Threadkillah!!!

but all that would be is, "debits must = credits."

(when the "structural defect test," and the "arbitrary and capricious standard" are discussed and defined is where the real fun begins!)

"Hi, would you like to fuck me?"

Ha ha very funny. This pseudo-ironic post-Christian hatin' on the Bible is sooo lame. Have you even read the damn thing? Thought not.

The verb of a proposition cannot be „is true” or „is false”, but whatever is true or false must already contain the verb.

Deductions only proceed according to the laws of deduction, but these laws cannot justify the deduction.

One reason for supposing that not all propositions which have more than one argument are relational propositions is that if they were, the relations of judgement and inference would have to hold between an arbitrary number of things.

Every proposition which seems to be about a complex can be analysed into a proposition about its constituents and about the proposition which describes the complex perfectly; i.e., that proposition which is equivalent to saying the complex exists.

The idea that propositions are names of complexes suggests that whatever is not a proper name is a sign for a relation. Because spatial complexes1 consist of Things and Relations only and the idea of a complex is taken from space.

In a proposition convert all its indefinables into variables; there then remains a class of propositions which is not all propositions but a type.

A2 There are thus two ways in which signs are similar. The names Socrates and Plato are similar: they are both names. But whatever they have in common must not be introduced before Socrates and Plato are introduced. The same applies to subject-predicate form etc. Therefore, thing, proposition, subject-predicate form, etc., are not indefinables, i.e., types are not indefinables.
When we say A judges that etc., then we have to mention a whole proposition which A judges. It will not do either to mention only its constituents, or its constituents and form, but not in the proper order. This shows that a proposition itself must occur in the statement that it is judged; however, for instance, „not-p” may be explained. The question „What is negated” must have a meaning.

To understand a proposition p it is not enough to know that p implies ‚„p” is true’, but we must also know that ~p implies „p is false”. This shows the bipolarity of the proposition.

W-F = Wahr-Falsch

To every molecular function a WF scheme corresponds. Therefore we may use the WF scheme itself instead of the function. Now what the WF scheme does is, it correlates the letters W and F with each proposition. These two letters are the poles of atomic propositions. Then the scheme correlates another W and F to these poles. In this notation all that matters is the correlation of the outside poles to the pole of the atomic propositions. Therefore not-not-p is the same symbol as p. And therefore we shall never get two symbols for the same molecular function.

A3 The meaning of a proposition is the fact which actually corresponds to it.
As the ab functions of atomic propositions are bi-polar propositions again we can perform ab operations on them. We shall, by doing so, correlate two new outside poles via the old outside poles to the poles of the atomic propositions.

The symbolising fact in a-p-b is that, say2 a is on the left of p and b on the right of p; then the correlation of new poles is to be transitive, so that for instance if a new pole a in whatever way i.e. via whatever poles is correlated to the inside a, the symbol is not changed thereby. It is therefore possible to construct all possible ab functions by performing one ab operation repeatedly, and we can therefore talk of all ab functions as of all those functions which can be obtained by performing this ab operation repeatedly.

[Note by Bertrand Russell]

[NB. ab means the same as WF, which means true-false.]

Naming is like pointing. A function is like a line dividing points of a plane into right and left ones; then „p or not-p” has no meaning because it does not divide the plane.

But though a particular proposition „p or not-p” has no meaning, a general proposition „for all p's, p or not-p” has a meaning because this does not contain the nonsensical function „p or not-p” but the function „p or not-q” just as „for all x's xRx” contains the function „xRy”.

A4 A proposition is a standard to which all facts behave, with names it is otherwise; it is thus bi-polarity and sense comes in; just as one arrow behaves to another arrow by being in the same sense or the opposite, so a fact behaves to a proposition.
The form of a proposition has meaning in the following way. Consider a symbol „xRy”. To symbols of this form correspond couples of things whose names are respectively „x” and „y”. The things x y stand to one another in all sorts of relations, amongst others some stand in the relation R, and some not; just as I single out a particular thing by a particular name I single out all behaviours of the points x and y with respect to the relation R. I say that if an x stands in the relation R to a y the sign „x R y” is to be called true to the fact and otherwise false. This is a definition of sense.

In my theory p has the same meaning as not-p but opposite sense. The meaning is the fact. The proper theory of judgment must make it impossible to judge nonsense.

It is not strictly true to say that we understand a proposition p if we know that p is equivalent to „p is true” for this would be the case if accidentally both were true or false. What is wanted is the formal equivalence with respect to the forms of the proposition, i.e., all the general indefinables involved. The sense of an ab function of a proposition is a function of its sense. There are only unasserted propositions. ¦ Assertion is merely psychological. In not-p, p is exactly the same as if it stands alone; this point is absolutely fundamental. Among the facts which make „p or q” true there are also facts which make „p and q” true; if propositions have only meaning, we ought, in such a case, to say that these two propositions are identical, but in fact, their sense is different for we have introduced sense by talking of all p's and all q's. Consequently the molecular propositions will only be used in cases where there ab function stands under a generality sign or enters into another function such as „I believe that, etc.,” because then the sense enters.

A5 In „a judges p” p cannot be replaced by a proper name. This appears if we substitute „a judges that p is true and not p is false”. The proposition „a judges p” consists of the proper name a, the proposition p with its 2 poles, and a being related to both of these poles in a certain way. This is obviously not a relation in the ordinary sense.
The ab notation makes it clear that not and or are dependent on one another and we can therefore not use them as simultaneous indefinables. Same objections in the case of apparent variables to old indefinables, as in the case of molecular functions: The application of the ab notation to apparent-variable propositions becomes clear if we consider that, for instance, the proposition „for all x, x” is to be true when x is true for all x's and false when jx is false for some x's. We see that some and all occur simultaneously in the proper apparent variable notation.

A6 The Notation is:
for (x) x ; a - (x) - a x b - ( x) - b

and

for (x) x : a - (x) - a x b - (x) - b

Old definitions now become tautologous.

In aRb it is not the complex that symbolises but the fact that the symbol a stands in a certain relation to the symbol b. Thus facts are symbolised by facts, or more correctly: that a certain thing is the case in the symbol says that a certain thing is the case in the world.

Judgment, question and command are all on the same level. What interests logic in them is only the unasserted proposition. Facts cannot be named.

A proposition cannot occur in itself. This is the fundamental truth of the theory of types.

Every proposition that says something about one thing is a subject-predicate proposition, and so on.

Therefore we can recognize a subject-predicate proposition if we know it contains only one name and one form, etc. This gives the construction of types. Hence the type of a proposition can be recognized by its symbol alone.

A7 What is essential in a correct apparent-variable notation is this:- (1) it must mention a type of propositions; (2) it must show which components of a proposition of this type are constants.
[Components are forms and constituents.]

Take ().!x. Then if we describe the kind of symbols, for which ! stands and which, by the above, is enough to determine the type, then automatically „().!x” cannot be fitted by this description, because it contains „!x” and the description is to describe all that symbolizes in symbols of the ! - kind. If the description is thus complete vicious circles can just as little occur as for instance in ().(x) (where (x) is a subject-predicate proposition).

B1 First MS.
Indefinables are of two sorts: names, and forms. Propositions cannot consist of names alone; they cannot be classes of names. A name can not only occur in two different propositions, but can occur in the same way in both.

Propositions [which are symbols having reference to facts] are themselves facts: that this inkpot is on this table may express that I sit in this chair.

It can never express the common characteristic of two objects that we designate them by the same name but by two different ways of designation, for, since names are arbitrary, we might also choose different names, and where then would be the common element in the designations? Nevertheless one is always tempted, in a difficulty, to take refuge in different ways of designation.

Frege said „propositions are names”; Russell said „propositions correspond to complexes”. Both are false; and especially false is the statement „propositions are names of complexes”.

It is easy to suppose that only such symbols are complex as contain names of objects, and that accordingly „(x,).x” or „(x,R,y).xRy” must be simple. It is then natural to call the first of these the name of a form, the second the name of a relation. But in that case what is the meaning of (e.g.) „˜(x,y).xRy”? Can we put „not” before a name?

B2 The reason why „˜Socrates” means nothing is that „˜x” does not express a property of x.
There are positive and negative facts: if the proposition „this rose is not red” is true, then what it signifies is negative. But the occurrence of the word „not” does not indicate this unless we know that the signification of the proposition „this rose is red” (when it is true) is positive. It is only from both, the negation and the negated proposition, that we can conclude to a characteristic of the significance of the whole proposition. (We are not here speaking of negations of general propositions, i.e. of such as contain apparent variables. Negative facts only justify the negations of atomic propositions.)

Positive and negative facts there are, but not true and false facts.

If we overlook the fact that propositions have a sense which is independent of their truth or falsehood, it easily seems as if true and false were two equally justified relations between the sign and what is signified. (We might then say e.g. that „q” signifies in the true way what „not-q” signifies in the false way). But are not true and false in fact equally justified? Could we not express ourselves by means of false propositions just as well as hitherto with true ones, so long as we know that they are meant falsely? ¦ No! For a proposition is then true when it is as we assert in this proposition; and accordingly if by „q” we mean „not-q”, and it is as we mean to assert, then in the new interpretation „q” is actually true and not false. But it is important that we can mean the same by „q” as by „not-q”, for it shows that neither to the symbol „not” nor to the manner of its combination with „q” does a characteristic of the denotation of „q” correspond.

B4 2nd MS.
We must be able to understand propositions which we have never heard before. But every proposition is a new symbol. Hence we must have general indefinable symbols; these are unavoidable if propositions are not all indefinable.

Whatever corresponds in reality to compound propositions must not be more than what corresponds to their several atomic propositions.

Not only must logic not deal with [particular] things, but just as little with relations and predicates.

There are no propositions containing real variables.

What corresponds in reality to a proposition depends upon whether it is true or false. But we must be able to understand a proposition without knowing if it is true or false.

What we know when we understand a proposition is this: We know what is the case if the proposition is true, and what is the case if it is false. But we do not know [necessarily] whether it is true or false.

Propositions are not names.

We can never distinguish one logical type from another by attributing a property to members of the one which we deny to members of the other.

Symbols are not what they seem to be. In „aRb”, „R” looks like a substantive, but is not one. What symbolizes in „aRb” is that R occurs between a and b. Hence „R” is not the indefinable in „aRb”. Similarly in „x”, „” looks like a substantive but is not one; in „˜p”, „˜” looks like „” but is not like it. This is the first thing that indicates that there may not be logical constants. A reason against them is the generality of logic: logic cannot treat a special set of things.

B5 Molecular propositions contain nothing beyond what is contained in their atoms; they add no material information above that contained in their atoms.
All that is essential about molecular functions is their T-F schema [i.e. the statement of the cases when they are true and the cases when they are false].

