 Section 4 of Tractatus Logicofilosophicus. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Recording by Jeffrey Edwards. Tractatus Logicofilosophicus by Ludwig Wittgenstein. Translated by C.K. Ogden. Section 4. 5.502. Therefore, I write instead of, quote, bracket, dash, dash, dash, dash, dash, capital T, close bracket, bracket, xi, comma, dot, dot, dot, dot, close bracket, close quote, quote, capital N, bracket, line over xi, close bracket, close quote, capital N, bracket, line over xi, close bracket is the negation of all the values of the propositional variable xi. 5.503. As it is obviously easy to express how propositions can be constructed by means of this operation, and how propositions are not to be constructed by means of it, this must be capable of exact expression. If xi has only one value, then capital N, bracket, line over xi, close bracket, equals not symbol p, bracket, not p, close bracket. If it has two values, then capital N, bracket, line over xi, close bracket, equals not symbol p, and symbol not symbol q. Bracket, neither p nor q, close bracket. 5.511. How can the all-embracing logic which mirrors the world use such special catches and manipulations? Only because all these are connected into an infinitely fine network to the great mirror. 5.512. Quotes not symbol p is true if quotes p is false. Therefore, in the true proposition quotes not symbol p, quotes p is a false proposition. How then can the stroke quotes not symbol bring it into agreement with reality? That which denies in quotes not symbol p is however not quotes not symbol, but at which all signs of this notation which deny p have in common. Hence, the common rule according to which quotes not symbol p, quotes not symbol not symbol p, quotes not symbol p, not symbol p, quotes not symbol p, and symbol not symbol p, etc., etc., bracket to infinity, close bracket, are constructed. And this which is common to them all mirrors denial. 5.513. We could say what is common to all symbols which asserts both p and q is the proposition, quote p and symbol q, close quote. What is common to all symbols which asserts either p or q is the proposition, quote p or symbol q, close quote. And similarly, we can say two propositions are opposed to one another when they have nothing in common with one another. And every proposition has only one negative because there's only one proposition which lies all together outside it. Thus, in Russell's notation, also it appears evident that quote q colon p or symbol not symbol p, close quote, says the same thing as quotes q, that quote p or symbol not symbol p, close quote, says nothing. 5.514. If a notation is fixed, there is in it a rule according to which all the propositions denying p are constructed. A rule according to which all the propositions asserting p are constructed. A rule according to which all the propositions asserting p or q are constructed, and so on. These rules are equivalent to the symbols, and in them their sense is mirrored. 5.515. It must be recognized in our symbols that what is connected by quotes or symbol, quotes and symbol, etc. must be propositions. And this is the case for the symbols quotes p and quotes q, presuppose quotes or symbol, quotes not symbol, etc. If the sign quotes p in quote p or symbol q, close quote, does not stand for a complex sign, then by itself it cannot have sense. But then also the signs quote p or symbol p, close quote, quote p and symbol p, close quote, etc. which have the same sense as quotes p have no sense. If however quote p or symbol p, close quote has no sense, then also quote p or symbol q, close quote can have no sense. 5.5151. Must the sign of the negative proposition be constructed by means of the sign of the positive? Why should one not be able to express the negative proposition by means of a negative fact? Bracket, like if quotes a does not stand in a certain relation to quotes b, it could express that a capital R, b is not the case. Close bracket, but here also the negative proposition is indirectly constructed with the positive. The positive proposition must presuppose the existence of the negative proposition, and conversely. 5.52. If the values of xi are the total values of a function, fx for all values of x, then capital N, bracket, line over xi. Close bracket equals, not symbol, bracket, there exists symbol x, close bracket, and symbol fx. 5.521. I separate the concept all from the truth function. Frigga and Russell have introduced generality in connection with the logical product of the logical sum. Then it would be difficult to understand the propositions, quote, bracket, there exists symbol x, close bracket, and symbol fx. Close quote, and quote, bracket, x, close bracket, and symbol fx. Close quote, in which both ideas lie concealed. 5.522. That which is peculiar to the quote, symbolism of generality, close quote, is firstly that it refers to a logical prototype, and secondly that it makes constants prominent. 5.523. The generality symbol occurs as an argument. 5.524. If the objects are given, therewith are all objects also given. If the elementary propositions are given, then therewith all elementary propositions are also given. 5.525. It is not correct to render the proposition, quote, bracket, there exists symbol x, close bracket, and symbol fx. Close quote, as Russell does in the words, quote, fx is possible. Close quote. Certainty, possibility, or impossibility of a state of affairs are not expressed by a proposition, but by the fact that an expression is a tautology, a significant proposition, or a contradiction. That precedent to which one must always appeal must be present in the symbol itself. 5.526. One can describe the world completely by completely generalized propositions. In essence, without from the outset coordinating any name with a definite object. In order then to arrive at the customary way of expression, we need simply say, after an expression, quote, there is one and only one x, which, dot, dot, dot, close quote, the words, quote, and that x is a close quote. 5.5261. A completely generalized proposition is, like every other proposition, composite, bracket. This is shown by the fact that in quote, bracket, there exists symbol x, phi, close bracket, and symbol phi, x, close quote. We must mention quotes, phi, and quotes, x, separately. Both stand independently in signifying relations to the world as in the ungeneralized proposition. Close bracket. A characteristic of a composite symbol, it has something in common with other symbols. 