Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| The general form of a truth-function is [p, E, N(E)] |
| What we cannot speak about we must pass over in silence. |
| What is the case--a fact--is the existence of states of affairs. |
| A thought is a proposition with a sense. |
| Propositions represent the existence and non-existence of states of affairs. |
| The sense of a proposition is its agreement and disagreement with |
| possibilities of existence and nonexistence of states of affairs. |
| Truth-possibilities of elementary propositions mean Possibilities of |
| existence and non-existence of states of affairs |
| We can represent truth-possibilities by schemata of the following |
| kind ('T' means 'true', 'F' means 'false'; the rows of 'T's' and 'F's' under the row of |
| elementary propositions symbolize their truthpossibilities in a way that can easily be |
| A proposition is an expression of agreement and disagreement with |
| truth-possibilities of elementary propositions. |
| It now seems possible to give the most general propositional form |
| Suppose that I am given all elementary propositions: then I can |
| simply ask what propositions I can construct out of them. And there I have all propositions, |
| and that fixes their limits. |
| Propositions comprise all that follows from the totality of all |
| elementary propositions (and, of course, from its being the totality of them all). (Thus, in |
| a certain sense, it could be said that all propositions were generalizations of elementary |
| The general propositional form is a variable. |
| A proposition is a truth-function of elementary propositions. |
| Elementary propositions are the truth-arguments of propositions. |
| The arguments of functions are readily confused with the affixes of |
| Truth-functions can be arranged in series. That is the foundation of the |
| The certainty of logical inference is a limiting case of probability |
| A logical picture of facts is a thought. |
| The world is all that is the case. |
| The world divides into facts. |
| Each item can be the case or not the case while everything else |
| The world is the totality of facts, not of things. |
| The world is determined by the facts, and by their being all the |
| For the totality of facts determines what is the case, and also |
| whatever is not the case. |
| The facts in logical space are the world. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| A state of affairs (a state of things) is a combination of objects (things) |
| It is essential to things that they should be possible constituents of states of |
| In logic nothing is accidental: if a thing can occur in a state of affairs, the |
| possibility of the state of affairs must be written into the thing itself. |
| It would seem to be a sort of accident, if it turned out that a situation |
| would fit a thing that could already exist entirely on its own. If things can occur in |
| states of affairs, this possibility must be in them from the beginning. (Nothing in the |
| province of logic can be merely possible. Logic deals with every possibility and all |
| possibilities are its facts.) Just as we are quite unable to imagine spatial objects outside |
| space or temporal objects outside time, so too there is no object that we can imagine |
| excluded from the possibility of combining with others. If I can imagine objects combined in |
| states of affairs, I cannot imagine them excluded from the possibility of such combinations. |
| Things are independent in so far as they can occur in all possible |
| situations, but this form of independence is a form of connexion with states of affairs, a |
| form of dependence. (It is impossible for words to appear in two different roles: by |
| themselves, and in propositions.) |
| If I know an object I also know all its possible occurrences in states of |
| affairs. (Every one of these possibilities must be part of the nature of the object.) A new |
| possibility cannot be discovered later. |
| If I am to know an object, thought I need not know its external |
| properties, I must know all its internal properties. |
| If all objects are given, then at the same time all possible states of |
| Each thing is, as it were, in a space of possible states of affairs. This space I |
| can imagine empty, but I cannot imagine the thing without the space. |
| A spatial object must be situated in infinite space. (A spatial point is an |
| argumentplace.) A speck in the visual field, thought it need not be red, must have some |
| colour: it is, so to speak, surrounded by colour-space. Notes must have some pitch, objects |
| of the sense of touch some degree of hardness, and so on. |
| Objects contain the possibility of all situations. |
| The possibility of its occurring in states of affairs is the form of an |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| Every statement about complexes can be resolved into a statement about their |
| constituents and into the propositions that describe the complexes completely. |
| Objects make up the substance of the world. That is why they cannot be composite. |
| If they world had no substance, then whether a proposition had sense would |
| depend on whether another proposition was true. |
| In that case we could not sketch any picture of the world (true or false). |
| It is obvious that an imagined world, however difference it may be from the real |
| one, must have something-- a form--in common with it. |
| Objects are just what constitute this unalterable form. |
| The substance of the world can only determine a form, and not any material |
| properties. For it is only by means of propositions that material properties are |
| represented--only by the configuration of objects that they are produced. |
| In a manner of speaking, objects are colourless. |
| If two objects have the same logical form, the only distinction between |
| them, apart from their external properties, is that they are different. |
| Either a thing has properties that nothing else has, in which case |
| we can immediately use a description to distinguish it from the others and refer to it; or, |
| on the other hand, there are several things that have the whole set of their properties in |
| common, in which case it is quite impossible to indicate one of them. For it there is |
| nothing to distinguish a thing, I cannot distinguish it, since otherwise it would be |
| The substance is what subsists independently of what is the case. |
| Space, time, colour (being coloured) are forms of objects. |
| There must be objects, if the world is to have unalterable form. |
| Objects, the unalterable, and the subsistent are one and the same. |
| Objects are what is unalterable and subsistent; their configuration is what |
| is changing and unstable. |
| The configuration of objects produces states of affairs. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| The fact that the propositions of logic are tautologies shows the |
| formal--logical--properties of language and the world. |
| It is clear that one could achieve the same purpose by using contradictions |
| In order to recognize an expression as a tautology, in cases where no |
| generalitysign occurs in it, one can employ the following intuitive method: instead of 'p', |
| 'q', 'r', etc. I write 'TpF', 'TqF', 'TrF', etc. Truth-combinations I express by means of |
| brackets, e.g. and I use lines to express the correlation of the truth or falsity of the |
| whole proposition with the truth-combinations of its truth-arguments, in the following way |
