TRACTATUS LOGICO-PHILOSOPHICUS

By Ludwig Wittgenstein

Perhaps this book will be understood only by someone who has himselfalready had the thoughts that are expressed in it--or at least similarthoughts. --So it is not a textbook. --Its purpose would be achieved if itgave 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 ourlanguage is misunderstood. The whole sense of the book might be summedup the following words: what can be said at all can be said clearly, andwhat 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--notto thought, but to the expression of thoughts: for in order to be ableto draw a limit to thought, we should have to find both sides of thelimit thinkable (i. E. We should have to be able to think what cannot bethought). 

It will therefore only be in language that the limit can be drawn, andwhat 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 otherphilosophers. Indeed, what I have written here makes no claim to noveltyin detail, and the reason why I give no sources is that it is a matterof indifference to me whether the thoughts that I have had have beenanticipated by someone else. 

I will only mention that I am indebted to Frege's great works and of thewritings of my friend Mr Bertrand Russell for much of the stimulation ofmy thoughts. 

If this work has any value, it consists in two things: the first is thatthoughts are expressed in it, and on this score the better the thoughtsare expressed--the more the nail has been hit on the head--the greaterwill be its value. --Here I am conscious of having fallen a long wayshort of what is possible. Simply because my powers are too slight forthe accomplishment of the task. --May others come and do it better. 

On the other hand the truth of the thoughts that are here communicatedseems to me unassailable and definitive. I therefore believe myself tohave 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 whichthe of this work consists is that it shows how little is achieved whenthese 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 thefacts. 

1. 12 For the totality of facts determines what is the case, and alsowhatever 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 elseremains 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 possibleconstituents of states of affairs. 

2. 012 In logic nothing is accidental: if a thing can occur in a stateof affairs, the possibility of the state of affairs must be written intothe thing itself. 

2. 0121 It would seem to be a sort of accident, if it turned out thata situation would fit a thing that could already exist entirely on itsown. If things can occur in states of affairs, this possibility mustbe in them from the beginning. (Nothing in the province of logic canbe merely possible. Logic deals with every possibility and allpossibilities are its facts. ) Just as we are quite unable to imaginespatial objects outside space or temporal objects outside time, so toothere is no object that we can imagine excluded from the possibility ofcombining with others. If I can imagine objects combined in states ofaffairs, I cannot imagine them excluded from the possibility of suchcombinations. 

2. 0122 Things are independent in so far as they can occur in allpossible situations, but this form of independence is a form ofconnexion with states of affairs, a form of dependence. (It isimpossible 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 instates of affairs. (Every one of these possibilities must be part of thenature 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 externalproperties, I must know all its internal properties. 

2. 0124 If all objects are given, then at the same time all possiblestates of affairs are also given. 

2. 013 Each thing is, as it were, in a space of possible states ofaffairs. This space I can imagine empty, but I cannot imagine the thingwithout the space. 

2. 0131 A spatial object must be situated in infinite space. (A spatialpoint is an argument-place. ) A speck in the visual field, thought itneed not be red, must have some colour: it is, so to speak, surroundedby colour-space. Notes must have some pitch, objects of the sense oftouch 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 formof an object. 

2. 02 Objects are simple. 

2. 0201 Every statement about complexes can be resolved into a statementabout their constituents and into the propositions that describe thecomplexes completely. 

2. 021 Objects make up the substance of the world. That is why theycannot be composite. 

2. 0211 If they world had no substance, then whether a proposition hadsense would depend on whether another proposition was true. 

2. 0212 In that case we could not sketch any picture of the world (trueor false). 

2. 022 It is obvious that an imagined world, however difference it may befrom 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 notany material properties. For it is only by means of propositions thatmaterial properties are represented--only by the configuration ofobjects 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 distinctionbetween them, apart from their external properties, is that they aredifferent. 

2. 02331 Either a thing has properties that nothing else has, in whichcase we can immediately use a description to distinguish it from theothers and refer to it; or, on the other hand, there are several thingsthat have the whole set of their properties in common, in which case itis quite impossible to indicate one of them. For it there is nothing todistinguish a thing, I cannot distinguish it, since otherwise it wouldbe 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; theirconfiguration 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 linksof a chain. 

2. 031 In a state of affairs objects stand in a determinate relation toone another. 

2. 032 The determinate way in which objects are connected in a state ofaffairs 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 ofaffairs. 

2. 04 The totality of existing states of affairs is the world. 

2. 05 The totality of existing states of affairs also determines whichstates of affairs do not exist. 

2. 06 The existence and non-existence of states of affairs is reality. (We call the existence of states of affairs a positive fact, and theirnon-existence a negative fact. )

2. 061 States of affairs are independent of one another. 

2. 062 From the existence or non-existence of one state of affairs it isimpossible to infer the existence or non-existence of another. 

2. 063 The sum-total of reality is the world. 

2. 1 We picture facts to ourselves. 

2. 11 A picture presents a situation in logical space, the existence andnon-existence 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 correspondingto them. 

2. 131 In a picture the elements of the picture are the representativesof objects. 

2. 14 What constitutes a picture is that its elements are related to oneanother 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 anotherin a determinate way represents that things are related to one anotherin the same way. Let us call this connexion of its elements thestructure of the picture, and let us call the possibility of thisstructure the pictorial form of the picture. 

2. 151 Pictorial form is the possibility that things are related to oneanother in the same way as the elements of the picture. 

2. 1511 That is how a picture is attached to reality; it reaches rightout to it. 

2. 1512 It is laid against reality like a measure. 

2. 15121 Only the end-points of the graduating lines actually touch theobject that is to be measured. 

2. 1514 So a picture, conceived in this way, also includes the pictorialrelationship, which makes it into a picture. 

2. 1515 These correlations are, as it were, the feelers of the picture'selements, with which the picture touches reality. 

2. 16 If a fact is to be a picture, it must have something in common withwhat it depicts. 

2. 161 There must be something identical in a picture and what itdepicts, 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 beable to depict it--correctly or incorrectly--in the way that it does, isits pictorial form. 

2. 171 A picture can depict any reality whose form it has. A spatialpicture can depict anything spatial, a coloured one anything coloured, etc. 

2. 172 A picture cannot, however, depict its pictorial form: it displaysit. 

2. 173 A picture represents its subject from a position outside it. (Its standpoint is its representational form. ) That is why a picturerepresents its subject correctly or incorrectly. 

2. 174 A picture cannot, however, place itself outside itsrepresentational form. 

2. 18 What any picture, of whatever form, must have in common withreality, in order to be able to depict it--correctly or incorrectly--inany 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 logicalpicture. 

2. 182 Every picture is at the same time a logical one. (On the otherhand, not every picture is, for example, a spatial one. )

2. 19 Logical pictures can depict the world. 

2. 2 A picture has logico-pictorial form in common with what it depicts. 

2. 201 A picture depicts reality by representing a possibility ofexistence and non-existence of states of affairs. 

2. 202 A picture contains the possibility of the situation that itrepresents. 

2. 203 A picture agrees with reality or fails to agree; it is correct orincorrect, true or false. 

2. 22 What a picture represents it represents independently of its truthor 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 realityconstitutes its truth or falsity. 

2. 223 In order to tell whether a picture is true or false we mustcompare it with reality. 

2. 224 It is impossible to tell from the picture alone whether it is trueor 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 canpicture 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 isthe thought. What is thinkable is possible too. 

3. 03 Thought can never be of anything illogical, since, if it were, weshould have to think illogically. 

3. 031 It used to be said that God could create anything except whatwould be contrary to the laws of logic. The truth is that we could notsay 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 coordinatesa figure that contradicts the laws of space, or to give the coordinatesof a point that does not exist. 

3. 0321 Though a state of affairs that would contravene the laws ofphysics can be represented by us spatially, one that would contravenethe laws of geometry cannot. 

3. 04 It a thought were correct a priori, it would be a thought whosepossibility ensured its truth. 

3. 05 A priori knowledge that a thought was true would be possible onlyit its truth were recognizable from the thought itself (without anythinga to compare it with). 

3. 1 In a proposition a thought finds an expression that can be perceivedby 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 projectionis to think of the sense of the proposition. 

3. 12 I call the sign with which we express a thought a propositionalsign. And a proposition is a propositional sign in its projectiverelation to the world. 

3. 13 A proposition, therefore, does not actually contain its sense, but does contain the possibility of expressing it. ('The content ofa proposition' means the content of a proposition that has sense. ) Aproposition contains the form, but not the content, of its sense. 

3. 14 What constitutes a propositional sign is that in its elements (thewords) stand in a determinate relation to one another. A propositionalsign is a fact. 

3. 141 A proposition is not a blend of words. (Just as a theme in music isnot 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 bythe usual form of expression in writing or print. For in a printedproposition, for example, no essential difference is apparent between apropositional sign and a word. (That is what made it possible for Fregeto call a proposition a composite name. )

3. 1431 The essence of a propositional sign is very clearly seen if weimagine one composed of spatial objects (such as tables, chairs, andbooks) instead of written signs. 

3. 1432 Instead of, 'The complex sign "aRb" says that a stands to b inthe relation R' we ought to put, 'That "a" stands to "b" in a certainrelation 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 thatelements of the propositional sign correspond to the objects of thethought. 

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 thesame sign as 'A'. )

3. 21 The configuration of objects in a situation corresponds to theconfiguration of simple signs in the propositional sign. 

3. 221 Objects can only be named. Signs are their representatives. I canonly speak about them: I cannot put them into words. Propositions canonly say how things are, not what they are. 

3. 23 The requirement that simple signs be possible is the requirementthat sense be determinate. 

3. 24 A proposition about a complex stands in an internal relation to aproposition about a constituent of the complex. A complex can be givenonly by its description, which will be right or wrong. A propositionthat mentions a complex will not be nonsensical, if the complex doesnot exits, but simply false. When a propositional element signifies acomplex, this can be seen from an indeterminateness in the propositionsin which it occurs. In such cases we know that the proposition leavessomething undetermined. (In fact the notation for generality containsa prototype. ) The contraction of a symbol for a complex into a simplesymbol can be expressed in a definition. 

3. 25 A proposition cannot be dissected any further by means of adefinition: it is a primitive sign. 

3. 261 Every sign that has a definition signifies via the signs thatserve to define it; and the definitions point the way. Two signs cannotsignify in the same manner if one is primitive and the other is definedby means of primitive signs. Names cannot be anatomized by means ofdefinitions. (Nor can any sign that has a meaning independently and onits own. )

3. 262 What signs fail to express, their application shows. What signsslur over, their application says clearly. 

3. 263 The meanings of primitive signs can be explained by means ofelucidations. Elucidations are propositions that stood if the meaningsof those signs are already known. 

3. 3 Only propositions have sense; only in the nexus of a propositiondoes a name have meaning. 

3. 31 I call any part of a proposition that characterizes its sensean expression (or a symbol). (A proposition is itself an expression. )Everything essential to their sense that propositions can have in commonwith one another is an expression. An expression is the mark of a formand a content. 

3. 311 An expression presupposes the forms of all the propositions inwhich it can occur. It is the common characteristic mark of a class ofpropositions. 