5.5262. The truth or falsehood of every proposition alters something in the general structure of the world, and the range which is allowed to its structure by the totality of elementary propositions is exactly that which the completely general propositions delimit. Bracket. If an elementary proposition is true, then, at any rate, there is one more elementary proposition true. Close bracket. 5.53. Identity of the object are expressed by identity of the sign, and not by means of a sign of identity. Difference of the objects by difference of the signs. 5.5301. That identity is not a relation between objects is obvious. This becomes very clear if, for example, one considers the proposition, quote, bracket, x, close bracket, colon, f, x, and symbol, imply symbol, and symbol, x equals a. Close quote. What this proposition says is simply that only a satisfies the function f, and not that only such things satisfy the function f, which have a certain relation to a. One could, of course, say that, in fact, only a has this relation to a, but in order to express this, we should need the sign of identity itself. 5.5302. Russell's definition of quotes equals symbol won't do, because, according to it, one cannot say that two objects have all their properties in common. Bracket, even if this proposition is never true, it is nevertheless significant. Close bracket. 5.5303. Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing. 5.531. I write, therefore, not, quote, f bracket, a, b, close bracket, and symbol, a equals b, close quote. But, quote, f bracket, a comma, a, close bracket, close quote. Bracket, or, quote, f bracket, b, comma, b, close bracket, close quote, close bracket, and not quote f bracket a comma b close bracket and symbol not symbol a equals b close quote but quote f bracket a comma b close bracket close quote 5.532 and analogously not quote bracket there exists symbol x comma y close bracket and symbol f bracket x comma y close bracket and symbol x equals y close quote but quote bracket there exists symbol x close bracket and symbol f bracket x comma x close bracket close quote and not quote bracket there exists symbol x comma y close bracket and symbol f bracket x comma y close bracket and symbol not symbol x equals y close quote but quote bracket there exists symbol x comma y close bracket and symbol f bracket x comma y close bracket close quote therefore instead of Russell's quote bracket there exists symbol x comma y close bracket and symbol f bracket x comma y close bracket close quote we get quote bracket there exists symbol x comma y close bracket and symbol f bracket x comma y close bracket and symbol or symbol and symbol, bracket, there exists symbol, x, closed bracket, and symbol, f bracket, x, comma, x, closed bracket, close quote. 5.5321. Instead of quote, bracket, x, closed bracket, colon, f, x, implies symbol, x equals a, close quote, we therefore write, for example, quote, bracket, there exists symbol, x, closed bracket, and symbol, f, x, and symbol, implies symbol, and symbol, f, a, colon, not symbol, bracket, there exists symbol, x, comma, y, closed bracket, and symbol, f, x, and symbol, f, y, close quote. And if the proposition, quote, only one x satisfies f, bracket, closed bracket, closed quote, reads, quote, bracket, there exists symbol, x, closed bracket, and symbol, f, x, colon, not symbol, bracket, there exists symbol, x, comma, y, closed bracket, and symbol, f, x, and symbol, f, y, close quote. 5.533. The identity sign is therefore not an essential constituent of logical notation. 5.534. And we see that the apparent propositions like, quote, a equals a, close quote, quote, a equals b, and symbol, b equals c, and symbol, implies symbol, a equals c, close quote, quote, bracket, x, closed bracket, and symbol, x equals x, close quote, quote, bracket, there exists symbol, x, closed bracket, and symbol, x equals a, close quote, etc., cannot be written in a correct logical notation at all. 5.535. So all such problems disappear, which are connected with such pseudo propositions. This is the place to solve all the problems, which arise through Russell's, quote, axiom of infinity, close quote. What the axiom of infinity is meant to say would be expressed in language by the fact that there is an infinite number of names with different meanings. 5.5351. There are certain cases in which one is tempted to use expressions of the form, quote, a equals a, close quote, or quote, p implies symbol, p, close quote, as for instance, when one would speak of the archetype, proposition, thing, etc. So Russell in the Principles of Mathematics has rendered the nonsense, quote, p is a proposition, close quote, in symbols by, quote, p implies symbol, p, close quote, and has put it as a hypothesis before certain propositions to show that their places for arguments could only be occupied by propositions. Bracket, it is nonsense to place the hypothesis, p implies symbol, p, before a proposition in order to ensure that its arguments have the right form. Because the hypothesis for a non-proposition as argument becomes not false but meaningless, and because the proposition itself becomes senseless for arguments of the wrong kind, and therefore it survives the wrong arguments no better and no worse than the senseless hypothesis attached for this purpose, close bracket. 5.5352. Similarly, it was proposed to express, quote, there are no things, close quote, by, quote, not symbol, bracket, there exists symbol x, close bracket, and symbol x equals x, close quote. But even if this were a proposition, would it not be true if indeed, quote, there were things, close quote, but these were not identical with themselves? 5.54. In the general propositional form, propositions occur in a proposition only as basis of the truth operations. 5.541. At first sight it appears as if there were also a different way in which one proposition could occur in another, especially in certain propositional forms of psychology like quote, capital A thinks that p is the case, close quote, or quote, capital A thinks p, close quote, etc. Here it appears superficially as if the proposition p stood to the object capital A in a kind of relation, bracket, and in modern epistemology, bracket, Russell, Moore, etc., close bracket, those propositions have been conceived in this way, close bracket. 5.542. But it is clear that quote, capital A believes that p, close quote, quote, capital A thinks p, close quote, quote, capital A says p, close quote, or of the form, quote, quotes p says p, close quote. And here we have no coordination of a fact and an object, but a coordination of facts by means of a coordination of their objects. 