Alternative indefinability shows that the indefinables have not been reached.

Every proposition is essentially true-false: to understand it, we must know both what must be the case if it is true, and what must be the case if it is false. Thus a proposition has two poles, corresponding to the case of its truth and the case of its falsehood. We call this the sense of a proposition.

In regard to notation, it is important to note that not every feature of a symbol symbolizes. In two molecular functions which have the same T-F schema, what symbolizes must be the same. In „not-not-p”, „not-p” does not occur; for „not-not-p” is the same as ”p”, and therefore, if „not-p” occurred in „not-not-p”, it would occur in „p”.

Logical indefinables cannot be predicates or relations, because propositions, owing to sense, cannot have predicates or relations. Nor are „not” and „or”, like judgment, analogous to predicates or relations, because they do not introduce anything new.

Propositions are always complex even if they contain no names.

B6 A proposition must be understood when all its indefinables are understood. The indefinables in „aRb” are introduced as follows:
„a” is indefinable;

„b” is indefinable;

Whatever „x” and „y” may mean, „xRy” says something indefinable about their meanings.

A complex symbol must never be introduced as a single indefinable. (Thus e.g. no proposition is indefinable.) For if one of its parts occurs also in another connection, it must there be re-introduced. And would it then mean the same?

The ways by which we introduce our indefinables must permit us to construct all propositions that have sense from these indefinables alone. It is easy to introduce „all” and „some” in a way that will make the construction of (say) „(x,y).xRy” possible from „all” and „xRy” as introduced before.

B7 3rd MS.
An analogy for the theory of truth: Consider a black patch on white paper; then we can describe the form of the patch by mentioning, for each point of the surface, whether it is white or black. To the fact that a point is black corresponds a positive fact, to the fact that a point is white (not black) corresponds a negative fact. If I designate a point of the surface (one of Frege's „truth-values”), this is as if I set up an assumption to be decided upon. But in order to be able to say of a point that it is black or that it is white, I must first know when a point is to be called black and when it is to be called white. In order to be able to say that „p” is true (or false), I must first have determined under what circumstances I call a proposition true, and thereby I determine the sense of a proposition. The point in which the analogy fails is this: I can indicate a point of the paper that is white and black, but to a proposition without sense nothing corresponds, for it does not designate a thing (truth-value), whose properties might be called „false” or „true”; the verb of a proposition is not „is true” or „is false”, as Frege believes, but what is true must already contain the verb.

The comparison of language and reality is like that of retinal image and visual image: to the blind spot nothing in the visual image seems to correspond, and thereby the boundaries of the blind spot determine the visual image - as true negations of atomic propositions determine reality.

B8 Logical inferences can, it is true, be made in accordance with Frege's or Russell's laws of deduction, but this cannot justify the inference; and therefore they are not primitive propositions of logic. If p follows from q, it can also be inferred from q, and the „manner of deduction” is indifferent.
Those symbols which are called propositions in which „variables occur” are in reality not propositions at all, but only schemes of propositions, which only become propositions when we replace the variables by constants. There is no proposition which is expressed by „x = x”, for „x” has no signification; but there is a proposition „(x).x = x” and propositions such as „Socrates = Socrates” etc.

In books on logic, no variables ought to occur, but only the general propositions which justify the use of variables. It follows that the so-called definitions of logic are not definitions, but only schemes of definitions, and instead of these we ought to put general propositions; and similarly the so-called primitive ideas (Urzeichen) of logic are not primitive ideas, but the schemes of them. The mistaken idea that there are things called facts or complexes and relations easily leads to the opinion that there must be a relation of questioning to the facts, and then the question arises whether a relation can hold between an arbitrary number of things, since a fact can follow from arbitrary cases. It is a fact that the proposition which e.g. expresses that q follows from p and pq is this: p.pq.p.q.q.

B9 At a pinch, one is tempted to interpret „not-p” as „everything else, only not p”. That from a single fact p an infinity of others, not-not-p etc., follow, is hardly credible. Man possesses an innate capacity for constructing symbols with which some sense can be expressed, without having the slightest idea what each word signifies. The best example of this is mathematics, for man has until lately used the symbols for numbers without knowing what they signify or that they signify nothing.
Russell's „complexes” were to have the useful property of being compounded, and were to combine with this the agreeable property that they could be treated like „simples”. But this alone made them unserviceable as logical types, since there would have been significance in asserting, of a simple, that it was complex. But a property cannot be a logical type.

Every statement about apparent complexes can be resolved into the logical sum of a statement about the constituents and a statement about the proposition which describes the complex completely. How, in each case, the resolution is to be made, is an important question, but its answer is not unconditionally necessary for the construction of logic.

B10 That „or” and „not” etc. are not relations in the same sense as „right” and „left” etc., is obvious to the plain man. The possibility of cross-definitions in the old logical indefinables shows, of itself, that these are not the right indefinables, and, even more conclusively, that they do not denote relations.
If we change a constituent a of a proposition (a) into a variable, then there is a class

˜p {(x).(x) = p}.

This class in general still depends upon what, by an arbitrary convention, we mean by „(x)”. But if we change into variables all those symbols whose significance was arbitrarily determined, there is still such a class. But this is now not dependent upon any convention, but only upon the nature of the symbol „(x)”. It corresponds to a logical type.

Types can never be distinguished from each other by saying (as is often done) that one has these but the other has that properties, for this presupposes that there is a meaning in asserting all these properties of both types. But from this it follows that, at best, these properties may be types, but certainly not the objects of which they are asserted.

B11 At a pinch, we are always inclined to explanations of logical functions of propositions which aim at introducing into the function either only the constituents of these propositions, or only their forms, etc. etc.; and we overlook that ordinary language would not contain the whole propositions if it did not need them: However, e.g., „not-p” may be explained, there must always be a meaning given to the question „what is denied?”
The very possibility of Frege's explanations of „not-p” and „if p then q”, from which it follows that not-not-p denotes the same as p, makes it probable that there is some method of designation in which „not-not-p” corresponds to the same symbol as „p”. But if this method of designation suffices for logic, it must be the right one.

Names are points, propositions arrows - they have sense. The sense of a proposition is determined by the two poles true and false. The form of a proposition is like a straight line, which divides all points of a plane into right and left. The line does this automatically, the form of proposition only by convention.

B12 Just as little as we are concerned, in logic, with the relation of a name to its meaning, just so little are we concerned with the relation of a proposition to reality, but we want to know the meaning of names and the sense of propositions - as we introduce an indefinable concept „A” by saying: „‚A’ denotes something indefinable”, so we introduce e.g. the form of propositions aRb by saying: „For all meanings of „x” and „y”, „xRy” expresses something indefinable about x and y”.
In place of every proposition „p”, let us write „abp”. Let every correlation of propositions to each other or of names to propositions be effected by a correlation of their poles „a” and „b”. Let this correlation be transitive. Then accordingly „a-ab-bp” is the same symbol as „abp”. Let n propositions be given. I then call a „class of poles” of these propositions every class of n members, of which each is a pole of one of the n propositions, so that one member corresponds to each proposition. I then correlate with each class of poles one of two poles (a and b). The sense of the symbolizing fact thus constructed I cannot define, but I know it.

If p = not-not-p etc., this shows that the traditional method of symbolism is wrong, since it allows a plurality of symbols with the same sense; and thence it follows that, in analyzing such propositions, we must not be guided by Russell's method of symbolizing.

B13 It is to be remembered that names are not things, but classes: „A” is the same letter as „A”. This has the most important consequences for every symbolic language.
Neither the sense nor the meaning of a proposition is a thing. These words are incomplete symbols.

It is impossible to dispense with propositions in which the same argument occurs in different positions. It is obviously useless to replace (a,a) by (a,b).a = b.

Since the ab-functions of p are again bi-polar propositions, we can form ab-functions of them, and so on. In this way a series of propositions will arise, in which in general the symbolizing facts will be the same in several members. If now we find an ab-function of such a kind that by repeated application of it every ab-function can be generated, then we can introduce the totality of ab-functions as the totality of those that are generated by application of this function. Such a function is ˜pv˜q.

B14 It is easy to suppose a contradiction in the fact that on the one hand every possible complex proposition is a simple ab-function of simple propositions, and that on the other hand the repeated application of one ab-function suffices to generate all these propositions. If e.g. an affirmation can be generated by double negation, is negation in any sense contained in affirmation? Does „p” deny „not-p” or assert „p”, or both? And how do matters stand with the definition of „” by „v” and „·”, or of „v” by „·” and „”? And how e.g. shall we introduce p|q (i.e. ˜pv˜q), if not by saying that this expression says something indefinable about all arguments p and q? But the ab-functions must be introduced as follows: The function p|q is merely a mechanical instrument for constructing all possible symbols of ab-functions. The symbols arising by repeated application of the symbol „¦” do not contain the symbol „p¦q”. We need a rule according to which we can form all symbols of ab-functions, in order to be able to speak of the class of them; and we now speak of them e.g. as those symbols of functions which can be generated by repeated application of the operation „¦”. And we say now: For all p's and q's, „p¦q” says something indefinable about the sense of those simple propositions which are contained in p and q.
B15 The assertion-sign is logically quite without significance. It only shows, in Frege and Whitehead and Russell, that these authors hold the propositions so indicated to be true. „” therefore belongs as little to the proposition as (say) the number of the proposition. A proposition cannot possibly assert of itself that it is true.
Every right theory of judgment must make it impossible for me to judge that this table penholders the book. Russell's theory does not satisfy this requirement.

It is clear that we understand propositions without knowing whether they are true or false. But we can only know the meaning of a proposition when we know if it is true or false. What we understand is the sense of the proposition.

The assumption of the existence of logical objects makes it appear remarkable that in the sciences propositions of the form „pvq”, „pq”, etc. are only then not provisional when „v” and ”” stand within the scope of a generality-sign [apparent variable].

B16 4th MS.
If we formed all possible atomic propositions, the world would be completely described if we declared the truth or falsehood of each. [I doubt this.]

The chief characteristic of my theory is that, in it, p has the same meaning as not-p.

A false theory of relations makes it easily seem as if the relation of fact and constituent were the same as that of fact and fact which follows from it. But the similarity of the two may be expressed thus: a..,a a = a.

If a word creates a world so that in it the principles of logic are true, it thereby creates a world in which the whole of mathematics holds; and similarly it could not create a world in which a proposition was true, without creating its constituents.

Signs of the form „pv˜p” are senseless, but not the proposition „(p).p v ˜p”. If I know that this rose is either red or not red, I know nothing. The same holds of all ab-functions.

To understand a proposition means to know what is the case if it is true. Hence we can understand it without knowing if it is true. We understand it when we understand its constituents and forms. If we know the meaning of „a” and „b”, and if we know what „xRy” means for all x's and y's, then we also understand „aRb”.