| So this sign, for instance, would represent the proposition p z q. Now, by way of example, I |
| wish to examine the proposition P(p .Pp) (the law of contradiction) in order to determine |
| whether it is a tautology. In our notation the form 'PE' is written as and the form 'E . n' |
| as Hence the proposition P(p . Pp). reads as follows If we here substitute 'p' for 'q' and |
| examine how the outermost T and F are connected with the innermost ones, the result will be |
| that the truth of the whole proposition is correlated with all the truth-combinations of its |
| argument, and its falsity with none of the truth-combinations. |
| The propositions of logic demonstrate the logical properties of propositions by |
| combining them so as to form propositions that say nothing. |
| It follows from this that we can actually do without logical propositions |
| 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. For example, we see from the two |
| propositions themselves that 'q' follows from 'p z q . p', but it is also possible to show |
| it in this way: we combine them to form 'p z q . p :z: q', and then show that this is a |
| This throws some light on the question why logical propositions cannot be |
| confirmed by experience any more than they can be refuted by it. Not only must a proposition |
| of logic be irrefutable by any possible experience, but it must also be unconfirmable by any |
| Now it becomes clear why people have often felt as if it were for us to |
| 'postulate ' the 'truths of logic'. The reason is that we can postulate them in so far as we |
| can postulate an adequate notation. |
| It also becomes clear now why logic was called the theory of forms and of |
| Clearly the laws of logic cannot in their turn be subject to laws of logic. |
| The mark of a logical proposition is not general validity. To be general |
| means no more than to be accidentally valid for all things. An ungeneralized proposition can |
| be tautological just as well as a generalized one. |
| The general validity of logic might be called essential, in contrast with |
| the accidental general validity of such propositions as 'All men are mortal'. Propositions |
| like Russell's 'axiom of reducibility' are not logical propositions, and this explains our |
| feeling that, even if they were true, their truth could only be the result of a fortunate |
| It is possible to imagine a world in which the axiom of reducibility is not |
| valid. It is clear, however, that logic has nothing to do with the question whether our |
| world really is like that or not. |
| The propositions of logic describe the scaffolding of the world, or rather they |
| It is possible--indeed possible even according to the old conception of logic--to |
| give in advance a description of all 'true' logical propositions. |
| Hence there can never be surprises in logic. |
| One can calculate whether a proposition belongs to logic, by calculating the logical |
| properties of the symbol. |
| In logic process and result are equivalent. (Hence the absence of surprise.) |
| Proof in logic is merely a mechanical expedient to facilitate the |
| recognition of tautologies in complicated cases. |
| Indeed, it would be altogether too remarkable if a proposition that had |
| sense could be proved logically from others, and so too could a logical proposition. It is |
| clear from the start that a logical proof of a proposition that has sense and a proof in |
| logic must be two entirely different things. |
| A proposition that has sense states something, which is shown by its 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. (And one cannot express the modus ponens by means of a |
| It is always possible to construe logic in such a way that every proposition |
| All the propositions of logic are of equal status |
| It is clear that the number of the 'primitive propositions of logic' is |
| arbitrary, since one could derive logic from a single primitive proposition, e.g. by simply |
| constructing the logical product of Frege's primitive propositions. (Frege would perhaps say |
| that we should then no longer have an immediately self-evident primitive proposition. But it |
| is remarkable that a thinker as rigorous as Frege appealed to the degree of self-evidence as |
| the criterion of a logical proposition.) |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| If we are given the general form according to which propositions are constructed, then with |
| it we are also given the general form according to which one proposition can be generated |
| out of another by means of an operation. |
| 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 one proposition to another. |
| And this is how we arrive at numbers. I give the following definitions x = /0x Def., |
| /'/v'x = /v+1'x Def. So, in accordance with these 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., |
| A number is the exponent of an operation. |
| The concept of number is simply what is common to all numbers, the general |
| form of a number. The concept of number is the variable number. And the concept of numerical |
| equality is the general form of all particular cases of numerical equality. |
| The general form of an integer is [0, E, E +1]. |
| The theory of classes is completely superfluous in mathematics. This is |
| connected with the fact that the generality required in mathematics is not accidental |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| Mathematics is a logical method. The propositions of mathematics are equations, and |
| therefore pseudopropositions. |
| A proposition of mathematics does not express a thought. |
| Indeed in real life a mathematical proposition is never what we want. |
| Rather, we make use of mathematical propositions only in inferences from propositions that |
| do not belong to mathematics to others that likewise do not belong to mathematics. (In |
| philosophy the question, 'What do we actually use this word or this proposition for?' |
| repeatedly leads to valuable insights.) |
| The logic of the world, which is shown in tautologies by the propositions of logic, |
| is shown in equations by mathematics. |
| 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 be manifest in the two expressions |
| themselves whether this is the case or not. When two expressions can be substituted for one |
| another, that characterizes their logical form. |
| It is a property of affirmation that it can be construed as double negation. |
| It is a property of'1 + 1 + 1 + 1' that it can be construed as '(1 + 1) + (1 + 1)'. |
| Frege says that the two expressions have the same meaning but different |
| senses. But the essential point about an equation is that it is not necessary in order to |
| show that the two expressions connected by the sign of equality have the same meaning, since |
| this can be seen from the two expressions themselves. |
| And the possibility of proving the propositions of mathematics means |
| simply that their correctness can be perceived without its being necessary that what they |
| express should itself be compared with the facts in order to determine its correctness. |
| It is impossible to assert the identity of meaning of two |
| expressions. For in order to be able to assert anything about their meaning, I must know |
| their meaning, and I cannot know their meaning without knowing whether what they mean is the |
| An equation merely marks the point of view from which I consider the |
| two expressions: it marks their equivalence in meaning. |
| The question whether intuition is needed for the solution of mathematical |