5.542. This shows that there is no such thing as the soul, the subject, etc., as it is conceived in superficial psychology. A composite soul would not be a soul any longer. 5.542. The correct explanation of the form of the proposition, quote, capital A judges p, close quote, must show that it is impossible to judge a nonsense. Bracket, Russell's theory does not satisfy this condition. Close bracket. 5.542. To perceive a complex means to perceive that its constituents are combined in such and such a way. Readers note, a necker cube is drawn with the top left corner labeled b, the top right corner labeled b, the upper middle left corner labeled a, the upper middle right corner labeled a, the lower middle left corner labeled b, the lower middle right corner labeled b, and the bottom left corner labeled a, and the bottom right corner labeled a. This perhaps explains that the figure can be seen in two ways, as a cube, and all similar phenomena. For, we really see two different facts. Bracket, if I fix my eyes first on the corners a and only glance at b, a appears in front and b behind, and vice versa. Close bracket. 5.55. We must now answer a priori, the question as to all possible forms of the elementary propositions. The elementary proposition consists of names. Since we cannot give the number of names with different meanings, we cannot give the composition of the elementary proposition. 5.551. Our fundamental principle is that every question which can be decided at all by logic can be decided offhand. Bracket, and if we get into a situation where we need to answer such a problem by looking at the world, this shows that we are on a fundamentally wrong track. Close bracket. 5.552. The quotes experience which we need to understand logic is not that such and such is the case, but that something is. But that is no experience. Logic precedes every experience. That something is so. It is before the how, not before the what. 5.5521. And if this were not the case, how could we apply logic? We could say, if there were a logic, even if there were no world, how then could there be a logic, since there is a world? 5.553. Russell said that there were simple relations between different numbers of things, bracket, individuals, close bracket. But between what numbers, and how should this be decided? By experience. Bracket, there is no preeminent number, close bracket. 5.554. The enumeration of any special forms would be entirely arbitrary. 5.5541. How could we decide a priori whether, for example, I can get into a situation in which I need to symbolize with a sign of a 27-term relation? 5.5542. May we then ask this at all? Can we set out a sign form and not know whether anything can correspond to it? As the question sense, what must there be in order that anything can be the case? 5.555. It is clear that we have a concept of the elementary proposition apart from its special logical form. Where, however, we can build symbols according to a system, there this system is the logically important thing, and not the single symbols. And how would it be possible that I should have to deal with forms and logic which I can invent? But I must have to deal with that which makes it possible for me to invent them. 5.556. There cannot be a hierarchy of the forms of the elementary propositions, only that which we ourselves construct can we foresee. 5.5561. Empirical reality is limited by the totality of objects. The boundary appears again in the totality of elementary propositions. The hierarchies are and must be independent of reality. 5.5562. If we know on purely logical grounds that there must be elementary propositions, then this must be known by everyone who understands propositions in their unanalyzed form. 5.5563. All propositions of our colloquial language are actually, just as they are, logically completely in order. The simple thing which we ought to give here is not a model of the truth, but the complete truth itself. Bracket. Our problems are not abstract, but perhaps the most concrete that there are. Close bracket. 5.557. The application of logic decides what elementary propositions there are. What lies in its application logic cannot anticipate. It is clear that logic may not conflict with its application, but logic must have contact with its application. Therefore, logic and its application may not overlap one another. 5.5571. If I cannot give elementary propositions a priori, then it must lead to obvious nonsense to try to give them. 5.6. The limits of my language means the limits of my world. 5.61. Logic fills the world. The limits of the world are also its limits. We cannot therefore say in logic, this and this there is in the world, that there is not. For the world apparently presupposes that we exclude certain possibilities, and this cannot be the case, since otherwise logic must get outside the limits of the world, that is, if it could consider these limits from the other side also. What we cannot think, that we cannot think. We cannot therefore say what we cannot think. 5.62. This remark provides a key to the question to what extent solipsism is a truth. In fact, what solipsism means is quite correct. Only it cannot be said, but it shows itself. That the world is my world, shows itself in the fact that the limits of the language, bracket, the language which I understand, close bracket, means the limits of my world. 5.621. The world and life are one. 5.63. I am my world, bracket the microcosm, close bracket. 5.631. The thinking, presenting subject, there is no such thing. If I wrote a book, quote, the world as I found it, close quote, I should also have therein to report on my body and say which members obey my will and which do not, etc. This then would be a method of isolating the subject or rather of showing that in an important sense there is no subject. That is to say, if it alone in this book mentioned could not be made. 5.632. The subject does not belong to the world, but it is a limit of the world. 5.633. Where in the world is a metaphysical subject to be noted? You say that this case is altogether like that of the eye and the field of sight, but you do not really see the eye. And from nothing in the field of sight, can it be concluded that it is seen from an eye? 5.6331. For the field of sight has not a form like this. Readers note, an illustration depicts a chicken egg shaped oval with a tiny circle representing the eye at the small end of the oval. The eye is inside the oval. And readers note, 5.634. This is connected with the fact that no part of our experience is also a priori. Everything we see could also be otherwise. Everything we describe at all can also be otherwise. There is no order of things a priori. 