I understand the proposition „aRb” when I know that either the fact that aRb or the fact that not aRb corresponds to it; but this is not to be confused with the false opinion that I understand „aRb” when I know that „aRb or not-aRb” is the case.

B17 But the form of a proposition symbolizes in the following way: Let us consider symbols of the form „xRy”; to these correspond primarily pairs of objects, of which one has the name „x”, the other the name „y”. The x's and y's stand in various relations to each other, among others the relation R holds between some, but not between others. I now determine the sense of „xRy” by laying down: when the facts behave in regard to „xRy” so that the meaning of „x” stands in the relation R to the meaning of „y”, then I say that they [the facts] are „of like sense” [„gleichsinnig”] with the proposition „xRy”; otherwise, „of opposite sense” [„entgegengesetzt”]; I correlate the facts to the symbol „xRy” by thus dividing them into those of like sense and those of opposite sense. To this correlation corresponds the correlation of name and meaning. Both are psychological. Thus I understand the form „xRy” when I know that it discriminates the behaviour of x and y according as these stand in the relation R or not. In this way I extract from all possible relations the relation R, as, by a name, I extract its meaning from among all possible things.
Strictly speaking, it is incorrect to say: We understand the proposition p when we know that ‚„p” is true’ p; for this would naturally always be the case if accidentally the propositions to right and left of the symbol „” were both true or both false. We require not only an equivalence, but a formal equivalence, which is bound up with the introduction of the form of p.

The sense of an ab-function of p is a function of the sense of p.

B18 The ab-functions use the discrimination of facts, which their arguments bring forth, in order to generate new discriminations.
Only facts can express sense, a class of names cannot. This is easily shown.

There is no thing which is the form of a proposition, and no name which is the name of a form. Accordingly we can also not say that a relation which in certain cases holds between things holds sometimes between forms and things. This goes against Russell's theory of judgment.

It is very easy to forget that, tho. the propositions of a form can be either true or false, each one of these propositions can only be either true or false, not both.

Among the facts which make „p or q” true, there are some which make „p and q” true; but the class which makes „p or q” true is different from the class which makes „p and q” true; and only this is what matters. For we introduce this class, as it were, when we introduce ab-functions.

A very natural objection to the way in which I have introduced e.g. propositions of the form xRy is that by it propositions such as (x,y).xRy and similar ones are not explained, which yet obviously have in common with aRb what cRd has in common with aRb. But when we introduced propositions of the form xRy we mentioned no one particular proposition of this form; and we only need to introduce (x,y).(x,y) for all 's in any way which makes the sense of these propositions dependent on the sense of all propositions of the form (a,b), and thereby the justification of our procedure is proved.

B19 The indefinables of logic must be independent of each other. If an indefinable is introduced, it must be introduced in all combinations in which it can occur. We cannot therefore introduce it first for one combination, then for another; e.g., if the form xRy has been introduced, it must henceforth be understood in propositions of the form aRb just in the same way as in propositions such as (x,y). xRy and others. We must not introduce it first for one class of cases, then for the other; for it would remain doubtful if its meaning was the same in both cases, and there would be no ground for using the same manner of combining symbols in both cases. In short, for the introduction of indefinable symbols and combinations of symbols the same holds, mutatis mutandis, that Frege has said for the introduction of symbols by definitions.
It is a priori likely that the introduction of atomic propositions is fundamental for the understanding of all other kinds of propositions. In fact the understanding of general propositions obviously depends on that of atomic propositions.

Cross-definability in the realm of general propositions leads to the quite similar questions to those in the realm of ab-functions.

B20 When we say „A believes p”, this sounds, it is true, as if here we could substitute a proper name for „p”; but we can see that here a sense, not a meaning, is concerned, if we say „A believes that ‚p’ is true”; and in order to make the direction of p even more explicit, we might say „A believes that ‚p’ is true and ‚not-p’ is false”. Here the bi-polarity of p is expressed, and it seems that we shall only be able to express the proposition „A believes p” correctly by the ab-notation; say by making „A” have a relation to the poles „a” and „b” of a-p-b.

The epistemological questions concerning the nature of judgment and belief cannot be solved without a correct apprehension of the form of the proposition.

The ab-notation shows the dependence of or and not, and thereby that they are not to be employed as simultaneous indefinables.

Not: „The complex sign ‚aRb’” says that a stands in the relation R to b; but that ‚a’ stands in a certain relation to ‚b’ says that aRb.

In philosophy there are no deductions: it is purely descriptive.

Philosophy gives no pictures of reality.

Philosophy can neither confirm nor confute scientific investigation.

B21 Philosophy consists of logic and metaphysics: logic is its basis.
Epistemology is the philosophy of psychology.

Distrust of grammar is the first requisite for philosophizing.

Propositions can never be indefinables, for they are always complex. That also words like „ambulo” are complex appears in the fact that their root with a different termination gives a different sense.

Only the doctrine of general indefinables permits us to understand the nature of functions. Neglect of this doctrine leads to an impenetrable thicket.

Philosophy is the doctrine of the logical form of scientific propositions (not only of primitive propositions).

The word „philosophy” ought always to designate something over or under, but not beside, the natural sciences.

Judgment, command and question all stand on the same level; but all have in common the propositional form, which does interests us.

The structure of the proposition must be recognized, the rest comes of itself. But ordinary language conceals the structure of the proposition: in it, relations look like predicates, predicates like names, etc.

Facts cannot be named.

B22 It is easy to suppose that „individual”, „particular”, „complex” etc. are primitive ideas of logic. Russell e.g. says „individual” and „matrix” are „primitive ideas”. This error presumably is to be explained by the fact that, by employment of variables instead of the generality-sign, it comes to seem as if logic dealt with things which have been deprived of all properties except thing-hood, and with propositions deprived of all properties except complexity. We forget that the indefinables of symbols [Urbilder von Zeichen] only occur under the generality-sign, never outside it.
Just as people used to struggle to bring all propositions into the subject-predicate form, so now it is natural to conceive every proposition as expressing a relation, which is just as incorrect. What is justified in this desire is fully satisfied by Russell's theory of manufactured relations.

One of the most natural attempts at solution consists in regarding „not-p” as „the opposite of p”, where then „opposite” would be the indefinable relation. But it is easy to see that every such attempt to replace the ab-functions by descriptions must fail.

B23 The false assumption that propositions are names leads us to believe that there must be logical objects: for the meanings of logical propositions will have to be such things.
A correct explanation of logical propositions must give them a unique position as against all other propositions.

No proposition can say anything about itself, because the symbol of the proposition cannot be contained in itself; this must be the basis of the theory of logical types.

Every proposition which says something indefinable about a thing is a subject-predicate proposition; every proposition which says something indefinable about two things expresses a dual relation between these things, and so on. Thus every proposition which contains only one name and one indefinable form is a subject-predicate proposition, and so on. An indefinable simple symbol can only be a name, and therefore we can know, by the symbol of an atomic proposition, whether it is a subject-predicate proposition.

B24 I. Bi-polarity of propositions: sense and meaning, truth and falsehood.
II. Analysis of atomic propositions: general indefinables, predicates, etc.

III. Analysis of molecular propositions: ab-functions.

IV. Analysis of general propositions.

V. Principles of symbolism: what symbolizes in a symbol. Facts for facts.

VI. Types.

B25

This is the symbol for

˜p v ˜q

B26

1 you for instance imagine every fact as a spatial complex.
2 This is quite arbitrary but if we such have fixed on which sides the poles have to stand we must of course stick to our convention. If for instance „apb” says p then bpa says nothing. (It does not say ˜p.) But a-apb-b is the same symbol as apb the ab function vanishes automatically for here the new poles are related to the same side of p as the old ones. The question is always: how are the new poles correlated to p compared with the way the old poles are correlated to p.

The verb of a proposition cannot be „is true” or „is false”, but whatever is true or false must already contain the verb.

Deductions only proceed according to the laws of deduction, but these laws cannot justify the deduction.

One reason for supposing that not all propositions which have more than one argument are relational propositions is that if they were, the relations of judgement and inference would have to hold between an arbitrary number of things.

Every proposition which seems to be about a complex can be analysed into a proposition about its constituents and about the proposition which describes the complex perfectly; i.e., that proposition which is equivalent to saying the complex exists.

The idea that propositions are names of complexes suggests that whatever is not a proper name is a sign for a relation. Because spatial complexes1 consist of Things and Relations only and the idea of a complex is taken from space.

In a proposition convert all its indefinables into variables; there then remains a class of propositions which is not all propositions but a type.

A2 There are thus two ways in which signs are similar. The names Socrates and Plato are similar: they are both names. But whatever they have in common must not be introduced before Socrates and Plato are introduced. The same applies to subject-predicate form etc. Therefore, thing, proposition, subject-predicate form, etc., are not indefinables, i.e., types are not indefinables.
When we say A judges that etc., then we have to mention a whole proposition which A judges. It will not do either to mention only its constituents, or its constituents and form, but not in the proper order. This shows that a proposition itself must occur in the statement that it is judged; however, for instance, „not-p” may be explained. The question „What is negated” must have a meaning.

To understand a proposition p it is not enough to know that p implies ‚„p” is true’, but we must also know that ~p implies „p is false”. This shows the bipolarity of the proposition.

W-F = Wahr-Falsch

To every molecular function a WF scheme corresponds. Therefore we may use the WF scheme itself instead of the function. Now what the WF scheme does is, it correlates the letters W and F with each proposition. These two letters are the poles of atomic propositions. Then the scheme correlates another W and F to these poles. In this notation all that matters is the correlation of the outside poles to the pole of the atomic propositions. Therefore not-not-p is the same symbol as p. And therefore we shall never get two symbols for the same molecular function.

A3 The meaning of a proposition is the fact which actually corresponds to it.
As the ab functions of atomic propositions are bi-polar propositions again we can perform ab operations on them. We shall, by doing so, correlate two new outside poles via the old outside poles to the poles of the atomic propositions.

The symbolising fact in a-p-b is that, say2 a is on the left of p and b on the right of p; then the correlation of new poles is to be transitive, so that for instance if a new pole a in whatever way i.e. via whatever poles is correlated to the inside a, the symbol is not changed thereby. It is therefore possible to construct all possible ab functions by performing one ab operation repeatedly, and we can therefore talk of all ab functions as of all those functions which can be obtained by performing this ab operation repeatedly.

[Note by Bertrand Russell]

[NB. ab means the same as WF, which means true-false.]

Naming is like pointing. A function is like a line dividing points of a plane into right and left ones; then „p or not-p” has no meaning because it does not divide the plane.