| problems must be given the answer that in this case language itself provides the necessary |
| The process of calculating serves to bring about that intuition. |
| Calculation is not an experiment. |
| Mathematics is a method of logic. |
| It is the essential characteristic of mathematical method that it |
| employs equations. For it is because of this method that every proposition of mathematics |
| The method by which mathematics arrives at its equations is the method of |
| substitution. For equations express the substitutability of two expressions and, starting |
| from a number of equations, we advance to new equations by substituting different |
| expressions in accordance with the equations. |
| 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. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| The propositions of logic are tautologies. |
| Therefore the propositions of logic say nothing. (They are the analytic |
| All theories that make a proposition of logic appear to have content are |
| false. One might think, for example, that the words 'true' and 'false' signified two |
| properties among other properties, and then it would seem to be a remarkable fact that every |
| proposition possessed one of these properties. On this theory it seems to be anything but |
| obvious, just as, for instance, the proposition, 'All roses are either yellow or red', would |
| not sound obvious even if it were true. Indeed, the logical proposition acquires all the |
| characteristics of a proposition of natural science and this is the sure sign that it has |
| The correct explanation of the propositions of logic must assign to them a |
| unique status among all propositions. |
| It is the peculiar mark of logical propositions that one can recognize that |
| they are true from the symbol alone, and this fact contains in itself the whole philosophy |
| of logic. And so too it is a very important fact that the truth or falsity of non-logical |
| propositions cannot be recognized from the propositions alone. |
| The fact that the propositions of logic are tautologies shows the |
| formal--logical--properties of language and the world. The fact that a tautology is yielded |
| by this particular way of connecting its constituents characterizes the logic of its |
| constituents. If propositions are to yield a tautology when they are connected in a certain |
| way, they must have certain structural properties. So their yielding a tautology when |
| combined in this shows that they possess these structural properties. |
| Logic is not a body of doctrine, but a mirror-image of the world. Logic is |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| All such propositions, including the principle of sufficient reason, 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 the propositions of science can be cast. |
| Newtonian mechanics, for example, imposes a unified form on the description of the |
| world. Let us imagine a white surface with irregular black spots on it. We then say that |
| whatever kind of picture these make, I can always approximate as closely as I wish to the |
| description of it by covering the surface with a sufficiently fine square mesh, and then |
| saying of every square whether it is black or white. In this way I shall have imposed a |
| unified form on the description of the surface. The form is optional, since I could have |
| achieved the same result by using a net with a triangular or hexagonal mesh. Possibly the |
| use of a triangular mesh would have made the description simpler: that is to say, it might |
| be that we could describe the surface more accurately with a coarse triangular mesh than |
| with a fine square mesh (or conversely), and so on. The different nets correspond to |
| different systems for describing the world. Mechanics determines one form of description of |
| the world by saying that all propositions used in the description of the world must be |
| obtained in a given way from a given set of propositions--the axioms of mechanics. It thus |
| supplies the bricks for building the edifice of science, and it says, 'Any building that you |
| want to erect, whatever it may be, must somehow be constructed with these bricks, and with |
| these alone.' (Just as with the number-system we must be able to write down any number we |
| wish, so with the system of mechanics we must be able to write down any proposition of |
| And now we can see the relative position of logic and mechanics. (The net might also |
| consist of more than one kind of mesh: e.g. we could use both triangles and hexagons.) The |
| possibility of describing a picture like the one mentioned above with a net of a given form |
| tells us nothing about the picture. (For that is true of all such pictures.) But what does |
| characterize the picture is that it can be described completely by a particular net with a |
| particular size of mesh. Similarly the possibility of describing the world by means of |
| Newtonian mechanics tells us nothing about the world: but what does tell us something about |
| it is the precise way in which it is possible to describe it by these means. We are also |
| told something about the world by the fact that it can be described more simply with one |
| system of mechanics than with another. |
| Mechanics is an attempt to construct according to a single plan all the true |
| propositions that we need for the description of the world. |
| The laws of physics, with all their logical apparatus, still speak, however |
| indirectly, about the objects of the world. |
| We ought not to forget that any description of the world by means of |
| mechanics will be of the completely general kind. For example, it will never mention |
| particular pointmasses: it will only talk about any point-masses whatsoever. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| If there were a law of causality, it might be put in the following way: There are laws of |
| nature. But of course that cannot be said: it makes itself manifest. |
| One might say, using Hertt:'s terminology, that only connexions that are subject to |
| We cannot compare a process with 'the passage of time'--there is no such |
| thing-- but only with another process (such as the working of a chronometer). Hence we can |
| describe the lapse of time only by relying on some 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 rather than the |
| other, it is really a matter of our being unable to describe one of the two events unless |
| there is some sort of asymmetry to be found. And if such an asymmetry is to be found, we can |
| regard it as the cause of the occurrence of the one and the nonoccurrence of the other. |
| Kant's problem about the right hand and the left hand, which cannot |
| be made to coincide, exists even in two dimensions. Indeed, it exists in onedimensional |
| space in which the two congruent figures, a and b, cannot be made to coincide unless they |
| are moved out of this space. The right hand and the left hand are in fact completely |
| congruent. 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 turned round in four-dimensional space. |
| What can be described can happen too: and what the law of causality is meant to |
| exclude cannot even be described. |
| The procedure of induction consists in accepting as true the simplest law that can |
| be reconciled with our experiences. |
| This procedure, however, has no logical justification but only a |
| psychological one. It is clear that there are no grounds for believing that the simplest |
| eventuality will in fact be realized. |