5.64. Here we see that solipsism strictly carried out coincides with pure realism. The eye in solipsism shrinks to an extensionless point and there remains the reality coordinated with it. 5.641. There is therefore really a sense in which philosophy can talk of a non-psychological eye. The eye occurs in philosophy through the fact that the quote, world is my world, close quote. The philosophical eye is not the man, not the human body or the human soul of which psychology treats, but the metaphysical subject, the limit, not a part of the world. End of section 4, recording by Jeffrey Edwards. Section 5 of Tractatus logico philosophicus. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Recording in memory of Mitchell Edwards. Tractatus logico philosophicus by Ludwig Wittgenstein. Translated by C. K. Ogden. Section 5. 6. The general form of truth function is square bracket line over p comma line over xi comma capital N bracket line over xi close bracket close square bracket. This is the general form of proposition. 6.001. This says nothing else than that every proposition is the result of successive applications of the operation capital N prime bracket line over xi close bracket to the elementary propositions. 6.002. If we are given the general form of the way in which a proposition is constructed, then thereby we are also given the general form of the way in which by an operation out of one proposition another can be created. 6.01. The general form of the operation omega prime bracket line over eta close bracket is therefore square bracket line over xi comma capital N bracket line over xi close bracket close square bracket prime bracket line over eta close bracket bracket equal symbol square bracket line over eta comma line over xi comma capital N bracket line over eta close bracket close square bracket close bracket. This is the most general form of transition from one proposition to another. 6.02. And thus we come to numbers. I define x equals omega superscript zero prime regular script x def and omega prime omega superscript v prime regular script x equals omega superscript v plus one prime regular script x def. According then to these symbolic rules, we write the series x comma omega prime x comma omega prime omega prime x comma omega prime omega prime omega prime x dot dot dot as omega superscript zero prime regular script x comma omega superscript zero plus one prime regular script x comma omega superscript zero plus one plus one prime regular script x comma omega superscript 0 plus 1 plus 1 plus 1 prime regular script x dot dot dot therefore I write in place of quote square bracket x comma xi comma omega prime xi close square bracket close quote quote square bracket omega superscript 0 comma omega superscript v prime regular script x comma omega superscript v plus 1 prime regular script x close square bracket close quote and I define 0 plus 1 equals 1 def 0 plus 1 plus 1 equals 2 def 0 plus 1 plus 1 plus 1 equals 3 def and so on 6.021 a number is the exponent of an operation 6.022 the concept number is nothing else than that which is common to all numbers the general form of a number 6.03 the general form of the cardinal number is square bracket 0 comma xi comma xi plus 1 close square bracket 6.031 the theory of classes is altogether superfluous in mathematics this is connected with the fact that the generality which we need in mathematics is not the accidental one 6.1 the propositions of logic are tautologies 6.11 the propositions of logic therefore say nothing bracket they are the analytical propositions close bracket 6.111 theories which make a proposition of logic appear substantial are always false one could for example believe that the words quotes true and quotes false signify two properties among other properties and then it would appear as a remarkable fact that every proposition possesses one of these properties this now by no means appear self-evident no more so then the proposition quote all roses are either yellow or red close quote would seem even if it were true indeed our proposition now gets quite the character of a proposition of natural science and this is a certain symptom of its being falsely understood 6.112 the correct explanation of logical propositions must give them a peculiar position among all propositions 6.113 it is the characteristic mark of logical propositions that one can perceive in the symbol alone that they are true and this fact contains in itself the whole philosophy of logic and so also it is one of the most important facts that the truth or falsehood of non-logical propositions cannot be recognized from the propositions alone 6.12 the fact that the propositions of logic are tautologies shows the formal logical properties of language of the world that its constituent parts connected together in this way give a tautology characterizes the logic of its constituent parts in order that propositions connected together in a definite way may give a tautology they must have definite properties of structure that they give a tautology when so connected shows therefore that they possess these properties of structure 6.1201 that for example the propositions quotes p and quotes not symbol p in the connection quote not symbol p and symbol not symbol p close quote give a tautology shows that they contradict one another that the propositions quote p implies symbol q close quote quotes p and quotes q connected together in the form quote bracket p implies symbol q close bracket n symbol bracket p close bracket colon implies symbol colon bracket q close bracket close quote give a tautology shows that q follows from p and p implies q that quote bracket x close bracket n symbol f x colon implies symbol f a close quote is a tautology shows that f a follows from bracket x close bracket and symbol f x etc etc 6.1202 it is clear that we could have used for this purpose contradictions instead of tautologies 6.1203 in order to recognize a tautology as such we can in cases in which no sign of generality occurs in the tautology make use of the following intuitive method i write instead of quotes p quotes q quotes r etc quote capital t p capital f close quote quote capital t q capital f close quote quote capital t r capital f close quote etc the truth combinations i express by brackets e.g. readers note the following is a description of an image the top horizontal bracket connects the t in capital t p capital f to the f in capital t q capital f the next horizontal