
TRACTATUS LOGICOPHILOSOPHICUS
By Ludwig Wittgenstein
Perhaps this book will be understood only by someone who has himself already had the thoughts that are expressed in it—or at least similar thoughts.—So it is not a textbook.—Its purpose would be achieved if it gave pleasure to one person who read and understood it.
The book deals with the problems of philosophy, and shows, I believe, that the reason why these problems are posed is that the logic of our language is misunderstood. The whole sense of the book might be summed up the following words: what can be said at all can be said clearly, and what we cannot talk about we must pass over in silence.
Thus the aim of the book is to draw a limit to thought, or rather—not to thought, but to the expression of thoughts: for in order to be able to draw a limit to thought, we should have to find both sides of the limit thinkable (i.e. we should have to be able to think what cannot be thought).
It will therefore only be in language that the limit can be drawn, and what lies on the other side of the limit will simply be nonsense.
I do not wish to judge how far my efforts coincide with those of other philosophers. Indeed, what I have written here makes no claim to novelty in detail, and the reason why I give no sources is that it is a matter of indifference to me whether the thoughts that I have had have been anticipated by someone else.
I will only mention that I am indebted to Frege's great works and to the writings of my friend Mr Bertrand Russell for much of the stimulation of my thoughts.
If this work has any value, it consists in two things: the first is that thoughts are expressed in it, and on this score the better the thoughts are expressed—the more the nail has been hit on the head—the greater will be its value.—Here I am conscious of having fallen a long way short of what is possible. Simply because my powers are too slight for the accomplishment of the task.—May others come and do it better.
On the other hand the truth of the thoughts that are here communicated seems to me unassailable and definitive. I therefore believe myself to have found, on all essential points, the final solution of the problems. And if I am not mistaken in this belief, then the second thing in which the of this work consists is that it shows how little is achieved when these problems are solved.
L.W. Vienna, 1918
1. The world is all that is the case.
1.1 The world is the totality of facts, not of things.
1.11 The world is determined by the facts, and by their being all the facts.
1.12 For the totality of facts determines what is the case, and also whatever is not the case.
1.13 The facts in logical space are the world.
1.2 The world divides into facts.
1.21 Each item can be the case or not the case while everything else remains the same.
2. What is the case—a fact—is the existence of states of affairs.
2.01 A state of affairs (a state of things) is a combination of objects (things).
2.011 It is essential to things that they should be possible constituents of states of affairs.
2.012 In logic nothing is accidental: if a thing can occur in a state of affairs, the possibility of the state of affairs must be written into the thing itself.
2.0121 It would seem to be a sort of accident, if it turned out that a situation would fit a thing that could already exist entirely on its own. If things can occur in states of affairs, this possibility must be in them from the beginning. (Nothing in the province of logic can be merely possible. Logic deals with every possibility and all possibilities are its facts.) Just as we are quite unable to imagine spatial objects outside space or temporal objects outside time, so too there is no object that we can imagine excluded from the possibility of combining with others. If I can imagine objects combined in states of affairs, I cannot imagine them excluded from the possibility of such combinations.
2.0122 Things are independent in so far as they can occur in all possible situations, but this form of independence is a form of connexion with states of affairs, a form of dependence. (It is impossible for words to appear in two different roles: by themselves, and in propositions.)
2.0123 If I know an object I also know all its possible occurrences in states of affairs. (Every one of these possibilities must be part of the nature of the object.) A new possibility cannot be discovered later.
2.01231 If I am to know an object, thought I need not know its external properties, I must know all its internal properties.
2.0124 If all objects are given, then at the same time all possible states of affairs are also given.
2.013 Each thing is, as it were, in a space of possible states of affairs. This space I can imagine empty, but I cannot imagine the thing without the space.
2.0131 A spatial object must be situated in infinite space. (A spatial point is an argumentplace.) A speck in the visual field, thought it need not be red, must have some colour: it is, so to speak, surrounded by colourspace. Notes must have some pitch, objects of the sense of touch some degree of hardness, and so on.
2.014 Objects contain the possibility of all situations.
2.0141 The possibility of its occurring in states of affairs is the form of an object.
2.02 Objects are simple.
2.0201 Every statement about complexes can be resolved into a statement about their constituents and into the propositions that describe the complexes completely.
2.021 Objects make up the substance of the world. That is why they cannot be composite.
2.0211 If they world had no substance, then whether a proposition had sense would depend on whether another proposition was true.
2.0212 In that case we could not sketch any picture of the world (true or false).
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.
2.023 Objects are just what constitute this unalterable form.
2.0231 The substance of the world can only determine a form, and not any material properties. For it is only by means of propositions that material properties are represented—only by the configuration of objects that they are produced.
2.0232 In a manner of speaking, objects are colourless.
2.0233 If two objects have the same logical form, the only distinction between them, apart from their external properties, is that they are different.
2.02331 Either a thing has properties that nothing else has, in which case we can immediately use a description to distinguish it from the others and refer to it; or, on the other hand, there are several things that have the whole set of their properties in common, in which case it is quite impossible to indicate one of them. For it there is nothing to distinguish a thing, I cannot distinguish it, since otherwise it would be distinguished after all.
2.024 The substance is what subsists independently of what is the case.
2.025 It is form and content.
2.0251 Space, time, colour (being coloured) are forms of objects.
2.026 There must be objects, if the world is to have unalterable form.
2.027 Objects, the unalterable, and the subsistent are one and the same.
2.0271 Objects are what is unalterable and subsistent; their configuration is what is changing and unstable.
2.0272 The configuration of objects produces states of affairs.
2.03 In a state of affairs objects fit into one another like the links of a chain.
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.
