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

5.01 Elementary propositions are the truth-arguments of propositions.

5.02 The arguments of functions are readily confused with the affixes of names. 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.

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

5.101 The truth-functions of a given number of elementary propositions can always be set out in a schema of the following kind: (TTTT) (p, q) Tautology (If p then p, and if q then q.) (p z p . q z q) (FTTT) (p, q) In words : Not both p and q. (P(p . q)) (TFTT) (p, q) " : If q then p. (q z p) (TTFT) (p, q) " : If p then q. (p z q) (TTTF) (p, q) " : p or q. (p C q) (FFTT) (p, q) " : Not g. (Pq) (FTFT) (p, q) " : Not p. (Pp) (FTTF) (p, q) " : p or q, but not both. (p . Pq : C : q . Pp) (TFFT) (p, q) " : If p then p, and if q then p. (p + q) (TFTF) (p, q) " : p (TTFF) (p, q) " : q (FFFT) (p, q) " : Neither p nor q. (Pp . Pq or p | q) (FFTF) (p, q) " : p and not q. (p . Pq) (FTFF) (p, q) " : q and not p. (q . Pp) (TFFF) (p,q) " : q and p. (q . p) (FFFF) (p, q) Contradiction (p and not p, and q and not q.) (p . Pp . q . Pq) I will give the name truth-grounds of a proposition to those truth-possibilities of its truth-arguments that make it true.

5.11 If all the truth-grounds that are common to a number of propositions are at the same time truth-grounds of a certain proposition, then we say that the truth of that proposition follows from the truth of the others.

5.12 In particular, the truth of a proposition 'p' follows from the truth of another proposition 'q' is all the truth-grounds of the latter are truth-grounds of the former.

5.121 The truth-grounds of the one are contained in those of the other: p follows from q.

5.122 If p follows from q, the sense of 'p' is contained in the sense of 'q'.

5.123 If a god creates a world in which certain propositions are true, then by that very act he also creates a world in which all the propositions that follow from them come true. And similarly he could not create a world in which the proposition 'p' was true without creating all its objects.

5.124 A proposition affirms every proposition that follows from it.

5.1241 'p . q' is one of the propositions that affirm 'p' and at the same time one of the propositions that affirm 'q'. Two propositions are opposed to one another if there is no proposition with a sense, that affirms them both. Every proposition that contradicts another negate it.

5.13 When the truth of one proposition follows from the truth of others, we can see this from the structure of the proposition.

5.131 If the truth of one proposition follows from the truth of others, this finds expression in relations in which the forms of the propositions stand to one another: nor is it necessary for us to set up these relations between them, by combining them with one another in a single proposition; on the contrary, the relations are internal, and their existence is an immediate result of the existence of the propositions.

5.1311 When we infer q from p C q and Pp, the relation between the propositional forms of 'p C q' and 'Pp' is masked, in this case, by our mode of signifying. But if instead of 'p C q' we write, for example, 'p|q . | . p|q', and instead of 'Pp', 'p|p' (p|q = neither p nor q), then the inner connexion becomes obvious. (The possibility of inference from (x) . fx to fa shows that the symbol (x) . fx itself has generality in it.)

5.132 If p follows from q, I can make an inference from q to p, deduce p from q. The nature of the inference can be gathered only from the two propositions. They themselves are the only possible justification of the inference. 'Laws of inference', which are supposed to justify inferences, as in the works of Frege and Russell, have no sense, and would be superfluous.

5.133 All deductions are made a priori.

5.134 One elementary proposition cannot be deduced form another.

5.135 There is no possible way of making an inference form the existence of one situation to the existence of another, entirely different situation.

5.136 There is no causal nexus to justify such an inference.

5.1361 We cannot infer the events of the future from those of the present. 

5.1362 The freedom of the will consists in the impossibility of knowing actions that still lie in the future. We could know them only if causality were an inner necessity like that of logical inference.--The connexion between knowledge and what is known is that of logical necessity. ('A knows that p is the case', has no sense if p is a tautology.)

5.1363 If the truth of a proposition does not follow from the fact that it is self-evident to us, then its self-evidence in no way justifies our belief in its truth.