| It is an hypothesis that the sun will rise tomorrow: and this means |
| that we do not know whether it will rise. |
| We picture facts to ourselves. |
| A picture presents a situation in logical space, the existence and non-existence of |
| A picture is a model of reality. |
| In a picture objects have the elements of the picture corresponding to them. |
| In a picture the elements of the picture are the representatives of objects. |
| What constitutes a picture is that its elements are related to one another in a |
| The fact that the elements of a picture are related to one another in a determinate |
| way represents that things are related to one another in the same way. Let us call this |
| connexion of its elements the structure of the picture, and let us call the possibility of |
| this structure the pictorial form of the picture. |
| Pictorial form is the possibility that things are related to one another in |
| the same way as the elements of the picture. |
| That is how a picture is attached to reality; it reaches right out |
| It is laid against reality like a measure. |
| Only the end-points of the graduating lines actually touch |
| the object that is to be measured. |
| So a picture, conceived in this way, also includes the pictorial |
| relationship, which makes it into a picture. |
| These correlations are, as it were, the feelers of the picture's |
| elements, with which the picture touches reality. |
| If a fact is to be a picture, it must have something in common with what it depicts. |
| There must be something identical in a picture and what it depicts, to |
| enable the one to be a picture of the other at all. |
| What a picture must have in common with reality, in order to be able to depict |
| it--correctly or incorrectly--in the way that it does, is its pictorial form. |
| A picture can depict any reality whose form it has. A spatial picture can |
| depict anything spatial, a coloured one anything coloured, etc. |
| A picture cannot, however, depict its pictorial form: it displays it. |
| A picture represents its subject from a position outside it. (Its standpoint |
| is its representational form.) That is why a picture represents its subject correctly or |
| A picture cannot, however, place itself outside its representational form. |
| What any picture, of whatever form, must have in common with reality, in order to be |
| able to depict it--correctly or incorrectly--in any way at all, is logical form, i.e. the |
| A picture whose pictorial form is logical form is called a logical picture. |
| Every picture is at the same time a logical one. (On the other hand, not |
| every picture is, for example, a spatial one.) |
| Logical pictures can depict the world. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| What is the case--a fact--is the existence of states of affairs. |
| A state of affairs (a state of things) is a combination of objects (things). |
| In a state of affairs objects fit into one another like the links of a |
| In a state of affairs objects stand in a determinate relation to one |
| The determinate way in which objects are connected in a state of |
| affairs is the structure of the state of affairs. |
| Form is the possibility of structure. |
| The structure of a fact consists of the structures of states of |
| The totality of existing states of affairs is the world. |
| The totality of existing states of affairs also determines which states of |
| The existence and non-existence of states of affairs is reality. (We call |
| the existence of states of affairs a positive fact, and their non-existence a negative |
| States of affairs are independent of one another. |
| From the existence or non-existence of one state of affairs it is |
| impossible to infer the existence or non-existence of another. |
| The sum-total of reality is the world. |
| We picture facts to ourselves. |
| A picture has logico-pictorial form in common with what it depicts. |
| A picture depicts reality by representing a possibility of existence |
| and non-existence of states of affairs. |
| A picture contains the possibility of the situation that it |
| A picture agrees with reality or fails to agree; it is correct or |
| incorrect, true or false. |
| What a picture represents it represents independently of its truth or |
| falsity, by means of its pictorial form. |
| What a picture represents is its sense. |
| The agreement or disagreement or its sense with reality constitutes |
| In order to tell whether a picture is true or false we must compare |
| It is impossible to tell from the picture alone whether it is true |
| There are no pictures that are true a priori. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| In a proposition a thought finds an expression that can be perceived by the senses. |
| We use the perceptible sign of a proposition (spoken or written, etc.) as a |
| projection of a possible situation. The method of projection is to think of the sense of the |
| I call the sign with which we express a thought a propositional sign.And a |
| proposition is a propositional sign in its projective relation to the world. |
| A proposition, therefore, does not actually contain its sense, but does contain the |
| possibility of expressing it. ('The content of a proposition' means the content of a |
| proposition that has sense.) A proposition contains the form, but not the content, of its |
| What constitutes a propositional sign is that in its elements (the words) stand in a |
| determinate relation to one another. A propositional sign is a fact. |
| A proposition is not a blend of words.(Just as a theme in music is not a |
| blend of notes.) A proposition is articulate. |
| Only facts can express a sense, a set of names cannot. |
| Although a propositional sign is a fact, this is obscured by the usual form |
| of expression in writing or print. For in a printed proposition, for example, no essential |
| difference is apparent between a propositional sign and a word. (That is what made it |
| possible for Frege to call a proposition a composite name.) |
| The essence of a propositional sign is very clearly seen if we |
| imagine one composed of spatial objects (such as tables, chairs, and books) instead of |
| Instead of, 'The complex sign "aRb" says that a stands to b in the |
| relation R' we ought to put, 'That "a" stands to "b" in a certain relation says that aRb.' |
| Situations can be described but not given names. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| In a proposition a thought can be expressed in such a way that elements of the propositional |
| sign correspond to the objects of the thought. |
| I call such elements 'simple signs', and such a proposition 'complete |
| The simple signs employed in propositions are called names. |
| A name means an object. The object is its meaning. ('A' is the same sign as |
| The configuration of objects in a situation corresponds to the configuration of |
| simple signs in the propositional sign. |
| Objects can only be named. Signs are their representatives. I can only speak |
| about them: I cannot put them into words. Propositions can only say how things are, not what |
| The requirement that simple signs be possible is the requirement that sense be |
| A proposition about a complex stands in an internal relation to a proposition about |
| a constituent of the complex. A complex can be given only by its description, which will be |
| right or wrong. A proposition that mentions a complex will not be nonsensical, if the |
| complex does not exits, but simply false. When a propositional element signifies a complex, |