3. 312 It is therefore presented by means of the general form of thepropositions that it characterizes. In fact, in this form the expressionwill be constant and everything else variable. 

3. 313 Thus an expression is presented by means of a variable whosevalues are the propositions that contain the expression. (In thelimiting case the variable becomes a constant, the expression becomes aproposition. ) I call such a variable a 'propositional variable'. 

3. 314 An expression has meaning only in a proposition. All variables canbe construed as propositional variables. (Even variable names. )

3. 315 If we turn a constituent of a proposition into a variable, thereis a class of propositions all of which are values of the resultingvariable proposition. In general, this class too will be dependent onthe meaning that our arbitrary conventions have given to parts of theoriginal proposition. But if all the signs in it that have arbitrarilydetermined meanings are turned into variables, we shall still geta class of this kind. This one, however, is not dependent on anyconvention, but solely on the nature of the pro position. It correspondsto a logical form--a logical prototype. 

3. 316 What values a propositional variable may take is something that isstipulated. The stipulation of values is the variable. 

3. 317 To stipulate values for a propositional variable is to givethe propositions whose common characteristic the variable is. Thestipulation is a description of those propositions. The stipulation willtherefore be concerned only with symbols, not with their meaning. Andthe only thing essential to the stipulation is that it is merely adescription of symbols and states nothing about what is signified. Howthe description of the propositions is produced is not essential. 

3. 318 Like Frege and Russell I construe a proposition as a function ofthe 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 commonto two different symbols--in which case they will signify in differentways. 

3. 322 Our use of the same sign to signify two different objects cannever indicate a common characteristic of the two, if we use it with twodifferent modes of signification. For the sign, of course, is arbitrary. So we could choose two different signs instead, and then what would beleft in common on the signifying side?

3. 323 In everyday language it very frequently happens that the sameword has different modes of signification--and so belongs to differentsymbols--or that two words that have different modes of significationare employed in propositions in what is superficially the same way. Thusthe word 'is' figures as the copula, as a sign for identity, and as anexpression for existence; 'exist' figures as an intransitive verb like'go', and 'identical' as an adjective; we speak of something, but alsoof something's happening. (In the proposition, 'Green is green'--wherethe first word is the proper name of a person and the last anadjective--these words do not merely have different meanings: they aredifferent 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-languagethat excludes them by not using the same sign for different symbols andby not using in a superficially similar way signs that have differentmodes of signification: that is to say, a sign-language that is governedby logical grammar--by logical syntax. (The conceptual notation of Fregeand Russell is such a language, though, it is true, it fails to excludeall mistakes. )

3. 326 In order to recognize a symbol by its sign we must observe how itis used with a sense. 

3. 327 A sign does not determine a logical form unless it is takentogether with its logico-syntactical employment. 

3. 328 If a sign is useless, it is meaningless. That is the point ofOccam's maxim. (If everything behaves as if a sign had meaning, then itdoes 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 mentioningthe meaning of a sign: only the description of expressions may bepresupposed. 

3. 331 From this observation we turn to Russell's 'theory of types'. Itcan be seen that Russell must be wrong, because he had to mention themeaning of signs when establishing the rules for them. 

3. 332 No proposition can make a statement about itself, because apropositional sign cannot be contained in itself (that is the whole ofthe 'theory of types'). 

3. 333 The reason why a function cannot be its own argument is that thesign for a function already contains the prototype of its argument, andit 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 musthave different meanings, since the inner one has the form O(f(x)) andthe outer one has the form Y(O(fx)). Only the letter 'F' is common tothe two functions, but the letter by itself signifies nothing. Thisimmediately 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 knowhow each individual sign signifies. 

3. 34 A proposition possesses essential and accidental features. Accidental features are those that result from the particular way inwhich the propositional sign is produced. Essential features are thosewithout which the proposition could not express its sense. 

3. 341 So what is essential in a proposition is what all propositionsthat can express the same sense have in common. And similarly, ingeneral, what is essential in a symbol is what all symbols that canserve the same purpose have in common. 

3. 3411 So one could say that the real name of an object was what allsymbols that signified it had in common. Thus, one by one, all kinds ofcomposition would prove to be unessential to a name. 

3. 342 Although there is something arbitrary in our notations, this muchis not arbitrary--that when we have determined one thing arbitrarily, something else is necessarily the case. (This derives from the essenceof notation. )

3. 3421 A particular mode of signifying may be unimportant but it isalways important that it is a possible mode of signifying. And that isgenerally so in philosophy: again and again the individual case turnsout to be unimportant, but the possibility of each individual casediscloses something about the essence of the world. 

3. 343 Definitions are rules for translating from one language intoanother. Any correct sign-language must be translatable into any otherin 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 symbolsthat the rules of logical syntax allow us to substitute for it. 

3. 3441 For instance, we can express what is common to all notationsfor truth-functions in the following way: they have in common that, forexample, 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 theway in which something general can be disclosed by the possibility of aspecific notation. )

3. 3442 Nor does analysis resolve the sign for a complex in an arbitraryway, so that it would have a different resolution every time that it wasincorporated in a different proposition. 

3. 4 A proposition determines a place in logical space. The existenceof this logical place is guaranteed by the mere existence of theconstituents--by the existence of the proposition with a sense. 

3. 41 The propositional sign with logical coordinates--that is thelogical place. 

3. 411 In geometry and logic alike a place is a possibility: somethingcan 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 introducemore and more new elements in co-ordination. ) (The logical scaffoldingsurrounding a picture determines logical space. The force of aproposition 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 ofexpressing every sense, without having any idea how each word hasmeaning or what its meaning is--just as people speak without knowing howthe individual sounds are produced. Everyday language is a part of thehuman organism and is no less complicated than it. It is not humanlypossible to gather immediately from it what the logic of language is. Language disguises thought. So much so, that from the outward form ofthe clothing it is impossible to infer the form of the thought beneathit, because the outward form of the clothing is not designed to revealthe form of the body, but for entirely different purposes. The tacitconventions on which the understanding of everyday language depends areenormously complicated. 

4. 003 Most of the propositions and questions to be found inphilosophical works are not false but nonsensical. Consequently wecannot give any answer to questions of this kind, but can only pointout that they are nonsensical. Most of the propositions and questionsof philosophers arise from our failure to understand the logic of ourlanguage. (They belong to the same class as the question whether thegood is more or less identical than the beautiful. ) And it is notsurprising that the deepest problems are in fact not problems at all. 

4. 0031 All philosophy is a 'critique of language' (though not inMauthner's sense). It was Russell who performed the service of showingthat the apparent logical form of a proposition need not be its realone. 

4. 01 A proposition is a picture of reality. A proposition is a model ofreality as we imagine it. 

4. 011 At first sight a proposition--one set out on the printed page, forexample--does not seem to be a picture of the reality with which itis concerned. But neither do written notes seem at first sight to be apicture 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 bepictures, even in the ordinary sense, of what they represent. 

4. 012 It is obvious that a proposition of the form 'aRb' strikes us asa picture. In this case the sign is obviously a likeness of what issignified. 

4. 013 And if we penetrate to the essence of this pictorial character, wesee that it is not impaired by apparent irregularities (such as theuse [sharp] of and [flat] in musical notation). For even theseirregularities depict what they are intended to express; only they do itin a different way. 

4. 014 A gramophone record, the musical idea, the written notes, and thesound-waves, all stand to one another in the same internal relationof depicting that holds between language and the world. They are allconstructed according to a common logical pattern. (Like the two youthsin the fairy-tale, their two horses, and their lilies. They are all in acertain sense one. )

4. 0141 There is a general rule by means of which the musician can obtainthe symphony from the score, and which makes it possible to derive thesymphony from the groove on the gramophone record, and, using the firstrule, to derive the score again. That is what constitutes the innersimilarity between these things which seem to be constructed in suchentirely different ways. And that rule is the law of projection whichprojects the symphony into the language of musical notation. It isthe rule for translating this language into the language of gramophonerecords. 

4. 015 The possibility of all imagery, of all our pictorial modes ofexpression, is contained in the logic of depiction. 

4. 016 In order to understand the essential nature of a proposition, weshould consider hieroglyphic script, which depicts the facts that itdescribes. And alphabetic script developed out of it without losing whatwas essential to depiction. 

4. 02 We can see this from the fact that we understand the sense of apropositional sign without its having been explained to us. 

4. 021 A proposition is a picture of reality: for if I understand aproposition, I know the situation that it represents. And I understandthe proposition without having had its sense explained to me. 

4. 022 A proposition shows its sense. A proposition shows how thingsstand if it is true. And it says that they do so stand. 

4. 023 A proposition must restrict reality to two alternatives: yesor no. In order to do that, it must describe reality completely. A proposition is a description of a state of affairs. Just as adescription of an object describes it by giving its external properties, so a proposition describes reality by its internal properties. Aproposition constructs a world with the help of a logical scaffolding, so that one can actually see from the proposition how everythingstands logically if it is true. One can draw inferences from a falseproposition. 

4. 024 To understand a proposition means to know what is the case if itis true. (One can understand it, therefore, without knowing whether itis true. ) It is understood by anyone who understands its constituents. 

4. 025 When translating one language into another, we do not proceed bytranslating each proposition of the one into a proposition of the other, but merely by translating the constituents of propositions. (And thedictionary 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 ifwe are to understand them. With propositions, however, we make ourselvesunderstood. 

4. 027 It belongs to the essence of a proposition that it should be ableto communicate a new sense to us. 

4. 03 A proposition must use old expressions to communicate a newsense. A proposition communicates a situation to us, and so it must beessentially connected with the situation. And the connexion is preciselythat it is its logical picture. A proposition states something only inso far as it is a picture. 

4. 031 In a proposition a situation is, as it were, constructed by way ofexperiment. Instead of, 'This proposition has such and such a sense, wecan simply say, 'This proposition represents such and such a situation'. 

4. 0311 One name stands for one thing, another for another thing, andthey are combined with one another. In this way the whole group--like atableau vivant--presents a state of affairs. 

4. 0312 The possibility of propositions is based on the principle thatobjects have signs as their representatives. My fundamental idea is thatthe 'logical constants' are not representatives; that there can be norepresentatives of the logic of facts. 

4. 032 It is only in so far as a proposition is logically articulatedthat it is a picture of a situation. (Even the proposition, 'Ambulo', is composite: for its stem with a different ending yields a differentsense, and so does its ending with a different stem. )

4. 04 In a proposition there must be exactly as many distinguishableparts as in the situation that it represents. The two must possess thesame logical (mathematical) multiplicity. (Compare Hertz's Mechanics ondynamical models. )

4. 041 This mathematical multiplicity, of course, cannot itself be thesubject 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 wasbeing generalized. If we wanted to signalize it with an affix 'g'--forinstance by writing 'f(xg)'--that would not be adequate either: weshould not know the scope of the generality-sign. If we were to try todo it by introducing a mark into the argument-places--for instance bywriting '(G, G). F(G, G)' --it would not be adequate: we should not beable to establish the identity of the variables. And so on. All thesemodes of signifying are inadequate because they lack the necessarymathematical multiplicity. 