5.14 If one proposition follows from another, then the latter says more than the former, and the former less than the latter.

5.141 If p follows from q and q from p, then they are one and same proposition.

5.142 A tautology follows from all propositions: it says nothing.

5.143 Contradiction is that common factor of propositions which no proposition has in common with another. Tautology is the common factor of all propositions that have nothing in common with one another. Contradiction, one might say, vanishes outside all propositions: tautology vanishes inside them. Contradiction is the outer limit of propositions: tautology is the unsubstantial point at their centre.

5.15 If Tr is the number of the truth-grounds of a proposition 'r', and if Trs is the number of the truth-grounds of a proposition 's' that are at the same time truth-grounds of 'r', then we call the ratio Trs : Tr the degree of probability that the proposition 'r' gives to the proposition 's'.

5.151 In a schema like the one above in 5.101, let Tr be the number of 'T's' in the proposition r, and let Trs, be the number of 'T's' in the proposition s that stand in columns in which the proposition r has 'T's'. Then the proposition r gives to the proposition s the probability Trs : Tr.

5.1511 There is no special object peculiar to probability propositions.

5.152 When propositions have no truth-arguments in common with one another, we call them independent of one another. Two elementary propositions give one another the probability 1/2. If p follows from q, then the proposition 'q' gives to the proposition 'p' the probability 1. The certainty of logical inference is a limiting case of probability. (Application of this to tautology and contradiction.)

5.153 In itself, a proposition is neither probable nor improbable. Either an event occurs or it does not: there is no middle way.

5.154 Suppose that an urn contains black and white balls in equal numbers (and none of any other kind). I draw one ball after another, putting them back into the urn. By this experiment I can establish that the number of black balls drawn and the number of white balls drawn approximate to one another as the draw continues. So this is not a mathematical truth. Now, if I say, 'The probability of my drawing a white ball is equal to the probability of my drawing a black one', this means that all the circumstances that I know of (including the laws of nature assumed as hypotheses) give no more probability to the occurrence of the one event than to that of the other. That is to say, they give each the probability 1/2 as can easily be gathered from the above definitions. What I confirm by the experiment is that the occurrence of the two events is independent of the circumstances of which I have no more detailed knowledge.

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

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

5.2 The structures of propositions stand in internal relations to one another.

5.21 In order to give prominence to these internal relations we can adopt the following mode of expression: we can represent a proposition as the result of an operation that produces it out of other propositions (which are the bases of the operation).

5.22 An operation is the expression of a relation between the structures of its result and of its bases.

5.23 The operation is what has to be done to the one proposition in order to make the other out of it.

5.231 And that will, of course, depend on their formal properties, on the internal similarity of their forms.

5.232 The internal relation by which a series is ordered is equivalent to the operation that produces one term from another.

5.233 Operations cannot make their appearance before the point at which one proposition is generated out of another in a logically meaningful way; i.e. the point at which the logical construction of propositions begins.

5.234 Truth-functions of elementary propositions are results of operations with elementary propositions as bases. (These operations I call truth-operations.)

5.2341 The sense of a truth-function of p is a function of the sense of p. Negation, logical addition, logical multiplication, etc. etc. are operations. (Negation reverses the sense of a proposition.)

5.24 An operation manifests itself in a variable; it shows how we can get from one form of proposition to another. It gives expression to the difference between the forms. (And what the bases of an operation and its result have in common is just the bases themselves.)

5.241 An operation is not the mark of a form, but only of a difference between forms.

5.242 The operation that produces 'q' from 'p' also produces 'r' from 'q', and so on. There is only one way of expressing this: 'p', 'q', 'r', etc. have to be variables that give expression in a general way to certain formal relations.

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

5.251 A function cannot be its own argument, whereas an operation can take one of its own results as its base.

5.252 It is only in this way that the step from one term of a series of forms to another is possible (from one type to another in the hierarchies of Russell and Whitehead). (Russell and Whitehead did not admit the possibility of such steps, but repeatedly availed themselves of it.)