| this can be seen from an indeterminateness in the propositions in which it occurs. In such |
| cases we know that the proposition leaves something undetermined. (In fact the notation for |
| generality contains a prototype.) The contraction of a symbol for a complex into a simple |
| symbol can be expressed in a definition. |
| A proposition cannot be dissected any further by means of a definition: it is a |
| Every sign that has a definition signifies via the signs that serve to |
| define it; and the definitions point the way. Two signs cannot signify in the same manner if |
| one is primitive and the other is defined by means of primitive signs. Names cannot be |
| anatomized by means of definitions. (Nor can any sign that has a meaning independently and |
| What signs fail to express, their application shows. What signs slur over, |
| their application says clearly. |
| The meanings of primitive signs can be explained by means of elucidations. |
| Elucidations are propositions that stood if the meanings of those signs are already known. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| The exploration of logic means the exploration of everything that is subject to law . And |
| outside logic everything is accidental. |
| The so-called law of induction cannot possibly be a law of logic, since it is |
| obviously a proposition with sense.---Nor, therefore, can it be an a priori law. |
| The law of causality is not a law but the form of a law. |
| 'Law of causality'--that is a general name. And just as in mechanics, for |
| example, there are 'minimum-principles', such as the law of least action, so too in physics |
| there are causal laws, laws of the causal form. |
| Indeed people even surmised that there must be a 'law of least |
| action' before they knew exactly how it went. (Here, as always, what is certain a priori |
| proves to be something purely logical.) |
| We do not have an a priori belief in a law of conservation, but rather a priori |
| knowledge of the possibility of a logical form. |
| All such propositions, including the principle of sufficient reason, 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 the propositions of science can be cast. |
| Although the spots in our picture are geometrical figures, nevertheless geometry can |
| obviously say nothing at all about their actual form and position. The network, however, is |
| purely geometrical; all its properties can be given a priori. Laws like the principle of |
| sufficient reason, etc. are about the net and not about what the net describes. |
| If there were a law of causality, it might be put in the following way: There are |
| laws of nature. But of course that cannot be said: it makes itself manifest. |
| There is no compulsion making one thing happen because another has happened. The |
| only necessity that exists is logical necessity. |
| The whole modern conception of the world is founded on the illusion that the |
| so-called laws of nature are the explanations of natural phenomena. |
| Thus people today stop at the laws of nature, treating them as something |
| inviolable, just as God and Fate were treated in past ages. And in fact both are right and |
| both wrong: though the view of the ancients is clearer in so far as they have a clear and |
| acknowledged terminus, while the modern system tries to make it look as if everything were |
| The world is independent of my will. |
| Even if all that we wish for were to happen, still this would only be a |
| favour granted by fate, so to speak: for there is no logical connexion between the will and |
| the world, which would guarantee it, and the supposed physical connexion itself is surely |
| not something that we could will. |
| Just as the only necessity that exists is logical necessity, so too the only |
| impossibility that exists is logical impossibility. |
| For example, the simultaneous presence of two colours at the same |
| place in the visual field is impossible, in fact logically impossible, since it is ruled out |
| by the logical structure of colour. Let us think how this contradiction appears in physics: |
| more or less as follows--a particle cannot have two velocities at the same time; that is to |
| say, it cannot be in two places at the same time; that is to say, particles that are in |
| different places at the same time cannot be identical. (It is clear that the logical product |
| of two elementary propositions can neither be a tautology nor a contradiction. The statement |
| that a point in the visual field has two different colours at the same time is a |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| All propositions are of equal value. |
| The sense of the world must lie outside the world. In the world everything is as it |
| is, and everything happens as it does happen: in it no value exists--and if it did exist, it |
| would have no value. If there is any value that does have value, it must lie outside the |
| whole sphere of what happens and is the case. For all that happens and is the case is |
| accidental. What makes it non-accidental cannot lie within the world, since if it did it |
| would itself be accidental. It must lie outside the world. |
| So too it is impossible for there to be propositions of ethics. Propositions can |
| express nothing that is higher. |
| It is clear that ethics cannot be put into words. Ethics is transcendental. |
| (Ethics and aesthetics are one and the same.) |
| When an ethical law of the form, 'Thou shalt ...' is laid down, one's first |
| thought is, 'And what if I do, not do it?' It is clear, however, that ethics has nothing to |
| do with punishment and reward in the usual sense of the terms. So our question about the |
| consequences of an action must be unimportant.--At least those consequences should not be |
| events. For there must be something right about the question we posed. There must indeed be |
| some kind of ethical reward and ethical punishment, but they must reside in the action |
| itself. (And it is also clear that the reward must be something pleasant and the punishment |
| It is impossible to speak about the will in so far as it is the subject of |
| ethical attributes. And the will as a phenomenon is of interest only to psychology. |
| If the good or bad exercise of the will does alter the world, it can alter only the |
| limits of the world, not the facts--not what can be expressed by means of language. In short |
| the effect must be that it becomes an altogether different world. It must, so to speak, wax |
| and wane as a whole. The world of the happy man is a different one from that of the unhappy |
| So too at death the world does not alter, but comes to an end. |
| Death is not an event in life: we do not live to experience death. |
| If we take eternity to mean not infinite temporal duration but timelessness, then eternal |
| life belongs to those who live in the present. Our life has no end in just the way in which |
| our visual field has no limits. |
| Not only is there no guarantee of the temporal immortality of the |
| human soul, that is to say of its eternal survival after death; but, in any case, this |
| assumption completely fails to accomplish the purpose for which it has always been intended. |
| Or is some riddle solved by my surviving for ever? Is not this eternal life itself as much |
| of a riddle as our present life? The solution of the riddle of life in space and time lies |
| outside space and time. (It is certainly not the solution of any problems of natural science |