4. 0412 For the same reason the idealist's appeal to 'spatial spectacles'is inadequate to explain the seeing of spatial relations, because itcannot 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 apicture of reality. 

4. 061 It must not be overlooked that a proposition has a sense that isindependent of the facts: otherwise one can easily suppose that true andfalse are relations of equal status between signs and what they signify. In that case one could say, for example, that 'p' signified in the trueway what 'Pp' signified in the false way, etc. 

4. 062 Can we not make ourselves understood with false propositions justas we have done up till now with true ones?--So long as it is known thatthey are meant to be false. --No! For a proposition is true if we use itto say that things stand in a certain way, and they do; and if by 'p' wemean Pp and things stand as we mean that they do, then, construed in thenew way, 'p' is true and not false. 

4. 0621 But it is important that the signs 'p' and 'Pp' can say the samething. For it shows that nothing in reality corresponds to the sign'P'. The occurrence of negation in a proposition is not enough tocharacterize its sense (PPp = p). The propositions 'p' and 'Pp' haveopposite sense, but there corresponds to them one and the same reality. 

4. 063 An analogy to illustrate the concept of truth: imagine a blackspot 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 factthat a point is black there corresponds a positive fact, and to the factthat a point is white (not black), a negative fact. If I designatea point on the sheet (a truth-value according to Frege), then thiscorresponds 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, Imust first know when a point is called black, and when white: in orderto be able to say, '"p" is true (or false)', I must have determined inwhat circumstances I call 'p' true, and in so doing I determine thesense of the proposition. Now the point where the simile breaks down isthis: we can indicate a point on the paper even if we do not knowwhat black and white are, but if a proposition has no sense, nothingcorresponds to it, since it does not designate a thing (a truth-value)which might have properties called 'false' or 'true'. The verb of aproposition 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 asense by affirmation. Indeed its sense is just what is affirmed. And thesame applies to negation, etc. 

4. 0641 One could say that negation must be related to the logicalplace determined by the negated proposition. The negating propositiondetermines a logical place different from that of the negatedproposition. The negating proposition determines a logical placewith the help of the logical place of the negated proposition. For itdescribes it as lying outside the latter's logical place. The negatedproposition can be negated again, and this in itself shows that whatis negated is already a proposition, and not merely something that ispreliminary to a proposition. 

4. 1 Propositions represent the existence and non-existence of states ofaffairs. 

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 thenatural 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 philosophicalwork consists essentially of elucidations. Philosophy does not resultin 'philosophical propositions', but rather in the clarification ofpropositions. Without philosophy thoughts are, as it were, cloudyand indistinct: its task is to make them clear and to give them sharpboundaries. 

4. 1121 Psychology is no more closely related to philosophy thanany other natural science. Theory of knowledge is the philosophy ofpsychology. Does not my study of sign-language correspond to the studyof thought-processes, which philosophers used to consider so essentialto the philosophy of logic? Only in most cases they got entangled inunessential psychological investigations, and with my method too thereis an analogous risk. 

4. 1122 Darwin's theory has no more to do with philosophy than any otherhypothesis in natural science. 

4. 113 Philosophy sets limits to the much disputed sphere of naturalscience. 

4. 114 It must set limits to what can be thought; and, in doing so, towhat cannot be thought. It must set limits to what cannot be thought byworking outwards through what can be thought. 

4. 115 It will signify what cannot be said, by presenting clearly whatcan 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. 12Propositions can represent the whole of reality, but they cannotrepresent what they must have in common with reality in order to be ableto represent it--logical form. In order to be able to represent logicalform, we should have to be able to station ourselves with propositionssomewhere outside logic, that is to say outside the world. 

4. 121 Propositions cannot represent logical form: it is mirrored inthem. What finds its reflection in language, language cannot represent. What expresses itself in language, we cannot express by means oflanguage. Propositions show the logical form of reality. They displayit. 

4. 1211 Thus one proposition 'fa' shows that the object a occurs inits sense, two propositions 'fa' and 'ga' show that the same object ismentioned in both of them. If two propositions contradict one another, then their structure shows it; the same is true if one of them followsfrom 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 asign-language in which everything is all right, we already have acorrect logical point of view. 

4. 122 In a certain sense we can talk about formal properties of objectsand states of affairs, or, in the case of facts, about structuralproperties: and in the same sense about formal relations and structuralrelations. (Instead of 'structural property' I also say 'internalproperty'; instead of 'structural relation', 'internal relation'. Iintroduce these expressions in order to indicate the source of theconfusion between internal relations and relations proper (externalrelations), which is very widespread among philosophers. ) It isimpossible, however, to assert by means of propositions that suchinternal properties and relations obtain: rather, this makes itselfmanifest in the propositions that represent the relevant states ofaffairs and are concerned with the relevant objects. 

4. 1221 An internal property of a fact can also be bed a feature of thatfact (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 shouldnot possess it. (This shade of blue and that one stand, eo ipso, in theinternal relation of lighter to darker. It is unthinkable that these twoobjects should not stand in this relation. ) (Here the shifting useof 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 isnot expressed by means of a proposition: rather, it expresses itselfin the proposition representing the situation, by means of an internalproperty of that proposition. It would be just as nonsensical to assertthat a proposition had a formal property as to deny it. 

4. 1241 It is impossible to distinguish forms from one another bysaying that one has this property and another that property: for thispresupposes that it makes sense to ascribe either property to eitherform. 

4. 125 The existence of an internal relation between possible situationsexpresses itself in language by means of an internal relation betweenthe propositions representing them. 

4. 1251 Here we have the answer to the vexed question 'whether allrelations are internal or external'. 

4. 1252 I call a series that is ordered by an internal relation a seriesof forms. The order of the number-series is not governed by an externalrelation but by an internal relation. The same is true of the series ofpropositions 'aRb', '(d: c): aRx. XRb', '(d x, y): aRx. XRy. YRb', and soforth. (If b stands in one of these relations to a, I call b a successorof a. )

4. 126 We can now talk about formal concepts, in the same sense that wespeak of formal properties. (I introduce this expression in order toexhibit the source of the confusion between formal concepts and conceptsproper, which pervades the whole of traditional logic. ) When somethingfalls under a formal concept as one of its objects, this cannot beexpressed by means of a proposition. Instead it is shown in the verysign for this object. (A name shows that it signifies an object, a signfor 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 bymeans of functions. The expression for a formal property is a feature ofcertain symbols. So the sign for the characteristics of a formal conceptis a distinctive feature of all symbols whose meanings fall under theconcept. So the expression for a formal concept is a propositionalvariable in which this distinctive feature alone is constant. 

4. 127 The propositional variable signifies the formal concept, and itsvalues signify the objects that fall under the concept. 

4. 1271 Every variable is the sign for a formal concept. For everyvariable represents a constant form that all its values possess, andthis can be regarded as a formal property of those values. 

4. 1272 Thus the variable name 'x' is the proper sign for thepseudo-concept object. Wherever the word 'object' ('thing', etc. ) iscorrectly used, it is expressed in conceptual notation by a variablename. For example, in the proposition, 'There are 2 objects which.. . ', it is expressed by ' (dx, y)... '. Wherever it is used in a differentway, that is as a proper concept-word, nonsensical pseudo-propositionsare the result. So one cannot say, for example, 'There are objects', asone might say, 'There are books'. And it is just as impossible tosay, 'There are 100 objects', or, 'There are!0 objects'. And it isnonsensical to speak of the total number of objects. The same applies tothe words 'complex', 'fact', 'function', 'number', etc. They allsignify formal concepts, and are represented in conceptual notation byvariables, not by functions or classes (as Frege and Russell believed). '1 is a number', 'There is only one zero', and all similar expressionsare nonsensical. (It is just as nonsensical to say, 'There is only one1', 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 underit is given. It is not possible, therefore, to introduce as primitiveideas objects belonging to a formal concept and the formal conceptitself. So it is impossible, for example, to introduce as primitiveideas both the concept of a function and specific functions, as Russelldoes; or the concept of a number and particular numbers. 

4. 1273 If we want to express in conceptual notation the generalproposition, 'b is a successor of a', then we require an expression forthe general term of the series of forms 'aRb', '(d: c): aRx. XRb', '(dx, y) : aRx. XRy. YRb', ... , In order to express the general term of aseries of forms, we must use a variable, because the concept 'termof that series of forms' is a formal concept. (This is what Frege andRussell overlooked: consequently the way in which they want to expressgeneral propositions like the one above is incorrect; it contains avicious circle. ) We can determine the general term of a series of formsby giving its first term and the general form of the operation thatproduces the next term out of the proposition that precedes it. 

4. 1274 To ask whether a formal concept exists is nonsensical. For noproposition can be the answer to such a question. (So, for example, the question, 'Are there unanalysable subject-predicate propositions?'cannot be asked. )

4. 128 Logical forms are without number. Hence there are no pre-eminentnumbers in logic, and hence there is no possibility of philosophicalmonism or dualism, etc. 

4. 2 The sense of a proposition is its agreement and disagreement withpossibilities of existence and non-existence of states of affairs. 4. 21The simplest kind of proposition, an elementary proposition, asserts theexistence of a state of affairs. 

4. 211 It is a sign of a proposition's being elementary that there can beno elementary proposition contradicting it. 

4. 22 An elementary proposition consists of names. It is a nexus, aconcatenation, of names. 

4. 221 It is obvious that the analysis of propositions must bring us toelementary propositions which consist of names in immediate combination. This raises the question how such combination into propositions comesabout. 

4. 2211 Even if the world is infinitely complex, so that every factconsists of infinitely many states of affairs and every state of affairsis composed of infinitely many objects, there would still have to beobjects and states of affairs. 

4. 23 It is only in the nexus of an elementary proposition that a nameoccurs 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 bythe letters 'p', 'q', 'r'. 

4. 241 When I use two signs with one and the same meaning, I express thisby 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 introducea new sign 'b', laying down that it shall serve as a substitute fora sign a that is already known, then, like Russell, I write theequation--definition--in the form 'a = b Def. ' A definition is a ruledealing with signs. )

4. 242 Expressions of the form 'a = b' are, therefore, mererepresentational devices. They state nothing about the meaning of thesigns 'a' and 'b'. 

4. 243 Can we understand two names without knowing whether they signifythe same thing or two different things?--Can we understand a propositionin which two names occur without knowing whether their meaning is thesame or different? Suppose I know the meaning of an English word and ofa German word that means the same: then it is impossible for me to beunaware that they do mean the same; I must be capable of translatingeach into the other. Expressions like 'a = a', and those derived fromthem, are neither elementary propositions nor is there any other way inwhich 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 notexist. 

4. 26 If all true elementary propositions are given, the result is acomplete description of the world. The world is completely described bygiving all elementary propositions, and adding which of them are trueand which false. For n states of affairs, there are possibilities ofexistence and non-existence. Of these states of affairs any combinationcan exist and the remainder not exist. 

4. 28 There correspond to these combinations the same number ofpossibilities of truth--and falsity--for n elementary propositions. 

4. 3 Truth-possibilities of elementary propositions mean Possibilities ofexistence and non-existence of states of affairs. 