2.04 The totality of existing states of affairs is the world.
2.05 The totality of existing states of affairs also determines which states of affairs do not exist.
2.06 The existence and nonexistence of states of affairs is reality. (We call the existence of states of affairs a positive fact, and their nonexistence a negative fact.)
2.061 States of affairs are independent of one another.
2.062 From the existence or nonexistence of one state of affairs it is impossible to infer the existence or nonexistence of another.
2.063 The sumtotal of reality is the world.
2.1 We picture facts to ourselves.
2.11 A picture presents a situation in logical space, the existence and nonexistence of states of affairs.
2.12 A picture is a model of reality.
2.13 In a picture objects have the elements of the picture corresponding to them.
2.131 In a picture the elements of the picture are the representatives of objects.
2.14 What constitutes a picture is that its elements are related to one another in a determinate way.
2.141 A picture is a fact.
2.15 The fact that the elements of a picture are related to one another in a determinate way represents that things are related to one another in the same way. Let us call this connexion of its elements the structure of the picture, and let us call the possibility of this structure the pictorial form of the picture.
2.151 Pictorial form is the possibility that things are related to one another in the same way as the elements of the picture.
2.1511 That is how a picture is attached to reality; it reaches right out to it.
2.1512 It is laid against reality like a measure.
2.15121 Only the endpoints of the graduating lines actually touch the object that is to be measured.
2.1514 So a picture, conceived in this way, also includes the pictorial relationship, which makes it into a picture.
2.1515 These correlations are, as it were, the feelers of the picture's elements, with which the picture touches reality.
2.16 If a fact is to be a picture, it must have something in common with what it depicts.
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.
2.17 What a picture must have in common with reality, in order to be able to depict it—correctly or incorrectly—in the way that it does, is its pictorial form.
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.
2.18 What any picture, of whatever form, must have in common with reality, in order to be able to depict it—correctly or incorrectly—in any way at all, is logical form, i.e. the form of reality.
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.)
2.19 Logical pictures can depict the world.
2.2 A picture has logicopictorial form in common with what it depicts.
2.201 A picture depicts reality by representing a possibility of existence and nonexistence of states of affairs.
2.202 A picture contains the possibility of the situation that it represents.
2.203 A picture agrees with reality or fails to agree; it is correct or incorrect, true or false.
2.22 What a picture represents it represents independently of its truth or falsity, by means of its pictorial form.
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. A logical picture of facts is a thought.
3.001 'A state of affairs is thinkable': what this means is that we can picture it to ourselves.
3.01 The totality of true thoughts is a picture of the world.
3.02 A thought contains the possibility of the situation of which it is the thought. What is thinkable is possible too.
3.03 Thought can never be of anything illogical, since, if it were, we should have to think illogically.
3.031 It used to be said that God could create anything except what would be contrary to the laws of logic. The truth is that we could not say what an 'illogical' world would look like.
3.032 It is as impossible to represent in language anything that 'contradicts logic' as it is in geometry to represent by its coordinates a figure that contradicts the laws of space, or to give the coordinates of a point that does not exist.
3.0321 Though a state of affairs that would contravene the laws of physics can be represented by us spatially, one that would contravene the laws of geometry cannot.
3.04 It a thought were correct a priori, it would be a thought whose possibility ensured its truth.
3.05 A priori knowledge that a thought was true would be possible only it its truth were recognizable from the thought itself (without anything a to compare it with).
3.1 In a proposition a thought finds an expression that can be perceived by the senses.
3.11 We use the perceptible sign of a proposition (spoken or written, etc.) as a projection of a possible situation. The method of projection is to think of the sense of the proposition.
3.12 I call the sign with which we express a thought a propositional sign. And a proposition is a propositional sign in its projective relation to the world.
3.13 A proposition, therefore, does not actually contain its sense, but does contain the possibility of expressing it. ('The content of a proposition' means the content of a proposition that has sense.) A proposition contains the form, but not the content, of its sense.
3.14 What constitutes a propositional sign is that in its elements (the words) stand in a determinate relation to one another. A propositional sign is a fact.
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 articulate.
3.142 Only facts can express a sense, a set of names cannot.
3.143 Although a propositional sign is a fact, this is obscured by the usual form of expression in writing or print. For in a printed proposition, for example, no essential difference is apparent between a propositional sign and a word. (That is what made it possible for Frege to call a proposition a composite name.)
3.1431 The essence of a propositional sign is very clearly seen if we imagine one composed of spatial objects (such as tables, chairs, and books) instead of written signs.
3.1432 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.'
3.144 Situations can be described but not given names.
3.2 In a proposition a thought can be expressed in such a way that elements of the propositional sign correspond to the objects of the thought.
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'.)
3.21 The configuration of objects in a situation corresponds to the configuration of simple signs in the propositional sign.
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.
3.23 The requirement that simple signs be possible is the requirement that sense be determinate.
3.24 A proposition about a complex stands in an internal relation to a proposition about a constituent of the complex. A complex can be given only by its description, which will be right or wrong. A proposition that mentions a complex will not be nonsensical, if the complex does not exits, but simply false. When a propositional element signifies a complex, this can be seen from an indeterminateness in the propositions in which it occurs. In such cases we know that the proposition leaves something undetermined. (In fact the notation for generality contains a prototype.) The contraction of a symbol for a complex into a simple symbol can be expressed in a definition.
3.25 A proposition cannot be dissected any further by means of a definition: it is a primitive sign.
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.