| How things are in the world is a matter of complete indifference for what is |
| higher. God does not reveal himself in the world. |
| The facts all contribute only to setting the problem, not to its |
| It is not how things are in the world that is mystical, but that it exists. |
| To view the world sub specie aeterni is to view it as a whole--a limited whole. |
| Feeling the world as a limited whole--it is this that is mystical. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| I call any part of a proposition that characterizes its sense an expression (or a symbol). |
| (A proposition is itself an expression.) Everything essential to their sense that |
| propositions can have in common with one another is an expression. An expression is the mark |
| An expression presupposes the forms of all the propositions in which it can occur. |
| It is the common characteristic mark of a class of propositions. |
| It is therefore presented by means of the general form of the propositions that it |
| characterizes. In fact, in this form the expression will be constant and everything else |
| Thus an expression is presented by means of a variable whose values are the |
| propositions that contain the expression. (In the limiting case the variable becomes a |
| constant, the expression becomes a proposition.) I call such a variable a 'propositional |
| An expression has meaning only in a proposition. All variables can be construed as |
| propositional variables. (Even variable names.) |
| If we turn a constituent of a proposition into a variable, there is a class of |
| propositions all of which are values of the resulting variable proposition. In general, this |
| class too will be dependent on the meaning that our arbitrary conventions have given to |
| parts of the original proposition. But if all the signs in it that have arbitrarily |
| determined meanings are turned into variables, we shall still get a class of this kind. This |
| one, however, is not dependent on any convention, but solely on the nature of the pro |
| position. It corresponds to a logical form--a logical prototype. |
| What values a propositional variable may take is something that is stipulated. The |
| stipulation of values is the variable. |
| To stipulate values for a propositional variable is to give the propositions whose |
| common characteristic the variable is. The stipulation is a description of those |
| propositions. The stipulation will therefore be concerned only with symbols, not with their |
| meaning. And the only thing essential to the stipulation is that it is merely a description |
| of symbols and states nothing about what is signified. How the description of the |
| propositions is produced is not essential. |
| Like Frege and Russell I construe a proposition as a function of the expressions |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| A sign is what can be perceived of a symbol. |
| So one and the same sign (written or spoken, etc.) can be common to two different |
| symbols--in which case they will signify in different ways. |
| Our use of the same sign to signify two different objects can never indicate a |
| common characteristic of the two, if we use it with two different modes of signification. |
| For the sign, of course, is arbitrary. So we could choose two different signs instead, and |
| then what would be left in common on the signifying side? |
| In everyday language it very frequently happens that the same word has different |
| modes of signification--and so belongs to different symbols--or that two words that have |
| different modes of signification are employed in propositions in what is superficially the |
| same way. Thus the word 'is' figures as the copula, as a sign for identity, and as an |
| expression for existence; 'exist' figures as an intransitive verb like 'go', and 'identical' |
| as an adjective; we speak of something, but also of something's happening. (In the |
| proposition, 'Green is green'--where the first word is the proper name of a person and the |
| last an adjective--these words do not merely have different meanings: they are different |
| In this way the most fundamental confusions are easily produced (the whole of |
| philosophy is full of them). |
| In order to avoid such errors we must make use of a sign-language that excludes them |
| by not using the same sign for different symbols and by not using in a superficially similar |
| way signs that have different modes of signification: that is to say, a sign-language that |
| is governed by logical grammar--by logical syntax. (The conceptual notation of Frege and |
| Russell is such a language, though, it is true, it fails to exclude all mistakes.) |
| In order to recognize a symbol by its sign we must observe how it is used with a |
| A sign does not determine a logical form unless it is taken together with its |
| logicosyntactical employment. |
| If a sign is useless, it is meaningless. That is the point of Occam's maxim. (If |
| everything behaves as if a sign had meaning, then it does have meaning.) |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| In logical syntax the meaning of a sign should never play a role. It must be possible to |
| establish logical syntax without mentioning the meaning of a sign: only the description of |
| expressions may be presupposed. |
| From this observation we turn to Russell's 'theory of types'. It can be seen that |
| Russell must be wrong, because he had to mention the meaning of signs when establishing the |
| No proposition can make a statement about itself, because a propositional sign |
| cannot be contained in itself (that is the whole of the 'theory of types'). |
| The reason why a function cannot be its own argument is that the sign for a function |
| already contains the prototype of its argument, and it cannot contain itself. For let us |
| suppose that the function F(fx) could be its own argument: in that case there would be a |
| proposition 'F(F(fx))', in which the outer function F and the inner function F must have |
| different meanings, since the inner one has the form O(f(x)) and the outer one has the form |
| Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself |
| signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(do) : |
| F(Ou) . Ou = Fu'. That disposes of Russell's paradox. |
| The rules of logical syntax must go without saying, once we know how each individual |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| A proposition possesses essential and accidental features. Accidental features are those |
| that result from the particular way in which the propositional sign is produced. Essential |
| features are those without which the proposition could not express its sense. |
| So what is essential in a proposition is what all propositions that can express the |
| same sense have in common. And similarly, in general, what is essential in a symbol is what |
| all symbols that can serve the same purpose have in common. |
| So one could say that the real name of an object was what all symbols that |
| signified it had in common. Thus, one by one, all kinds of composition would prove to be |
| Although there is something arbitrary in our notations, this much is not |
| arbitrary--that when we have determined one thing arbitrarily, something else is necessarily |
| the case. (This derives from the essence of notation.) |
| A particular mode of signifying may be unimportant but it is always |
| important that it is a possible mode of signifying. And that is generally so in philosophy: |
| again and again the individual case turns out to be unimportant, but the possibility of each |