4. 31 We can represent truth-possibilities by schemata of the followingkind ('T' means 'true', 'F' means 'false'; the rows of 'T's' and'F's' under the row of elementary propositions symbolize theirtruth-possibilities in a way that can easily be understood):

4. 4 A proposition is an expression of agreement and disagreement withtruth-possibilities of elementary propositions. 

4. 41 Truth-possibilities of elementary propositions are the conditionsof the truth and falsity of propositions. 

4. 411 It immediately strikes one as probable that the introduction ofelementary propositions provides the basis for understanding all otherkinds of proposition. Indeed the understanding of general propositionspalpably depends on the understanding of elementary propositions. 

4. 42 For n elementary propositions there are ways in which a propositioncan agree and disagree with their truth possibilities. 

4. 43 We can express agreement with truth-possibilities by correlatingthe mark 'T' (true) with them in the schema. The absence of this markmeans disagreement. 

4. 431 The expression of agreement and disagreement with the truthpossibilities of elementary propositions expresses the truth-conditionsof a proposition. A proposition is the expression of itstruth-conditions. (Thus Frege was quite right to use them as a startingpoint when he explained the signs of his conceptual notation. But theexplanation of the concept of truth that Frege gives is mistaken: if'the true' and 'the false' were really objects, and were the argumentsin Pp etc. , then Frege's method of determining the sense of 'Pp' wouldleave it absolutely undetermined. )

4. 44 The sign that results from correlating the mark 'I' withtruth-possibilities 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 isnone corresponding to the horizontal and vertical lines or to thebrackets. --There are no 'logical objects'. Of course the same applies toall 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 worksof Frege (and Russell) it simply indicates that these authors hold thepropositions marked with this sign to be true. Thus '|-' is no more acomponent part of a proposition than is, for instance, the proposition'snumber. It is quite impossible for a proposition to state that it itselfis true. ) If the order or the truth-possibilities in a scheme is fixedonce and for all by a combinatory rule, then the last column by itselfwill be an expression of the truth-conditions. If we now write thiscolumn as a row, the propositional sign will become '(TT-T) (p, q)' ormore explicitly '(TTFT) (p, q)' (The number of places in the left-handpair of brackets is determined by the number of terms in the right-handpair. )

4. 45 For n elementary propositions there are Ln possible groups oftruth-conditions. The groups of truth-conditions that are obtainablefrom the truth-possibilities of a given number of elementarypropositions can be arranged in a series. 

4. 46 Among the possible groups of truth-conditions there are twoextreme cases. In one of these cases the proposition is true for allthe truth-possibilities of the elementary propositions. We say that thetruth-conditions are tautological. In the second case the propositionis false for all the truth-possibilities: the truth-conditions arecontradictory. In the first case we call the proposition a tautology; inthe second, a contradiction. 

4. 461 Propositions show what they say; tautologies and contradictionsshow that they say nothing. A tautology has no truth-conditions, sinceit is unconditionally true: and a contradiction is true on no condition. Tautologies and contradictions lack sense. (Like a point from which twoarrows go out in opposite directions to one another. ) (For example, Iknow nothing about the weather when I know that it is either raining ornot raining. )

4. 46211 Tautologies and contradictions are not, however, nonsensical. They are part of the symbolism, much as '0' is part of the symbolism ofarithmetic. 

4. 462 Tautologies and contradictions are not pictures of reality. Theydo not represent any possible situations. For the former admit allpossible situations, and latter none. In a tautology the conditions ofagreement with the world--the representational relations--cancel oneanother, so that it does not stand in any representational relation toreality. 

4. 463 The truth-conditions of a proposition determine the range that itleaves open to the facts. (A proposition, a picture, or a model is, in the negative sense, like a solid body that restricts the freedom ofmovement of others, and in the positive sense, like a space bounded bysolid substance in which there is room for a body. ) A tautology leavesopen to reality the whole--the infinite whole--of logical space: acontradiction fills the whole of logical space leaving no point of itfor reality. Thus neither of them can determine reality in any way. 

4. 464 A tautology's truth is certain, a proposition's possible, acontradiction's impossible. (Certain, possible, impossible: here wehave the first indication of the scale that we need in the theory ofprobability. )

4. 465 The logical product of a tautology and a proposition says the samething as the proposition. This product, therefore, is identical with theproposition. For it is impossible to alter what is essential to a symbolwithout altering its sense. 

4. 466 What corresponds to a determinate logical combination of signs isa determinate logical combination of their meanings. It is only to theuncombined signs that absolutely any combination corresponds. Inother words, propositions that are true for every situation cannot becombinations of signs at all, since, if they were, only determinatecombinations of objects could correspond to them. (And what is not alogical combination has no combination of objects corresponding toit. ) Tautology and contradiction are the limiting cases--indeed thedisintegration--of the combination of signs. 

4. 4661 Admittedly the signs are still combined with one another even intautologies and contradictions--i. E. They stand in certain relations toone another: but these relations have no meaning, they are not essentialto 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 sign-languagewhatsoever in such a way that every possible sense can be expressed bya symbol satisfying the description, and every symbol satisfying thedescription can express a sense, provided that the meanings of the namesare suitably chosen. It is clear that only what is essential to themost general propositional form may be included in its description--forotherwise it would not be the most general form. The existence of ageneral propositional form is proved by the fact that there cannot be aproposition 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 cansimply ask what propositions I can construct out of them. And there Ihave all propositions, and that fixes their limits. 

4. 52 Propositions comprise all that follows from the totality of allelementary propositions (and, of course, from its being the totalityof them all ). (Thus, in a certain sense, it could be said that allpropositions were generalizations of elementary propositions. )

4. 53 The general propositional form is a variable. 

5. A proposition is a truth-function of elementary propositions. 

(An elementary proposition is a truth-function of itself. )

5. 01 Elementary propositions are the truth-arguments of propositions. 

5. 02 The arguments of functions are readily confused with the affixes ofnames. For both arguments and affixes enable me to recognize the meaningof the signs containing them. For example, when Russell writes '+c', the 'c' is an affix which indicates that the sign as a whole is theaddition-sign for cardinal numbers. But the use of this sign is theresult of arbitrary convention and it would be quite possible to choosea simple sign instead of '+c'; in 'Pp' however, 'p' is not an affix butan 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 towhose name we attach it: e. G. The Caesar of the Julian gens. ) If Iam not mistaken, Frege's theory about the meaning of propositions andfunctions is based on the confusion between an argument and an affix. Frege regarded the propositions of logic as names, and their argumentsas the affixes of those names. 

5. 1 Truth-functions can be arranged in series. That is the foundation ofthe theory of probability. 

5. 101 The truth-functions of a given number of elementary propositionscan 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. (qz p) (TTFT) (p, q) ": If p then q. (p z q) (TTTF) (p, q) ": p or q. (p Cq) (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 pthen 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) ": pand 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 notq. ) (p. Pp. Q. Pq) I will give the name truth-grounds of a propositionto those truth-possibilities of its truth-arguments that make it true. 

5. 11 If all the truth-grounds that are common to a number ofpropositions are at the same time truth-grounds of a certainproposition, then we say that the truth of that proposition follows fromthe truth of the others. 

5. 12 In particular, the truth of a proposition 'p' follows from thetruth of another proposition 'q' is all the truth-grounds of the latterare truth-grounds of the former. 

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 aretrue, then by that very act he also creates a world in which all thepropositions that follow from them come true. And similarly he could notcreate a world in which the proposition 'p' was true without creatingall 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 thesame time one of the propositions that affirm 'q'. Two propositions areopposed to one another if there is no proposition with a sense, thataffirms 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 ofothers, this finds expression in relations in which the forms of thepropositions stand to one another: nor is it necessary for us to set upthese relations between them, by combining them with one another in asingle proposition; on the contrary, the relations are internal, and their existence is an immediate result of the existence of thepropositions. 

5. 1311 When we infer q from p C q and Pp, the relation between thepropositional forms of 'p C q' and 'Pp' is masked, in this case, by ourmode 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), thenthe inner connexion becomes obvious. (The possibility of inference from(x). Fx to fa shows that the symbol (x). Fx itself has generality init. )

5. 132 If p follows from q, I can make an inference from q to p, deducep from q. The nature of the inference can be gathered only from the twopropositions. They themselves are the only possible justification ofthe inference. 'Laws of inference', which are supposed to justifyinferences, as in the works of Frege and Russell, have no sense, andwould 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 existenceof one situation to the existence of another, entirely differentsituation. 

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 thepresent. Belief in the causal nexus is superstition. 

5. 1362 The freedom of the will consists in the impossibility of knowingactions that still lie in the future. We could know them only ifcausality were an inner necessity like that of logical inference. --Theconnexion between knowledge and what is known is that of logicalnecessity. ('A knows that p is the case', has no sense if p is atautology. )

5. 1363 If the truth of a proposition does not follow from the fact thatit is self-evident to us, then its self-evidence in no way justifies ourbelief in its truth. 

5. 14 If one proposition follows from another, then the latter says morethan 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 sameproposition. 

5. 142 A tautology follows from all propositions: it says nothing. 

5. 143 Contradiction is that common factor of propositions which noproposition has in common with another. Tautology is the common factorof 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 ofpropositions: tautology is the unsubstantial point at their centre. 

5. 15 If Tr is the number of the truth-grounds of a proposition 'r', andif Trs is the number of the truth-grounds of a proposition 's' that areat the same time truth-grounds of 'r', then we call the ratio Trs:Tr the degree of probability that the proposition 'r' gives to theproposition '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 inwhich the proposition r has 'T's'. Then the proposition r gives to theproposition s the probability Trs: Tr. 

5. 1511 There is no special object peculiar to probability propositions. 

5. 152 When propositions have no truth-arguments in common with oneanother, we call them independent of one another. Two elementarypropositions give one another the probability 1/2. If p follows from q, then the proposition 'q' gives to the proposition 'p' the probability1. 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 equalnumbers (and none of any other kind). I draw one ball after another, putting them back into the urn. By this experiment I can establish thatthe number of black balls drawn and the number of white balls drawnapproximate to one another as the draw continues. So this is not amathematical truth. Now, if I say, 'The probability of my drawing awhite ball is equal to the probability of my drawing a black one', thismeans that all the circumstances that I know of (including the laws ofnature assumed as hypotheses) give no more probability to the occurrenceof the one event than to that of the other. That is to say, they giveeach the probability 1/2 as can easily be gathered from the abovedefinitions. What I confirm by the experiment is that the occurrence ofthe two events is independent of the circumstances of which I have nomore detailed knowledge. 

5. 155 The minimal unit for a probability proposition is this: Thecircumstances--of which I have no further knowledge--give such and sucha degree of probability to the occurrence of a particular event. 

5. 156 It is in this way that probability is a generalization. Itinvolves a general description of a propositional form. We useprobability only in default of certainty--if our knowledge of a factis not indeed complete, but we do know something about its form. (Aproposition may well be an incomplete picture of a certain situation, but it is always a complete picture of something. ) A probabilityproposition is a sort of excerpt from other propositions. 

5. 2 The structures of propositions stand in internal relations to oneanother. 

5. 21 In order to give prominence to these internal relations we canadopt the following mode of expression: we can represent a propositionas 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 structuresof its result and of its bases. 