But though a particular proposition „p or not-p” has no meaning, a general proposition „for all p's, p or not-p” has a meaning because this does not contain the nonsensical function „p or not-p” but the function „p or not-q” just as „for all x's xRx” contains the function „xRy”.

A4 A proposition is a standard to which all facts behave, with names it is otherwise; it is thus bi-polarity and sense comes in; just as one arrow behaves to another arrow by being in the same sense or the opposite, so a fact behaves to a proposition.
The form of a proposition has meaning in the following way. Consider a symbol „xRy”. To symbols of this form correspond couples of things whose names are respectively „x” and „y”. The things x y stand to one another in all sorts of relations, amongst others some stand in the relation R, and some not; just as I single out a particular thing by a particular name I single out all behaviours of the points x and y with respect to the relation R. I say that if an x stands in the relation R to a y the sign „x R y” is to be called true to the fact and otherwise false. This is a definition of sense.

In my theory p has the same meaning as not-p but opposite sense. The meaning is the fact. The proper theory of judgment must make it impossible to judge nonsense.

It is not strictly true to say that we understand a proposition p if we know that p is equivalent to „p is true” for this would be the case if accidentally both were true or false. What is wanted is the formal equivalence with respect to the forms of the proposition, i.e., all the general indefinables involved. The sense of an ab function of a proposition is a function of its sense. There are only unasserted propositions. ¦ Assertion is merely psychological. In not-p, p is exactly the same as if it stands alone; this point is absolutely fundamental. Among the facts which make „p or q” true there are also facts which make „p and q” true; if propositions have only meaning, we ought, in such a case, to say that these two propositions are identical, but in fact, their sense is different for we have introduced sense by talking of all p's and all q's. Consequently the molecular propositions will only be used in cases where there ab function stands under a generality sign or enters into another function such as „I believe that, etc.,” because then the sense enters.

A5 In „a judges p” p cannot be replaced by a proper name. This appears if we substitute „a judges that p is true and not p is false”. The proposition „a judges p” consists of the proper name a, the proposition p with its 2 poles, and a being related to both of these poles in a certain way. This is obviously not a relation in the ordinary sense.
The ab notation makes it clear that not and or are dependent on one another and we can therefore not use them as simultaneous indefinables. Same objections in the case of apparent variables to old indefinables, as in the case of molecular functions: The application of the ab notation to apparent-variable propositions becomes clear if we consider that, for instance, the proposition „for all x, x” is to be true when x is true for all x's and false when jx is false for some x's. We see that some and all occur simultaneously in the proper apparent variable notation.

A6 The Notation is:
for (x) x ; a - (x) - a x b - ( x) - b

and

for (x) x : a - (x) - a x b - (x) - b

Old definitions now become tautologous.

In aRb it is not the complex that symbolises but the fact that the symbol a stands in a certain relation to the symbol b. Thus facts are symbolised by facts, or more correctly: that a certain thing is the case in the symbol says that a certain thing is the case in the world.

Judgment, question and command are all on the same level. What interests logic in them is only the unasserted proposition. Facts cannot be named.

A proposition cannot occur in itself. This is the fundamental truth of the theory of types.

Every proposition that says something about one thing is a subject-predicate proposition, and so on.

Therefore we can recognize a subject-predicate proposition if we know it contains only one name and one form, etc. This gives the construction of types. Hence the type of a proposition can be recognized by its symbol alone.

A7 What is essential in a correct apparent-variable notation is this:- (1) it must mention a type of propositions; (2) it must show which components of a proposition of this type are constants.
[Components are forms and constituents.]

Take ().!x. Then if we describe the kind of symbols, for which ! stands and which, by the above, is enough to determine the type, then automatically „().!x” cannot be fitted by this description, because it contains „!x” and the description is to describe all that symbolizes in symbols of the ! - kind. If the description is thus complete vicious circles can just as little occur as for instance in ().(x) (where (x) is a subject-predicate proposition).

B1 First MS.
Indefinables are of two sorts: names, and forms. Propositions cannot consist of names alone; they cannot be classes of names. A name can not only occur in two different propositions, but can occur in the same way in both.

Propositions [which are symbols having reference to facts] are themselves facts: that this inkpot is on this table may express that I sit in this chair.

It can never express the common characteristic of two objects that we designate them by the same name but by two different ways of designation, for, since names are arbitrary, we might also choose different names, and where then would be the common element in the designations? Nevertheless one is always tempted, in a difficulty, to take refuge in different ways of designation.

Frege said „propositions are names”; Russell said „propositions correspond to complexes”. Both are false; and especially false is the statement „propositions are names of complexes”.

It is easy to suppose that only such symbols are complex as contain names of objects, and that accordingly „(x,).x” or „(x,R,y).xRy” must be simple. It is then natural to call the first of these the name of a form, the second the name of a relation. But in that case what is the meaning of (e.g.) „˜(x,y).xRy”? Can we put „not” before a name?

B2 The reason why „˜Socrates” means nothing is that „˜x” does not express a property of x.
There are positive and negative facts: if the proposition „this rose is not red” is true, then what it signifies is negative. But the occurrence of the word „not” does not indicate this unless we know that the signification of the proposition „this rose is red” (when it is true) is positive. It is only from both, the negation and the negated proposition, that we can conclude to a characteristic of the significance of the whole proposition. (We are not here speaking of negations of general propositions, i.e. of such as contain apparent variables. Negative facts only justify the negations of atomic propositions.)

Positive and negative facts there are, but not true and false facts.

If we overlook the fact that propositions have a sense which is independent of their truth or falsehood, it easily seems as if true and false were two equally justified relations between the sign and what is signified. (We might then say e.g. that „q” signifies in the true way what „not-q” signifies in the false way). But are not true and false in fact equally justified? Could we not express ourselves by means of false propositions just as well as hitherto with true ones, so long as we know that they are meant falsely? ¦ No! For a proposition is then true when it is as we assert in this proposition; and accordingly if by „q” we mean „not-q”, and it is as we mean to assert, then in the new interpretation „q” is actually true and not false. But it is important that we can mean the same by „q” as by „not-q”, for it shows that neither to the symbol „not” nor to the manner of its combination with „q” does a characteristic of the denotation of „q” correspond.

B4 2nd MS.
We must be able to understand propositions which we have never heard before. But every proposition is a new symbol. Hence we must have general indefinable symbols; these are unavoidable if propositions are not all indefinable.

Whatever corresponds in reality to compound propositions must not be more than what corresponds to their several atomic propositions.

Not only must logic not deal with [particular] things, but just as little with relations and predicates.

There are no propositions containing real variables.

What corresponds in reality to a proposition depends upon whether it is true or false. But we must be able to understand a proposition without knowing if it is true or false.

What we know when we understand a proposition is this: We know what is the case if the proposition is true, and what is the case if it is false. But we do not know [necessarily] whether it is true or false.

Propositions are not names.

We can never distinguish one logical type from another by attributing a property to members of the one which we deny to members of the other.

Symbols are not what they seem to be. In „aRb”, „R” looks like a substantive, but is not one. What symbolizes in „aRb” is that R occurs between a and b. Hence „R” is not the indefinable in „aRb”. Similarly in „x”, „” looks like a substantive but is not one; in „˜p”, „˜” looks like „” but is not like it. This is the first thing that indicates that there may not be logical constants. A reason against them is the generality of logic: logic cannot treat a special set of things.

B5 Molecular propositions contain nothing beyond what is contained in their atoms; they add no material information above that contained in their atoms.
All that is essential about molecular functions is their T-F schema [i.e. the statement of the cases when they are true and the cases when they are false].

Alternative indefinability shows that the indefinables have not been reached.

Every proposition is essentially true-false: to understand it, we must know both what must be the case if it is true, and what must be the case if it is false. Thus a proposition has two poles, corresponding to the case of its truth and the case of its falsehood. We call this the sense of a proposition.

In regard to notation, it is important to note that not every feature of a symbol symbolizes. In two molecular functions which have the same T-F schema, what symbolizes must be the same. In „not-not-p”, „not-p” does not occur; for „not-not-p” is the same as ”p”, and therefore, if „not-p” occurred in „not-not-p”, it would occur in „p”.

Logical indefinables cannot be predicates or relations, because propositions, owing to sense, cannot have predicates or relations. Nor are „not” and „or”, like judgment, analogous to predicates or relations, because they do not introduce anything new.

Propositions are always complex even if they contain no names.

B6 A proposition must be understood when all its indefinables are understood. The indefinables in „aRb” are introduced as follows:
„a” is indefinable;

„b” is indefinable;

Whatever „x” and „y” may mean, „xRy” says something indefinable about their meanings.

A complex symbol must never be introduced as a single indefinable. (Thus e.g. no proposition is indefinable.) For if one of its parts occurs also in another connection, it must there be re-introduced. And would it then mean the same?

The ways by which we introduce our indefinables must permit us to construct all propositions that have sense from these indefinables alone. It is easy to introduce „all” and „some” in a way that will make the construction of (say) „(x,y).xRy” possible from „all” and „xRy” as introduced before.

B7 3rd MS.
An analogy for the theory of truth: Consider a black patch on white paper; then we can describe the form of the patch by mentioning, for each point of the surface, whether it is white or black. To the fact that a point is black corresponds a positive fact, to the fact that a point is white (not black) corresponds a negative fact. If I designate a point of the surface (one of Frege's „truth-values”), this is as if I set up an assumption to be decided upon. But in order to be able to say of a point that it is black or that it is white, I must first know when a point is to be called black and when it is to be called white. In order to be able to say that „p” is true (or false), I must first have determined under what circumstances I call a proposition true, and thereby I determine the sense of a proposition. The point in which the analogy fails is this: I can indicate a point of the paper that is white and black, but to a proposition without sense nothing corresponds, for it does not designate a thing (truth-value), whose properties might be called „false” or „true”; the verb of a proposition is not „is true” or „is false”, as Frege believes, but what is true must already contain the verb.

The comparison of language and reality is like that of retinal image and visual image: to the blind spot nothing in the visual image seems to correspond, and thereby the boundaries of the blind spot determine the visual image - as true negations of atomic propositions determine reality.

B8 Logical inferences can, it is true, be made in accordance with Frege's or Russell's laws of deduction, but this cannot justify the inference; and therefore they are not primitive propositions of logic. If p follows from q, it can also be inferred from q, and the „manner of deduction” is indifferent.
Those symbols which are called propositions in which „variables occur” are in reality not propositions at all, but only schemes of propositions, which only become propositions when we replace the variables by constants. There is no proposition which is expressed by „x = x”, for „x” has no signification; but there is a proposition „(x).x = x” and propositions such as „Socrates = Socrates” etc.