5.2521 If an operation is applied repeatedly to its own results, I speak of successive applications of it. ('O'O'O'a' is the result of three successive applications of the operation 'O'E' to 'a'.) In a similar sense I speak of successive applications of more than one operation to a number of propositions.

5.2522 Accordingly I use the sign '[a, x, O'x]' for the general term of the series of forms a, O'a, O'O'a, ... . This bracketed expression is a variable: the first term of the bracketed expression is the beginning of the series of forms, the second is the form of a term x arbitrarily selected from the series, and the third is the form of the term that immediately follows x in the series.

5.2523 The concept of successive applications of an operation is equivalent to the concept 'and so on'.

5.253 One operation can counteract the effect of another. Operations can cancel one another.

5.254 An operation can vanish (e.g. negation in 'PPp' : PPp = p).

5.3 All propositions are results of truth-operations on elementary propositions. A truth-operation is the way in which a truth-function is produced out of elementary propositions. It is of the essence of truth-operations that, just as elementary propositions yield a truth-function of themselves, so too in the same way truth-functions yield a further truth-function. When a truth-operation is applied to truth-functions of elementary propositions, it always generates another truth-function of elementary propositions, another proposition. When a truth-operation is applied to the results of truth-operations on elementary propositions, there is always a single operation on elementary propositions that has the same result. Every proposition is the result of truth-operations on elementary propositions.

5.31 The schemata in 4.31 have a meaning even when 'p', 'q', 'r', etc. are not elementary propositions. And it is easy to see that the propositional sign in 4.442 expresses a single truth-function of elementary propositions even when 'p' and 'q' are truth-functions of elementary propositions.

5.32 All truth-functions are results of successive applications to elementary propositions of a finite number of truth-operations.

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

5.41 The reason is that the results of truth-operations on truth-functions are always identical whenever they are one and the same truth-function of elementary propositions.

5.42 It is self-evident that C, z, etc. are not relations in the sense in which right and left etc. are relations. The interdefinability of Frege's and Russell's 'primitive signs' of logic is enough to show that they are not primitive signs, still less signs for relations. And it is obvious that the 'z' defined by means of 'P' and 'C' is identical with the one that figures with 'P' in the definition of 'C'; and that the second 'C' is identical with the first one; and so on.

5.43 Even at first sight it seems scarcely credible that there should follow from one fact p infinitely many others , namely PPp, PPPPp, etc. And it is no less remarkable that the infinite number of propositions of logic (mathematics) follow from half a dozen 'primitive propositions'. But in fact all the propositions of logic say the same thing, to wit nothing.

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

5.441 This vanishing of the apparent logical constants also occurs in the case of 'P(dx) . Pfx', which says the same as '(x) . fx', and in the case of '(dx) . fx . x = a', which says the same as 'fa'.

5.442 If we are given a proposition, then with it we are also given the results of all truth-operations that have it as their base.

5.45 If there are primitive logical signs, then any logic that fails to show clearly how they are placed relatively to one another and to justify their existence will be incorrect. The construction of logic out of its primitive signs must be made clear.

5.451 If logic has primitive ideas, they must be independent of one another. If a primitive idea has been introduced, it must have been introduced in all the combinations in which it ever occurs. It cannot, therefore, be introduced first for one combination and later reintroduced for another. For example, once negation has been introduced, we must understand it both in propositions of the form 'Pp' and in propositions like 'P(p C q)', '(dx) . Pfx', etc. We must not introduce it first for the one class of cases and then for the other, since it would then be left in doubt whether its meaning were the same in both cases, and no reason would have been given for combining the signs in the same way in both cases. (In short, Frege's remarks about introducing signs by means of definitions (in The Fundamental Laws of Arithmetic ) also apply, mutatis mutandis, to the introduction of primitive signs.)

5.452 The introduction of any new device into the symbolism of logic is necessarily a momentous event. In logic a new device should not be introduced in brackets or in a footnote with what one might call a completely innocent air. (Thus in Russell and Whitehead's Principia Mathematica there occur definitions and primitive propositions expressed in words. Why this sudden appearance of words? It would require a justification, but none is given, or could be given, since the procedure is in fact illicit.) But if the introduction of a new device has proved necessary at a certain point, we must immediately ask ourselves, 'At what points is the employment of this device now unavoidable ?' and its place in logic must be made clear.

