6 The general form of a truth-function is [p, ξ, N(ξ)]. This is the general form of a proposition.
6.001 What this says is just that every proposition is a result of successive applications to elementary propositions of the operation N'(ξ)
6.002 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.
6.01 Therefore the general form of an operation Ω'(η) is [ξ, N(ξ)]' (η) (= [η, ξ, N(ξ)]). This is the most general form of transition from one proposition to another.
6.02 And this is how we arrive at numbers. I give the following definitions
x = Ω0' x Def.
and:
Ω'Ωv'x = Ωv+1'x Def.
So, in accordance with these rules, which deal with signs, we write the series
x, Ω'x, Ω'Ω'x, Ω'Ω'Ω'x, …,
in the following way
Ω0'x, Ω0+1'x, Ω0+1+1'x, Ω0+1+1+1'x, … .
Therefore, instead of '[x, ξ, Ω'ξ]',
I write '[Ω0'x, Ωv'x, Ωv+1'x]'.
And I give the following definitions
0 + 1 = 1 Def.,
0 + 1 + 1 = 2 Def.,
0 + 1 + 1 + 1 = 3 Def.,
(and so on).
6.021 A number is the exponent of an operation.
6.022 The concept of number is simply what is common to all numbers, 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.
6.03 The general form of an integer is [0, ξ, ξ+1].
6.031 The theory of cla**es is completely superfluous in mathematics. This is connected with the fact that the generality required in mathematics is not accidental generality.
6.1 The propositions of logic are tautologies.
6.11 Therefore the propositions of logic say nothing. (They are the an*lytic propositions.)
6.111 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.
6.112 The correct explanation of the propositions of logic must a**ign to them a unique status among all propositions.
6.113 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.
6.12 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.
6.1201 For example, the fact that the propositions 'p' and '~p' in the combination '(p . ~p)' yield a tautology shows that they contradict one another. The fact that the propositions 'p ⊃ q', 'p', and 'q', combined with one another in the form '(p ⊃ q) . (p) :⊃: (q)', yield a tautology shows that q follows from p and p ⊃ q. The fact that '(x).fxx :⊃: fa' is a tautology shows that fa follows from (x).fx. Etc. etc.
6.1202 It is clear that one could achieve the same purpose by using contradictions instead of tautologies.
6.1203 In order to recognize an expression as a tautology, in cases where no generality-sign 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.
link
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
link
So this sign, for instance, would represent the proposition p ⊃ q. Now, by way of example, I wish to examine the proposition P(p .~p) (the law of contradiction) in order to determine whether it is a tautology. In our notation the form '~ξ' is written as
link
and the form 'ξ . η' as
link
Hence the proposition ~(p . ~p). reads as follows
link
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.
6.121 The propositions of logic demonstrate the logical properties of propositions by combining them so as to form propositions that say nothing. This method could also be called a zero-method. 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.
6.122 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.
6.1221 If, for example, two propositions 'p' and 'q' in the combination 'p ⊃ 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 ⊃ q . p', but it is also possible to show it in this way: we combine them to form 'p ⊃ q . p :⊃: q', and then show that this is a tautology.
6.1222 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.
6.1223 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.
6.1224 It also becomes clear now why logic was called the theory of forms and of inference.
6.123 Clearly the laws of logic cannot in their turn be subject to laws of logic. (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.)
6.1231 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.
6.1232 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.
6.1233 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.
6.124 The propositions of logic describe the scaffolding of the world, or rather they represent it. 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.
6.125 It is possible—indeed possible even according to the old conception of logic—to give in advance a description of all 'true' logical propositions.
6.1251 Hence there can never be surprises in logic.
6.126 One can calculate whether a proposition belongs to logic, by calculating the logical properties of the symbol. 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.
6.1261 In logic process and result are equivalent. (Hence the absence of surprise.)
6.1262 Proof in logic is merely a mechanical expedient to facilitate the recognition of tautologies in complicated cases.
6.1263 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.
6.1264 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.)
6.1265 It is always possible to construe logic in such a way that every proposition is its own proof.
6.127 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.
6.1271 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.)
6.13 Logic is not a body of doctrine, but a mirror-image of the world. Logic is transcendental.
6.2 Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-propositions.
6.21 A proposition of mathematics does not express a thought.
6.211 Indeed in real life a mathematical proposition is never 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.)
6.22 The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics.
6.23 If two expressions are combined by means of the sign of equality, that means that they can be substituted for one another. But it must 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.
6.231 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)'.
6.232 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.
6.2321 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.
6.2322 It is impossible to a**ert the identity of meaning of two expressions. For in order to be able to a**ert 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.
6.2323 An equation merely marks the point of view from which I consider the two expressions: it marks their equivalence in meaning.
6.233 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.