3.3 Only propositions have sense; only in the nexus of a proposition does a name have meaning.
3.31 I call any part of a proposition that characterizes its sense an expression (or a symbol). (A proposition is itself an expression.) Everything essential to their sense that propositions can have in common with one another is an expression. An expression is the mark of a form and a content.
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.
3.32 A sign is what can be perceived of a symbol.
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 signlanguage 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 signlanguage 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 logicosyntactical 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.)
3.33 In logical syntax the meaning of a sign should never play a role. It must be possible to establish logical syntax without mentioning the meaning of a sign: only the description of expressions may be presupposed.
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(Fu)' we write '(do): 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.
3.34 A proposition possesses essential and accidental features. Accidental features are those that result from the particular way in which the propositional sign is produced. Essential features are those without which the proposition could not express its sense.
3.341 So what is essential in a proposition is what all propositions that can express the same sense have in common. And similarly, in general, what is essential in a symbol is what all symbols that can serve the same purpose have in common.
3.3411 So one could say that the real name of an object was what all symbols that signified it had in common. Thus, one by one, all kinds of composition would prove to be unessential to a name.
3.342 Although there is something arbitrary in our notations, this much is not arbitrary—that when we have determined one thing arbitrarily, something else is necessarily the case. (This derives from the essence of notation.)
3.3421 A particular mode of signifying may be unimportant but it is always important that it is a possible mode of signifying. And that is generally so in philosophy: again and again the individual case turns out to be unimportant, but the possibility of each individual case discloses something about the essence of the world.
3.343 Definitions are rules for translating from one language into another. Any correct signlanguage must be translatable into any other in accordance with such rules: it is this that they all have in common.
3.344 What signifies in a symbol is what is common to all the symbols that the rules of logical syntax allow us to substitute for it.
3.3441 For instance, we can express what is common to all notations for truthfunctions in the following way: they have in common that, for example, the notation that uses 'Pp' ('not p') and 'p C g' ('p or g') can be substituted for any of them. (This serves to characterize the way in which something general can be disclosed by the possibility of a specific notation.)
3.3442 Nor does analysis resolve the sign for a complex in an arbitrary way, so that it would have a different resolution every time that it was incorporated in a different proposition.
3.4 A proposition determines a place in logical space. The existence of this logical place is guaranteed by the mere existence of the constituents—by the existence of the proposition with a sense.
3.41 The propositional sign with logical coordinates—that is the logical place.
3.411 In geometry and logic alike a place is a possibility: something can exist in it.
3.42 A proposition can determine only one place in logical space: nevertheless the whole of logical space must already be given by it. (Otherwise negation, logical sum, logical product, etc., would introduce more and more new elements in coordination.) (The logical scaffolding surrounding a picture determines logical space. The force of a proposition reaches through the whole of logical space.)
3.5 A propositional sign, applied and thought out, is a thought.
4. A thought is a proposition with a sense.
4.001 The totality of propositions is language.
4.022 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.
4.003 Most of the propositions and questions to be found in philosophical works are not false but nonsensical. Consequently we cannot give any answer to questions of this kind, but can only point out that they are nonsensical. Most of the propositions and questions of philosophers arise from our failure to understand the logic of our language. (They belong to the same class as the question whether the good is more or less identical than the beautiful.) And it is not surprising that the deepest problems are in fact not problems at all.
4.0031 All philosophy is a 'critique of language' (though not in Mauthner's sense). It was Russell who performed the service of showing that the apparent logical form of a proposition need not be its real one.
4.01 A proposition is a picture of reality. A proposition is a model of reality as we imagine it.
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 signlanguages 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.
4.014 A gramophone record, the musical idea, the written notes, and the soundwaves, all stand to one another in the same internal relation of depicting that holds between language and the world. They are all constructed according to a common logical pattern. (Like the two youths in the fairytale, their two horses, and their lilies. They are all in a certain sense one.)
4.0141 There is a general rule by means of which the musician can obtain the symphony from the score, and which makes it possible to derive the symphony from the groove on the gramophone record, and, using the first rule, to derive the score again. 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 which projects the symphony into the language of musical notation. It is the rule for translating this language into the language of gramophone records.
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.
4.02 We can see this from the fact that we understand the sense of a propositional sign without its having been explained to us.
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.
4.03 A proposition must use old expressions to communicate a new sense. A proposition communicates a situation to us, and so it must be essentially connected with the situation. And the connexion is precisely that it is its logical picture. A proposition states something only in so far as it is a picture.
4.031 In a proposition a situation is, as it were, constructed by way of experiment. Instead of, 'This proposition has such and such a sense, we can simply say, 'This proposition represents such and such a situation'.
4.0311 One name stands for one thing, another for another thing, and they are combined with one another. In this way the whole group—like a tableau vivant—presents a state of affairs.
4.0312 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.
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.)
4.04 In a proposition there must be exactly as many distinguishable parts as in the situation that it represents. The two must possess the same logical (mathematical) multiplicity. (Compare Hertz's Mechanics on dynamical models.)
4.041 This mathematical multiplicity, of course, cannot itself be the subject of depiction. One cannot get away from it when depicting.
4.0411. If, for example, we wanted to express what we now write as '(x). fx' by putting an affix in front of 'fx'—for instance by writing 'Gen. fx'—it would not be adequate: we should not know what was being generalized. If we wanted to signalize it with an affix 'g'—for instance by writing 'f(xg)'—that would not be adequate either: we should not know the scope of the generalitysign. If we were to try to do it by introducing a mark into the argumentplaces—for instance by writing '(G,G). F(G,G)' —it would not be adequate: we should not be able to establish the identity of the variables. And so on. All these modes of signifying are inadequate because they lack the necessary mathematical multiplicity.