In books on logic, no variables ought to occur, but only the general propositions which justify the use of variables. It follows that the so-called definitions of logic are not definitions, but only schemes of definitions, and instead of these we ought to put general propositions; and similarly the so-called primitive ideas (Urzeichen) of logic are not primitive ideas, but the schemes of them. The mistaken idea that there are things called facts or complexes and relations easily leads to the opinion that there must be a relation of questioning to the facts, and then the question arises whether a relation can hold between an arbitrary number of things, since a fact can follow from arbitrary cases. It is a fact that the proposition which e.g. expresses that q follows from p and pq is this: p.pq.p.q.q.

B9 At a pinch, one is tempted to interpret „not-p” as „everything else, only not p”. That from a single fact p an infinity of others, not-not-p etc., follow, is hardly credible. Man possesses an innate capacity for constructing symbols with which some sense can be expressed, without having the slightest idea what each word signifies. The best example of this is mathematics, for man has until lately used the symbols for numbers without knowing what they signify or that they signify nothing.
Russell's „complexes” were to have the useful property of being compounded, and were to combine with this the agreeable property that they could be treated like „simples”. But this alone made them unserviceable as logical types, since there would have been significance in asserting, of a simple, that it was complex. But a property cannot be a logical type.

Every statement about apparent complexes can be resolved into the logical sum of a statement about the constituents and a statement about the proposition which describes the complex completely. How, in each case, the resolution is to be made, is an important question, but its answer is not unconditionally necessary for the construction of logic.

B10 That „or” and „not” etc. are not relations in the same sense as „right” and „left” etc., is obvious to the plain man. The possibility of cross-definitions in the old logical indefinables shows, of itself, that these are not the right indefinables, and, even more conclusively, that they do not denote relations.
If we change a constituent a of a proposition (a) into a variable, then there is a class

˜p {(x).(x) = p}.

This class in general still depends upon what, by an arbitrary convention, we mean by „(x)”. But if we change into variables all those symbols whose significance was arbitrarily determined, there is still such a class. But this is now not dependent upon any convention, but only upon the nature of the symbol „(x)”. It corresponds to a logical type.

Types can never be distinguished from each other by saying (as is often done) that one has these but the other has that properties, for this presupposes that there is a meaning in asserting all these properties of both types. But from this it follows that, at best, these properties may be types, but certainly not the objects of which they are asserted.

B11 At a pinch, we are always inclined to explanations of logical functions of propositions which aim at introducing into the function either only the constituents of these propositions, or only their forms, etc. etc.; and we overlook that ordinary language would not contain the whole propositions if it did not need them: However, e.g., „not-p” may be explained, there must always be a meaning given to the question „what is denied?”
The very possibility of Frege's explanations of „not-p” and „if p then q”, from which it follows that not-not-p denotes the same as p, makes it probable that there is some method of designation in which „not-not-p” corresponds to the same symbol as „p”. But if this method of designation suffices for logic, it must be the right one.

Names are points, propositions arrows - they have sense. The sense of a proposition is determined by the two poles true and false. The form of a proposition is like a straight line, which divides all points of a plane into right and left. The line does this automatically, the form of proposition only by convention.

B12 Just as little as we are concerned, in logic, with the relation of a name to its meaning, just so little are we concerned with the relation of a proposition to reality, but we want to know the meaning of names and the sense of propositions - as we introduce an indefinable concept „A” by saying: „‚A’ denotes something indefinable”, so we introduce e.g. the form of propositions aRb by saying: „For all meanings of „x” and „y”, „xRy” expresses something indefinable about x and y”.
In place of every proposition „p”, let us write „abp”. Let every correlation of propositions to each other or of names to propositions be effected by a correlation of their poles „a” and „b”. Let this correlation be transitive. Then accordingly „a-ab-bp” is the same symbol as „abp”. Let n propositions be given. I then call a „class of poles” of these propositions every class of n members, of which each is a pole of one of the n propositions, so that one member corresponds to each proposition. I then correlate with each class of poles one of two poles (a and b). The sense of the symbolizing fact thus constructed I cannot define, but I know it.

If p = not-not-p etc., this shows that the traditional method of symbolism is wrong, since it allows a plurality of symbols with the same sense; and thence it follows that, in analyzing such propositions, we must not be guided by Russell's method of symbolizing.

B13 It is to be remembered that names are not things, but classes: „A” is the same letter as „A”. This has the most important consequences for every symbolic language.
Neither the sense nor the meaning of a proposition is a thing. These words are incomplete symbols.

It is impossible to dispense with propositions in which the same argument occurs in different positions. It is obviously useless to replace (a,a) by (a,b).a = b.

Since the ab-functions of p are again bi-polar propositions, we can form ab-functions of them, and so on. In this way a series of propositions will arise, in which in general the symbolizing facts will be the same in several members. If now we find an ab-function of such a kind that by repeated application of it every ab-function can be generated, then we can introduce the totality of ab-functions as the totality of those that are generated by application of this function. Such a function is ˜pv˜q.

B14 It is easy to suppose a contradiction in the fact that on the one hand every possible complex proposition is a simple ab-function of simple propositions, and that on the other hand the repeated application of one ab-function suffices to generate all these propositions. If e.g. an affirmation can be generated by double negation, is negation in any sense contained in affirmation? Does „p” deny „not-p” or assert „p”, or both? And how do matters stand with the definition of „” by „v” and „·”, or of „v” by „·” and „”? And how e.g. shall we introduce p|q (i.e. ˜pv˜q), if not by saying that this expression says something indefinable about all arguments p and q? But the ab-functions must be introduced as follows: The function p|q is merely a mechanical instrument for constructing all possible symbols of ab-functions. The symbols arising by repeated application of the symbol „¦” do not contain the symbol „p¦q”. We need a rule according to which we can form all symbols of ab-functions, in order to be able to speak of the class of them; and we now speak of them e.g. as those symbols of functions which can be generated by repeated application of the operation „¦”. And we say now: For all p's and q's, „p¦q” says something indefinable about the sense of those simple propositions which are contained in p and q.
B15 The assertion-sign is logically quite without significance. It only shows, in Frege and Whitehead and Russell, that these authors hold the propositions so indicated to be true. „” therefore belongs as little to the proposition as (say) the number of the proposition. A proposition cannot possibly assert of itself that it is true.
Every right theory of judgment must make it impossible for me to judge that this table penholders the book. Russell's theory does not satisfy this requirement.

It is clear that we understand propositions without knowing whether they are true or false. But we can only know the meaning of a proposition when we know if it is true or false. What we understand is the sense of the proposition.

The assumption of the existence of logical objects makes it appear remarkable that in the sciences propositions of the form „pvq”, „pq”, etc. are only then not provisional when „v” and ”” stand within the scope of a generality-sign [apparent variable].

B16 4th MS.
If we formed all possible atomic propositions, the world would be completely described if we declared the truth or falsehood of each. [I doubt this.]

The chief characteristic of my theory is that, in it, p has the same meaning as not-p.

A false theory of relations makes it easily seem as if the relation of fact and constituent were the same as that of fact and fact which follows from it. But the similarity of the two may be expressed thus: a..,a a = a.

If a word creates a world so that in it the principles of logic are true, it thereby creates a world in which the whole of mathematics holds; and similarly it could not create a world in which a proposition was true, without creating its constituents.

Signs of the form „pv˜p” are senseless, but not the proposition „(p).p v ˜p”. If I know that this rose is either red or not red, I know nothing. The same holds of all ab-functions.

To understand a proposition means to know what is the case if it is true. Hence we can understand it without knowing if it is true. We understand it when we understand its constituents and forms. If we know the meaning of „a” and „b”, and if we know what „xRy” means for all x's and y's, then we also understand „aRb”.

I understand the proposition „aRb” when I know that either the fact that aRb or the fact that not aRb corresponds to it; but this is not to be confused with the false opinion that I understand „aRb” when I know that „aRb or not-aRb” is the case.

B17 But the form of a proposition symbolizes in the following way: Let us consider symbols of the form „xRy”; to these correspond primarily pairs of objects, of which one has the name „x”, the other the name „y”. The x's and y's stand in various relations to each other, among others the relation R holds between some, but not between others. I now determine the sense of „xRy” by laying down: when the facts behave in regard to „xRy” so that the meaning of „x” stands in the relation R to the meaning of „y”, then I say that they [the facts] are „of like sense” [„gleichsinnig”] with the proposition „xRy”; otherwise, „of opposite sense” [„entgegengesetzt”]; I correlate the facts to the symbol „xRy” by thus dividing them into those of like sense and those of opposite sense. To this correlation corresponds the correlation of name and meaning. Both are psychological. Thus I understand the form „xRy” when I know that it discriminates the behaviour of x and y according as these stand in the relation R or not. In this way I extract from all possible relations the relation R, as, by a name, I extract its meaning from among all possible things.
Strictly speaking, it is incorrect to say: We understand the proposition p when we know that ‚„p” is true’ p; for this would naturally always be the case if accidentally the propositions to right and left of the symbol „” were both true or both false. We require not only an equivalence, but a formal equivalence, which is bound up with the introduction of the form of p.

The sense of an ab-function of p is a function of the sense of p.

B18 The ab-functions use the discrimination of facts, which their arguments bring forth, in order to generate new discriminations.
Only facts can express sense, a class of names cannot. This is easily shown.

There is no thing which is the form of a proposition, and no name which is the name of a form. Accordingly we can also not say that a relation which in certain cases holds between things holds sometimes between forms and things. This goes against Russell's theory of judgment.

It is very easy to forget that, tho. the propositions of a form can be either true or false, each one of these propositions can only be either true or false, not both.

Among the facts which make „p or q” true, there are some which make „p and q” true; but the class which makes „p or q” true is different from the class which makes „p and q” true; and only this is what matters. For we introduce this class, as it were, when we introduce ab-functions.

A very natural objection to the way in which I have introduced e.g. propositions of the form xRy is that by it propositions such as (x,y).xRy and similar ones are not explained, which yet obviously have in common with aRb what cRd has in common with aRb. But when we introduced propositions of the form xRy we mentioned no one particular proposition of this form; and we only need to introduce (x,y).(x,y) for all 's in any way which makes the sense of these propositions dependent on the sense of all propositions of the form (a,b), and thereby the justification of our procedure is proved.

B19 The indefinables of logic must be independent of each other. If an indefinable is introduced, it must be introduced in all combinations in which it can occur. We cannot therefore introduce it first for one combination, then for another; e.g., if the form xRy has been introduced, it must henceforth be understood in propositions of the form aRb just in the same way as in propositions such as (x,y). xRy and others. We must not introduce it first for one class of cases, then for the other; for it would remain doubtful if its meaning was the same in both cases, and there would be no ground for using the same manner of combining symbols in both cases. In short, for the introduction of indefinable symbols and combinations of symbols the same holds, mutatis mutandis, that Frege has said for the introduction of symbols by definitions.
It is a priori likely that the introduction of atomic propositions is fundamental for the understanding of all other kinds of propositions. In fact the understanding of general propositions obviously depends on that of atomic propositions.