5. 23 The operation is what has to be done to the one proposition inorder to make the other out of it. 

5. 231 And that will, of course, depend on their formal properties, onthe internal similarity of their forms. 

5. 232 The internal relation by which a series is ordered is equivalentto the operation that produces one term from another. 

5. 233 Operations cannot make their appearance before the point at whichone proposition is generated out of another in a logically meaningfulway; i. E. The point at which the logical construction of propositionsbegins. 

5. 234 Truth-functions of elementary propositions are results ofoperations with elementary propositions as bases. (These operations Icall truth-operations. )

5. 2341 The sense of a truth-function of p is a function of the senseof p. Negation, logical addition, logical multiplication, etc. Etc. Areoperations. (Negation reverses the sense of a proposition. )

5. 24 An operation manifests itself in a variable; it shows how we canget from one form of proposition to another. It gives expression to thedifference between the forms. (And what the bases of an operation andits result have in common is just the bases themselves. )

5. 241 An operation is not the mark of a form, but only of a differencebetween 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 tocertain formal relations. 

5. 25 The occurrence of an operation does not characterize the sense ofa proposition. Indeed, no statement is made by an operation, but only byits result, and this depends on the bases of the operation. (Operationsand functions must not be confused with each other. )

5. 251 A function cannot be its own argument, whereas an operation cantake one of its own results as its base. 

5. 252 It is only in this way that the step from one term of a seriesof forms to another is possible (from one type to another in thehierarchies of Russell and Whitehead). (Russell and Whitehead did notadmit the possibility of such steps, but repeatedly availed themselvesof it. )

5. 2521 If an operation is applied repeatedly to its own results, I speakof successive applications of it. ('O'O'O'a' is the result of threesuccessive applications of the operation 'O'E' to 'a'. ) In a similarsense I speak of successive applications of more than one operation to anumber of propositions. 

5. 2522 Accordingly I use the sign '[a, x, O'x]' for the general term ofthe series of forms a, O'a, O'O'a, .... This bracketed expression is avariable: the first term of the bracketed expression is the beginningof the series of forms, the second is the form of a term x arbitrarilyselected from the series, and the third is the form of the term thatimmediately follows x in the series. 

5. 2523 The concept of successive applications of an operation isequivalent to the concept 'and so on'. 

5. 253 One operation can counteract the effect of another. Operations cancancel one another. 

5. 254 An operation can vanish (e. G. Negation in 'PPp': PPp = p). 

5. 3 All propositions are results of truth-operations on elementarypropositions. A truth-operation is the way in which a truth-functionis produced out of elementary propositions. It is of the essenceof truth-operations that, just as elementary propositions yield atruth-function of themselves, so too in the same way truth-functionsyield a further truth-function. When a truth-operation is applied totruth-functions of elementary propositions, it always generates anothertruth-function of elementary propositions, another proposition. Whena truth-operation is applied to the results of truth-operationson elementary propositions, there is always a single operation onelementary propositions that has the same result. Every proposition isthe result of truth-operations 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 thatthe propositional sign in 4. 442 expresses a single truth-function ofelementary propositions even when 'p' and 'q' are truth-functions ofelementary propositions. 

5. 32 All truth-functions are results of successive applications toelementary propositions of a finite number of truth-operations. 

5. 4 At this point it becomes manifest that there are no 'logicalobjects' or 'logical constants' (in Frege's and Russell's sense). 

5. 41 The reason is that the results of truth-operations ontruth-functions are always identical whenever they are one and the sametruth-function of elementary propositions. 

5. 42 It is self-evident that C, z, etc. Are not relations in the sensein which right and left etc. Are relations. The interdefinability ofFrege's and Russell's 'primitive signs' of logic is enough to show thatthey are not primitive signs, still less signs for relations. And it isobvious that the 'z' defined by means of 'P' and 'C' is identical withthe one that figures with 'P' in the definition of 'C'; and that thesecond 'C' is identical with the first one; and so on. 

5. 43 Even at first sight it seems scarcely credible that there shouldfollow from one fact p infinitely many others, namely PPp, PPPPp, etc. And it is no less remarkable that the infinite number of propositions oflogic (mathematics) follow from half a dozen 'primitive propositions'. But in fact all the propositions of logic say the same thing, to witnothing. 

5. 44 Truth-functions are not material functions. For example, anaffirmation can be produced by double negation: in such a case does itfollow that in some sense negation is contained in affirmation? Does'PPp' negate Pp, or does it affirm p--or both? The proposition 'PPp' isnot about negation, as if negation were an object: on the other hand, the possibility of negation is already written into affirmation. Andif there were an object called 'P', it would follow that 'PPp' saidsomething different from what 'p' said, just because the one propositionwould then be about P and the other would not. 

5. 441 This vanishing of the apparent logical constants also occurs inthe case of 'P(dx). Pfx', which says the same as '(x). Fx', and in thecase 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 theresults of all truth-operations that have it as their base. 

5. 45 If there are primitive logical signs, then any logic that failsto show clearly how they are placed relatively to one another and tojustify their existence will be incorrect. The construction of logic outof its primitive signs must be made clear. 

5. 451 If logic has primitive ideas, they must be independent of oneanother. If a primitive idea has been introduced, it must have beenintroduced in all the combinations in which it ever occurs. Itcannot, therefore, be introduced first for one combination andlater reintroduced for another. For example, once negation has beenintroduced, we must understand it both in propositions of the form'Pp' and in propositions like 'P(p C q)', '(dx). Pfx', etc. We must notintroduce 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 samein both cases, and no reason would have been given for combining thesigns in the same way in both cases. (In short, Frege's remarks aboutintroducing signs by means of definitions (in The Fundamental Lawsof Arithmetic ) also apply, mutatis mutandis, to the introduction ofprimitive signs. )

5. 452 The introduction of any new device into the symbolism of logicis necessarily a momentous event. In logic a new device should notbe introduced in brackets or in a footnote with what one might calla completely innocent air. (Thus in Russell and Whitehead's PrincipiaMathematica there occur definitions and primitive propositions expressedin words. Why this sudden appearance of words? It would require ajustification, but none is given, or could be given, since the procedureis in fact illicit. ) But if the introduction of a new device has provednecessary at a certain point, we must immediately ask ourselves, 'Atwhat points is the employment of this device now unavoidable?' and itsplace 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 nopre-eminent numbers. 

5. 454 In logic there is no co-ordinate status, and there can be noclassification. In logic there can be no distinction between the generaland the specific. 

5. 4541 The solutions of the problems of logic must be simple, since theyset the standard of simplicity. Men have always had a presentimentthat there must be a realm in which the answers to questions aresymmetrically combined--a priori--to form a self-contained system. Arealm subject to the law: Simplex sigillum veri. 

5. 46 If we introduced logical signs properly, then we should also haveintroduced 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 haveintroduced at the same time the effect of all possible combinations ofbrackets. And thus it would have been made clear that the real generalprimitive signs are not 'p C q', '(dx). Fx', etc. But the most generalform of their combinations. 

5. 461 Though it seems unimportant, it is in fact significant that thepseudo-relations of logic, such as C and z, need brackets--unlike realrelations. Indeed, the use of brackets with these apparently primitivesigns is itself an indication that they are not primitive signs. Andsurely no one is going to believe brackets have an independent meaning. 5. 4611 Signs for logical operations are punctuation-marks. 

5. 47 It is clear that whatever we can say in advance about the formof all propositions, we must be able to say all at once. An elementaryproposition really contains all logical operations in itself. For'fa' says the same thing as '(dx). Fx. X = a' Wherever there iscompositeness, argument and function are present, and where these arepresent, we already have all the logical constants. One could say thatthe sole logical constant was what all propositions, by their verynature, had in common with one another. But that is the generalpropositional 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 ofall description, and thus the essence of the world. 

5. 472 The description of the most general propositional form is thedescription of the one and only general primitive sign in logic. 

5. 473 Logic must look after itself. If a sign is possible, then itis also capable of signifying. Whatever is possible in logic is alsopermitted. (The reason why 'Socrates is identical' means nothing is thatthere is no property called 'identical'. The proposition is nonsensicalbecause we have failed to make an arbitrary determination, and notbecause the symbol, in itself, would be illegitimate. ) In a certainsense, we cannot make mistakes in logic. 

5. 4731 Self-evidence, which Russell talked about so much, can becomedispensable in logic, only because language itself prevents everylogical mistake. --What makes logic a priori is the impossibility ofillogical 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 thatis justified by its success in practice: its point is that unnecessaryunits in a sign-language mean nothing. Signs that serve one purposeare logically equivalent, and signs that serve none are logicallymeaningless. 

5. 4733 Frege says that any legitimately constructed proposition musthave a sense. And I say that any possible proposition is legitimatelyconstructed, and, if it has no sense, that can only be because we havefailed to give a meaning to some of its constituents. (Even if we thinkthat we have done so. ) Thus the reason why 'Socrates is identical' saysnothing is that we have not given any adjectival meaning to the word'identical'. For when it appears as a sign for identity, it symbolizesin an entirely different way--the signifying relation is a differentone--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 dependssolely on our notation. 

5. 475 All that is required is that we should construct a system of signswith a particular number of dimensions--with a particular mathematicalmultiplicity. 

5. 476 It is clear that this is not a question of a number of primitiveideas that have to be signified, but rather of the expression of a rule. 

5. 5 Every truth-function is a result of successive applications toelementary propositions of the operation '(-----T)(E, .... )'. Thisoperation negates all the propositions in the right-hand pair ofbrackets, and I call it the negation of those propositions. 

5. 501 When a bracketed expression has propositions as its terms--and theorder of the terms inside the brackets is indifferent--then I indicateit by a sign of the form '(E)'. '(E)' is a variable whose valuesare terms of the bracketed expression and the bar over the variableindicates that it is the representative of all its values in thebrackets. (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 thevariable as their representative. How the description of the termsof the bracketed expression is produced is not essential. We candistinguish three kinds of description: 1. Direct enumeration, in whichcase we can simply substitute for the variable the constants that areits values; 2. Giving a function fx whose values for all values of x arethe propositions to be described; 3. Giving a formal law that governsthe construction of the propositions, in which case the bracketedexpression 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 thenegation of all the values of the propositional variable E. 

5. 503 It is obvious that we can easily express how propositions may beconstructed with this operation, and how they may not be constructedwith 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 twovalues, then N(E) = Pp. Pq. (neither p nor g). 

5. 511 How can logic--all-embracing logic, which mirrors the world--usesuch peculiar crotchets and contrivances? Only because they are allconnected with one another in an infinitely fine network, the greatmirror. 

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, itis rather what is common to all the signs of this notation that negatep. 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 factormirrors negation. 

5. 513 We might say that what is common to all symbols that affirm both pand q is the proposition 'p. Q'; and that what is common to all symbolsthat affirm either p or q is the proposition 'p C q'. And similarly wecan say that two propositions are opposed to one another if they havenothing in common with one another, and that every proposition has onlyone negative, since there is only one proposition that lies completelyoutside it. Thus in Russell's notation too it is manifest that 'q: p CPp' 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 rulegoverning the construction of all propositions that negate p, a rulegoverning the construction of all propositions that affirm p, and a rulegoverning the construction of all propositions that affirm p or q; andso on. These rules are equivalent to the symbols; and in them theirsense is mirrored. 