5.453 All numbers in logic stand in need of justification. Or rather, it must become evident that there are no numbers in logic. There are no pre-eminent numbers.

5.454 In logic there is no co-ordinate status, and there can be no classification. In logic there can be no distinction between the general and the specific.

5.4541 The solutions of the problems of logic must be simple, since they set the standard of simplicity. Men have always had a presentiment that there must be a realm in which the answers to questions are symmetrically combined--a priori--to form a self-contained system. A realm subject to the law: Simplex sigillum veri.

5.46 If we introduced logical signs properly, then we should also have introduced at the same time the sense of all combinations of them; i.e. not only 'p C q' but 'P(p C q)' as well, etc. etc. We should also have introduced at the same time the effect of all possible combinations of brackets. And thus it would have been made clear that the real general primitive signs are not ' p C q', '(dx) . fx', etc. but the most general form of their combinations.

5.461 Though it seems unimportant, it is in fact significant that the pseudo-relations of logic, such as C and z, need brackets--unlike real relations. Indeed, the use of brackets with these apparently primitive signs is itself an indication that they are not primitive signs. And surely no one is going to believe brackets have an independent meaning.

5.4611 Signs for logical operations are punctuation-marks,

5.47 It is clear that whatever we can say in advance about the form of all propositions, we must be able to say all at once . An elementary proposition really contains all logical operations in itself. For 'fa' says the same thing as '(dx) . fx . x = a' Wherever there is compositeness, argument and function are present, and where these are present, we already have all the logical constants. One could say that the sole logical constant was what all propositions, by their very nature, had in common with one another. But that is the general propositional form.

5.471 The general propositional form is the essence of a proposition.

5.4711 To give the essence of a proposition means to give the essence of all description, and thus the essence of the world.

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

5.473 Logic must look after itself. If a sign is possible , then it is also capable of signifying. Whatever is possible in logic is also permitted. (The reason why 'Socrates is identical' means nothing is that there is no property called 'identical'. The proposition is nonsensical because we have failed to make an arbitrary determination, and not because the symbol, in itself, would be illegitimate.) In a certain sense, we cannot make mistakes in logic.

5.4731 Self-evidence, which Russell talked about so much, can become dispensable in logic, only because language itself prevents every logical mistake.--What makes logic a priori is the impossibility of illogical thought.

5.4732 We cannot give a sign the wrong sense.

5,47321 Occam's maxim is, of course, not an arbitrary rule, nor one that is justified by its success in practice: its point is that unnecessary units in a sign-language mean nothing. Signs that serve one purpose are logically equivalent, and signs that serve none are logically meaningless.

5.4733 Frege says that any legitimately constructed proposition must have a sense. And I say that any possible proposition is legitimately constructed, and, if it has no sense, that can only be because we have failed to give a meaning to some of its constituents. (Even if we think that we have done so.) Thus the reason why 'Socrates is identical' says nothing is that we have not given any adjectival meaning to the word 'identical'. For when it appears as a sign for identity, it symbolizes in an entirely different way--the signifying relation is a different one--therefore the symbols also are entirely different in the two cases: the two symbols have only the sign in common, and that is an accident.

5.474 The number of fundamental operations that are necessary depends solely on our notation.

5.475 All that is required is that we should construct a system of signs with a particular number of dimensions--with a particular mathematical multiplicity

5.476 It is clear that this is not a question of a number of primitive ideas that have to be signified, but rather of the expression of a rule.

5.5 Every truth-function is a result of successive applications to elementary propositions of the operation '(-----T)(E, ....)'. This operation negates all the propositions in the right-hand pair of brackets, and I call it the negation of those propositions.