Cross-definability in the realm of general propositions leads to the quite similar questions to those in the realm of ab-functions.

B20 When we say „A believes p”, this sounds, it is true, as if here we could substitute a proper name for „p”; but we can see that here a sense, not a meaning, is concerned, if we say „A believes that ‚p’ is true”; and in order to make the direction of p even more explicit, we might say „A believes that ‚p’ is true and ‚not-p’ is false”. Here the bi-polarity of p is expressed, and it seems that we shall only be able to express the proposition „A believes p” correctly by the ab-notation; say by making „A” have a relation to the poles „a” and „b” of a-p-b.

The epistemological questions concerning the nature of judgment and belief cannot be solved without a correct apprehension of the form of the proposition.

The ab-notation shows the dependence of or and not, and thereby that they are not to be employed as simultaneous indefinables.

Not: „The complex sign ‚aRb’” says that a stands in the relation R to b; but that ‚a’ stands in a certain relation to ‚b’ says that aRb.

In philosophy there are no deductions: it is purely descriptive.

Philosophy gives no pictures of reality.

Philosophy can neither confirm nor confute scientific investigation.

B21 Philosophy consists of logic and metaphysics: logic is its basis.
Epistemology is the philosophy of psychology.

Distrust of grammar is the first requisite for philosophizing.

Propositions can never be indefinables, for they are always complex. That also words like „ambulo” are complex appears in the fact that their root with a different termination gives a different sense.

Only the doctrine of general indefinables permits us to understand the nature of functions. Neglect of this doctrine leads to an impenetrable thicket.

Philosophy is the doctrine of the logical form of scientific propositions (not only of primitive propositions).

The word „philosophy” ought always to designate something over or under, but not beside, the natural sciences.

Judgment, command and question all stand on the same level; but all have in common the propositional form, which does interests us.

The structure of the proposition must be recognized, the rest comes of itself. But ordinary language conceals the structure of the proposition: in it, relations look like predicates, predicates like names, etc.

Facts cannot be named.

B22 It is easy to suppose that „individual”, „particular”, „complex” etc. are primitive ideas of logic. Russell e.g. says „individual” and „matrix” are „primitive ideas”. This error presumably is to be explained by the fact that, by employment of variables instead of the generality-sign, it comes to seem as if logic dealt with things which have been deprived of all properties except thing-hood, and with propositions deprived of all properties except complexity. We forget that the indefinables of symbols [Urbilder von Zeichen] only occur under the generality-sign, never outside it.
Just as people used to struggle to bring all propositions into the subject-predicate form, so now it is natural to conceive every proposition as expressing a relation, which is just as incorrect. What is justified in this desire is fully satisfied by Russell's theory of manufactured relations.

One of the most natural attempts at solution consists in regarding „not-p” as „the opposite of p”, where then „opposite” would be the indefinable relation. But it is easy to see that every such attempt to replace the ab-functions by descriptions must fail.

B23 The false assumption that propositions are names leads us to believe that there must be logical objects: for the meanings of logical propositions will have to be such things.
A correct explanation of logical propositions must give them a unique position as against all other propositions.

No proposition can say anything about itself, because the symbol of the proposition cannot be contained in itself; this must be the basis of the theory of logical types.

Every proposition which says something indefinable about a thing is a subject-predicate proposition; every proposition which says something indefinable about two things expresses a dual relation between these things, and so on. Thus every proposition which contains only one name and one indefinable form is a subject-predicate proposition, and so on. An indefinable simple symbol can only be a name, and therefore we can know, by the symbol of an atomic proposition, whether it is a subject-predicate proposition.

B24 I. Bi-polarity of propositions: sense and meaning, truth and falsehood.
II. Analysis of atomic propositions: general indefinables, predicates, etc.

III. Analysis of molecular propositions: ab-functions.

IV. Analysis of general propositions.

V. Principles of symbolism: what symbolizes in a symbol. Facts for facts.

VI. Types.

B25

This is the symbol for

˜p v ˜q

B26

1 you for instance imagine every fact as a spatial complex.
2 This is quite arbitrary but if we such have fixed on which sides the poles have to stand we must of course stick to our convention. If for instance „apb” says p then bpa says nothing. (It does not say ˜p.) But a-apb-b is the same symbol as apb the ab function vanishes automatically for here the new poles are related to the same side of p as the old ones. The question is always: how are the new poles correlated to p compared with the way the old poles are correlated to p.

4
A thought is a proposition with a sense.

4.001
The totality of propositions is language.

4.022
Man possesses the ability to construct languages capable of expressing every sense, without having any idea how each word has meaning or what its meaning is--just as people speak without knowing how the individual sounds are produced. Everyday language is a part of the human organism and is no less complicated than it. It is not humanly possible to gather immediately from it what the logic of language is. Language disguises thought. So much so, that from the outward form of the clothing it is impossible to infer the form of the thought beneath it, because the outward form of the clothing is not designed to reveal the form of the body, but for entirely different purposes. The tacit conventions on which the understanding of everyday language depends are enormously complicated.

4.003
Most of the propositions and questions to be found in philosophical works are not false but nonsensical. Consequently we cannot give any answer to questions of this kind, but can only point out that they are nonsensical. Most of the propositions and questions of philosophers arise from our failure to understand the logic of our language. (They belong to the same class as the question whether the good is more or less identical than the beautiful.) And it is not surprising that the deepest problems are in fact not problems at all.

4.0031
All philosophy is a 'critique of language' (though not in Mauthner's sense). It was Russell who performed the service of showing that the apparent logical form of a proposition need not be its real one.

4.01
A proposition is a picture of reality. A proposition is a model of reality as we imagine it.

4.011
At first sight a proposition--one set out on the printed page, for example--does not seem to be a picture of the reality with which it is concerned. But neither do written notes seem at first sight to be a picture of a piece of music, nor our phonetic notation (the alphabet) to be a picture of our speech. And yet these sign-languages prove to be pictures, even in the ordinary sense, of what they represent.

4.012
It is obvious that a proposition of the form 'aRb' strikes us as a picture. In this case the sign is obviously a likeness of what is signified.

4.013
And if we penetrate to the essence of this pictorial character, we see that it is not impaired by apparent irregularities (such as the use [sharp] of and [flat] in musical notation). For even these irregularities depict what they are intended to express; only they do it in a different way.

4.014
A gramophone record, the musical idea, the written notes, and the sound-waves, all stand to one another in the same internal relation of depicting that holds between language and the world. They are all constructed according to a common logical pattern. (Like the two youths in the fairy-tale, their two horses, and their lilies. They are all in a certain sense one.)

4.0141
There is a general rule by means of which the musician can obtain the symphony from the score, and which makes it possible to derive the symphony from the groove on the gramophone record, and, using the first rule, to derive the score again. That is what constitutes the inner similarity between these things which seem to be constructed in such entirely different ways. And that rule is the law of projection which projects the symphony into the language of musical notation. It is the rule for translating this language into the language of gramophone records.

4.015
The possibility of all imagery, of all our pictorial modes of expression, is contained in the logic of depiction.

4.016
In order to understand the essential nature of a proposition, we should consider hieroglyphic script, which depicts the facts that it describes. And alphabetic script developed out of it without losing what was essential to depiction.

4.02
We can see this from the fact that we understand the sense of a propositional sign without its having been explained to us.

4.021
A proposition is a picture of reality: for if I understand a proposition, I know the situation that it represents. And I understand the proposition without having had its sense explained to me.

4.022
A proposition shows its sense. A proposition shows how things stand if it is true. And it says that they do so stand.

4.023
A proposition must restrict reality to two alternatives: yes or no. In order to do that, it must describe reality completely. A proposition is a description of a state of affairs. Just as a description of an object describes it by giving its external properties, so a proposition describes reality by its internal properties. A proposition constructs a world with the help of a logical scaffolding, so that one can actually see from the proposition how everything stands logically if it is true. One can draw inferences from a false proposition.

4.024
To understand a proposition means to know what is the case if it is true. (One can understand it, therefore, without knowing whether it is true.) It is understood by anyone who understands its constituents.

4.025
When translating one language into another, we do not proceed by translating each proposition of the one into a proposition of the other, but merely by translating the constituents of propositions. (And the dictionary translates not only substantives, but also verbs, adjectives, and conjunctions, etc.; and it treats them all in the same way.)

4.026
The meanings of simple signs (words) must be explained to us if we are to understand them. With propositions, however, we make ourselves understood.

4.027
It belongs to the essence of a proposition that it should be able to communicate a new sense to us.

4.03
A proposition must use old expressions to communicate a new sense. A proposition communicates a situation to us, and so it must be essentially connected with the situation. And the connexion is precisely that it is its logical picture. A proposition states something only in so far as it is a picture.

4.031
In a proposition a situation is, as it were, constructed by way of experiment. Instead of, 'This proposition has such and such a sense, we can simply say, 'This proposition represents such and such a situation'.

4.0311
One name stands for one thing, another for another thing, and they are combined with one another. In this way the whole group--like a tableau vivant--presents a state of affairs.

4.0312
The possibility of propositions is based on the principle that objects have signs as their representatives. My fundamental idea is that the 'logical constants' are not representatives; that there can be no representatives of the logic of facts.

4.032
It is only in so far as a proposition is logically articulated that it is a picture of a situation. (Even the proposition, 'Ambulo', is composite: for its stem with a different ending yields a different sense, and so does its ending with a different stem.)

4.04
In a proposition there must be exactly as many distinguishable parts as in the situation that it represents. The two must possess the same logical (mathematical) multiplicity. (Compare Hertz's Mechanics on dynamical models.)

4.041
This mathematical multiplicity, of course, cannot itself be the subject of depiction. One cannot get away from it when depicting.

4.0411
If, for example, we wanted to express what we now write as '(x) . fx' by putting an affix in front of 'fx'--for instance by writing 'Gen. fx'--it would not be adequate: we should not know what was being generalized. If we wanted to signalize it with an affix 'g'--for instance by writing 'f(xg)'--that would not be adequate either: we should not know the scope of the generality-sign. If we were to try to do it by introducing a mark into the argument-places--for instance by writing '(G,G) . F(G,G)' --it would not be adequate: we should not be able to establish the identity of the variables. And so on. All these modes of signifying are inadequate because they lack the necessary mathematical multiplicity.

4.0412
For the same reason the idealist's appeal to 'spatial spectacles' is inadequate to explain the seeing of spatial relations, because it cannot explain the multiplicity of these relations.