6.2331 The process of calculating serves to bring about that intuition. Calculation is not an experiment.
6.234 Mathematics is a method of logic.
6.2341 It is the essential characteristic of mathematical method that it employs equations. For it is because of this method that every proposition of mathematics must go without saying.
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, starting from a number of equations, we advance to new equations by substituting different expressions in accordance with the equations.
6.241 Thus the proof of the proposition 2 × 2 = 4 runs as follows:
(Ωv)μ'x = Ωv×μ'x Def.,
Ω2×2'x = (Ω2)2'x = (Ω2)1+1'x
= Ω2' Ω2'x = Ω1+1'Ω1+1'x= (Ω'Ω)'(Ω'Ω)'x
= Ω'Ω'Ω'Ω'x = Ω1+1+1+1'x = Ω4'x.
6.3 The exploration of logic means the exploration of everything that is subject to law. And outside logic everything is accidental.
6.31 The so-called law of induction cannot possibly be a law of logic, since it is obviously a proposition with sense.—-Nor, therefore, can it be an a priori law.
6.32 The law of causality is not a law but the form of a law.
6.321 'Law of causality'—that is a general name. And just as 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.
6.3211 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.)
6.33 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.
6.34 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.
6.341 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.)
6.342 And now we can see the relative position of logic and mechanics. (The net might also consist of more than one kind of mesh: e.g. 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.
6.343 Mechanics is an attempt to construct according to a single plan all the true propositions that we need for the description of the world.
6.3431 The laws of physics, with all their logical apparatus, still speak, however indirectly, about the objects of the world.
6.3432 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 point-ma**es: it will only talk about any point- ma**es whatsoever.
6.35 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.
6.36 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.
6.361 One might say, using Hertt:'s terminology, that only connexions that are subject to law are thinkable.
6.3611 We cannot compare a process with 'the pa**age 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 an*logous 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 non-occurrence of the other.
6.36111 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 one-dimensional 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.
6.362 What can be described can happen too: and what the law of causality is meant to exclude cannot even be described.
6.363 The procedure of induction consists in accepting as true the simplest law that can be reconciled with our experiences.
6.3631 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.
6.36311 It is an hypothesis that the sun will rise tomorrow: and this means that we do not know whether it will rise.
6.37 There is no compulsion making one thing happen because another has happened. The only necessity that exists is logical necessity.
6.371 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.
6.372 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.
6.373 The world is independent of my will.
6.374 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.
6.375 Just as the only necessity that exists is logical necessity, so too the only impossibility that exists is logical impossibility.
6.3751 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.)
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 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.
6.42 So too it is impossible for there to be propositions of ethics. Propositions can express nothing that is higher.
6.421 It is clear that ethics cannot be put into words. Ethics is transcendental. (Ethics and aesthetics are one and the same.)
6.422 When an ethical law of the form, 'Thou shalt …' is laid down, one's first thought is, 'And what if I do, not do it?' It is clear, however, that ethics has nothing to do with punishment and reward in 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.)
6.423 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.
6.43 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.
6.431 So too at d**h the world does not alter, but comes to an end.
6.4311 d**h is not an event in life: we do not live to experience d**h. 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.
6.4312 Not only is there no guarantee of the temporal immortality of the human soul, that is to say of its eternal survival after d**h; but, in any case, this a**umption 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.)
6.432 How things are in the world is a matter of complete indifference for what is higher. God does not reveal himself in the world.
6.4321 The facts all contribute only to setting the problem, not to its solution.
6.44 It is not how things are in the world that is mystical, but that it exists.
6.45 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.
6.5 When the answer cannot be put into words, neither can the question be put into words. The riddle does not exist. If a question can be framed at all, it is also possible to answer it.
6.51 Scepticism is not irrefutable, but obviously nonsensical, when it tries to raise doubts where no questions can be asked. For doubt can exist only where a question exists, a question only where an answer exists, and an answer only where something can be said.
6.52 We feel that even when all possible scientific questions have been answered, the problems of life remain completely untouched. Of course there are then no questions left, and this itself is the answer.
6.521 The solution of the problem of life is seen in the vanishing of the problem. (Is not this the reason why those who have found after a long period of doubt that the sense of life became clear to them have then been unable to say what constituted that sense?)
6.522 There are, indeed, things that cannot be put into words. They make themselves manifest. They are what is mystical.
6.53 The correct method in philosophy would really be the following: to say nothing except what can be said, i.e. propositions of natural science—i.e. something that has nothing to do with philosophy — and then, whenever someone else wanted to say something metaphysical, to demonstrate to him that he had failed to give a meaning to certain signs in his propositions. Although it would not be satisfying to the other person—he would not have the feeling that we were teaching him philosophy—this method would be the only strictly correct one.
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. (He must so to speak throw away the ladder, after he has climbed up on it.)