2.011 It is essential to things that they should be possible constituents of states of affairs.

2.022 It is obvious that an imagined world, however difference it may be from the real one, must have *something*--a form--in common with it.

2.024 The substance is what subsists independently of what is the case.

2.026 There must be objects, if the world is to have unalterable form.

2.031 In a state of affairs objects stand in a determinate relation to one another.

2.032 The determinate way in which objects are connected in a state of affairs is the structure of the state of affairs.

2.033 Form is the possibility of structure.

2.034 The structure of a fact consists of the structures of states of affairs.

2.061 States of affairs are independent of one another.

2.062 From the existence or non-existence of one state of affairs it is impossible to infer the existence or non-existence of another.

2.063 The sum-total of reality is the world.

2.131 In a picture the elements of the picture are the representatives of objects.

2.141 A picture is a fact.

2.161 There must be something identical in a picture and what it depicts, to enable the one to be a picture of the other at all.

2.171 A picture can depict any reality whose form it has. A spatial picture can depict anything spatial, a coloured one anything coloured, etc.

2.172 A picture cannot, however, depict its pictorial form: it displays it.

2.173 A picture represents its subject from a position outside it. (Its standpoint is its representational form.) That is why a picture represents its subject correctly or incorrectly.

2.174 A picture cannot, however, place itself outside its representational form.

2.181 A picture whose pictorial form is logical form is called a logical picture.

2.182 Every picture is *at the same time* a logical one. (On the other hand, not every picture is, for example, a spatial one.)

2.201 A picture depicts reality by representing a possibility of existence and non-existence of states of affairs.

2.202 A picture represents a possible situation in logical space.

2.203 A picture contains the possibility of the situation that it represents.

2.221 What a picture represents is its sense.

2.222 The agreement or disagreement or its sense with reality constitutes its truth or falsity.

2.223 In order to tell whether a picture is true or false we must compare it with reality.

2.224 It is impossible to tell from the picture alone whether it is true or false.

2.225 There are no pictures that are true a priori.

3.001 'A state of affairs is thinkable': what this means is that we can picture it to ourselves.

3.031 It used to be said that God could create anything except what would be contrary to the laws of logic. The reason being that we could not *say* what an 'illogical' world would look like.

3.141 A proposition is not a blend of words.(Just as a theme in music is not a blend of notes.) A proposition is articulated.

3.142 Only facts can express a sense, a set of names cannot.

3.144 Situations can be described but not *given names*. (Names are like points; propositions like arrows--they have sense.)

3.201 I call such elements 'simple signs', and such a proposition 'complete analysed'.

3.202 The simple signs employed in propositions are called names.

3.203 A name means an object. The object is its meaning. ('A' is the same sign as 'A'.)

3.221 Objects can only be *named*. Signs are their representatives. I can only speak *about* them: I cannot *put them into words*. Propositions can only say *how* things are, not *what* they are.

3.251 What a proposition expresses it expresses in a determinate manner, which can be set out clearly: A proposition is articulated.

3.261 Every sign that has a definition signifies *via* the signs that serve to define it; and the definitions point the way. Two signs cannot signify in the same manner if one is primitive and the other is defined by means of primitive signs. Names *cannot* be anatomized by means of definitions. (Nor can any sign that has a meaning independently and on its own.)

3.262 What signs fail to express, their application shows. What signs slur over, their application says clearly.

3.263 The meanings of primitive signs can be explained by means of elucidations. Elucidations are propositions that stood if the meanings of those signs are already known.

3.311 An expression presupposes the forms of all the propositions in which it can occur. It is the common characteristic mark of a class of propositions.

3.312 It is therefore presented by means of the general form of the propositions that it characterizes. In fact, in this form the expression will be *constant* and everything else *variable*.

3.313 Thus an expression is presented by means of a variable whose values are the propositions that contain the expression. (In the limiting case the variable becomes a constant, the expression becomes a proposition.) I call such a variable a 'propositional variable'.

3.314 An expression has meaning only in a proposition. All variables can be construed as propositional variables. (Even variable names.)

3.315 If we turn a constituent of a proposition into a variable, there is a class of propositions all of which are values of the resulting variable proposition. In general, this class too will be dependent on the meaning that our arbitrary conventions have given to parts of the original proposition. But if all the signs in it that have arbitrarily determined meanings are turned into variables, we shall still get a class of this kind. This one, however, is not dependent on any convention, but solely on the nature of the pro position. It corresponds to a logical form--a logical prototype.