| individual case discloses something about the essence of the world. |
| Definitions are rules for translating from one language into another. Any correct |
| signlanguage must be translatable into any other in accordance with such rules: it is this |
| that they all have in common. |
| What signifies in a symbol is what is common to all the symbols that the rules of |
| logical syntax allow us to substitute for it. |
| For instance, we can express what is common to all notations for |
| truth-functions in the following way: they have in common that, for example, the notation |
| that uses 'Pp' ('not p') and 'p C g' ('p or g') can be substituted for any of them. (This |
| serves to characterize the way in which something general can be disclosed by the |
| possibility of a specific notation.) |
| Nor does analysis resolve the sign for a complex in an arbitrary way, so |
| that it would have a different resolution every time that it was incorporated in a different |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| A proposition is a picture of reality. |
| At first sight a proposition--one set out on the printed page, for example--does not |
| seem to be a picture of the reality with which it is concerned. But neither do written notes |
| seem at first sight to be a picture of a piece of music, nor our phonetic notation (the |
| alphabet) to be a picture of our speech. And yet these sign-languages prove to be pictures, |
| even in the ordinary sense, of what they represent. |
| It is obvious that a proposition of the form 'aRb' strikes us as a picture. In this |
| case the sign is obviously a likeness of what is signified. |
| And if we penetrate to the essence of this pictorial character, we see that it is |
| not impaired by apparent irregularities (such as the use [sharp] of and [flat] in musical |
| notation). For even these irregularities depict what they are intended to express; only they |
| do it in a different way. |
| A gramophone record, the musical idea, the written notes, and the sound-waves, all |
| stand to one another in the same internal relation of depicting that holds between language |
| and the world. They are all constructed according to a common logical pattern. (Like the two |
| youths in the fairy-tale, their two horses, and their lilies. They are all in a certain |
| There is a general rule by means of which the musician can obtain the |
| symphony from the score, and which makes it possible to derive the symphony from the groove |
| on the gramophone record, and, using the first rule, to derive the score again. That is what |
| constitutes the inner similarity between these things which seem to be constructed in such |
| entirely different ways. And that rule is the law of projection which projects the symphony |
| into the language of musical notation. It is the rule for translating this language into the |
| language of gramophone records. |
| The possibility of all imagery, of all our pictorial modes of expression, is |
| contained in the logic of depiction. |
| In order to understand the essential nature of a proposition, we should consider |
| hieroglyphic script, which depicts the facts that it describes. And alphabetic script |
| developed out of it without losing what was essential to depiction. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| We can see this from the fact that we understand the sense of a propositional sign without |
| its having been explained to us. |
| A proposition is a picture of reality: for if I understand a proposition, I know the |
| situation that it represents. And I understand the proposition without having had its sense |
| A proposition shows its sense. A proposition shows how things stand if it is true. |
| And it says that they do so stand. |
| A proposition must restrict reality to two alternatives: yes or no. In order to do |
| that, it must describe reality completely. A proposition is a description of a state of |
| affairs. Just as a description of an object describes it by giving its external properties, |
| so a proposition describes reality by its internal properties. A proposition constructs a |
| world with the help of a logical scaffolding, so that one can actually see from the |
| proposition how everything stands logically if it is true. One can draw inferences from a |
| To understand a proposition means to know what is the case if it is true. (One can |
| understand it, therefore, without knowing whether it is true.) It is understood by anyone |
| who understands its constituents. |
| When translating one language into another, we do not proceed by translating each |
| proposition of the one into a proposition of the other, but merely by translating the |
| constituents of propositions. (And the dictionary translates not only substantives, but also |
| verbs, adjectives, and conjunctions, etc.; and it treats them all in the same way.) |
| The meanings of simple signs (words) must be explained to us if we are to understand |
| them. With propositions, however, we make ourselves understood. |
| It belongs to the essence of a proposition that it should be able to communicate a |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| A proposition can be true or false only in virtue of being a picture of reality. |
| It must not be overlooked that a proposition has a sense that is independent of the |
| facts: otherwise one can easily suppose that true and false are relations of equal status |
| between signs and what they signify. In that case one could say, for example, that 'p' |
| signified in the true way what 'Pp' signified in the false way, etc. |
| Can we not make ourselves understood with false propositions just as we have done up |
| till now with true ones?--So long as it is known that they are meant to be false.--No! For a |
| proposition is true if we use it to say that things stand in a certain way, and they do; and |
| if by 'p' we mean Pp and things stand as we mean that they do, then, construed in the new |
| way, 'p' is true and not false. |
| But it is important that the signs 'p' and 'Pp' can say the same thing. For |
| it shows that nothing in reality corresponds to the sign 'P'. The occurrence of negation in |
| a proposition is not enough to characterize its sense (PPp = p). The propositions 'p' and |
| 'Pp' have opposite sense, but there corresponds to them one and the same reality. |
| An analogy to illustrate the concept of truth: imagine a black spot on white paper: |
| you can describe the shape of the spot by saying, for each point on the sheet, whether it is |
| black or white. To the fact that a point is black there corresponds a positive fact, and to |
| the fact that a point is white (not black), a negative fact. If I designate a point on the |
| sheet (a truth-value according to Frege), then this corresponds to the supposition that is |
| put forward for judgement, etc. etc. But in order to be able to say that a point is black or |
| white, I must first know when a point is called black, and when white: in order to be able |
| to say,'"p" is true (or false)', I must have determined in what circumstances I call 'p' |
| true, and in so doing I determine the sense of the proposition. Now the point where the |
| simile breaks down is this: we can indicate a point on the paper even if we do not know what |
| black and white are, but if a proposition has no sense, nothing corresponds to it, since it |
| does not designate a thing (a truth-value) which might have properties called 'false' or |