5. 515 It must be manifest in our symbols that it can only bepropositions that are combined with one another by 'C', '. ', etc. And this is indeed the case, since the symbol in 'p' and 'q' itselfpresupposes 'C', 'P', etc. If the sign 'p' in 'p C q' does not stand fora complex sign, then it cannot have sense by itself: but in that casethe signs 'p C p', 'p. P', etc. , which have the same sense as p, mustalso lack sense. But if 'p C p' has no sense, then 'p C q' cannot have asense either. 

5. 5151 Must the sign of a negative proposition be constructed with thatof the positive proposition? Why should it not be possible to express anegative 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 usedto say that aRb was not the case. ) But really even in this case thenegative proposition is constructed by an indirect use of the positive. The positive proposition necessarily presupposes the existence of thenegative proposition and vice versa. 

5. 52 If E has as its values all the values of a function fx for allvalues of x, then N(E) = P(dx). Fx. 

5. 521 I dissociate the concept all from truth-functions. Frege andRussell introduced generality in association with logical productorlogical 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 generality-sign is first, that itindicates a logical prototype, and secondly, that it gives prominence toconstants. 

5. 523 The generality-sign occurs as an argument. 

5. 524 If objects are given, then at the same time we are given allobjects. If elementary propositions are given, then at the same time allelementary propositions are given. 

5. 525 It is incorrect to render the proposition '(dx). Fx' in thewords, 'fx is possible' as Russell does. The certainty, possibility, orimpossibility of a situation is not expressed by a proposition, butby an expression's being a tautology, a proposition with a sense, ora contradiction. The precedent to which we are constantly inclined toappeal must reside in the symbol itself. 

5. 526 We can describe the world completely by means of fully generalizedpropositions, i. E. Without first correlating any name with a particularobject. 

5. 5261 A fully generalized proposition, like every other proposition, iscomposite. (This is shown by the fact that in '(dx, O). Ox' we haveto mention 'O' and 's' separately. They both, independently, stand insignifying relations to the world, just as is the case in ungeneralizedpropositions. ) It is a mark of a composite symbol that it has somethingin common with other symbols. 

5. 5262 The truth or falsity of every proposition does make somealteration in the general construction of the world. And the range thatthe totality of elementary propositions leaves open for its constructionis exactly the same as that which is delimited by entirely generalpropositions. (If an elementary proposition is true, that means, at anyrate, one more true elementary proposition. )

5. 53 Identity of object I express by identity of sign, and not by usinga sign for identity. Difference of objects I express by difference ofsigns. 

5. 5301 It is self-evident that identity is not a relation betweenobjects. This becomes very clear if one considers, for example, theproposition '(x) : fx. Z. X = a'. What this proposition says is simplythat only a satisfies the function f, and not that only things that havea certain relation to a satisfy the function, Of course, it might thenbe said that only a did have this relation to a; but in order to expressthat, we should need the identity-sign itself. 

5. 5302 Russell's definition of '=' is inadequate, because according toit we cannot say that two objects have all their properties in common. (Even if this proposition is never correct, it still has sense. )

5. 5303 Roughly speaking, to say of two things that they are identical isnonsense, and to say of one thing that it is identical with itself is tosay nothing at all. 

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). Pa = b', but 'f(a, b)'. 

5. 532 And analogously I do not write '(dx, y). F(x, y). X = y', but'(dx) . F(x, x)'; and not '(dx, y). F(x, y). Px = y', but '(dx, y). F(x, y)'. 5. 5321 Thus, for example, instead of '(x): fx z x = a' we write'(dx). Fx . Z: (dx, y). Fx. Fy'. And the proposition, 'Only one xsatisfies f( )', will read '(dx). Fx: P(dx, y). Fx. Fy'. 

5. 533 The identity-sign, therefore, is not an essential constituent ofconceptual notation. 

5. 534 And now we see that in a correct conceptual notationpseudo-propositions like 'a = a', 'a = b. B = c. Z a = c', '(x). X = x', '(dx). X = a', etc. Cannot even be written down. 

5. 535 This also disposes of all the problems that were connected withsuch pseudo-propositions. All the problems that Russell's 'axiom ofinfinity' brings with it can be solved at this point. What the axiom ofinfinity is intended to say would express itself in language through theexistence of infinitely many names with different meanings. 

5. 5351 There are certain cases in which one is tempted to useexpressions of the form 'a = a' or 'p z p' and the like. In fact, thishappens when one wants to talk about prototypes, e. G. About proposition, thing, etc. Thus in Russell's Principles of Mathematics 'p is aproposition'--which is nonsense--was given the symbolic rendering 'p zp' and placed as an hypothesis in front of certain propositions in orderto exclude from their argument-places everything but propositions. (Itis nonsense to place the hypothesis 'p z p' in front of a proposition, in order to ensure that its arguments shall have the right form, if onlybecause with a non-proposition as argument the hypothesis becomes notfalse but nonsensical, and because arguments of the wrong kind make theproposition itself nonsensical, so that it preserves itself from wrongarguments just as well, or as badly, as the hypothesis without sensethat was appended for that purpose. )

5. 5352 In the same way people have wanted to express, 'There areno things ', by writing 'P(dx). X = x'. But even if this were aproposition, would it not be equally true if in fact 'there were things'but they were not identical with themselves?

5. 54 In the general propositional form propositions occur in otherpropositions only as bases of truth-operations. 

5. 541 At first sight it looks as if it were also possible for oneproposition to occur in another in a different way. Particularly withcertain forms of proposition in psychology, such as 'A believes that pis the case' and A has the thought p', etc. For if these are consideredsuperficially, it looks as if the proposition p stood in some kind ofrelation to an object A. (And in modern theory of knowledge (Russell, Moore, etc. ) these propositions have actually been construed in thisway. )

5. 542 It is clear, however, that 'A believes that p', 'A has the thoughtp', and 'A says p' are of the form '"p" says p': and this doesnot involve a correlation of a fact with an object, but rather thecorrelation of facts by means of the correlation of their objects. 

5. 5421 This shows too that there is no such thing as the soul--thesubject, etc. --as it is conceived in the superficial psychology of thepresent day. Indeed a composite soul would no longer be a soul. 

5. 5422 The correct explanation of the form of the proposition, 'A makesthe judgement p', must show that it is impossible for a judgement tobe a piece of nonsense. (Russell's theory does not satisfy thisrequirement. )

5. 5423 To perceive a complex means to perceive that its constituentsare related to one another in such and such a way. This no doubt alsoexplains why there are two possible ways of seeing the figure as a cube;and all similar phenomena. For we really see two different facts. (If Ilook in the first place at the corners marked a and only glance at theb's, then the a's appear to be in front, and vice versa). 

5. 55 We now have to answer a priori the question about all the possibleforms of elementary propositions. Elementary propositions consist ofnames. Since, however, we are unable to give the number of names withdifferent meanings, we are also unable to give the composition ofelementary propositions. 

5. 551 Our fundamental principle is that whenever a question can bedecided by logic at all it must be possible to decide it without moreado. (And if we get into a position where we have to look at the worldfor an answer to such a problem, that shows that we are on a completelywrong track. )

5. 552 The 'experience' that we need in order to understand logic is notthat something or other is the state of things, but that somethingis: that, however, is not an experience. Logic is prior to everyexperience--that something is so. It is prior to the question 'How?' notprior to the question 'What?'

5. 5521 And if this were not so, how could we apply logic? We might putit in this way: if there would be a logic even if there were no world, how then could there be a logic given that there is a world?

5. 553 Russell said that there were simple relations between differentnumbers of things (individuals). But between what numbers? And how isthis supposed to be decided?--By experience? (There is no pre-eminentnumber. )

5. 554 It would be completely arbitrary to give any specific form. 

5. 5541 It is supposed to be possible to answer a priori the questionwhether I can get into a position in which I need the sign for a27-termed relation in order to signify something. 

5. 5542 But is it really legitimate even to ask such a question? Can weset up a form of sign without knowing whether anything can correspond toit? Does it make sense to ask what there must be in order that somethingcan be the case?

5. 555 Clearly we have some concept of elementary propositions quiteapart from their particular logical forms. But when there is a system bywhich we can create symbols, the system is what is important for logicand not the individual symbols. And anyway, is it really possible thatin logic I should have to deal with forms that I can invent? What I haveto deal with must be that which makes it possible for me to invent them. 

5. 556 There cannot be a hierarchy of the forms of elementarypropositions. We can foresee only what we ourselves construct. 

5. 5561 Empirical reality is limited by the totality of objects. The limit also makes itself manifest in the totality of elementarypropositions. Hierarchies are and must be independent of reality. 

5. 5562 If we know on purely logical grounds that there must beelementary propositions, then everyone who understands propositions intheir C form must know It. 

5. 5563 In fact, all the propositions of our everyday language, just asthey stand, are in perfect logical order. --That utterly simple thing, which we have to formulate here, is not a likeness of the truth, butthe truth itself in its entirety. (Our problems are not abstract, butperhaps the most concrete that there are. )

5. 557 The application of logic decides what elementary propositionsthere are. What belongs to its application, logic cannot anticipate. Itis clear that logic must not clash with its application. But logichas to be in contact with its application. Therefore logic and itsapplication must not overlap. 

5. 5571 If I cannot say a priori what elementary propositions there are, then the attempt to do so must lead to obvious nonsense. 5. 6 The limitsof my language mean the limits of my world. 

5. 61 Logic pervades the world: the limits of the world are also itslimits. So we cannot say in logic, 'The world has this in it, andthis, but not that. ' For that would appear to presuppose that we wereexcluding certain possibilities, and this cannot be the case, since itwould require that logic should go beyond the limits of the world; foronly in that way could it view those limits from the other side as well. We cannot think what we cannot think; so what we cannot think we cannotsay either. 

5. 62 This remark provides the key to the problem, how much truth thereis in solipsism. For what the solipsist means is quite correct; only itcannot be said, but makes itself manifest. The world is my world: thisis manifest in the fact that the limits of language (of that languagewhich alone I understand) mean the limits of my world. 

5. 621 The world and life are one. 

5. 63 I am my world. (The microcosm. )

5. 631 There is no such thing as the subject that thinks or entertainsideas. If I wrote a book called The World as l found it, I should haveto include a report on my body, and should have to say which parts weresubordinate to my will, and which were not, etc. , this being a method ofisolating the subject, or rather of showing that in an important sensethere 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 ofthe world. 

5. 633 Where in the world is a metaphysical subject to be found? You willsay that this is exactly like the case of the eye and the visual field. But really you do not see the eye. And nothing in the visual fieldallows you to infer that it is seen by an eye. 

5. 6331 For the form of the visual field is surely not like this

5. 634 This is connected with the fact that no part of our experience isat 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 apriori order of things. 

5. 64 Here it can be seen that solipsism, when its implications arefollowed out strictly, coincides with pure realism. The self ofsolipsism shrinks to a point without extension, and there remains thereality co-ordinated with it. 