3.316 What values a propositional variable may take is something that is stipulated. The stipulation of values *is* the variable.

3.317 To stipulate values for a propositional variable is *to give the propositions* whose common characteristic the variable is. The stipulation is a description of those propositions. The stipulation will therefore be concerned only with symbols, not with their meaning. And the *only* thing essential to the stipulation is *that it is merely a description of symbols and states nothing about what is signified*. How the description of the propositions is produced is not essential.

3.318 Like Frege and Russell I construe a proposition as a function of the expressions contained in it.

3.321 So one and the same sign (written or spoken, etc.) can be common to two different symbols--in which case they will signify in different ways.

3.322 Our use of the same sign to signify two different objects can never indicate a common characteristic of the two, if we use it with two different *modes of signification*. For the sign, of course, is arbitrary. So we could choose two different signs instead, and then what would be left in common on the signifying side?

3.323 In everyday language it very frequently happens that the same word has different modes of signification--and so belongs to different symbols--or that two words that have different modes of signification are employed in propositions in what is superficially the same way. Thus the word 'is' figures as the copula, as a sign for identity, and as an expression for existence; 'exist' figures as an intransitive verb like 'go', and 'identical' as an adjective; we speak of *something*, but also of *something's* happening. (In the proposition, 'Green is green'--where the first word is the proper name of a person and the last an adjective--these words do not merely have different meanings: they are *different symbols*.)

3.324 In this way the most fundamental confusions are easily produced (the whole of philosophy is full of them).

3.325 In order to avoid such errors we must make use of a sign-language that excludes them by not using the same sign for different symbols and by not using in a superficially similar way signs that have different modes of signification: that is to say, a sign-language that is governed by *logical* grammar--by logical syntax. (The conceptual notation of Frege and Russell is such a language, though, it is true, it fails to exclude all mistakes.)

3.326 In order to recognize a symbol by its sign we must observe how it is used with a sense.

3.327 A sign does not determine a logical form unless it is taken together with its logico-syntactical employment.

3.328 If a sign is *useless*, it is meaningless. That is the point of Occam's maxim. (If everything behaves as if a sign had meaning, then it does have meaning.)

3.331 From this observation we turn to Russell's 'theory of types'. It can be seen that Russell must be wrong, because he had to mention the meaning of signs when establishing the rules for them.

3.332 No proposition can make a statement about itself, because a propositional sign cannot be contained in itself (that is the whole of the 'theory of types').

3.333 The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition 'F(F(fx))', in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(F(fu))' we write '(exists O): F(Ou). Ou = Fu'. That disposes of Russell's paradox.

3.334 The rules of logical syntax must go without saying, once we know how each individual sign signifies.

3.343 Definitions are rules for translating from one language into another. Any correct sign-language must be translatable into any other in accordance with such rules: it is *this* that they all have in common.

3.411 In geometry and logic alike a place is a possibility: something can exist in it.

4.001 The totality of propositions is language.

4.002 Man possesses the ability to construct languages capable of expressing every sense, without having any idea how each word has meaning or what its meaning is--just as people speak without knowing how the individual sounds are produced. Everyday language is a part of the human organism and is no less complicated than it. It is not humanly possible to gather immediately from it what the logic of language is. Language disguises thought. So much so, that from the outward form of the clothing it is impossible to infer the form of the thought beneath it, because the outward form of the clothing is not designed to reveal the form of the body, but for entirely different purposes. The tacit conventions on which the understanding of everyday language depends are enormously complicated.

4.011 At first sight a proposition--one set out on the printed page, for example--does not seem to be a picture of the reality with which it is concerned. But neither do written notes seem at first sight to be a picture of a piece of music, nor our phonetic notation (the alphabet) to be a picture of our speech. And yet these sign-languages prove to be pictures, even in the ordinary sense, of what they represent.

4.012 It is obvious that a proposition of the form 'aRb' strikes us as a picture. In this case the sign is obviously a likeness of what is signified.

4.013 And if we penetrate to the essence of this pictorial character, we see that it is *not* impaired by apparent irregularities (such as the use sharp of and flat in musical notation). For even these irregularities depict what they are intended to express; only they do it in a different way.

4.015 The possibility of all imagery, of all our pictorial modes of expression, is contained in the logic of depiction.

4.016 In order to understand the essential nature of a proposition, we should consider hieroglyphic script, which depicts the facts that it describes. And alphabetic script developed out of it without losing what was essential to depiction.