4.0412 For the same reason the idealist's appeal to 'spatial spectacles' is inadequate to explain the seeing of spatial relations, because it cannot explain the multiplicity of these relations.
4.05 Reality is compared with propositions.
4.06 A proposition can be true or false only in virtue of being a picture of reality.
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 'Pp' signified in the false way, etc.
4.062 Can we not make ourselves understood with false propositions just as we have done up till now with true ones?—So long as it is known that they are meant to be false.—No! For a proposition is true if we use it to say that things stand in a certain way, and they do; and if by 'p' we mean Pp and things stand as we mean that they do, then, construed in the new way, 'p' is true and not false.
4.0621 But it is important that the signs 'p' and 'Pp' can say the same thing. For it shows that nothing in reality corresponds to the sign 'P'. The occurrence of negation in a proposition is not enough to characterize its sense (PPp = p). The propositions 'p' and 'Pp' have opposite sense, but there corresponds to them one and the same reality.
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 truthvalue 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 truthvalue) 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.
4.064 Every proposition must already have a sense: it cannot be given a sense by affirmation. Indeed its sense is just what is affirmed. And the same applies to negation, etc.
4.0641 One could say that negation must be related to the logical place determined by the negated proposition. The negating proposition determines a logical place different from that of the negated proposition. The negating proposition determines a logical place with the help of the logical place of the negated proposition. For it describes it as lying outside the latter's logical place. The negated proposition can be negated again, and this in itself shows that what is negated is already a proposition, and not merely something that is preliminary to a proposition.
4.1 Propositions represent the existence and nonexistence of states of affairs.
4.11 The totality of true propositions is the whole of natural science (or the whole corpus of the natural sciences).
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.)
4.112 Philosophy aims at the logical clarification of thoughts. Philosophy is not a body of doctrine but an activity. A philosophical work consists essentially of elucidations. Philosophy does not result in 'philosophical propositions', but rather in the clarification of propositions. Without philosophy thoughts are, as it were, cloudy and indistinct: its task is to make them clear and to give them sharp boundaries.
4.1121 Psychology is no more closely related to philosophy than any other natural science. Theory of knowledge is the philosophy of psychology. Does not my study of signlanguage correspond to the study of thoughtprocesses, which philosophers used to consider so essential to the philosophy of logic? Only in most cases they got entangled in unessential psychological investigations, and with my method too there is an analogous risk.
4.1122 Darwin's theory has no more to do with philosophy than any other hypothesis in natural science.
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. 4.12 Propositions can represent the whole of reality, but they cannot represent what they must have in common with reality in order to be able to represent it—logical form. In order to be able to represent logical form, we should have to be able to station ourselves with propositions somewhere outside logic, that is to say outside the world.
4.121 Propositions cannot represent logical form: it is mirrored in them. What finds its reflection in language, language cannot represent. What expresses itself in language, we cannot express by means of language. Propositions show the logical form of reality. They display it.
4.1211 Thus one proposition 'fa' shows that the object a occurs in its sense, two propositions 'fa' and 'ga' show that the same object is mentioned in both of them. If two propositions contradict one another, then their structure shows it; the same is true if one of them follows from the other. And so on.
4.1212 What can be shown, cannot be said.
4.1213 Now, too, we understand our feeling that once we have a signlanguage in which everything is all right, we already have a correct logical point of view.
4.122 In a certain sense we can talk about formal properties of objects and states of affairs, or, in the case of facts, about structural properties: and in the same sense about formal relations and structural relations. (Instead of 'structural property' I also say 'internal property'; instead of 'structural relation', 'internal relation'. I introduce these expressions in order to indicate the source of the confusion between internal relations and relations proper (external relations), which is very widespread among philosophers.) It is impossible, however, to assert by means of propositions that such internal properties and relations obtain: rather, this makes itself manifest in the propositions that represent the relevant states of affairs and are concerned with the relevant objects.
4.1221 An internal property of a fact can also be bed a feature of that fact (in the sense in which we speak of facial features, for example).
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'.)
4.124 The existence of an internal property of a possible situation is not expressed by means of a proposition: rather, it expresses itself in the proposition representing the situation, by means of an internal property of that proposition. It would be just as nonsensical to assert that a proposition had a formal property as to deny it.
4.1241 It is impossible to distinguish forms from one another by saying that one has this property and another that property: for this presupposes that it makes sense to ascribe either property to either form.
4.125 The existence of an internal relation between possible situations expresses itself in language by means of an internal relation between the propositions representing them.
4.1251 Here we have the answer to the vexed question 'whether all relations are internal or external'.
4.1252 I call a series that is ordered by an internal relation a series of forms. The order of the numberseries is not governed by an external relation but by an internal relation. The same is true of the series of propositions 'aRb', '(d: c): aRx. xRb', '(d x,y): aRx. xRy. yRb', and so forth. (If b stands in one of these relations to a, I call b a successor of a.)
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.
4.127 The propositional variable signifies the formal concept, and its values signify the objects that fall under the concept.
4.1271 Every variable is the sign for a formal concept. For every variable represents a constant form that all its values possess, and this can be regarded as a formal property of those values.
4.1272 Thus the variable name 'x' is the proper sign for the pseudoconcept object. Wherever the word 'object' ('thing', etc.) is correctly used, it is expressed in conceptual notation by a variable name. For example, in the proposition, 'There are 2 objects which.. .', it is expressed by ' (dx,y)... '. Wherever it is used in a different way, that is as a proper conceptword, nonsensical pseudopropositions are the result. So one cannot say, for example, 'There are objects', as one might say, 'There are books'. And it is just as impossible to say, 'There are 100 objects', or, 'There are!0 objects'. And it is nonsensical to speak of the total number of objects. The same applies to the words 'complex', 'fact', 'function', 'number', etc. They all signify formal concepts, and are represented in conceptual notation by variables, not by functions or classes (as Frege and Russell believed). '1 is a number', 'There is only one zero', and all similar expressions are nonsensical. (It is just as nonsensical to say, 'There is only one 1', as it would be to say, '2 + 2 at 3 o'clock equals 4'.)