5. 641 Thus there really is a sense in which philosophy can talkabout the self in a non-psychological way. What brings the self intophilosophy is the fact that 'the world is my world'. The philosophicalself is not the human being, not the human body, or the human soul, withwhich psychology deals, but rather the metaphysical subject, the limitof the world--not a part of it. 

6. The general form of a truth-function is [p, E, N(E)]. 

This is the general form of a proposition. 

6. 001 What this says is just that every proposition is a result ofsuccessive applications to elementary propositions of the operation N(E)

6. 002 If we are given the general form according to which propositionsare constructed, then with it we are also given the general formaccording to which one proposition can be generated out of another bymeans of an operation. 

6. 01 Therefore the general form of an operation /'(n) is [E, N(E)]' (n)( = [n, E, N(E)]). This is the most general form of transition from oneproposition to another. 

6. 02 And this is how we arrive at numbers. I give the followingdefinitions x = /0x Def. , /'/v'x = /v+1'x Def. So, in accordance withthese rules, which deal with signs, we write the series x, /'x, /'/'x, /'/'/'x, ... , in the following way /0'x, /0+1'x, /0+1+1'x, /0+1+1+1'x, .... Therefore, instead of '[x, E, /'E]', I write '[/0'x, /v'x, /v+1'x]'. And I give the following definitions 0 + 1 = 1 Def. , 0 +1 + 1 = 2 Def. , 0 + 1 + 1 +1 = 3 Def. , (and so on). 

6. 021 A number is the exponent of an operation. 

6. 022 The concept of number is simply what is common to all numbers, thegeneral form of a number. The concept of number is the variable number. And the concept of numerical equality is the general form of allparticular cases of numerical equality. 

6. 03 The general form of an integer is [0, E, E +1]. 

6. 031 The theory of classes is completely superfluous in mathematics. This is connected with the fact that the generality required inmathematics is not accidental generality. 

6. 1 The propositions of logic are tautologies. 

6. 11 Therefore the propositions of logic say nothing. (They are theanalytic propositions. )

6. 111 All theories that make a proposition of logic appear to havecontent are false. One might think, for example, that the words 'true'and 'false' signified two properties among other properties, and then itwould seem to be a remarkable fact that every proposition possessed oneof these properties. On this theory it seems to be anything but obvious, just as, for instance, the proposition, 'All roses are either yellow orred', would not sound obvious even if it were true. Indeed, the logicalproposition acquires all the characteristics of a proposition of naturalscience and this is the sure sign that it has been construed wrongly. 

6. 112 The correct explanation of the propositions of logic must assignto them a unique status among all propositions. 

6. 113 It is the peculiar mark of logical propositions that one canrecognize that they are true from the symbol alone, and this factcontains in itself the whole philosophy of logic. And so too it isa very important fact that the truth or falsity of non-logicalpropositions cannot be recognized from the propositions alone. 

6. 12 The fact that the propositions of logic are tautologies shows theformal--logical--properties of language and the world. The fact thata tautology is yielded by this particular way of connecting itsconstituents characterizes the logic of its constituents. Ifpropositions are to yield a tautology when they are connected in acertain way, they must have certain structural properties. So theiryielding a tautology when combined in this shows that they possess thesestructural properties. 

6. 1201 For example, the fact that the propositions 'p' and 'Pp' in thecombination '(p. Pp)' yield a tautology shows that they contradict oneanother. The fact that the propositions 'p z q', 'p', and 'q', combinedwith one another in the form '(p z q). (p):z: (q)', yield a tautologyshows that q follows from p and p z q. The fact that '(x). Fxx:z: fa' isa tautology shows that fa follows from (x). Fx. Etc. Etc. 

6. 1202 It is clear that one could achieve the same purpose by usingcontradictions instead of tautologies. 

6. 1203 In order to recognize an expression as a tautology, in caseswhere no generality-sign occurs in it, one can employ the followingintuitive method: instead of 'p', 'q', 'r', etc. I write 'TpF', 'TqF', 'TrF', etc. Truth-combinations I express by means of brackets, e. G. AndI use lines to express the correlation of the truth or falsity of thewhole proposition with the truth-combinations of its truth-arguments, in the following way So this sign, for instance, would representthe proposition p z q. Now, by way of example, I wish to examine theproposition P(p. Pp) (the law of contradiction) in order to determinewhether it is a tautology. In our notation the form 'PE' is written asand the form 'E. N' as Hence the proposition P(p. Pp). Reads as followsIf we here substitute 'p' for 'q' and examine how the outermost T and Fare connected with the innermost ones, the result will be that the truthof the whole proposition is correlated with all the truth-combinationsof its argument, and its falsity with none of the truth-combinations. 

6. 121 The propositions of logic demonstrate the logical propertiesof propositions by combining them so as to form propositions that saynothing. This method could also be called a zero-method. In a logicalproposition, propositions are brought into equilibrium with oneanother, and the state of equilibrium then indicates what the logicalconstitution of these propositions must be. 

6. 122 It follows from this that we can actually do without logicalpropositions; for in a suitable notation we can in fact recognize theformal properties of propositions by mere inspection of the propositionsthemselves. 

6. 1221 If, for example, two propositions 'p' and 'q' in the combination'p z q' yield a tautology, then it is clear that q follows from p. Forexample, we see from the two propositions themselves that 'q' followsfrom 'p z q. P', but it is also possible to show it in this way: wecombine them to form 'p z q. P:z: q', and then show that this is atautology. 

6. 1222 This throws some light on the question why logical propositionscannot be confirmed by experience any more than they can be refuted byit. Not only must a proposition of logic be irrefutable by anypossible experience, but it must also be unconfirmable by any possibleexperience. 

6. 1223 Now it becomes clear why people have often felt as if it werefor us to 'postulate' the 'truths of logic'. The reason is that we canpostulate them in so far as we can postulate an adequate notation. 

6. 1224 It also becomes clear now why logic was called the theory offorms and of inference. 

6. 123 Clearly the laws of logic cannot in their turn be subject tolaws of logic. (There is not, as Russell thought, a special law ofcontradiction for each 'type'; one law is enough, since it is notapplied to itself. )

6. 1231 The mark of a logical proposition is not general validity. To begeneral means no more than to be accidentally valid for all things. An ungeneralized proposition can be tautological just as well as ageneralized one. 

6. 1232 The general validity of logic might be called essential, incontrast with the accidental general validity of such propositionsas 'All men are mortal'. Propositions like Russell's 'axiom ofreducibility' are not logical propositions, and this explains ourfeeling that, even if they were true, their truth could only be theresult of a fortunate accident. 

6. 1233 It is possible to imagine a world in which the axiom ofreducibility is not valid. It is clear, however, that logic has nothingto do with the question whether our world really is like that or not. 

6. 124 The propositions of logic describe the scaffolding of the world, or rather they represent it. They have no 'subject-matter'. Theypresuppose that names have meaning and elementary propositions sense;and that is their connexion with the world. It is clear that somethingabout the world must be indicated by the fact that certain combinationsof symbols--whose essence involves the possession of a determinatecharacter--are tautologies. This contains the decisive point. We havesaid that some things are arbitrary in the symbols that we use and thatsome things are not. In logic it is only the latter that express: butthat means that logic is not a field in which we express what we wishwith the help of signs, but rather one in which the nature of theabsolutely necessary signs speaks for itself. If we know the logicalsyntax of any sign-language, then we have already been given all thepropositions of logic. 

6. 125 It is possible--indeed possible even according to the oldconception of logic--to give in advance a description of all 'true'logical propositions. 

6. 1251 Hence there can never be surprises in logic. 

6. 126 One can calculate whether a proposition belongs to logic, bycalculating the logical properties of the symbol. And this is what wedo when we 'prove' a logical proposition. For, without bothering aboutsense or meaning, we construct the logical proposition out of othersusing only rules that deal with signs. The proof of logical propositionsconsists in the following process: we produce them out of other logicalpropositions by successively applying certain operations that alwaysgenerate further tautologies out of the initial ones. (And in fact onlytautologies follow from a tautology. ) Of course this way of showing thatthe propositions of logic are tautologies is not at all essential tologic, if only because the propositions from which the proof starts mustshow without any proof that they are tautologies. 

6. 1261 In logic process and result are equivalent. (Hence the absence ofsurprise. )

6. 1262 Proof in logic is merely a mechanical expedient to facilitate therecognition of tautologies in complicated cases. 

6. 1263 Indeed, it would be altogether too remarkable if a propositionthat had sense could be proved logically from others, and so too could alogical proposition. It is clear from the start that a logical proof ofa proposition that has sense and a proof in logic must be two entirelydifferent things. 

6. 1264 A proposition that has sense states something, which is shown byits proof to be so. In logic every proposition is the form of a proof. Every proposition of logic is a modus ponens represented in signs. (Andone cannot express the modus ponens by means of a proposition. )

6. 1265 It is always possible to construe logic in such a way that everyproposition is its own proof. 

6. 127 All the propositions of logic are of equal status: it is notthe case that some of them are essentially derived propositions. Everytautology itself shows that it is a tautology. 

6. 1271 It is clear that the number of the 'primitive propositionsof logic' is arbitrary, since one could derive logic from a singleprimitive proposition, e. G. By simply constructing the logical productof Frege's primitive propositions. (Frege would perhaps say thatwe should then no longer have an immediately self-evident primitiveproposition. But it is remarkable that a thinker as rigorous as Fregeappealed to the degree of self-evidence as the criterion of a logicalproposition. )

6. 13 Logic is not a body of doctrine, but a mirror-image of the world. Logic is transcendental. 

6. 2 Mathematics is a logical method. The propositions of mathematics areequations, and therefore pseudo-propositions. 

6. 21 A proposition of mathematics does not express a thought. 

6. 211 Indeed in real life a mathematical proposition is never whatwe want. Rather, we make use of mathematical propositions only ininferences from propositions that do not belong to mathematics to othersthat likewise do not belong to mathematics. (In philosophy the question, 'What do we actually use this word or this proposition for?' repeatedlyleads to valuable insights. )

6. 22 The logic of the world, which is shown in tautologies by thepropositions of logic, is shown in equations by mathematics. 

6. 23 If two expressions are combined by means of the sign of equality, that means that they can be substituted for one another. But it must bemanifest in the two expressions themselves whether this is the caseor not. When two expressions can be substituted for one another, thatcharacterizes their logical form. 

6. 231 It is a property of affirmation that it can be construed as doublenegation. It is a property of '1 + 1 + 1 + 1' that it can be construedas '(1 + 1) + (1 + 1)'. 

6. 232 Frege says that the two expressions have the same meaning butdifferent senses. But the essential point about an equation is that itis not necessary in order to show that the two expressions connected bythe sign of equality have the same meaning, since this can be seen fromthe two expressions themselves. 

6. 2321 And the possibility of proving the propositions of mathematicsmeans simply that their correctness can be perceived without its beingnecessary that what they express should itself be compared with thefacts in order to determine its correctness. 

6. 2322 It is impossible to assert the identity of meaning of twoexpressions. For in order to be able to assert anything about theirmeaning, I must know their meaning, and I cannot know their meaningwithout knowing whether what they mean is the same or different. 