4.021 A proposition is a picture of reality: for if I understand a proposition, I know the situation that it represents. And I understand the proposition without having had its sense explained to me.

4.022 A proposition *shows* its sense. A proposition shows how things stand *if* it is true. And it *says that* they do so stand.

4.023 A proposition must restrict reality to two alternatives: yes or no. In order to do that, it must describe reality completely. A proposition is a description of a state of affairs. Just as a description of an object describes it by giving its external properties, so a proposition describes reality by its internal properties. A proposition constructs a world with the help of a logical scaffolding, so that one can actually see from the proposition how everything stands logically *if* it is true. One can *draw inferences* from a false proposition.

4.024 To understand a proposition means to know what is the case if it is true. (One can understand it, therefore, without knowing whether it is true.) It is understood by anyone who understands its constituents.

4.025 When translating one language into another, we do not proceed by translating each *proposition* of the one into a *proposition* of the other, but merely by translating the constituents of propositions. (And the dictionary translates not only substantives, but also verbs, adjectives, and conjunctions, etc.; and it treats them all in the same way.)

4.026 The meanings of simple signs (words) must be explained to us if we are to understand them. With propositions, however, we make ourselves understood.

4.027 It belongs to the essence of a proposition that it should be able to communicate a *new* sense to us.

4.032 It is only in so far as a proposition is logically articulated that it is a picture of a situation. (Even the proposition, 'Ambulo', is composite: for its stem with a different ending yields a different sense, and so does its ending with a different stem.)

4.061 It must not be overlooked that a proposition has a sense that is independent of the facts: otherwise one can easily suppose that true and false are relations of equal status between signs and what they signify. In that case one could say, for example, that 'p' signified in the true way what '~p' signified in the false way, etc.

4.063 An analogy to illustrate the concept of truth: imagine a black spot on white paper: you can describe the shape of the spot by saying, for each point on the sheet, whether it is black or white. To the fact that a point is black there corresponds a positive fact, and to the fact that a point is white (not black), a negative fact. If I designate a point on the sheet (a truth-value according to Frege), then this corresponds to the supposition that is put forward for judgement, etc. etc. But in order to be able to say that a point is black or white, I must first know when a point is called black, and when white: in order to be able to say,'"p" is true (or false)', I must have determined in what circumstances I call 'p' true, and in so doing I determine the sense of the proposition. Now the point where the simile breaks down is this: we can indicate a point on the paper even if we do not know what black and white are, but if a proposition has no sense, nothing corresponds to it, since it does not designate a thing (a truth-value) which might have properties called 'false' or 'true'. The verb of a proposition is not 'is true' or 'is false', as Frege thought: rather, that which 'is true' must already contain the verb.

4.111 Philosophy is not one of the natural sciences. (The word 'philosophy' must mean something whose place is above or below the natural sciences, not beside them.)

4.113 Philosophy sets limits to the much disputed sphere of natural science.

4.114 It must set limits to what can be thought; and, in doing so, to what cannot be thought. It must set limits to what cannot be thought by working outwards through what can be thought.

4.115 It will signify what cannot be said, by presenting clearly what can be said.

4.116 Everything that can be thought at all can be thought clearly. Everything that can be put into words can be put clearly.

4.123 A property is internal if it is unthinkable that its object should not possess it. (This shade of blue and that one stand, eo ipso, in the internal relation of lighter to darker. It is unthinkable that *these* two objects should not stand in this relation.) (Here the shifting use of the word 'object' corresponds to the shifting use of the words 'property' and 'relation'.)

4.126 We can now talk about formal concepts, in the same sense that we speak of formal properties. (I introduce this expression in order to exhibit the source of the confusion between formal concepts and concepts proper, which pervades the whole of traditional logic.) When something falls under a formal concept as one of its objects, this cannot be expressed by means of a proposition. Instead it is shown in the very sign for this object. (A name shows that it signifies an object, a sign for a number that it signifies a number, etc.) Formal concepts cannot, in fact, be represented by means of a function, as concepts proper can. For their characteristics, formal properties, are not expressed by means of functions. The expression for a formal property is a feature of certain symbols. So the sign for the characteristics of a formal concept is a distinctive feature of all symbols whose meanings fall under the concept. So the expression for a formal concept is a propositional variable in which this distinctive feature alone is constant.

4.128 Logical forms are *without* number. Hence there are no pre-eminent numbers in logic, and hence there is no possibility of philosophical monism or dualism, etc.

