| The general form of a truth-function is [p, E, N(E)] |
| >>Note: This is the general form of a proposition. |
| 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 understood) |
| 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 |
| >>Note: It now seems possible to give the most general propositional form: that is, to give a description of the propositions of any sign-language whatsoever in such a way that every possible sense can be expressed by a symbol satisfying the description, and every symbol satisfying the description can express a sense, provided that the meanings of the names are suitably chosen. It is clear that only what is essential to the most general propositional form may be included in its description--for otherwise it would not be the most general form. The existence of a general propositional form is proved by the fact that there cannot be a proposition whose form could not have been foreseen (i.e. constructed). The general form of a proposition is: This is how things stand. |
| 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 propositions.) |
| The general propositional form is a variable. |
| A proposition is a truth-function of elementary propositions. |
| >>Note: (An elementary proposition is a truth-function of itself.) |
| Elementary propositions are the truth-arguments of propositions. |
| The arguments of functions are readily confused with the affixes of names. |
| >>Note: For both arguments and affixes enable me to recognize the meaning of the signs containing them. For example, when Russell writes '+c', the 'c' is an affix which indicates that the sign as a whole is the addition-sign for cardinal numbers. But the use of this sign is the result of arbitrary convention and it would be quite possible to choose a simple sign instead of '+c'; in 'Pp' however, 'p' is not an affix but an argument: the sense of 'Pp' cannot be understood unless the sense of 'p' has been understood already. (In the name Julius Caesar 'Julius' is an affix. An affix is always part of a description of the object to whose name we attach it: e.g. the Caesar of the Julian gens.) If I am not mistaken, Frege's theory about the meaning of propositions and functions is based on the confusion between an argument and an affix. Frege regarded the propositions of logic as names, and their arguments as the affixes of those names. |
| Truth-functions can be arranged in series. That is the foundation of the theory of probability. |
| The certainty of logical inference is a limiting case of probability |
| >>Note: (Application of this to tautology and contradiction.) |
| 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 remains the same. |
| The world is the totality of facts, not of things. |
| The world is determined by the facts, and by their being all the facts. |
| 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 affairs. |
| 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 affairs are also given. |
| 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 object. |
| 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 distinguished after all. |
| 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. |
| >>Note: 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. |
| >>Note: For example, the fact that the propositions 'p' and 'Pp' in the combination '(p . Pp)' yield a tautology shows that they contradict one another. The fact that the propositions 'p z q', 'p', and 'q', combined with one another in the form '(p z q) . (p) :z: (q)', yield a tautology shows that q follows from p and p z q. The fact that '(x) . fxx :z: fa' is a tautology shows that fa follows from (x) . fx. Etc. etc. |
| It is clear that one could achieve the same purpose by using contradictions instead of tautologies. |
| 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. |
| >>Note: This method could also be called a zeromethod. In a logical proposition, propositions are brought into equilibrium with one another, and the state of equilibrium then indicates what the logical constitution of these propositions must be. |
| It follows from this that we can actually do without logical propositions |
| >>Note: It follows from this that we can actually do without logical propositions; for in a suitable notation we can in fact recognize the formal properties of propositions by mere inspection of the propositions themselves. |
| 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 tautology. |
| 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 possible experience. |
| 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 inference. |
| Clearly the laws of logic cannot in their turn be subject to laws of logic. |
| >>Note: There is not, as Russell thought, a special law of contradiction for each 'type'; one law is enough, since it is not applied to itself. |
| 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 accident. |
| 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 represent it. |
| >>Note: They have no 'subject-matter'. They presuppose that names have meaning and elementary propositions sense; and that is their connexion with the world. It is clear that something about the world must be indicated by the fact that certain combinations of symbols--whose essence involves the possession of a determinate character--are tautologies. This contains the decisive point. We have said that some things are arbitrary in the symbols that we use and that some things are not. In logic it is only the latter that express: but that means that logic is not a field in which we express what we wish with the help of signs, but rather one in which the nature of the absolutely necessary signs speaks for itself. If we know the logical syntax of any sign-language, then we have already been given all the propositions of logic. |
| 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. |
| >>Note: And this is what we do when we 'prove' a logical proposition. For, without bothering about sense or meaning, we construct the logical proposition out of others using only rules that deal with signs . The proof of logical propositions consists in the following process: we produce them out of other logical propositions by successively applying certain operations that always generate further tautologies out of the initial ones. (And in fact only tautologies follow from a tautology.) Of course this way of showing that the propositions of logic are tautologies is not at all essential to logic, if only because the propositions from which the proof starts must show without any proof that they are tautologies. |
| 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 proposition.) |
| It is always possible to construe logic in such a way that every proposition is its own proof. |
| All the propositions of logic are of equal status |
| >>Note: All the propositions of logic are of equal status: it is not the case that some of them are essentially derived propositions. Every tautology itself shows that it is a tautology. |
| 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., (and so on). |
| 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 generality. |
| 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 same or different. |
| 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 intuition. |
| 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 must go without saying. |
| 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 propositions.) |
| All theories that make a proposition of logic appear to have content are false. One might think, for example, that the words 'true' and 'false' signified two properties among other properties, and then it would seem to be a remarkable fact that every proposition possessed one of these properties. On this theory it seems to be anything but obvious, just as, for instance, the proposition, 'All roses are either yellow or red', would not sound obvious even if it were true. Indeed, the logical proposition acquires all the characteristics of a proposition of natural science and this is the sure sign that it has been construed wrongly. |
| 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 transcendental. |
| 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 physics that we wish.) |
| 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 law are thinkable. |
| 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 states of affairs. |
| 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 determinate way. |
| 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 to it. |
| 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 incorrectly. |
| 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 form of reality. |
| 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 chain. |
| In a state of affairs objects stand in a determinate relation to one another. |
| 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 affairs. |
| The totality of existing states of affairs is the world. |
| The totality of existing states of affairs also determines which states of affairs do not exist. |
| 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 fact.) |
| 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 represents. |
| 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 its truth or falsity. |
| In order to tell whether a picture is true or false we must compare it with reality. |
| It is impossible to tell from the picture alone whether it is true or false. |
| 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 proposition. |
| 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 sense. |
| 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 written signs. |
| 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 analysed'. |
| 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 'A'.) |
| 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 they are. |
| The requirement that simple signs be possible is the requirement that sense be determinate. |
| 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 primitive sign. |
| Every sign that has a definition signifies via the signs that serve to define it; and the definitions point the way. Two signs cannot signify in the same manner if one is primitive and the other is defined by means of primitive signs. Names cannot be anatomized by means of definitions. (Nor can any sign that has a meaning independently and on its own.) |
| 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 explained. |
| 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 contradiction.) |
| 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 something unpleasant.) |
| 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 man. |
| 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 that is required.) |
| 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 solution. |
| 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 of a form and a content. |
| 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 variable. |
| Thus an expression is presented by means of a variable whose values are the propositions that contain the expression. (In the limiting case the variable becomes a constant, the expression becomes a proposition.) I call such a variable a 'propositional variable'. |
| 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 contained in it. |
| 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 symbols.) |
| 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 sense. |
| 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 rules for them. |
| 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 sign signifies. |
| 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 unessential to a name. |
| 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 proposition. |
| Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Published (1922) |
| A proposition is a picture of reality. |
| >>Note: A proposition is a model of reality as we imagine it. |
| 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 sense one.) |
| 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 explained to me. |
| 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 false proposition. |
| 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 new sense to us. |
| 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, etc. |
| 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 the natural sciences). |
| 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 in natural science. |
| 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 be thought. |
| 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 those values. |
| 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 ob |
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