6. 2323 An equation merely marks the point of view from which I considerthe two expressions: it marks their equivalence in meaning. 

6. 233 The question whether intuition is needed for the solution ofmathematical problems must be given the answer that in this caselanguage itself provides the necessary intuition. 

6. 2331 The process of calculating serves to bring about that intuition. Calculation is not an experiment. 

6. 234 Mathematics is a method of logic. 

6. 2341 It is the essential characteristic of mathematical method thatit employs equations. For it is because of this method that everyproposition of mathematics must go without saying. 

6. 24 The method by which mathematics arrives at its equations is themethod of substitution. For equations express the substitutability oftwo expressions and, starting from a number of equations, we advance tonew equations by substituting different expressions in accordance withthe equations. 

6. 241 Thus the proof of the proposition 2 t 2 = 4 runs as follows:(/v)n'x = /v x u'x Def. , /2 x 2'x = (/2)2'x = (/2)1 + 1'x = /2' /2'x =/1 + 1'/1 + 1'x = (/'/)'(/'/)'x =/'/'/'/'x = /1 + 1 + 1 + 1'x = /4'x. 6. 3 The exploration of logic means the exploration of everything that issubject to law. And outside logic everything is accidental. 

6. 31 The so-called law of induction cannot possibly be a law of logic, since it is obviously a proposition with sense. ---Nor, therefore, can itbe an a priori law. 

6. 32 The law of causality is not a law but the form of a law. 

6. 321 'Law of causality'--that is a general name. And just as inmechanics, for example, there are 'minimum-principles', such as the lawof least action, so too in physics there are causal laws, laws of thecausal form. 

6. 3211 Indeed people even surmised that there must be a 'law of leastaction' before they knew exactly how it went. (Here, as always, what iscertain a priori proves to be something purely logical. )

6. 33 We do not have an a priori belief in a law of conservation, butrather a priori knowledge of the possibility of a logical form. 

6. 34 All such propositions, including the principle of sufficientreason, tile laws of continuity in nature and of least effort in nature, etc. Etc. --all these are a priori insights about the forms in which thepropositions of science can be cast. 

6. 341 Newtonian mechanics, for example, imposes a unified form on thedescription of the world. Let us imagine a white surface with irregularblack 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 itby covering the surface with a sufficiently fine square mesh, and thensaying of every square whether it is black or white. In this way I shallhave imposed a unified form on the description of the surface. The formis optional, since I could have achieved the same result by using a netwith a triangular or hexagonal mesh. Possibly the use of a triangularmesh would have made the description simpler: that is to say, it mightbe that we could describe the surface more accurately with a coarsetriangular mesh than with a fine square mesh (or conversely), and so on. The different nets correspond to different systems for describing theworld. Mechanics determines one form of description of the world bysaying that all propositions used in the description of the world mustbe obtained in a given way from a given set of propositions--the axiomsof mechanics. It thus supplies the bricks for building the edifice ofscience, and it says, 'Any building that you want to erect, whatever itmay be, must somehow be constructed with these bricks, and with thesealone. ' (Just as with the number-system we must be able to write downany number we wish, so with the system of mechanics we must be able towrite 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. Wecould use both triangles and hexagons. ) The possibility of describing apicture like the one mentioned above with a net of a given form tells usnothing about the picture. (For that is true of all such pictures. )But what does characterize the picture is that it can be describedcompletely by a particular net with a particular size of mesh. Similarlythe possibility of describing the world by means of Newtonian mechanicstells us nothing about the world: but what does tell us something aboutit is the precise way in which it is possible to describe it by thesemeans. We are also told something about the world by the fact that itcan be described more simply with one system of mechanics than withanother. 

6. 343 Mechanics is an attempt to construct according to a single planall the true propositions that we need for the description of the world. 

6. 3431 The laws of physics, with all their logical apparatus, stillspeak, however indirectly, about the objects of the world. 

6. 3432 We ought not to forget that any description of the world by meansof mechanics will be of the completely general kind. For example, itwill never mention particular point-masses: it will only talk about anypoint-masses whatsoever. 

6. 35 Although the spots in our picture are geometrical figures, nevertheless geometry can obviously say nothing at all about theiractual form and position. The network, however, is purely geometrical;all its properties can be given a priori. Laws like the principle ofsufficient reason, etc. Are about the net and not about what the netdescribes. 

6. 36 If there were a law of causality, it might be put in the followingway: There are laws of nature. But of course that cannot be said: itmakes itself manifest. 

6. 361 One might say, using Hertt:'s terminology, that only connexionsthat are subject to law are thinkable. 

6. 3611 We cannot compare a process with 'the passage of time'--there isno such thing--but only with another process (such as the working of achronometer). Hence we can describe the lapse of time only by relying onsome other process. Something exactly analogous applies to space: e. G. When people say that neither of two events (which exclude one another)can occur, because there is nothing to cause the one to occur ratherthan the other, it is really a matter of our being unable to describeone of the two events unless there is some sort of asymmetry to befound. And if such an asymmetry is to be found, we can regard it as thecause of the occurrence of the one and the non-occurrence of the other. 

6. 36111 Kant's problem about the right hand and the left hand, whichcannot be made to coincide, exists even in two dimensions. Indeed, itexists in one-dimensional space in which the two congruent figures, a and b, cannot be made to coincide unless they are moved out ofthis space. The right hand and the left hand are in fact completelycongruent. It is quite irrelevant that they cannot be made to coincide. A right-hand glove could be put on the left hand, if it could be turnedround in four-dimensional space. 

6. 362 What can be described can happen too: and what the law ofcausality is meant to exclude cannot even be described. 

6. 363 The procedure of induction consists in accepting as true thesimplest law that can be reconciled with our experiences. 

6. 3631 This procedure, however, has no logical justification but only apsychological one. It is clear that there are no grounds for believingthat the simplest eventuality will in fact be realized. 

6. 36311 It is an hypothesis that the sun will rise tomorrow: and thismeans that we do not know whether it will rise. 

6. 37 There is no compulsion making one thing happen because another hashappened. The only necessity that exists is logical necessity. 

6. 371 The whole modern conception of the world is founded on theillusion that the so-called laws of nature are the explanations ofnatural phenomena. 

6. 372 Thus people today stop at the laws of nature, treating them assomething inviolable, just as God and Fate were treated in past ages. And in fact both are right and both wrong: though the view of theancients is clearer in so far as they have a clear and acknowledgedterminus, while the modern system tries to make it look as if everythingwere explained. 

6. 373 The world is independent of my will. 

6. 374 Even if all that we wish for were to happen, still this wouldonly be a favour granted by fate, so to speak: for there is no logicalconnexion between the will and the world, which would guarantee it, andthe supposed physical connexion itself is surely not something that wecould will. 

6. 375 Just as the only necessity that exists is logical necessity, sotoo the only impossibility that exists is logical impossibility. 

6. 3751 For example, the simultaneous presence of two colours at the sameplace in the visual field is impossible, in fact logically impossible, since it is ruled out by the logical structure of colour. Let us thinkhow this contradiction appears in physics: more or less as follows--aparticle cannot have two velocities at the same time; that is to say, itcannot be in two places at the same time; that is to say, particles thatare in different places at the same time cannot be identical. (Itis clear that the logical product of two elementary propositions canneither be a tautology nor a contradiction. The statement that a pointin the visual field has two different colours at the same time is acontradiction. )

6. 4 All propositions are of equal value. 

6. 41 The sense of the world must lie outside the world. In the worldeverything is as it is, and everything happens as it does happen: in itno value exists--and if it did exist, it would have no value. If thereis any value that does have value, it must lie outside the whole sphereof what happens and is the case. For all that happens and is the caseis accidental. What makes it non-accidental cannot lie within the world, since if it did it would itself be accidental. It must lie outside theworld. 

6. 42 So too it is impossible for there to be propositions of ethics. Propositions can express nothing that is higher. 

6. 421 It is clear that ethics cannot be put into words. Ethics istranscendental. (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 theusual sense of the terms. So our question about the consequences of anaction must be unimportant. --At least those consequences should not beevents. 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 thatthe reward must be something pleasant and the punishment somethingunpleasant. )

6. 423 It is impossible to speak about the will in so far as it isthe subject of ethical attributes. And the will as a phenomenon is ofinterest only to psychology. 

6. 43 If the good or bad exercise of the will does alter the world, itcan alter only the limits of the world, not the facts--not what canbe expressed by means of language. In short the effect must be that itbecomes an altogether different world. It must, so to speak, wax andwane as a whole. The world of the happy man is a different one from thatof the unhappy man. 

6. 431 So too at death the world does not alter, but comes to an end. 

6. 4311 Death is not an event in life: we do not live to experiencedeath. If we take eternity to mean not infinite temporal durationbut timelessness, then eternal life belongs to those who live in thepresent. Our life has no end in just the way in which our visual fieldhas no limits. 

6. 4312 Not only is there no guarantee of the temporal immortality of thehuman soul, that is to say of its eternal survival after death; but, inany case, this assumption completely fails to accomplish the purposefor which it has always been intended. Or is some riddle solved by mysurviving for ever? Is not this eternal life itself as much of a riddleas our present life? The solution of the riddle of life in space andtime lies outside space and time. (It is certainly not the solution ofany problems of natural science that is required. )

6. 432 How things are in the world is a matter of complete indifferencefor what is higher. God does not reveal himself in the world. 

6. 4321 The facts all contribute only to setting the problem, not to itssolution. 

6. 44 It is not how things are in the world that is mystical, but that itexists. 

6. 45 To view the world sub specie aeterni is to view it as a whole--alimited whole. Feeling the world as a limited whole--it is this that ismystical. 

6. 5 When the answer cannot be put into words, neither can the questionbe put into words. The riddle does not exist. If a question can beframed at all, it is also possible to answer it. 

6. 51 Scepticism is not irrefutable, but obviously nonsensical, when ittries to raise doubts where no questions can be asked. For doubt canexist only where a question exists, a question only where an answerexists, and an answer only where something can be said. 

6. 52 We feel that even when all possible scientific questions have beenanswered, the problems of life remain completely untouched. Of coursethere are then no questions left, and this itself is the answer. 

6. 521 The solution of the problem of life is seen in the vanishing ofthe problem. (Is not this the reason why those who have found after along period of doubt that the sense of life became clear to them havethen been unable to say what constituted that sense?)

6. 522 There are, indeed, things that cannot be put into words. They makethemselves manifest. They are what is mystical. 

6. 53 The correct method in philosophy would really be the following:to say nothing except what can be said, i. E. Propositions of naturalscience--i. E. Something that has nothing to do with philosophy--andthen, whenever someone else wanted to say something metaphysical, todemonstrate to him that he had failed to give a meaning to certain signsin his propositions. Although it would not be satisfying to theother person--he would not have the feeling that we were teaching himphilosophy--this method would be the only strictly correct one. 

6. 54 My propositions are elucidatory in this way: he who understands mefinally recognizes them as senseless, when he has climbed out throughthem, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it. ) He must transcend these propositions, and then he will see the world aright. 

7. What we cannot speak about we must pass over in silence.