4.211 It is a sign of a proposition's being elementary that there can be no elementary proposition contradicting it.

4.241 When I use two signs with one and the same meaning, I express this by putting the sign '=' between them. So 'a = b' means that the sign 'b' can be substituted for the sign 'a'. (If I use an equation to introduce a new sign 'b', laying down that it shall serve as a substitute for a sign a that is already known, then, like Russell, I write the equation--definition--in the form 'a = b Def.' A definition is a rule dealing with signs.)

4.242 Expressions of the form 'a = b' are, therefore, mere representational devices. They state nothing about the meaning of the signs 'a' and 'b'.

4.243 Can we understand two names without knowing 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? Suppose I know the meaning of an English word and of a German word that means the same: then it is impossible for me to be unaware that they do mean the same; I must be capable of translating each into the other. Expressions like 'a = a', and those derived from them, are neither elementary propositions nor is there any other way in which they have sense. (This will become evident later.)

4.411 It immediately strikes one as probable that the introduction of elementary propositions provides the basis for understanding all other kinds of proposition. Indeed the understanding of general propositions *palpably* depends on the understanding of elementary propositions.

4.431 The expression of agreement and disagreement with the truth possibilities of elementary propositions expresses the truth-conditions of a proposition. A proposition is the expression of its truth-conditions. (Thus Frege was quite right to use them as a starting point when he explained the signs of his conceptual notation. But the explanation of the concept of truth that Frege gives is mistaken: if 'the true' and 'the false' were really objects, and were the arguments in ~p etc., then Frege's method of determining the sense of '~p' would leave it absolutely undetermined.)

4.441 It is clear that a complex of the signs 'F' and 'T' has no object (or complex of objects) corresponding to it, just as there is none corresponding to the horizontal and vertical lines or to the brackets.--There are no 'logical objects'. Of course the same applies to all signs that express what the schemata of 'T's' and 'F's' express.

4.442 For example, the following is a propositional sign: [Truth table for implication] (Frege's 'judgement stroke' '|-' is logically quite meaningless: in the works of Frege (and Russell) it simply indicates that these authors hold the propositions marked with this sign to be true. Thus '|-' is no more a component part of a proposition than is, for instance, the proposition's number. It is quite impossible for a proposition to state that it itself is true.) If the order or the truth-possibilities in a scheme is fixed once and for all by a combinatory rule, then the last column by itself will be an expression of the truth-conditions. If we now write this column as a row, the propositional sign will become '(TT-T) (p,q)' or more explicitly '(TTFT) (p,q)' (The number of places in the left-hand pair of brackets is determined by the number of terms in the right-hand pair.)

4.462 Tautologies and contradictions are not pictures of reality. They do not represent any possible situations. For the former admit *all* possible situations, and latter *none*. In a tautology the conditions of agreement with the world--the representational relations--cancel one another, so that it does not stand in any representational relation to reality.

4.463 The truth-conditions of a proposition determine the range that it leaves open to the facts. (A proposition, a picture, or a model is, in the negative sense, like a solid body that restricts the freedom of movement of others, and in the positive sense, like a space bounded by solid substance in which there is room for a body.) A tautology leaves open to reality the whole--the infinite whole--of logical space: a contradiction fills the whole of logical space leaving no point of it for reality. Thus neither of them can determine reality in any way.

4.464 A tautology's truth is certain, a proposition's possible, a contradiction's impossible. (Certain, possible, impossible: here we have the first indication of the scale that we need in the theory of probability.)

4.465 The logical product of a tautology and a proposition says the same thing as the proposition. This product, therefore, is identical with the proposition. For it is impossible to alter what is essential to a symbol without altering its sense.

5.101 The truth-functions of a given number of elementary propositions can always be set out in a schema of the following kind: (TTTT) (p, q) Tautology (If p then p, and if q then q.) (p → p. q → q) (FTTT) (p, q) In words: Not both p and q. (~(p. q)) (TFTT) (p, q) " ": If q then p. (q → p) (TTFT) (p, q) " ": If p then q. (p → q) (TTTF) (p, q) " ": p or q. (p v q) (FFTT) (p, q) " ": Not q. (~q) (FTFT) (p, q) " ": Not p. (~p) (FTTF) (p, q) " ": p or q, but not both. (p. ~q: v : q. ~p) (TFFT) (p, q) " ": If p then q, and if q then p. (p ↔ ­ q) (TFTF) (p, q) " ": p (TTFF) (p, q) " ": q (FFFT) (p, q) " ": Neither p nor q. (~p. ~q or p | q) (FFTF) (p, q) " ": p and not q. (p. ~q) (FTFF) (p, q) " ": q and not p. (q. ~p) (TFFF) (p, q) " ": q and p. (q. p) (FFFF) (p, q) Contradiction (p and not p, and q and not q.) (p. ~p. q. ~q) I will give the name *truth-grounds* of a proposition to those truth-possibilities of its truth-arguments that make it true.

