nonsense in this context.)
In 6.01 Wittgenstein had said that the general form of operation is ‘the most general form of transition from one proposition to another’. The result of any transition of this form would be a proposition, not a number. The general form of operation does enter somehow into the formal series in which numbers are generated, apparently. But the first idea is that numbers are exponents of the successive application of an operation—not that they are generated by it. The formal series in which they are generated or constructed is a series of arithmetical operations.
In 5.2523 the Tractatus says: ‘Der Begriff der successiven Anwendung der Operation ist äquivalent mit dem Begriff “und so weiter”.’9 The general form of operation is not itself an operation of a formal series. It is what makes possible the development of formal series. And it is what makes mathematical constructions possible. So that in arithmetic we can calculate, and we do not wonder whether the same calculation will always have the same result. We can see that that is how it goes; just as with a periodic decimal, when we see that the remainder is the same as the dividend we can see that it goes on like that.
In the Tractatus number is a formal concept or a form. We do not learn the meaning of a form as we learn the meaning of a name or a phrase; and I could not explain a form to you as I might explain a general concept. ‘Form’ and ‘construction’ go together; and you can understand a construction that is carried out; just as you can understand a sentence, without having anyone tell you what it is. But we could not define a propositional form without being circular (the expression for ‘the general form of proposition’ contains ‘elementary proposition’). Neither can we define numbers, in that sense. I say ‘in that sense’ because Wittgenstein’s constructions in 6.02 are definitions in a different sense.
When we are given a formal series we can ‘see that it must go on like that’. This is what underlies recursive proofs and definitions. Wittgenstein gives a formal series of definitions by writing the definitions of 1 and 2 and 3 and then writing ‘(and so on)’. The series develops by the repetition of ‘+1’ and in this way it shows not only that every number after 1 includes the number which precedes it, but also that the rules for addition—the associative and the commutative laws, for instance—hold for all natural numbers. This means that Wittgenstein is assuming arithmetic in order to define number; but this need not be an objection.
He recognized later that he needed brackets if the succession of +1’s was to be a formal series. Suppose /, //, ///, ////, /////, with nothing in the signs suggesting the operation by which we get from one of them to another. These signs would not be terms in a formal series. There would be no general term or rule of the series determining the development of it. ‘And so on’ would mean nothing. But if we write 1+1+1+1+1 this is just as formless. Unless we have the brackets ((((1) + 1) + 1)+ 1) how shall we know to what we are adding the next 1?
If Wittgenstein had used brackets in 6.02 the connexion with the repetition of an operation might have seemed less direct. He may also have thought that the 1’s should be written without differences in order to show that the brackets are justified—to show how repetition of an operation provides for the use of brackets. In 6.231 he says ‘It is a property of “1 + 1 +1 + 1” that it can be construed as “(1 + 1)+(1 + 1)”.’ It is as though the grammar of ‘+1’ were fundamental for all numerals of the natural numbers. And sometimes when we want to show that it is the same number on each side of the equation, we may feel that to make the demonstration complete we should resolve each numeral into a sum of 1’s. This is all right for small numbers. If we write (1 + 1) + (1 + 1 + 1) = (1 + 1 + 1) + (1 + 1) and then drop the brackets we can see that it is the same sign on each side. But if we did this for 18 + 17 = 17 + 18 the substitution of ‘1+1+1…’ would clarify nothing. We should have to count the 1’s after the brackets were removed and rely on our original equation. This equation (18 + 17 = 17 + 18) is obvious anyway. So why were we inclined to substitute the +1’s? Is it a way of showing the general form of addition? of showing why the rules of addition hold for all natural numbers? As though that way of writing numerals would show how arithmetic springs from the general form of operation.
In 1923 Skolem spoke of recursive proof of the associative law for addition, for instance. Is Wittgenstein assuming something of the sort when he writes ‘and so on’ in his definitions with +1? Is he assuming that the general form of operation provides it? A little later he would say he was not. But perhaps when he wrote the Tractatus he was not clear about it. I suppose the general form of operation (if we want to speak of it) would come in when we have given the recursive proof for a + (b + 2), a + (b + 3), … and we see that this is a series of proofs having a particular form—a form which holds for all natural numbers. This is not the form of the recursive proof; it is what is shown in the recursive proof.—If we speak here of drawing a conclusion, then we draw the conclusion from the particular model: the paradigm transition from a + (b + 1) = (a + b) + 1 to the corresponding rule for a + (b + 2), say. We do not base anything on the form of calculation in general. And when I said just now that the general form of operation ‘would come in when…’ this was not correct; it does not come in at all.
Wittgenstein wanted to show a connexion between arithmetic and the possibility of symbolism. What makes it possible for a symbolism to have sense.
He wanted to forestall the idea that an operation might yield a rigmarole of meaningless signs. Later he wrote of pseudo-operations—what look like mathematical operations but are not. It is a pseudo-operation if we cannot see in the signs written down the law or the rule of development which determines them; if the development of a decimal, for instance, is not completely determined by a rule of operation which we know at the start. (In what sense would this be a ‘development’?) If it were generally like this—if there were no difference between operation and pseudo-operation—we could not understand any operation. We should not understand the instruction: ‘work out the calculation’. The ‘so’ in ‘and so on’ would have no meaning.—If this is what ‘There is a general form of operation’ means, it does not follow that we can ask to have the general form of operation written down. Wittgenstein dropped the whole way of speaking when (in 1929) he gave up speaking of the general form of proposition. But the distinction of operations and truth functions was important in discussing Russell’s logical notation for arithmetic. It was perhaps one step towards recognizing that mathematical and logical operations cannot be run together.
1 Max Black, A Companion 10 Wittgenstein’s Tractatus, Cambridge, 1964, pp. 4, 7 and 8.
2 ‘The possibility of propositions is based on the principle that objects have signs as their representatives.
‘My fundamental idea is that the “logical constants” are not representatives; that there can be no representatives of the logic of facts.’
3 C. S. Peirce, Collected Papers, Vol. IV, §§ 4.537, 4.538, 4.544.
4 Cf. 5.47: ‘ … Wo Zusammengesetztheit ist, da ist Argument und Funktion, und wo diesc sind, sind bereits alle logischen Konstanten.’ Or 4.0141: ‘Daß es eine allgemeine Regel gibt … darin besteht eben die innere Ähnlichkeit dieser scheinbar ganz verschiedenen Gebilde. Und jene Regel ist das Gesetz der Pro-jektion….’(‘Wherever there is compositeness, argument and function are present, and where these are present, we already have all the logical constants.’ ‘There is a general rule … That is what constitutes the inner similarity between these things which seem to be constructed in such entirely different ways. And that rule is the law of projection. …’)
5 ‘Only in the nexus of a proposition does a name have meaning.’
6 F. P. Ramsey, The Foundations of Mathematics, p. 53.
7 Pbiiosophische Btmerkungen, p. 129: ‘This is in line with what I once meant when I said: numbers come into logic with the system of calculation.’
8 Leçons de Philosophie de Simone Weil, présentées par Anne Reynaud, Paris, 1959, p. 65.
‘The concept of successive applications of an operation is equivalent to the concept “and so on”.’