4.12721 A formal concept is given immediately any object falling under it is given. It is not possible, therefore, to introduce as primitive ideas objects belonging to a formal concept and the formal concept itself. So it is impossible, for example, to introduce as primitive ideas both the concept of a function and specific functions, as Russell does; or the concept of a number and particular numbers.
4.1273 If we want to express in conceptual notation the general proposition, 'b is a successor of a', then we require an expression for the general term of the series of forms 'aRb', '(d: c): aRx. xRb', '(d x,y) : aRx. xRy. yRb',..., In order to express the general term of a series of forms, we must use a variable, because the concept 'term of that series of forms' is a formal concept. (This is what Frege and Russell overlooked: consequently the way in which they want to express general propositions like the one above is incorrect; it contains a vicious circle.) We can determine the general term of a series of forms by giving its first term and the general form of the operation that produces the next term out of the proposition that precedes it.
4.1274 To ask whether a formal concept exists is nonsensical. For no proposition can be the answer to such a question. (So, for example, the question, 'Are there unanalysable subjectpredicate propositions?' cannot be asked.)
4.128 Logical forms are without number. Hence there are no preeminent numbers in logic, and hence there is no possibility of philosophical monism or dualism, etc.
4.2 The sense of a proposition is its agreement and disagreement with possibilities of existence and nonexistence of states of affairs. 4.21 The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs.
4.211 It is a sign of a proposition's being elementary that there can be no elementary proposition contradicting it.
4.22 An elementary proposition consists of names. It is a nexus, a concatenation, of names.
4.221 It is obvious that the analysis of propositions must bring us to elementary propositions which consist of names in immediate combination. This raises the question how such combination into propositions comes about.
4.2211 Even if the world is infinitely complex, so that every fact consists of infinitely many states of affairs and every state of affairs is composed of infinitely many objects, there would still have to be objects and states of affairs.
4.23 It is only in the nexus of an elementary proposition that a name occurs in a proposition.
4.24 Names are the simple symbols: I indicate them by single letters ('x', 'y', 'z'). I write elementary propositions as functions of names, so that they have the form 'fx', 'O (x,y)', etc. Or I indicate them by the letters 'p', 'q', 'r'.
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.)
4.25 If an elementary proposition is true, the state of affairs exists: if an elementary proposition is false, the state of affairs does not exist.
4.26 If all true elementary propositions are given, the result is a complete description of the world. The world is completely described by giving all elementary propositions, and adding which of them are true and which false. For n states of affairs, there are possibilities of existence and nonexistence. Of these states of affairs any combination can exist and the remainder not exist.
4.28 There correspond to these combinations the same number of possibilities of truth—and falsity—for n elementary propositions.
4.3 Truthpossibilities of elementary propositions mean Possibilities of existence and nonexistence of states of affairs.
4.31 We can represent truthpossibilities by schemata of the following kind ('T' means 'true', 'F' means 'false'; the rows of 'T's' and 'F's' under the row of elementary propositions symbolize their truthpossibilities in a way that can easily be understood):
4.4 A proposition is an expression of agreement and disagreement with truthpossibilities of elementary propositions.
4.41 Truthpossibilities of elementary propositions are the conditions of the truth and falsity of propositions.
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.
4.42 For n elementary propositions there are ways in which a proposition can agree and disagree with their truth possibilities.
4.43 We can express agreement with truthpossibilities by correlating the mark 'T' (true) with them in the schema. The absence of this mark means disagreement.
4.431 The expression of agreement and disagreement with the truth possibilities of elementary propositions expresses the truthconditions of a proposition. A proposition is the expression of its truthconditions. (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 Pp etc., then Frege's method of determining the sense of 'Pp' would leave it absolutely undetermined.)
4.44 The sign that results from correlating the mark 'I' with truthpossibilities is a propositional sign.
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: (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 truthpossibilities 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 truthconditions. If we now write this column as a row, the propositional sign will become '(TTT) (p,q)' or more explicitly '(TTFT) (p,q)' (The number of places in the lefthand pair of brackets is determined by the number of terms in the righthand pair.)
4.45 For n elementary propositions there are Ln possible groups of truthconditions. The groups of truthconditions that are obtainable from the truthpossibilities of a given number of elementary propositions can be arranged in a series.
4.46 Among the possible groups of truthconditions there are two extreme cases. In one of these cases the proposition is true for all the truthpossibilities of the elementary propositions. We say that the truthconditions are tautological. In the second case the proposition is false for all the truthpossibilities: the truthconditions are contradictory. In the first case we call the proposition a tautology; in the second, a contradiction.
4.461 Propositions show what they say; tautologies and contradictions show that they say nothing. A tautology has no truthconditions, since it is unconditionally true: and a contradiction is true on no condition. Tautologies and contradictions lack sense. (Like a point from which two arrows go out in opposite directions to one another.) (For example, I know nothing about the weather when I know that it is either raining or not raining.)
4.46211 Tautologies and contradictions are not, however, nonsensical. They are part of the symbolism, much as '0' is part of the symbolism of arithmetic.
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 truthconditions 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.