4.05
Reality is compared with propositions.

4.06
A proposition can be true or false only in virtue of being a picture of reality.

4.061
It must not be overlooked that a proposition has a sense that is independent of the facts: otherwise one can easily suppose that true and false are relations of equal status between signs and what they signify. In that case one could say, for example, that 'p' signified in the true way what 'Pp' signified in the false way, etc.

4.062
Can we not make ourselves understood with false propositions just as we have done up till now with true ones?--So long as it is known that they are meant to be false.--No! For a proposition is true if we use it to say that things stand in a certain way, and they do; and if by 'p' we mean Pp and things stand as we mean that they do, then, construed in the new way, 'p' is true and not false.

4.0621
But it is important that the signs 'p' and 'Pp' can say the same thing. For it shows that nothing in reality corresponds to the sign 'P'. The occurrence of negation in a proposition is not enough to characterize its sense (PPp = p). The propositions 'p' and 'Pp' have opposite sense, but there corresponds to them one and the same reality.

4.063
An analogy to illustrate the concept of truth: imagine a black spot on white paper: you can describe the shape of the spot by saying, for each point on the sheet, whether it is black or white. To the fact that a point is black there corresponds a positive fact, and to the fact that a point is white (not black), a negative fact. If I designate a point on the sheet (a truth-value according to Frege), then this corresponds to the supposition that is put forward for judgement, etc. etc. But in order to be able to say that a point is black or white, I must first know when a point is called black, and when white: in order to be able to say,'"p" is true (or false)', I must have determined in what circumstances I call 'p' true, and in so doing I determine the sense of the proposition. Now the point where the simile breaks down is this: we can indicate a point on the paper even if we do not know what black and white are, but if a proposition has no sense, nothing corresponds to it, since it does not designate a thing (a truth-value) which might have properties called 'false' or 'true'. The verb of a proposition is not 'is true' or 'is false', as Frege thought: rather, that which 'is true' must already contain the verb.

4.064
Every proposition must already have a sense: it cannot be given a sense by affirmation. Indeed its sense is just what is affirmed. And the same applies to negation, etc.

4.0641
One could say that negation must be related to the logical place determined by the negated proposition. The negating proposition determines a logical place different from that of the negated proposition. The negating proposition determines a logical place with the help of the logical place of the negated proposition. For it describes it as lying outside the latter's logical place. The negated proposition can be negated again, and this in itself shows that what is negated is already a proposition, and not merely something that is prelimary to a proposition.

4.1
Propositions represent the existence and non-existence of states of affairs.

4.11
The totality of true propositions is the whole of natural science (or the whole corpus of the natural sciences).

4.111
Philosophy is not one of the natural sciences. (The word 'philosophy' must mean something whose place is above or below the natural sciences, not beside them.)

4.112
Philosophy aims at the logical clarification of thoughts. Philosophy is not a body of doctrine but an activity. A philosophical work consists essentially of elucidations. Philosophy does not result in 'philosophical propositions', but rather in the clarification of propositions. Without philosophy thoughts are, as it were, cloudy and indistinct: its task is to make them clear and to give them sharp boundaries.

4.1121
Psychology is no more closely related to philosophy than any other natural science. Theory of knowledge is the philosophy of psychology. Does not my study of sign-language correspond to the study of thought-processes, which philosophers used to consider so essential to the philosophy of logic? Only in most cases they got entangled in unessential psychological investigations, and with my method too there is an analogous risk.

4.1122
Darwin's theory has no more to do with philosophy than any other hypothesis in natural science.

4.113
Philosophy sets limits to the much disputed sphere of natural science.

4.114
It must set limits to what can be thought; and, in doing so, to what cannot be thought. It must set limits to what cannot be thought by working outwards through what can be thought.

4.115
It will signify what cannot be said, by presenting clearly what can be said.

4.116
Everything that can be thought at all can be thought clearly. Everything that can be put into words can be put clearly.

4.12
Propositions can represent the whole of reality, but they cannot represent what they must have in common with reality in order to be able to represent it--logical form. In order to be able to represent logical form, we should have to be able to station ourselves with propositions somewhere outside logic, that is to say outside the world.

4.121
Propositions cannot represent logical form: it is mirrored in them. What finds its reflection in language, language cannot represent. What expresses itself in language, we cannot express by means of language. Propositions show the logical form of reality. They display it.

4.1211
Thus one proposition 'fa' shows that the object a occurs in its sense, two propositions 'fa' and 'ga' show that the same object is mentioned in both of them. If two propositions contradict one another, then their structure shows it; the same is true if one of them follows from the other. And so on.

4.1212
What can be shown, cannot be said.

4.1213
Now, too, we understand our feeling that once we have a sign-language in which everything is all right, we already have a correct logical point of view.

4.122
In a certain sense we can talk about formal properties of objects and states of affairs, or, in the case of facts, about structural properties: and in the same sense about formal relations and structural relations. (Instead of 'structural property' I also say 'internal property'; instead of 'structural relation', 'internal relation'. I introduce these expressions in order to indicate the source of the confusion between internal relations and relations proper (external relations), which is very widespread among philosophers.) It is impossible, however, to assert by means of propositions that such internal properties and relations obtain: rather, this makes itself manifest in the propositions that represent the relevant states of affairs and are concerned with the relevant objects.

4.1221
An internal property of a fact can also be bed a feature of that fact (in the sense in which we speak of facial features, for example).

4.123
A property is internal if it is unthinkable that its object should not possess it. (This shade of blue and that one stand, eo ipso, in the internal relation of lighter to darker. It is unthinkable that these two objects should not stand in this relation.) (Here the shifting use of the word 'object' corresponds to the shifting use of the words 'property' and 'relation'.)

4.124
The existence of an internal property of a possible situation is not expressed by means of a proposition: rather, it expresses itself in the proposition representing the situation, by means of an internal property of that proposition. It would be just as nonsensical to assert that a proposition had a formal property as to deny it.

4.1241
It is impossible to distinguish forms from one another by saying that one has this property and another that property: for this presupposes that it makes sense to ascribe either property to either form.

4.125
The existence of an internal relation between possible situations expresses itself in language by means of an internal relation between the propositions representing them.

4.1251
Here we have the answer to the vexed question 'whether all relations are internal or external'.

4.1252
I call a series that is ordered by an internal relation a series of forms. The order of the number-series is not governed by an external relation but by an internal relation. The same is true of the series of propositions
'aRb',
'(d : c) : aRx . xRb',
'(d x,y) : aRx . xRy . yRb',

and so forth. (If b stands in one of these relations to a, I call b a successor of a.)

4.126
We can now talk about formal concepts, in the same sense that we speak of formal properties. (I introduce this expression in order to exhibit the source of the confusion between formal concepts and concepts proper, which pervades the whole of traditional logic.) When something falls under a formal concept as one of its objects, this cannot be expressed by means of a proposition. Instead it is shown in the very sign for this object. (A name shows that it signifies an object, a sign for a number that it signifies a number, etc.) Formal concepts cannot, in fact, be represented by means of a function, as concepts proper can. For their characteristics, formal properties, are not expressed by means of functions. The expression for a formal property is a feature of certain symbols. So the sign for the characteristics of a formal concept is a distinctive feature of all symbols whose meanings fall under the concept. So the expression for a formal concept is a propositional variable in which this distinctive feature alone is constant.

4.127
The propositional variable signifies the formal concept, and its values signify the objects that fall under the concept.

4.1271
Every variable is the sign for a formal concept. For every variable represents a constant form that all its values possess, and this can be regarded as a formal property of those values.

4.1272
Thus the variable name 'x' is the proper sign for the pseudo-concept object. Wherever the word 'object' ('thing', etc.) is correctly used, it is expressed in conceptual notation by a variable name. For example, in the proposition, 'There are 2 objects which. . .', it is expressed by ' (dx,y) ... '. Wherever it is used in a different way, that is as a proper concept-word, nonsensical pseudo-propositions are the result. So one cannot say, for example, 'There are objects', as one might say, 'There are books'. And it is just as impossible to say, 'There are 100 objects', or, 'There are !0 objects'. And it is nonsensical to speak of the total number of objects. The same applies to the words 'complex', 'fact', 'function', 'number', etc. They all signify formal concepts, and are represented in conceptual notation by variables, not by functions or classes (as Frege and Russell believed). '1 is a number', 'There is only one zero', and all similar expressions are nonsensical. (It is just as nonsensical to say, 'There is only one 1', as it would be to say, '2 + 2 at 3 o'clock equals 4'.)

4.12721
A formal concept is given immediately any object falling under it is given. It is not possible, therefore, to introduce as primitive ideas objects belonging to a formal concept and the formal concept itself. So it is impossible, for example, to introduce as primitive ideas both the concept of a function and specific functions, as Russell does; or the concept of a number and particular numbers.

4.1273
If we want to express in conceptual notation the general proposition, 'b is a successor of a', then we require an expression for the general term of the series of forms
'aRb',
'(d : c) : aRx . xRb',
'(d x,y) : aRx . xRy . yRb',
... ,
In order to express the general term of a series of forms, we must use a variable, because the concept 'term of that series of forms' is a formal concept. (This is what Frege and Russell overlooked: consequently the way in which they want to express general propositions like the one above is incorrect; it contains a vicious circle.)
We can determine the general term of a series of forms by giving its first term and the general form of the operation that produces the next term out of the proposition that precedes it.

4.1274
To ask whether a formal concept exists is nonsensical. For no proposition can be the answer to such a question. (So, for example, the question, 'Are there unanalysable subject-predicate propositions?' cannot be asked.)

4.128
Logical forms are without number. Hence there are no preeminent numbers in logic, and hence there is no possibility of philosophical monism or dualism, etc.

4.2
The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs.

4.21
The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs.

4.211
It is a sign of a proposition's being elementary that there can be no elementary proposition contradicting it.

4.22
An elementary proposition consists of names. It is a nexus, a concatenation, of names.

4.221
It is obvious that the analysis of propositions must bring us to elementary propositions which consist of names in immediate combination. This raises the question how such combination into propositions comes about.

4.2211
Even if the world is infinitely complex, so that every fact consists of infinitely many states of affairs and every state of affairs is composed of infinitely many objects, there would still have to be objects and states of affairs.

4.23
It is only in the nexus of an elementary proposition that a name occurs in a proposition.

4.24
Names are the simple symbols: I indicate them by single letters ('x', 'y', 'z'). I write elementary propositions as functions of names, so that they have the form 'fx', 'O (x,y)', etc. Or I indicate them by the letters 'p', 'q', 'r'.

4.241
When I use two signs with one and the same meaning, I express this by putting the sign '=' between them. So 'a = b' means that the sign 'b' can be substituted for the sign 'a'. (If I use an equation to introduce a new sign 'b', laying down that it shall serve as a substitute for a sign a that is already known, then, like Russell, I write the equation-- definition--in the form 'a = b Def.' A definition is a rule dealing with signs.)