bracket connects the t in capital t p capital f to the t in capital t q capital f the next horizontal bracket connects the f in capital t p capital f to the t in capital t q capital f the bottom horizontal bracket connects the f in capital t p capital f to the f in capital t q capital f end of description of image and the coordination of the truth or falsity of the whole proposition with the truth combinations of the truth arguments by lines in the following way readers note the following is a description of an image the top horizontal bracket which connects the t in capital t p capital f to the f in capital t q capital f is labeled with an f the next horizontal bracket which connects the t in capital t p capital f to the t in capital t q capital f is labeled with a t the next horizontal bracket which connects the f in capital t p capital f to the t in capital t q capital f is labeled with a t the bottom horizontal bracket which connects the f in capital t p capital f to the f in capital t q capital f is labeled with a t end of description of image this sign for example would therefore present the proposition p implies symbol q now i will proceed to inquire whether such a proposition as not symbol bracket p end symbol not symbol p close bracket bracket the law of contradiction close bracket is a tautology the form quote not symbol xi close quote is written in our notation readers note the following is a description of an image capital t xi capital f inside of quotes the capital t in the expression is labeled by a line with an f and the capital f in the expression is labeled by a line with a t end of description of image the form quote xi end symbol eta close quote thus readers note the following is a description of an image the top horizontal bracket which connects the f in capital t xi capital f to the f in capital t eta capital f is labeled with an f the next horizontal bracket which connects the f in capital t xi capital f to the t in capital t eta capital f is labeled with an f the next horizontal bracket which connects the t in capital t xi capital f to the t in capital t eta capital f is labeled with a t the bottom horizontal bracket which connects the t in capital t xi capital f to The F in capital T, eta capital F, is labeled with an F. End of description of image. Hence the proposition, not symbol, bracket, P and symbol, not symbol, Q, close bracket, runs thus. Readers note, the following is a description of an image. The top bracket connects the T stemming from the F in capital T, Q, capital F, to the F in capital T, P, capital F, and is labeled with an F, which is in turn labeled with a T. The next bracket contains the T stemming from the F in capital T, Q, capital F, to the T in capital T, P, capital F. It is labeled with a T, which is in turn labeled with an F. The next bracket connects the F stemming from the T in capital T, Q, capital F, to the T in capital T, P, capital F. It is labeled with an F, which is in turn labeled with a T. The bottom bracket connects the F stemming from the T in capital T, Q, capital F, to the F in capital T, P, capital F. It is also labeled with an F, which is in turn labeled with a T. End of description of image. If we put quotes P instead of quotes Q and examine the combination of the outermost capital T and capital F with the innermost, it is seen that the truth of the whole proposition is coordinated with all the truth combinations of its argument. It's falsity with none of the truth combinations. 6.121 The propositions of logic demonstrate the logical properties of propositions. By combining them into propositions which say nothing. This method could be called a zero method. In a logical proposition, propositions are brought into equilibrium with one another. And the state of equilibrium then shows how these propositions must be logically constructed. 6.122 Once it follows that we can get on without logical propositions, 4 we can recognize in an adequate notation the formal properties of the propositions by mere inspection. 6.121 If, for example, two propositions, quotes P and quotes Q, give a tautology in the connection, quote P implies symbol Q, close quote, then it is clear that Q follows from P. For example, that quotes Q follows from quote P implies symbol Q and symbol P, close quote. We see from these two propositions themselves. But we can also show it by combining them to quote P implies symbol Q and symbol P, colon implies symbol, colon Q, close quote. And then showing that this is a tautology. 6.1222 This throws light on the question why logical propositions can no more be empirically confirmed than they can be empirically refuted. Not only must a proposition of logic be incapable of being contradicted by any possible experience, but it must also be incapable of being confirmed by any such. 6.1223 It now becomes clear why we often feel as though quote logical truths, close quote, must be quotes postulated by us. We can in fact postulate them insofar as we can postulate an adequate notation. 6.1224 It also becomes clear why logic has been called the theory of forms and of inference. 6.123 It is clear that the laws of logic cannot themselves obey further logical laws. Bracket. There is not, as Russell supposed, for every quote's type, a special law of contradiction. But one is sufficient, since it is not applied to itself. Close bracket. 6.1231 The mark of logical propositions is not their general validity. To be general is only to be accidentally valid for all things. An ungeneralized proposition can be tautologist, just as well as a generalized one. 6.1232 Logical general validity we could call essential, as opposed to accidental general validity. For example, of the proposition, quote, all men are mortal, close quote. Propositions like Russell's quote, axiom of reducibility, close quote, are not logical propositions, and this explains our feeling that, if true, they can only be true by a happy chance. 6.1233 We can imagine a world in which the axiom of reducibility is not valid, but it is clear that logic has nothing to do with the question whether our world is really of this kind or not. 6.124 The logical propositions describe the scaffolding of the world, or rather they present it. They, quotes, treat of nothing. They presuppose that names have meaning, and that elementary propositions have sense. And this is their connection with the world. It is clear that it must show something about the world that certain combinations of symbols, which essentially have a definite character, are tautologies. Herein lies the decisive point. We said that in the symbols, which we use, something is arbitrary, something not. In logic, only this expresses. But this means that in logic, it is not we who express by means of signs what we want, but in logic, the nature of the essentially necessary signs itself asserts. That is to say, if we know the logical syntax of any sign language, then all the propositions of logic are already given. 