5.501 When a bracketed expression has propositions as its terms--and the order of the terms inside the brackets is indifferent--then I indicate it by a sign of the form '(E)'. '(E)' is a variable whose values are terms of the bracketed expression and the bar over the variable indicates that it is the representative of ali its values in the brackets. (E.g. if E has the three values P,Q, R, then (E) = (P, Q, R). ) What the values of the variable are is something that is stipulated. The stipulation is a description of the propositions that have the variable as their representative. How the description of the terms of the bracketed expression is produced is not essential. We can distinguish three kinds of description: 1.Direct enumeration, in which case we can simply substitute for the variable the constants that are its values; 2. giving a function fx whose values for all values of x are the propositions to be described; 3. giving a formal law that governs the construction of the propositions, in which case the bracketed expression has as its members all the terms of a series of forms.

5.502 So instead of '(-----T)(E, ....)', I write 'N(E)'. N(E) is the negation of all the values of the propositional variable E.

5.503 It is obvious that we can easily express how propositions may be constructed with this operation, and how they may not be constructed with it; so it must be possible to find an exact expression for this.

5.51 If E has only one value, then N(E) = Pp (not p); if it has two values, then N(E) = Pp . Pq. (neither p nor g).

5.511 How can logic--all-embracing logic, which mirrors the world--use such peculiar crotchets and contrivances? Only because they are all connected with one another in an infinitely fine network, the great mirror.

5.512 'Pp' is true if 'p' is false. Therefore, in the proposition 'Pp', when it is true, 'p' is a false proposition. How then can the stroke 'P' make it agree with reality? But in 'Pp' it is not 'P' that negates, it is rather what is common to all the signs of this notation that negate p. That is to say the common rule that governs the construction of 'Pp', 'PPPp', 'Pp C Pp', 'Pp . Pp', etc. etc. (ad inf.). And this common factor mirrors negation.

5.513 We might say that what is common to all symbols that affirm both p and q is the proposition 'p . q'; and that what is common to all symbols that affirm either p or q is the proposition 'p C q'. And similarly we can say that two propositions are opposed to one another if they have nothing in common with one another, and that every proposition has only one negative, since there is only one proposition that lies completely outside it. Thus in Russell's notation too it is manifest that 'q : p C Pp' says the same thing as 'q', that 'p C Pq' says nothing.

5.514 Once a notation has been established, there will be in it a rule governing the construction of all propositions that negate p, a rule governing the construction of all propositions that affirm p, and a rule governing the construction of all propositions that affirm p or q; and so on. These rules are equivalent to the symbols; and in them their sense is mirrored.

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

5.5151 Must the sign of a negative proposition be constructed with that of the positive proposition? Why should it not be possible to express a negative proposition by means of a negative fact? (E.g. suppose that "a' does not stand in a certain relation to 'b'; then this might be used to say that aRb was not the case.) But really even in this case the negative proposition is constructed by an indirect use of the positive. The positive proposition necessarily presupposes the existence of the negative proposition and vice versa.

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

5.521 I dissociate the concept all from truth-functions. Frege and Russell introduced generality in association with logical productor logical sum. This made it difficult to understand the propositions '(dx) . fx' and '(x) . fx', in which both ideas are embedded.

5.522 What is peculiar to the generality-sign is first, that it indicates a logical prototype, and secondly, that it gives prominence to constants.

5.523 The generality-sign occurs as an argument.

5.524 If objects are given, then at the same time we are given all objects. If elementary propositions are given, then at the same time all elementary propositions are given.

5.525 It is incorrect to render the proposition '(dx) . fx' in the words, 'fx is possible ' as Russell does. The certainty, possibility, or impossibility of a situation is not expressed by a proposition, but by an expression's being a tautology, a proposition with a sense, or a contradiction. The precedent to which we are constantly inclined to appeal must reside in the symbol itself.

5.526 We can describe the world completely by means of fully generalized propositions, i.e. without first correlating any name with a particular object.

5.5261 A fully generalized proposition, like every other proposition, is composite. (This is shown by the fact that in '(dx, O) . Ox' we have to mention 'O' and 's' separately. They both, independently, stand in signifying relations to the world, just as is the case in ungeneralized propositions.) It is a mark of a composite symbol that it has something in common with other symbols.

5.5262 The truth or falsity of every proposition does make some alteration in the general construction of the world. And the range that the totality of elementary propositions leaves open for its construction is exactly the same as that which is delimited by entirely general propositions. (If an elementary proposition is true, that means, at any rate, one more true elementary proposition.)