4.466 What corresponds to a determinate logical combination of signs is a determinate logical combination of their meanings. It is only to the uncombined signs that absolutely any combination corresponds. In other words, propositions that are true for every situation cannot be combinations of signs at all, since, if they were, only determinate combinations of objects could correspond to them. (And what is not a logical combination has no combination of objects corresponding to it.) Tautology and contradiction are the limiting cases—indeed the disintegration—of the combination of signs.
4.4661 Admittedly the signs are still combined with one another even in tautologies and contradictions—i.e. they stand in certain relations to one another: but these relations have no meaning, they are not essential to the symbol.
4.5 It now seems possible to give the most general propositional form: that is, to give a description of the propositions of any signlanguage whatsoever in such a way that every possible sense can be expressed by a symbol satisfying the description, and every symbol satisfying the description can express a sense, provided that the meanings of the names are suitably chosen. It is clear that only what is essential to the most general propositional form may be included in its description—for otherwise it would not be the most general form. The existence of a general propositional form is proved by the fact that there cannot be a proposition whose form could not have been foreseen (i.e. constructed). The general form of a proposition is: This is how things stand.
4.51 Suppose that I am given all elementary propositions: then I can simply ask what propositions I can construct out of them. And there I have all propositions, and that fixes their limits.
4.52 Propositions comprise all that follows from the totality of all elementary propositions (and, of course, from its being the totality of them all ). (Thus, in a certain sense, it could be said that all propositions were generalizations of elementary propositions.)
4.53 The general propositional form is a variable.
5. A proposition is a truthfunction of elementary propositions.
(An elementary proposition is a truthfunction of itself.)
5.01 Elementary propositions are the trutharguments of propositions.
5.02 The arguments of functions are readily confused with the affixes of names. For both arguments and affixes enable me to recognize the meaning of the signs containing them. For example, when Russell writes '+c', the 'c' is an affix which indicates that the sign as a whole is the additionsign for cardinal numbers. But the use of this sign is the result of arbitrary convention and it would be quite possible to choose a simple sign instead of '+c'; in 'Pp' however, 'p' is not an affix but an argument: the sense of 'Pp' cannot be understood unless the sense of 'p' has been understood already. (In the name Julius Caesar 'Julius' is an affix. An affix is always part of a description of the object to whose name we attach it: e.g. the Caesar of the Julian gens.) If I am not mistaken, Frege's theory about the meaning of propositions and functions is based on the confusion between an argument and an affix. Frege regarded the propositions of logic as names, and their arguments as the affixes of those names.
5.1 Truthfunctions can be arranged in series. That is the foundation of the theory of probability.
5.101 The truthfunctions 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 z p. q z q) (FTTT) (p, q) In words: Not both p and q. (P(p. q)) (TFTT) (p, q) ": If q then p. (q z p) (TTFT) (p, q) ": If p then q. (p z q) (TTTF) (p, q) ": p or q. (p C q) (FFTT) (p, q) ": Not g. (Pq) (FTFT) (p, q) ": Not p. (Pp) (FTTF) (p, q) " : p or q, but not both. (p. Pq: C: q. Pp) (TFFT) (p, q) ": If p then p, and if q then p. (p + q) (TFTF) (p, q) ": p (TTFF) (p, q) ": q (FFFT) (p, q) ": Neither p nor q. (Pp. Pq or p q) (FFTF) (p, q) ": p and not q. (p. Pq) (FTFF) (p, q) ": q and not p. (q. Pp) (TFFF) (p,q) ": q and p. (q. p) (FFFF) (p, q) Contradiction (p and not p, and q and not q.) (p. Pp. q. Pq) I will give the name truthgrounds of a proposition to those truthpossibilities of its trutharguments that make it true.
5.11 If all the truthgrounds that are common to a number of propositions are at the same time truthgrounds of a certain proposition, then we say that the truth of that proposition follows from the truth of the others.
5.12 In particular, the truth of a proposition 'p' follows from the truth of another proposition 'q' is all the truthgrounds of the latter are truthgrounds of the former.
5.121 The truthgrounds 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.
5.124 A proposition affirms every proposition that follows from it.
5.1241 'p. q' is one of the propositions that affirm 'p' and at the same time one of the propositions that affirm 'q'. Two propositions are opposed to one another if there is no proposition with a sense, that affirms them both. Every proposition that contradicts another negate it.
5.13 When the truth of one proposition follows from the truth of others, we can see this from the structure of the proposition.
5.131 If the truth of one proposition follows from the truth of others, this finds expression in relations in which the forms of the propositions stand to one another: nor is it necessary for us to set up these relations between them, by combining them with one another in a single proposition; on the contrary, the relations are internal, and their existence is an immediate result of the existence of the propositions.
5.1311 When we infer q from p C q and Pp, the relation between the propositional forms of 'p C q' and 'Pp' is masked, in this case, by our mode of signifying. But if instead of 'p C q' we write, for example, 'p q. . p q', and instead of 'Pp', 'p p' (p q = neither p nor q), then the inner connexion becomes obvious. (The possibility of inference from (x). fx to fa shows that the symbol (x). fx itself has generality in it.)
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 form the existence of one situation to the existence of another, entirely different situation.
5.136 There is no causal nexus to justify such an inference.
5.1361 We cannot infer the events of the future from those of the present. Belief in the causal nexus is superstition.
5.1362 The freedom of the will consists in the impossibility of knowing actions that still lie in the future. We could know them only if causality were an inner necessity like that of logical inference.—The connexion between knowledge and what is known is that of logical necessity. ('A knows that p is the case', has no sense if p is a tautology.)
5.1363 If the truth of a proposition does not follow from the fact that it is selfevident to us, then its selfevidence in no way justifies our belief in its truth.