6.125 It is possible, also with the old conception of logic, to give at the outset a description of all, quotes, true, logical propositions. 6.1251 Hence, there can never be surprises in logic. 6.126 Whether a proposition belongs to logic can be calculated by calculating the logical properties of the symbol. In this we do when we prove a logical proposition. For, without troubling ourselves about a sense and a meaning, we form the logical propositions out of others by mere symbolic rules. We prove a logical proposition by creating it out of other logical propositions by applying in succession certain operations, which again generate tautologies out of the first. Bracket and firmatotology only tautologies follow. Close bracket. Naturally, this way of showing that its propositions are tautologies is quite unessential to logic, because the propositions from which the proof starts must show without proof that they are tautologies. 6.1261 In logic, process and result are equivalent. Bracket, therefore, no surprises. Close bracket. 6.1262 Proof in logic is only a mechanical expedient to facilitate the recognition of tautology, where it is complicated. 6.1263 It would be too remarkable if one could prove a significant proposition logically from another, and a logical proposition also. It is clear from the beginning that the logical proof of a significant proposition and the proof in logic must be two quite different things. 6.1264 The significant proposition asserts something, and its proof shows that it is so. In logic, every proposition is the form of a proof. Every proposition of logic is a modus ponens present in science. Bracket and the modus ponens cannot be expressed by a proposition. Close bracket. 6.1265 Logic can always be conceived to be such that every proposition is its own proof. 6.127 All propositions of logic are of equal rank. There are not some which are essentially primitive and others deduced from there. Every tautology itself shows that it is a tautology. 6.1271 It is clear that the number of, quote, primitive propositions of logic, close quote, is arbitrary. For, we could deduce logic from one primitive proposition by simply forming, for example, the logical produce of Frigga's primitive propositions. Bracket. Frigga would perhaps say that this would no longer be immediately self-evident. But it is remarkable that so exact a thinker as Frigga should have appealed to the degree of self-evidence as the criterion of a logical proposition. Close bracket. 6.13 Logic is not a theory, but a reflection of the world. Logic is transcendental. 6.2 Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-propositions. 6.21 Mathematical propositions express no thoughts. 6.211 In life, it is never a mathematical proposition which we need. But we use mathematical propositions only in order to infer from propositions which do not belong to mathematics, to others which equally do not belong to mathematics. In philosophy, the question, quote, why do we really use that word? That proposition, close quote, constantly leads to valuable results. 6.22 The logic of the world which the propositions of logic show in tautologies, mathematics shows in equations. 6.23 If two expressions are connected by the sign of equality, this means that they can be substituted for one another. But whether this is the case must show itself in the two expressions themselves. It characterizes the logical form of two expressions that they can be substituted for one another. 6.231 It is a property of affirmation that it can be conceived as double denial. It is a property of, quote, one plus one plus one plus one, close quote, that it can be conceived as, quote, bracket one plus one, close bracket plus, bracket one plus one, close bracket, close quote. 6.232 Frigga says that these expressions have the same meaning but different senses. But what is essential about equations is that it is not necessary in order to show that both expressions which are connected by the sign of equality have the same meaning. For this can be perceived from the two expressions themselves. 6.2321 And that the propositions of mathematics can be proved means nothing else than that the correctness can be seen without our having to compare what they express with facts as regards correctness. 6.2322 The identity of the meaning of two expressions cannot be asserted For in order to be able to assert anything about their meaning I must know their meaning and if I know their meaning I know whether they mean the same or something different. 6.2323 The equation characterizes only the standpoint from which I consider the two expressions. That is to say from the standpoint of their equality of meaning. 6.233 To the question whether we need intuition for the solution of mathematical problems it must be answered that language itself here supplies the necessary intuition. 6.2331 The process of calculation brings about just this intuition. Calculation is not an experiment. 6.234 Mathematics is a method of logic. 6.2341 The essential of mathematical method is working with equations. On this method depends the fact that every proposition of mathematics must be self-evident. 6.24 The method by which mathematics arrives at its equations is the method of substitution. For equations express the substitutability of two expressions and we proceed from a number of equations to new equations replacing expressions by others in accordance with the equations. 6.241 Thus the proof of the proposition 2 times 2 equals 4 runs bracket omega superscript v close bracket superscript mu prime regular script x equals omega superscript v times mu prime regular script x def omega superscript 2 times 2 prime regular script x equals bracket omega superscript 2 close bracket superscript 2 prime regular script x equals bracket omega superscript 2 close bracket superscript 1 plus 1 prime regular script x equals omega superscript 2 prime regular script omega superscript 2 prime regular script x equals omega superscript 1 plus 1 prime regular script omega superscript 1 plus 1 prime regular script x equals bracket omega prime omega close bracket prime bracket omega prime omega close bracket prime x equals omega prime omega prime omega prime omega prime omega prime x equals omega superscript 1 plus 1 plus 1 plus 1 prime regular script x equals omega superscript 4 prime regular script x. 6.3. Logical research means the investigation of all regularity and outside logic all is accident. 