5.121 The truth-grounds of the one are contained in those of the other: p follows from q.

5.122 If p follows from q, the sense of 'p' is contained in the sense of 'q'.

5.123 If a god creates a world in which certain propositions are true, then by that very act he also creates a world in which all the propositions that follow from them come true. And similarly he could not create a world in which the proposition 'p' was true without creating all its objects.

5.132 If p follows from q, I can make an inference from q to p, deduce p from q. The nature of the inference can be gathered only from the two propositions. They themselves are the only possible justification of the inference. 'Laws of inference', which are supposed to justify inferences, as in the works of Frege and Russell, have no sense, and would be superfluous.

5.133 All deductions are made a priori.

5.134 One elementary proposition cannot be deduced form another.

5.135 There is no possible way of making an inference from the existence of one situation to the existence of another, entirely different situation.

5.141 If p follows from q and q from p, then they are one and same proposition.

5.142 A tautology follows from all propositions: it says nothing.

5.143 Contradiction is that common factor of propositions which *no* proposition has in common with another. Tautology is the common factor of all propositions that have nothing in common with one another. Contradiction, one might say, vanishes outside all propositions: tautology vanishes inside them. Contradiction is the outer limit of propositions: tautology is the unsubstantial point at their centre.

5.152 When propositions have no truth-arguments in common with one another, we call them independent of one another. Two elementary propositions give one another the probability ½. If p follows from q, then the proposition 'q' gives to the proposition 'p' the probability 1. The certainty of logical inference is a limiting case of probability. (Application of this to tautology and contradiction.)

5.153 In itself, a proposition is neither probable nor improbable. Either an event occurs or it does not: there is no middle way.

5.154 Suppose that an urn contains black and white balls in equal numbers (and none of any other kind). I draw one ball after another, putting them back into the urn. By this experiment I can establish that the number of black balls drawn and the number of white balls drawn approximate to one another as the draw continues. So *this* is not a mathematical truth. Now, if I say, 'The probability of my drawing a white ball is equal to the probability of my drawing a black one', this means that all the circumstances that I know of (including the laws of nature assumed as hypotheses) give no *more* probability to the occurrence of the one event than to that of the other. That is to say, they give each the probability ½, as can easily be gathered from the above definitions. What I confirm by the experiment is that the occurrence of the two events is independent of the circumstances of which I have no more detailed knowledge.

5.155 The minimal unit for a probability proposition is this: The circumstances--of which I have no further knowledge--give such and such a degree of probability to the occurrence of a particular event.

5.156 It is in this way that probability is a generalization. It involves a general description of a propositional form. We use probability only in default of certainty--if our knowledge of a fact is not indeed complete, but we do know *something* about its form. (A proposition may well be an incomplete picture of a certain situation, but it is always a complete picture of *something*.) A probability proposition is a sort of excerpt from other propositions.

5.231 And that will, of course, depend on their formal properties, on the internal similarity of their forms.

5.232 The internal relation by which a series is ordered is equivalent to the operation that produces one term from another.

5.233 Operations cannot make their appearance before the point at which one proposition is generated out of another in a logically meaningful way; i.e. the point at which the logical construction of propositions begins.

5.241 An operation is not the mark of a form, but only of a difference between forms.

5.242 The operation that produces 'q' from 'p' also produces 'r' from 'q', and so on. There is only one way of expressing this: 'p', 'q', 'r', etc. have to be variables that give expression in a general way to certain formal relations.

5.251 A function cannot be its own argument, whereas an operation can take one of its own results as its base.

5.253 One operation can counteract the effect of another. Operations can cancel one another.

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

5.441 This vanishing of the apparent logical constants also occurs in the case of '~(exists x). ~fx', which says the same as '(x). fx', and in the case of '(exists x). fx. x = a', which says the same as 'fa'.

5.442 If we are given a proposition, then *with it* we are also given the results of all truth-operations that have it as their base.