5.53 Identity of object I express by identity of sign, and not by using a sign for identity. Difference of objects I express by difference of signs.

5.5301 It is self-evident that identity is not a relation between objects. This becomes very clear if one considers, for example, the proposition '(x) : fx . z . x = a'. What this proposition says is simply that only a satisfies the function f, and not that only things that have a certain relation to a satisfy the function, Of course, it might then be said that only a did have this relation to a; but in order to express that, we should need the identity-sign itself.

5.5302 Russell's definition of '=' is inadequate, because according to it we cannot say that two objects have all their properties in common. (Even if this proposition is never correct, it still has sense .)

5.5303 Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all.

5.531 Thus I do not write 'f(a, b) . a = b', but 'f(a, a)' (or 'f(b, b)); and not 'f(a,b) . Pa = b', but 'f(a, b)'.

5.532 And analogously I do not write '(dx, y) . f(x, y) . x = y', but '(dx) . f(x, x)'; and not '(dx, y) . f(x, y) . Px = y', but '(dx, y) . f(x, y)'.

5.5321 Thus, for example, instead of '(x) : fx z x = a' we write '(dx) . fx . z : (dx, y) . fx. fy'. And the proposition, 'Only one x satisfies f( )', will read '(dx) . fx : P(dx, y) . fx . fy'.

5.533 The identity-sign, therefore, is not an essential constituent of conceptual notation.

5.534 And now we see that in a correct conceptual notation pseudo-propositions like 'a = a', 'a = b . b = c . z a = c', '(x) . x = x', '(dx) . x = a', etc. cannot even be written down.

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

5.5351 There are certain cases in which one is tempted to use expressions of the form 'a = a' or 'p z p' and the like. In fact, this happens when one wants to talk about prototypes, e.g. about proposition, thing, etc. Thus in Russell's Principles of Mathematics 'p is a proposition'--which is nonsense--was given the symbolic rendering 'p z p' and placed as an hypothesis in front of certain propositions in order to exclude from their argument-places everything but propositions. (It is nonsense to place the hypothesis 'p z p' in front of a proposition, in order to ensure that its arguments shall have the right form, if only because with a non-proposition as argument the hypothesis becomes not false but nonsensical, and because arguments of the wrong kind make the proposition itself nonsensical, so that it preserves itself from wrong arguments just as well, or as badly, as the hypothesis without sense that was appended for that purpose.)

5.5352 In the same way people have wanted to express, 'There are no things ', by writing 'P(dx) . x = x'. But even if this were a proposition, would it not be equally true if in fact 'there were things' but they were not identical with themselves?

5.54 In the general propositional form propositions occur in other propositions only as bases of truth-operations.

5.541 At first sight it looks as if it were also possible for one proposition to occur in another in a different way. Particularly with certain forms of proposition in psychology, such as 'A believes that p is the case' and A has the thought p', etc. For if these are considered superficially, it looks as if the proposition p stood in some kind of relation to an object A. (And in modern theory of knowledge (Russell, Moore, etc.) these propositions have actually been construed in this way.)

5.542 It is clear, however, that 'A believes that p', 'A has the thought p', and 'A says p' are of the form '"p" says p': and this does not involve a correlation of a fact with an object, but rather the correlation of facts by means of the correlation of their objects.

5.5421 This shows too that there is no such thing as the soul--the subject, etc.--as it is conceived in the superficial psychology of the present day. Indeed a composite soul would no longer be a soul.

5.5422 The correct explanation of the form of the proposition, 'A makes the judgement p', must show that it is impossible for a judgement to be a piece of nonsense. (Russell's theory does not satisfy this requirement.)

5.5423 To perceive a complex means to perceive that its constituents are related to one another in such and such a way. This no doubt also explains why there are two possible ways of seeing the figure as a cube; and all similar phenomena. For we really see two different facts. (If I look in the first place at the corners marked a and only glance at the b's, then the a's appear to be in front, and vice versa).