6.31. The so-called law of induction cannot in any case be a logical law, for it is obviously a significant proposition, and therefore it cannot be a law a priori either. 6.32. The law of causality is not a law, but the form of a law. 6.321. Quote, law of causality, close quote, is a class name, and as in mechanics there are, for instance, minimum laws such as that of least actions, so in physics there are causal laws, laws of the causality of form. 6.3211. Men had indeed an idea that there must be a, quote, law of least action, close quote, before they knew exactly how it ran. Bracket, here, as always, the a priori certain proves to be something purely logical. Close bracket. 6.33. We do not believe a priori in a law of conservation, but we know a priori, the possibility of a logical form. 6.34. All propositions, such as the law of causation, the law of continuity in nature, the law of least expenditure in nature, etc., etc., all these are a priori, intuitions of possible forms of the propositions of science. 6.341. Newtonian mechanics, for example, brings the description of the universe to a unified form. Let us imagine a white surface with irregular black spots. We now say whatever kind of picture these make, I can always get as near as I like to its description if I cover the surface with a sufficiently fine square network and now say of every square that it is white or black. In this way I shall have brought the description of the surface to a unified form. This form is arbitrary because I could have applied with equal success a net of triangular or hexagonal mesh. It can happen that the description would have been simpler with the aid of a triangular mesh. That is to say we might have described the surface more accurately with a triangular and coarser than with the finer square mesh or vice versa and so on. 2. The different networks correspond different systems of describing the world. Mechanics determine a form of description by saying all propositions in the description of the world must be obtained in a given way from a number of given propositions. The mechanical axioms. It thus provides the bricks for building the edifice of science and says whatever building thou wouldst erect, thou shalt construct in some manner with these bricks and these alone. Bracket. As with the system of numbers, one must be able to write down any arbitrary number. So with the system of mechanics, one must be able to write down any arbitrary physical proposition. Close bracket. 6.342. And now we see the relative position of logic and mechanics. Bracket. We could construct the network out of figures of different kinds as out of triangles and hexagons together. Close bracket. That a picture like that instance above can be described by a network of a given form asserts nothing about the picture. Bracket for this holds of every picture of this kind. Close bracket. But this does characterize the picture. The fact, namely, that it can be completely described by a definite net of definite fineness. So too, the fact that it can be described by Newtonian mechanics asserts nothing about the world. But this asserts something, namely, that it can be described in that particular way, in which, as a matter of fact, it is described. The fact, too, that it can be described more simply by one system of mechanics than by another says something about the world. 6.343. Mechanics is an attempt to construct, according to a single plan, all true propositions which we need for the description of the world. 6.3431. Through their whole logical apparatus, the physical laws still speak of the objects of the world. 6.3432. We must not forget that the description of the world by mechanics is always quite general. There is, for example, never any mention of particular material points in it, but always only of some points or others. 6.35. Although the spots in our picture are geometrical figures, geometry can obviously say nothing about their actual form and position. But the network is purely geometrical, and all its properties can be given a priori. Laws, like the law of causation, etc., treat of the network and not what the network describes. 6.36. If there were a law of causality, it might run, quote, there are natural laws, close quote, but that can clearly not be said, it shows itself. 6.361. In their terminology of Hertz, we might say, only uniform connections are thinkable. 6.3611. We cannot compare any process with the quote passage of time, close quote. There is no such thing, but only with another process. Bracket, say, with the movement of the chronometer, close bracket. Hence the description of the temporal sequence of events is only possible if we support ourselves on another process. It is exactly analogous for space. When, for example, we say that neither of two events, bracket, which mutually exclude one another, close bracket, can occur because there is no cause why the one should 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. And if there is such an asymmetry, we can regard this as the cause of the occurrence of the one and of the non-occurrence of the other. 6.3611. The Kantian problem of the right and left hand, which cannot be made to cover one another, already exists in the plane. And even in one-dimensional space, where the two congruent figures, A and B, cannot be made to cover one another without reader's note. There is an image depicting three dashes, a zero, a line, an X, two dashes, an X, a line, a zero, and four dashes. The first line is labeled A, and the second line is labeled B. Moving them out of this space, the right and left hand are in fact completely congruent, and the fact that they cannot be made to cover one another has nothing to do with it. A right hand glove could be put on a left hand if it could be turned round in four-dimensional space. 6.362. What can be described can happen too, and what is excluded by the law of causality cannot be described. 6.363. The process of induction is the process of assuming the simplest law that can be made to harmonize with our experience. 6.3631. This process, however, has no logical foundation, but only a psychological one. It is clear that there are no grounds for believing that the simplest course of events will really happen. 