5.451 If logic has primitive ideas, they must be independent of one another. If a primitive idea has been introduced, it must have been introduced in all the combinations in which it ever occurs. It cannot, therefore, be introduced first for *one* combination and later reintroduced for another. For example, once negation has been introduced, we must understand it both in propositions of the form '~p' and in propositions like '~(p v q)', '(exists x). ~fx', etc. We must not introduce it first for the one class of cases and then for the other, since it would then be left in doubt whether its meaning were the same in both cases, and no reason would have been given for combining the signs in the same way in both cases. (In short, Frege's remarks about introducing signs by means of definitions (in 'The Fundamental Laws of Arithmetic') also apply, mutatis mutandis, to the introduction of primitive signs.)

5.452 The introduction of any new device into the symbolism of logic is necessarily a momentous event. In logic a new device should not be introduced in brackets or in a footnote with what one might call a completely innocent air. (Thus in Russell and Whitehead's 'Principia Mathematica' there occur definitions and primitive propositions expressed in words. Why this sudden appearance of words? It would require a justification, but none is given, or could be given, since the procedure is in fact illicit.) But if the introduction of a new device has proved necessary at a certain point, we must immediately ask ourselves, 'At what points is the employment of this device now *unavoidable*?' and its place in logic must be made clear.

5.453 All numbers in logic stand in need of justification. Or rather, it must become evident that there are no numbers in logic. There are no pre-eminent numbers.

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

5.474 The number of fundamental operations that are necessary depends *solely* on our notation.

5.475 All that is required is that we should construct a system of signs with a particular number of dimensions--with a particular mathematical multiplicity.

5.476 It is clear that this is not a question of a *number of primitive ideas* that have to be signified, but rather of the expression of a rule.

5.501 When a bracketed expression has propositions as its terms--and the order of the terms inside the brackets is indifferent--then I indicate it by a sign of the form '(x*)'. 'x' is a variable whose values are terms of the bracketed expression and the star after the variable indicates that it is the representative of all its values in the brackets. (E.g. if x has the three values P,Q, R, then (x*) = (P, Q, R). ) What the values of the variable are is something that is stipulated. The stipulation is a description of the propositions that have the variable as their representative. How the description of the terms of the bracketed expression is produced is not essential. We *can* distinguish three kinds of description: 1. Direct enumeration, in which case we can simply substitute for the variable the constants that are its values; 2. Giving a function fx whose values for all values of x are the propositions to be described; 3. Giving a formal law that governs the construction of the propositions, in which case the bracketed expression has as its members all the terms of a series of forms.

5.502 So instead of '(-----T)(x,....)', I write 'N(x*)'. N(x*) is the negation of all the values of the propositional variable x.

5.503 It is obvious that we can easily express how propositions may be constructed with this operation, and how they may not be constructed with it; so it must be possible to find an exact expression for this.

5.511 How can logic--all-embracing logic, which mirrors the world--use such peculiar crotchets and contrivances? Only because they are all connected with one another in an infinitely fine network, the great mirror.

5.512 '~p' is true if 'p' is false. Therefore, in the proposition '~p', when it is true, 'p' is a false proposition. How then can the stroke '~' make it agree with reality? But in '~p' it is not '~' that negates, it is rather what is common to all the signs of this notation that negate p. That is to say the common rule that governs the construction of '~p', '~~~p', '~p v ~p', '~p. ~p', etc. etc. (ad inf.). And this common factor mirrors negation.

5.513 We might say that what is common to all symbols that affirm both p and q is the proposition 'p. q'; and that what is common to all symbols that affirm either p or q is the proposition 'p v q'. And similarly we can say that two propositions are opposed to one another if they have nothing in common with one another, and that every proposition has only one negative, since there is only one proposition that lies completely outside it. Thus in Russell's notation too it is manifest that 'q: p v ~p' says the same thing as 'q', that 'p v ~p' says nothing.

5.514 Once a notation has been established, there will be in it a rule governing the construction of all propositions that negate p, a rule governing the construction of all propositions that affirm p, and a rule governing the construction of all propositions that affirm p or q; and so on. These rules are equivalent to the symbols; and in them their sense is mirrored.

5.521 I dissociate the concept *all* from truth-functions. Frege and Russell introduced generality in association with logical product or logical sum. This made it difficult to understand the propositions '(exists x). fx' and '(x) . fx', in which both ideas are embedded.

5.522 What is peculiar to the generality-sign is first, that it indicates a logical prototype, and secondly, that it gives prominence to constants.

5.523 The generality-sign occurs as an argument.

5.524 If objects are given, then at the same time we are given *all* objects. If elementary propositions are given, then at the same time *all* elementary propositions are given.