5.55 We now have to answer a priori the question about all the possible forms of elementary propositions. Elementary propositions consist of names. Since, however, we are unable to give the number of names with different meanings, we are also unable to give the composition of elementary propositions.

5.551 Our fundamental principle is that whenever a question can be decided by logic at all it must be possible to decide it without more ado. (And if we get into a position where we have to look at the world for an answer to such a problem, that shows that we are on a completely wrong track.)

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

5.5521 And if this were not so, how could we apply logic? We might put it in this way: if there would be a logic even if there were no world, how then could there be a logic given that there is a world?

5.553 Russell said that there were simple relations between different numbers of things (individuals). But between what numbers? And how is this supposed to be decided?--By experience? (There is no pre-eminent number.)

5.554 It would be completely arbitrary to give any specific form.

5.5541 It is supposed to be possible to answer a priori the question whether I can get into a position in which I need the sign for a 27-termed relation in order to signify something.

5.5542 But is it really legitimate even to ask such a question? Can we set up a form of sign without knowing whether anything can correspond to it? Does it make sense to ask what there must be in order that something can be the case?

5.555 Clearly we have some concept of elementary propositions quite apart from their particular logical forms. But when there is a system by which we can create symbols, the system is what is important for logic and not the individual symbols. And anyway, is it really possible that in logic I should have to deal with forms that I can invent? What I have to deal with must be that which makes it possible for me to invent them.

5.556 There cannot be a hierarchy of the forms of elementary propositions. We can foresee only what we ourselves construct.

5.5561 Empirical reality is limited by the totality of objects. The limit also makes itself manifest in the totality of elementary propositions. Hierarchies are and must be independent of reality.

5.5562 If we know on purely logical grounds that there must be elementary propositions, then everyone who understands propositions in their C form must know It.

5.5563 In fact, all the propositions of our everyday language, just as they stand, are in perfect logical order.--That utterly simple thing, which we have to formulate here, is not a likeness of the truth, but the truth itself in its entirety. (Our problems are not abstract, but perhaps the most concrete that there are.)

5.557 The application of logic decides what elementary propositions there are. What belongs to its application, logic cannot anticipate. It is clear that logic must not clash with its application. But logic has to be in contact with its application. Therefore logic and its application must not overlap.

5.5571 If I cannot say a priori what elementary propositions there are, then the attempt to do so must lead to obvious nonsense.

5.6 The limits of my language mean the limits of my world.

5.61 Logic pervades the world: the limits of the world are also its limits. So we cannot say in logic, 'The world has this in it, and this, but not that.' For that would appear to presuppose that we were excluding certain possibilities, and this cannot be the case, since it would require that logic should go beyond the limits of the world; for only in that way could it view those limits from the other side as well. We cannot think what we cannot think; so what we cannot think we cannot say either.

5.62 This remark provides the key to the problem, how much truth there is in solipsism. For what the solipsist means is quite correct; only it cannot be said , but makes itself manifest. The world is my world: this is manifest in the fact that the limits of language (of that language which alone I understand) mean the limits of my world.

5.621 The world and life are one.

5.63 I am my world. (The microcosm.)

5.631 There is no such thing as the subject that thinks or entertains ideas. If I wrote a book called The World as l found it , I should have to include a report on my body, and should have to say which parts were subordinate to my will, and which were not, etc., this being a method of isolating the subject, or rather of showing that in an important sense there is no subject; for it alone could not be mentioned in that book.--

5.632 The subject does not belong to the world: rather, it is a limit of the world.

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

5.6331 For the form of the visual field is surely not like this

5.634 This is connected with the fact that no part of our experience is at the same time a priori. Whatever we see could be other than it is. Whatever we can describe at all could be other than it is. There is no a priori order of things.

5.64 Here it can be seen that solipsism, when its implications are followed out strictly, coincides with pure realism. The self of solipsism shrinks to a point without extension, and there remains the reality co-ordinated with it.

5.641 Thus there really is a sense in which philosophy can talk about the self in a non-psychological way. What brings the self into philosophy is the fact that 'the world is my world'. The philosophical self is not the human being, not the human body, or the human soul, with which psychology deals, but rather the metaphysical subject, the limit of the world--not a part of it.