| 'true'. The verb of a proposition is not 'is true' or 'is false', as Frege thought: rather, |
| that which 'is true' must already contain the verb. |
| Every proposition must already have a sense: it cannot be given a sense by |
| affirmation. Indeed its sense is just what is affirmed. And the same applies to negation, |
| One could say that negation must be related to the logical place determined |
| by the negated proposition. The negating proposition determines a logical place different |
| from that of the negated proposition. The negating proposition determines a logical place |
| with the help of the logical place of the negated proposition. For it describes it as lying |
| outside the latter's logical place. The negated proposition can be negated again, and this |
| in itself shows that what is negated is already a proposition, and not merely something that |
| is prelimary to a proposition. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| The totality of true propositions is the whole of natural science (or the whole corpus of |
| Philosophy is not one of the natural sciences. (The word 'philosophy' must mean |
| something whose place is above or below the natural sciences, not beside them.) |
| Philosophy aims at the logical clarification of thoughts. Philosophy is not a body |
| of doctrine but an activity. A philosophical work consists essentially of elucidations. |
| Philosophy does not result in 'philosophical propositions', but rather in the clarification |
| of propositions. Without philosophy thoughts are, as it were, cloudy and indistinct: its |
| task is to make them clear and to give them sharp boundaries. |
| Psychology is no more closely related to philosophy than any other natural |
| science. Theory of knowledge is the philosophy of psychology. Does not my study of |
| sign-language correspond to the study of thought-processes, which philosophers used to |
| consider so essential to the philosophy of logic? Only in most cases they got entangled in |
| unessential psychological investigations, and with my method too there is an analogous risk. |
| Darwin's theory has no more to do with philosophy than any other hypothesis |
| Philosophy sets limits to the much disputed sphere of natural science. |
| It must set limits to what can be thought; and, in doing so, to what cannot be |
| thought. It must set limits to what cannot be thought by working outwards through what can |
| It will signify what cannot be said, by presenting clearly what can be said. |
| Everything that can be thought at all can be thought clearly. Everything that can be |
| put into words can be put clearly. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| The propositional variable signifies the formal concept, and its values signify the objects |
| that fall under the concept. |
| Every variable is the sign for a formal concept. For every variable represents a |
| constant form that all its values possess, and this can be regarded as a formal property of |
| Thus the variable name 'x' is the proper sign for the pseudo-concept object. |
| Wherever the word 'object' ('thing', etc.) is correctly used, it is expressed in conceptual |
| notation by a variable name. For example, in the proposition, 'There are 2 objects which. . |
| .', it is expressed by ' (dx,y) ... '. Wherever it is used in a different way, that is as a |
| proper concept-word, nonsensical pseudo-propositions are the result. So one cannot say, for |
| example, 'There are objects', as one might say, 'There are books'. And it is just as |
| impossible to say, 'There are 100 objects', or, 'There are !0 objects'. And it is |
| nonsensical to speak of the total number of objects. The same applies to the words |
| 'complex', 'fact', 'function', 'number', etc. They all signify formal concepts, and are |
| represented in conceptual notation by variables, not by functions or classes (as Frege and |
| Russell believed). '1 is a number', 'There is only one zero', and all similar expressions |
| are nonsensical. (It is just as nonsensical to say, 'There is only one 1', as it would be to |
| say, '2 + 2 at 3 o'clock equals 4'.) |
| A formal concept is given immediately any object falling under it is given. |
| It is not possible, therefore, to introduce as primitive ideas objects belonging to a formal |
| concept and the formal concept itself. So it is impossible, for example, to introduce as |
| primitive ideas both the concept of a function and specific functions, as Russell does; or |
| the concept of a number and particular numbers. |
| If we want to express in conceptual notation the general proposition, 'b is a |
| successor of a', then we require an expression for the general term of the series of forms |
| 'aRb', '(d : c) : aRx . xRb', '(d x,y) : aRx . xRy . yRb', ... , In order to express the |
| general term of a series of forms, we must use a variable, because the concept 'term of that |
| series of forms' is a formal concept. (This is what Frege and Russell overlooked: |
| consequently the way in which they want to express general propositions like the one above |
| is incorrect; it contains a vicious circle.) We can determine the general term of a series |
| of forms by giving its first term and the general form of the operation that produces the |
| next term out of the proposition that precedes it. |
| To ask whether a formal concept exists is nonsensical. For no proposition can be the |
| answer to such a question. (So, for example, the question, 'Are there unanalysable |
| subject-predicate propositions?' cannot be asked.) |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| The sense of a proposition is its agreement and disagreement with possibilities of existence |
| and nonexistence of states of affairs. |
| The simplest kind of proposition, an elementary proposition, asserts the existence |
| It is a sign of a proposition's being elementary that there can be no |
| elementary proposition contradicting it. |
| An elementary proposition consists of names. It is a nexus, a concatenation, of |
| It is obvious that the analysis of propositions must bring us to elementary |
| propositions which consist of names in immediate combination. This raises the question how |
| such combination into propositions comes about. |
| Even if the world is infinitely complex, so that every fact consists |
| of infinitely many states of affairs and every state of affairs is composed of infinitely |
| many objects, there would still have to be objects and states of affairs. |
| It is only in the nexus of an elementary proposition that a name occurs in a |
| Names are the simple symbols: I indicate them by single letters ('x', 'y', 'z'). I |
| write elementary propositions as functions of names, so that they have the form 'fx', 'O |
| (x,y)', etc. Or I indicate them by the letters 'p', 'q', 'r'. |
| When I use two signs with one and the same meaning, I express this by |
| putting the sign '=' between them. So 'a = b' means that the sign 'b' can be substituted for |
| the sign 'a'. (If I use an equation to introduce a new sign 'b', laying down that it shall |
| serve as a substitute for a sign a that is already known, then, like Russell, I write the |
| equation-- definition--in the form 'a = b Def.' A definition is a rule dealing with signs.) |
| Expressions of the form 'a = b' are, therefore, mere representational |
| devices. They state nothing about the meaning of the signs 'a' and 'b'. |
| Can we understand two names without knowing whether they signify the same |
| thing or two different things?--Can we understand a proposition in which two names occur |
| |