5.525 It is incorrect to render the proposition '(exists x). fx' in the words, 'fx is *possible*' as Russell does. The certainty, possibility, or impossibility of a situation is not expressed by a proposition, but by an expression's being a tautology, a proposition with a sense, or a contradiction. The precedent to which we are constantly inclined to appeal must reside in the symbol itself.

5.531 Thus I do not write 'f(a, b). a = b', but 'f(a, a)' (or 'f(b, b)); and not 'f(a,b). ~a = b', but 'f(a, b)'.

5.533 The identity-sign, therefore, is not an essential constituent of conceptual notation.

5.534 And now we see that in a correct conceptual notation pseudo-propositions like 'a = a', 'a = b. b = c. → a = c', '(x). x = x', '(exists x). x = a', etc. cannot even be written down.

5.541 At first sight it looks as if it were also possible for one proposition to occur in another in a different way. Particularly with certain forms of proposition in psychology, such as 'A believes that p is the case' and 'A has the thought p', etc. For if these are considered superficially, it looks as if the proposition p stood in some kind of relation to an object A. (And in modern theory of knowledge (Russell, Moore, etc.) these propositions have actually been construed in this way.)

5.551 Our fundamental principle is that whenever a question can be decided by logic at all it must be possible to decide it without more ado. (And if we get into a position where we have to look at the world for an answer to such a problem, that shows that we are on a completely wrong track.)

5.553 Russell said that there were simple relations between different numbers of things (individuals). But between what numbers? And how is this supposed to be decided?--By experience? (There is no pre-eminent number.)

5.555 Clearly we have some concept of elementary propositions quite apart from their particular logical forms. But when there is a system by which we can create symbols, the system is what is important for logic and not the individual symbols. And anyway, is it really possible that in logic I should have to deal with forms that I can invent? What I have to deal with must be that which makes it possible for me to invent them.

5.621 The world and life are one.

5.631 There is no such thing as the subject that thinks or entertains ideas. If I wrote a book called 'The World as l found it,' I should have to include a report on my body, and should have to say which parts were subordinate to my will, and which were not, etc., this being a method of isolating the subject, or rather of showing that in an important sense there is no subject; for it alone could not be mentioned in that book.--

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

5.634 This is connected with the fact that no part of our experience is at the same time a priori. Whatever we see could be other than it is. Whatever we can describe at all could be other than it is. There is no a priori order of things.

5.641 Thus there really is a sense in which philosophy can talk about the self in a non-psychological way. What brings the self into philosophy is the fact that 'the world is my world'. The philosophical self is not the human being, not the human body, or the human soul, with which psychology deals, but rather the metaphysical subject, the limit of the world--not a part of it.

6.001 What this says is just that every proposition is a result of successive applications to elementary propositions of the operation N(x*)

6.002 If we are given the general form according to which propositions are constructed, then with it we are also given the general form according to which one proposition can be generated out of another by means of an operation.

6.021 A number is the exponent of an operation.

6.022 The concept of number is simply what is common to all numbers, the general form of a number. The concept of number is the variable number. And the concept of numerical equality is the general form of all particular cases of numerical equality.

6.031 The theory of classes is completely superfluous in mathematics. This is connected with the fact that the generality required in mathematics is not *accidental* generality.

6.111 All theories that make a proposition of logic appear to have content are false. One might think, for example, that the words 'true' and 'false' signified two properties among other properties, and then it would seem to be a remarkable fact that every proposition possessed one of these properties. On this theory it seems to be anything but obvious, just as, for instance, the proposition, 'All roses are either yellow or red', would not sound obvious even if it were true. Indeed, the logical proposition acquires all the characteristics of a proposition of natural science and this is the sure sign that it has been construed wrongly.

6.112 The correct explanation of the propositions of logic must assign to them a unique status among all propositions.

6.113 It is the peculiar mark of logical propositions that one can recognize that they are true from the symbol alone, and this fact contains in itself the whole philosophy of logic. And so too it is a very important fact that the truth or falsity of non-logical propositions *cannot* be recognized from the propositions alone.

6.121 The propositions of logic demonstrate the logical properties of propositions by combining them so as to form propositions that say nothing. This method could also be called a zero-method. In a logical proposition, propositions are brought into equilibrium with one another, and the state of equilibrium then indicates what the logical constitution of these propositions must be.

6.124 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 propositions sense; and that is their connexion with the world. 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. 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. In logic it is only the latter that express: 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 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.