6.36311. That the sun will rise tomorrow is a hypothesis, and that means that we do not know whether it will rise. 6.37. A necessity for one thing to happen because another has happened does not exist. There is only logical necessity. 6.371. At the basis of the whole modern view of the world lies the illusion that the so-called laws of nature are the explanations of natural phenomena. 6.372. So people stop short at natural laws as something unassailable, as did the ancients at God and fate. And they are both right and wrong, but the ancients were clear in so far as they recognized one clear terminus, whereas the modern system makes it appear as though everything were explained. 6.373. The world is independent of my will. 6.374. Even if everything we wished were to happen, this would only be, so to speak, a favor of fate. For there is no logical connection between will and world, which would guarantee this, and the assumed physical connection itself we could not against will. 6.375. As there is only a logical necessity, so there is only a logical impossibility. 6.3751. For two colors, for example, to be at one place in the visual field is impossible, logically impossible, for it is excluded by the logical structure of color. Let us consider how this contradiction presents itself in physics, somewhat as follows. That a particle cannot at the same time have two different velocities, in essence, that at the same time it cannot be in two places, in essence, that particles 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 assertion that a point in the visual fields has two different colors at the same time is a contradiction. 6.4 All propositions are of equal value. 6.41 The sense of the world must lie outside the world. In the world, everything is as it is, and happens as it does happen. In it, there is no value, and if there were, it would be of no value. If there is a value which is of value, it must lie outside all happening and being so. For all happening and being so is accidental. What makes it non-accidental cannot lie in the world, for otherwise this would again be accidental. It must lie outside the world. 6.42 Hence also there can be no ethical propositions. Propositions cannot express anything higher. 6.421 It is clear that ethics cannot be expressed. Ethics is transcendental. Bracket, ethics, and aesthetics are one. Close bracket. 6.422 The first thought in setting up an ethical law of the form, quote, thou shalt, close quote, is, and what if I do not do it? But it is clear that ethics has nothing to do with punishment and reward in the ordinary sense. This question as to consequences of an action must therefore be irrelevant. At least these consequences will not be events, for there must be something right in that formulation of the question. There must be some sort of ethical reward and ethical punishment, but this must lie in the action itself. Bracket, and this is clear also, but the reward must be something acceptable, and punishment something unacceptable. Close bracket. 6.423 Of the will as the subject of the ethical, we cannot speak, and the will as a phenomenon is only of interest to psychology. 6.43 If good or bad willing changes the world, it can only change the limits of the world, not the facts, not the things that can be expressed in language. In brief, the world must thereby become quite another. It must, so to speak, wax or wane as a whole. The world of the happy is quite another than that of the unhappy. 6.431 As in death, too, the world does not change, but ceases. 6.4311 Death is not an event of life. Death is not lived through. If, by eternity, is understood not endless temporal duration, but timelessness, then he lives eternally, who lives in the present. Our life is endless in the way that our visual field is without limit. 6.4312 The temporal immortality of the human soul, that is to say, its eternal survival after death, is not only in no way guaranteed, but this assumption in the first place will not do for us what we always tried to make it do. Is a riddle solved by the fact that I survive forever? Is this eternal life not as enigmatic as our present one? The solution of the riddle of life in space and time lies outside space and time. Bracket, it is not problems of natural science which have to be solved. Close bracket. 6.432 How the world is, is completely indifferent for what is higher. God does not reveal himself in the world. 6.4321 The facts all belong only to the task and not to its performance. 6.44 Not how the world is, is the mystical, but that it is. 6.45 The contemplation of the world, subspecchiae aeternae, is its contemplation as a limited whole. The feeling that the world is a limited whole is the mystical feeling. 6.5 For an answer which cannot be expressed, the question too cannot be expressed. The riddle does not exist. If a question can be put at all, then it can also be answered. 6.51 Skepticism is not irrefutable, but palpably senseless. If it would doubt where a question cannot be asked. 4. Doubt can only exist where there is a question. A question only where there is an answer, and this only where something can be said. 6.52 We feel that even if all possible scientific questions be answered, the problems of life have still not been touched at all. Of course, there is then no question left, and just this is the answer. 6.521 The solution of the problem of life is seen in the vanishing of this problem. Bracket is not this the reason why men to whom, after long doubting the sense of life became clear, could not then say where in this sense consisted. 6.522 There is indeed the inexpressible. This shows itself. It is the mystical. 6.53 The right method of philosophy would be this, to say nothing except what can be said. In essence, the propositions of natural science. In essence, something that has nothing to do with philosophy. And then always, when someone else wished to say something metaphysical, to demonstrate to him that he had given no meaning to certain signs in his propositions. This method would be unsatisfying to the other. He would not have the feeling that we were teaching him philosophy, but it would be the only strictly correct method. 6.54 My propositions are elucidatory in this way. He who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. Bracket, he must so to speak throw away the latter, after he has climbed up on it, close bracket. He must surmount these propositions, then he sees the world rightly. 7. Whereof one cannot speak, thereof one must be silent. End of Section 5. End of Tractatus Logico Philosophicus by Ludwig Wittgenstein, translated by C. K. Ogden. Recording in memory of Mitchell Edwards.