5.14 If one proposition follows from another, then the latter says more than the former, and the former less than the latter.
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.
5.15 If Tr is the number of the truthgrounds of a proposition 'r', and if Trs is the number of the truthgrounds of a proposition 's' that are at the same time truthgrounds of 'r', then we call the ratio Trs: Tr the degree of probability that the proposition 'r' gives to the proposition 's'. 5.151 In a schema like the one above in
5.101, let Tr be the number of 'T's' in the proposition r, and let Trs, be the number of 'T's' in the proposition s that stand in columns in which the proposition r has 'T's'. Then the proposition r gives to the proposition s the probability Trs: Tr.
5.1511 There is no special object peculiar to probability propositions.
5.152 When propositions have no trutharguments in common with one another, we call them independent of one another. Two elementary propositions give one another the probability 1/2. 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 1/2 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.
5.2 The structures of propositions stand in internal relations to one another.
5.21 In order to give prominence to these internal relations we can adopt the following mode of expression: we can represent a proposition as the result of an operation that produces it out of other propositions (which are the bases of the operation).
5.22 An operation is the expression of a relation between the structures of its result and of its bases.
5.23 The operation is what has to be done to the one proposition in order to make the other out of it.
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.
5.234 Truthfunctions of elementary propositions are results of operations with elementary propositions as bases. (These operations I call truthoperations.)
5.2341 The sense of a truthfunction of p is a function of the sense of p. Negation, logical addition, logical multiplication, etc. etc. are operations. (Negation reverses the sense of a proposition.)
5.24 An operation manifests itself in a variable; it shows how we can get from one form of proposition to another. It gives expression to the difference between the forms. (And what the bases of an operation and its result have in common is just the bases themselves.)
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.
5.25 The occurrence of an operation does not characterize the sense of a proposition. Indeed, no statement is made by an operation, but only by its result, and this depends on the bases of the operation. (Operations and functions must not be confused with each other.)
5.251 A function cannot be its own argument, whereas an operation can take one of its own results as its base.
5.252 It is only in this way that the step from one term of a series of forms to another is possible (from one type to another in the hierarchies of Russell and Whitehead). (Russell and Whitehead did not admit the possibility of such steps, but repeatedly availed themselves of it.)
5.2521 If an operation is applied repeatedly to its own results, I speak of successive applications of it. ('O'O'O'a' is the result of three successive applications of the operation 'O'E' to 'a'.) In a similar sense I speak of successive applications of more than one operation to a number of propositions.
5.2522 Accordingly I use the sign '[a, x, O'x]' for the general term of the series of forms a, O'a, O'O'a,.... This bracketed expression is a variable: the first term of the bracketed expression is the beginning of the series of forms, the second is the form of a term x arbitrarily selected from the series, and the third is the form of the term that immediately follows x in the series.
5.2523 The concept of successive applications of an operation is equivalent to the concept 'and so on'.
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 'PPp': PPp = p).
5.3 All propositions are results of truthoperations on elementary propositions. A truthoperation is the way in which a truthfunction is produced out of elementary propositions. It is of the essence of truthoperations that, just as elementary propositions yield a truthfunction of themselves, so too in the same way truthfunctions yield a further truthfunction. When a truthoperation is applied to truthfunctions of elementary propositions, it always generates another truthfunction of elementary propositions, another proposition. When a truthoperation is applied to the results of truthoperations on elementary propositions, there is always a single operation on elementary propositions that has the same result. Every proposition is the result of truthoperations on elementary propositions.
5.31 The schemata in 4.31 have a meaning even when 'p', 'q', 'r', etc. are not elementary propositions. And it is easy to see that the propositional sign in 4.442 expresses a single truthfunction of elementary propositions even when 'p' and 'q' are truthfunctions of elementary propositions.
5.32 All truthfunctions are results of successive applications to elementary propositions of a finite number of truthoperations.
5.4 At this point it becomes manifest that there are no 'logical objects' or 'logical constants' (in Frege's and Russell's sense).
5.41 The reason is that the results of truthoperations on truthfunctions are always identical whenever they are one and the same truthfunction of elementary propositions.
5.42 It is selfevident that C, z, etc. are not relations in the sense in which right and left etc. are relations. The interdefinability of Frege's and Russell's 'primitive signs' of logic is enough to show that they are not primitive signs, still less signs for relations. And it is obvious that the 'z' defined by means of 'P' and 'C' is identical with the one that figures with 'P' in the definition of 'C'; and that the second 'C' is identical with the first one; and so on.
5.43 Even at first sight it seems scarcely credible that there should follow from one fact p infinitely many others, namely PPp, PPPPp, etc. And it is no less remarkable that the infinite number of propositions of logic (mathematics) follow from half a dozen 'primitive propositions'. But in fact all the propositions of logic say the same thing, to wit nothing.
5.44 Truthfunctions are not material functions. For example, an affirmation can be produced by double negation: in such a case does it follow that in some sense negation is contained in affirmation? Does 'PPp' negate Pp, or does it affirm p—or both? The proposition 'PPp' is not about negation, as if negation were an object: on the other hand, the possibility of negation is already written into affirmation. And if there were an object called 'P', it would follow that 'PPp' said something different from what 'p' said, just because the one proposition would then be about P and the other would not.
5.441 This vanishing of the apparent logical constants also occurs in the case of 'P(dx). Pfx', which says the same as '(x). fx', and in the case of '(dx). 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 truthoperations that have it as their base.
5.45 If there are primitive logical signs, then any logic that fails to show clearly how they are placed relatively to one another and to justify their existence will be incorrect. The construction of logic out of its primitive signs must be made clear.