6.211 Indeed in real life a mathematical proposition is never what we want. Rather, we make use of mathematical propositions *only* in inferences from propositions that do not belong to mathematics to others that likewise do not belong to mathematics. (In philosophy the question, 'What do we actually use this word or this proposition for?' repeatedly leads to valuable insights.)

6.231 It is a property of affirmation that it can be construed as double negation. It is a property of '1 + 1 + 1 + 1' that it can be construed as '(1 + 1) + (1 + 1)'.

6.241 Thus the proof of the proposition 2 X 2 = 4 runs as follows: ((F^k)^l)'x = (F^(k X l))'x Def., (F^(2 X 2))'x = ((F^2)^2)'x = ((F^2)^(1 + 1))'x = (F^2)'(F^2)'x = (F^(1 + 1))'(F^(1 + 1))'x = (F'F)'(F'F)'x =F'F'F'F'x = (F^(1 + 1 + 1 + 1))'x = (F^4)'x.

6.341 Newtonian mechanics, for example, imposes a unified form on the description of the world. Let us imagine a white surface with irregular black spots on it. We then say that whatever kind of picture these make, I can always approximate as closely as I wish to the description of it by covering the surface with a sufficiently fine square mesh, and then saying of every square whether it is black or white. In this way I shall have imposed a unified form on the description of the surface. The form is optional, since I could have achieved the same result by using a net with a triangular or hexagonal mesh. Possibly the use of a triangular mesh would have made the description simpler: that is to say, it might be that we could describe the surface more accurately with a coarse triangular mesh than with a fine square mesh (or conversely), and so on. The different nets correspond to different systems for describing the world. Mechanics determines one form of description of the world by saying that all propositions used in the description of the world must be obtained in a given way from a given set of propositions--the axioms of mechanics. It thus supplies the bricks for building the edifice of science, and it says, 'Any building that you want to erect, whatever it may be, must somehow be constructed with these bricks, and with these alone.' (Just as with the number-system we must be able to write down any number we wish, so with the system of mechanics we must be able to write down any proposition of physics that we wish.)

6.342 And now we can see the relative position of logic and mechanics. (The net might also consist of more than one kind of mesh: e.g. we could use both triangles and hexagons.) The possibility of describing a picture like the one mentioned above with a net of a given form tells us *nothing* about the picture. (For that is true of all such pictures.) But what *does* characterize the picture is that it can be described *completely* by a particular net with a *particular* size of mesh. Similarly the possibility of describing the world by means of Newtonian mechanics tells us nothing about the world: but what does tell us something about it is the precise *way* in which it is possible to describe it by these means. We are also told something about the world by the fact that it can be described more simply with one system of mechanics than with another.

6.362 What can be described can happen too: and what the law of causality is meant to exclude cannot even be described.

6.371 The whole modern conception of the world is founded on the illusion that the so-called laws of nature are the explanations of natural phenomena.

6.372 Thus people today stop at the laws of nature, treating them as something inviolable, just as God and Fate were treated in past ages. And in fact both are right and both wrong: though the view of the ancients is clearer in so far as they have a clear and acknowledged terminus, while the modern system tries to make it look as if *everything* were explained.

6.373 The world is independent of my will.

6.374 Even if all that we wish for were to happen, still this would only be a favour granted by fate, so to speak: for there is no *logical* connexion between the will and the world, which would guarantee it, and the supposed physical connexion itself is surely not something that we could will.

6.421 It is clear that ethics cannot be put into words. Ethics is transcendental. (Ethics and aesthetics are one and the same.)

6.422 When an ethical law of the form, 'Thou shalt...' is laid down, one's first thought is, 'And what if I do, not do it?' It is clear, however, that ethics has nothing to do with punishment and reward in the usual sense of the terms. So our question about the *consequences* of an action must be unimportant.--At least those consequences should not be events. For there must be something right about the question we posed. There must indeed be some kind of ethical reward and ethical punishment, but they must reside in the action itself. (And it is also clear that the reward must be something pleasant and the punishment something unpleasant.)

6.423 It is impossible to speak about the will in so far as it is the subject of ethical attributes. And the will as a phenomenon is of interest only to psychology.

6.521 The solution of the problem of life is seen in the vanishing of the problem. (Is not this the reason why those who have found after a long period of doubt that the sense of life became clear to them have then been unable to say what constituted that sense?)

6.522 There are, indeed, things that cannot be put into words. *They make themselves manifest*. They are what is mystical.