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 'Pp' and in propositions like 'P(p C q)', '(dx). Pfx', 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 preeminent numbers.
5.454 In logic there is no coordinate status, and there can be no classification. In logic there can be no distinction between the general and the specific.
5.4541 The solutions of the problems of logic must be simple, since they set the standard of simplicity. Men have always had a presentiment that there must be a realm in which the answers to questions are symmetrically combined—a priori—to form a selfcontained system. A realm subject to the law: Simplex sigillum veri.
5.46 If we introduced logical signs properly, then we should also have introduced at the same time the sense of all combinations of them; i.e. not only 'p C q' but 'P(p C q)' as well, etc. etc. We should also have introduced at the same time the effect of all possible combinations of brackets. And thus it would have been made clear that the real general primitive signs are not 'p C q', '(dx). fx', etc. but the most general form of their combinations.
5.461 Though it seems unimportant, it is in fact significant that the pseudorelations of logic, such as C and z, need brackets—unlike real relations. Indeed, the use of brackets with these apparently primitive signs is itself an indication that they are not primitive signs. And surely no one is going to believe brackets have an independent meaning. 5.4611 Signs for logical operations are punctuationmarks.
5.47 It is clear that whatever we can say in advance about the form of all propositions, we must be able to say all at once. An elementary proposition really contains all logical operations in itself. For 'fa' says the same thing as '(dx). fx. x = a' Wherever there is compositeness, argument and function are present, and where these are present, we already have all the logical constants. One could say that the sole logical constant was what all propositions, by their very nature, had in common with one another. But that is the general propositional form.
5.471 The general propositional form is the essence of a proposition.
5.4711 To give the essence of a proposition means to give the essence of all description, and thus the essence of the world.
5.472 The description of the most general propositional form is the description of the one and only general primitive sign in logic.
5.473 Logic must look after itself. If a sign is possible, then it is also capable of signifying. Whatever is possible in logic is also permitted. (The reason why 'Socrates is identical' means nothing is that there is no property called 'identical'. The proposition is nonsensical because we have failed to make an arbitrary determination, and not because the symbol, in itself, would be illegitimate.) In a certain sense, we cannot make mistakes in logic.
5.4731 Selfevidence, which Russell talked about so much, can become dispensable in logic, only because language itself prevents every logical mistake.—What makes logic a priori is the impossibility of illogical thought.
5.4732 We cannot give a sign the wrong sense.
5,47321 Occam's maxim is, of course, not an arbitrary rule, nor one that is justified by its success in practice: its point is that unnecessary units in a signlanguage mean nothing. Signs that serve one purpose are logically equivalent, and signs that serve none are logically meaningless.
5.4733 Frege says that any legitimately constructed proposition must have a sense. And I say that any possible proposition is legitimately constructed, and, if it has no sense, that can only be because we have failed to give a meaning to some of its constituents. (Even if we think that we have done so.) Thus the reason why 'Socrates is identical' says nothing is that we have not given any adjectival meaning to the word 'identical'. For when it appears as a sign for identity, it symbolizes in an entirely different way—the signifying relation is a different one—therefore the symbols also are entirely different in the two cases: the two symbols have only the sign in common, and that is an accident.
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.
5.5 Every truthfunction is a result of successive applications to elementary propositions of the operation '(——T)(E,....)'. This operation negates all the propositions in the righthand pair of brackets, and I call it the negation of those propositions.
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 '(E)'. '(E)' is a variable whose values are terms of the bracketed expression and the bar over the variable indicates that it is the representative of all its values in the brackets. (E.g. if E has the three values P,Q, R, then (E) = (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)(E,....)', I write 'N(E)'. N(E) is the negation of all the values of the propositional variable E.
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.
5.51 If E has only one value, then N(E) = Pp (not p); if it has two values, then N(E) = Pp. Pq. (neither p nor g).
5.511 How can logic—allembracing 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 'Pp' is true if 'p' is false. Therefore, in the proposition 'Pp', when it is true, 'p' is a false proposition. How then can the stroke 'P' make it agree with reality? But in 'Pp' it is not 'P' 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 'Pp', 'PPPp', 'Pp C Pp', 'Pp. Pp', 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 C 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 C Pp' says the same thing as 'q', that 'p C Pq' 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.
5.515 It must be manifest in our symbols that it can only be propositions that are combined with one another by 'C', '.', etc. And this is indeed the case, since the symbol in 'p' and 'q' itself presupposes 'C', 'P', etc. If the sign 'p' in 'p C q' does not stand for a complex sign, then it cannot have sense by itself: but in that case the signs 'p C p', 'p. p', etc., which have the same sense as p, must also lack sense. But if 'p C p' has no sense, then 'p C q' cannot have a sense either.
5.5151 Must the sign of a negative proposition be constructed with that of the positive proposition? Why should it not be possible to express a negative proposition by means of a negative fact? (E.g. suppose that "a' does not stand in a certain relation to 'b'; then this might be used to say that aRb was not the case.) But really even in this case the negative proposition is constructed by an indirect use of the positive. The positive proposition necessarily presupposes the existence of the negative proposition and vice versa.
5.52 If E has as its values all the values of a function fx for all values of x, then N(E) = P(dx). fx.
5.521 I dissociate the concept all from truthfunctions. Frege and Russell introduced generality in association with logical productor logical sum. This made it difficult to understand the propositions '(dx). fx' and '(x) . fx', in which both ideas are embedded.
5.522 What is peculiar to the generalitysign is first, that it indicates a logical prototype, and secondly, that it gives prominence to constants.
5.523 The generalitysign 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 '(dx). 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. 