4.242
Expressions of the form 'a = b' are, therefore, mere representational devices. They state nothing about the meaning of the signs 'a' and 'b'.

4.243
Can we understand two names without knowing whether they signify the same thing or two different things?--Can we understand a proposition in which two names occur without knowing whether their meaning is the same or different? Suppose I know the meaning of an English word and of a German word that means the same: then it is impossible for me to be unaware that they do mean the same; I must be capable of translating each into the other. Expressions like 'a = a', and those derived from them, are neither elementary propositions nor is there any other way in which they have sense. (This will become evident later.)

4.25
If an elementary proposition is true, the state of affairs exists: if an elementary proposition is false, the state of affairs does not exist.

4.26
If all true elementary propositions are given, the result is a complete description of the world. The world is completely described by giving all elementary propositions, and adding which of them are true and which false. For n states of affairs, there are possibilities of existence and non-existence. Of these states of affairs any combination can exist and the remainder not exist.

4.28
There correspond to these combinations the same number of possibilities of truth--and falsity--for n elementary propositions.

4.3
Truth-possibilities of elementary propositions mean Possibilities of existence and non-existence of states of affairs.

4.31
We can represent truth-possibilities by schemata of the following kind ('T' means 'true', 'F' means 'false'; the rows of 'T's' and 'F's' under the row of elementary propositions symbolize their truth-possibilities in a way that can easily be understood):

4.4
A proposition is an expression of agreement and disagreement with truth-possibilities of elementary propositions.

4.41
Truth-possibilities of elementary propositions are the conditions of the truth and falsity of propositions.

4.411
It immediately strikes one as probable that the introduction of elementary propositions provides the basis for understanding all other kinds of proposition. Indeed the understanding of general propositions palpably depends on the understanding of elementary propositions.

4.42
For n elementary

5.01
Elementary propositions are the truth-arguments of propositions.

5.02
The arguments of functions are readily confused with the aff ixes 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 proposi tions 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. Belief in the causal nexus is superstition.

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.

6
The general form of a truth-function is [p, E, N(E)]. 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(E)

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 /'(n) is [E, N(E)] ' (n) ( = [n, E, N(E)]). 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 = /0x Def.,
/'/v'x = /v+1'x Def.

So, in accordance with these rules, which deal with signs, we write the series
x, /'x, /'/'x, /'/'/'x, ... ,
in the following way
/0'x, /0+1'x, /0+1+1'x, /0+1+1+1'x, ... .
Therefore, instead of '[x, E, /'E]',
I write '[/0'x, /v'x, /v+1'x]'.
And I give the following definitions
0 + 1 = 1 Def.,
0 + 1 + 1 = 2 Def.,
0 + 1 + 1 +1 = 3 Def.,
(and so on).

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, E, E +1].

6.031
The theory of classes is completely superfluous in mathematics. This is connected with the fact that the generality required in mathematics is not accidental generality.

6.1
The propositions of logic are tautologies.

6.11
Therefore the propositions of logic say nothing. (They are the analytic 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 assign 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 'Pp' in the combination '(p . Pp)' yield a tautology shows that they contradict one another. The fact that the propositions 'p z q', 'p', and 'q', combined with one another in the form '(p z q) . (p) :z: (q)', yield a tautology shows that q follows from p and p z q. The fact that '(x) . fxx :z: fa' is a tautology shows that fa follows from (x) . fx. Etc. etc.

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. and I use lines to express the correlation of the truth or falsity of the whole proposition with the truth-combinations of its truth-arguments, in the following way
So this sign, for instance, would represent the proposition p z q. Now, by way of example, I wish to examine the proposition P(p .Pp) (the law of contradiction) in order to determine whether it is a tautology. In our notation the form 'PE' is written as and the form 'E . n' as Hence the proposition P(p . Pp). reads as follows

If we here substitute 'p' for 'q' and examine how the outermost T and F are connected with the innermost ones, the result will be that the truth of the whole proposition is correlated with all the truth-combinations of its argument, and its falsity with none of the truth-combinations.

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 z q' yield a tautology, then it is clear that q follows from p. For example, we see from the two propositions themselves that 'q' follows from 'p z q . p', but it is also possible to show it in this way: we combine them to form 'p z q . p :z: q', and then show that this is a tautology.

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 assert the identity of meaning of two expressions. For in order to be able to assert anything about their meaning, I must know their meaning, and I cannot know their meaning without knowing whether what they mean is the same or different.

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 t 2 = 4 runs as follows:
(/v)n'x = /v x u'x Def.,
/2 x 2'x = (/2)2'x = (/2)1 + 1'x
= /2' /2'x = /1 + 1'/1 + 1'x = (/'/)'(/'/)'x
=/'/'/'/'x = /1 + 1 + 1 + 1'x = /4'x.

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 ature, 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-masses: it will only talk about any point-masses 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 passage of time'--there is no such thing--but only with another process (such as the working of a chronometer). Hence we can describe the lapse of time only by relying on some other process. Something exactly analogous applies to space: e.g. when people say that neither of two events (which exclude one another) can occur, because there is nothing to cause the one to occur rather than the other, it is really a matter of our being unable to describe one of the two events unless there is some sort of asymmetry to be found. And if such an asymmetry is to be found, we can regard it as the cause of the occurrence of the one and the 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 pu

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 death the world does not alter, but comes to an end.

6.4311
Death is not an event in life: we do not live to experience death. If we take eternity to mean not infinite temporal duration but timelessness, then eternal life belongs to those who live in the present. Our life has no end in just the way in which our visual field has no limits.

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 death; but, in any case, this assumption completely fails to accomplish the purpose for which it has always been intended. Or is some riddle solved by my surviving for ever? Is not this eternal life itself as much of a riddle as our present life? The solution of the riddle of life in space and time lies outside space and time. (It is certainly not the solution of any problems of natural science that is required.)

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.)

7
What we cannot speak about we must pass over in silence.

Anyway, I thought "what we cannot speak about we must pass over in silence" would've been a great ending to this thread. Apparently not.

its always killed threads in the past

-- kate (masonicboo...), July 8th, 2003.
Answers
*shoots thread with 12 bore shotgun*

-- C J (CJ_The_Unrul...), July 8th, 2003.
I'm sad and no one understands me.

-- kate (masonicboo...), July 8th, 2003.
(sorry, Kate)

-- C J (CJ_The_Unrul...), July 8th, 2003.
I'm a reed-thin recovered alcoholic and I love my puppy to bits.

-- estela (estelaisale...), July 8th, 2003.
the mustard-yellow pimp!

-- Tad (llamasfu...), July 8th, 2003.
visit this thread later for my in-depth workshop in accountancy

-- mark s (mar...), July 8th, 2003.
Well, what the Bible says is this...

-- kate (masonicboo...), July 8th, 2003.
visit this thread later for my in-depth workshop in accountancy

but all that would be is, "debits must = credits."

-- Tad (llamasfu...), July 8th, 2003.
::Your Boss:: posts on thread.

-- kate (masonicboo...), July 8th, 2003.
Have I told you all about my teapot collection?

-- C J (CJ_The_Unrul...), July 8th, 2003.
"Under section 404, as under the similar provisions of section 302(c)(5) of the Labor Management Relations (Taft-Hartley) Act, 29 U.S.C. § 186(c)(5), trustees with the discretion to establish benefit provisions must act solely in the interests of 'participants and beneficiaries.' We have implemented this requirement with the 'structural defect' test." Harm v. Bay Area Pipe Trades Pension Plan Trust Fund, 701 F.2d 1301 (9th Cir. 1983).

(when the "structural defect test," and the "arbitrary and capricious standard" are discussed and defined is where the real fun begins!)

-- Tad (llamasfu...), July 8th, 2003.

"Hi, would you like to fuck me?"

-- Tuomas (tuomas.alh...), July 8th, 2003.
i was five and staying at my friend H's house........

-- Chris Radford (radford_chri...), July 8th, 2003.
.....and we were munching on witchedy grubs

-- Tad (llamasfu...), July 8th, 2003.
post after me if you're a piss fetishist

-- anon_anon (ano...), July 8th, 2003.
Num num piss. (I prefer white to red mind - get less of a hangover).

-- Pete (pb1...), July 8th, 2003.
does anyone know that tune with the big dark bassline is that marco zaffarano used to play? it sounds kind of frankfurt-esque, but i dont think its on harthouse. thanks

-- gareth (garet...), July 8th, 2003.

-- kate (masonicboo...), July 8th, 2003.
(At this point, I would post the goatse picture if I knew no restraint.)

-- Colin Meeder (amisrau...), July 8th, 2003.
Well, what the Bible says is this...

Ha ha very funny. This pseudo-ironic post-Christian hatin' on the Bible is sooo lame. Have you even read the damn thing? Thought not.

-- Sommermute (wintermute_v0.3...), July 8th, 2003.
I have, actually.

-- kate (masonicboo...), July 8th, 2003.
Summary
One reason for thinking the old notation wrong is that it is very unlikely that from every proposition p an infinite number of other propositions not-not-p, not-not-not-not-p, etc., should follow.
If only those signs which contain proper names were complex then propositions containing nothing but apparent variables would be simple. Then what about their denials?

The verb of a proposition cannot be „is true” or „is false”, but whatever is true or false must already contain the verb.

Deductions only proceed according to the laws of deduction, but these laws cannot justify the deduction.

One reason for supposing that not all propositions which have more than one argument are relational propositions is that if they were, the relations of judgement and inference would have to hold between an arbitrary number of things.

Every proposition which seems to be about a complex can be analysed into a proposition about its constituents and about the proposition which describes the complex perfectly; i.e., that proposition which is equivalent to saying the complex exists.

The idea that propositions are names of complexes suggests that whatever is not a proper name is a sign for a relation. Because spatial complexes1 consist of Things and Relations only and the idea of a complex is taken from space.

In a proposition convert all its indefinables into variables; there then remains a class of propositions which is not all propositions but a type.

A2 There are thus two ways in which signs are similar. The names Socrates and Plato are similar: they are both names. But whatever they have in common must not be introduced before Socrates and Plato are introduced. The same applies to subject-predicate form etc. Therefore, thing, proposition, subject-predicate form, etc., are not indefinables, i.e., types are not indefinables.
When we say A judges that etc., then we have to mention a whole proposition which A judges. It will not do either to mention only its constituents, or its constituents and form, but not in the proper order. This shows that a proposition itself must occur in the statement that it is judged; however, for instance, „not-p” may be explained. The question „What is negated” must have a meaning.

To understand a proposition p it is not enough to know that p implies ‚„p” is true’, but we must also know that ~p implies „p is false”. This shows the bipolarity of the proposition.

W-F = Wahr-Falsch

To every molecular function a WF scheme corresponds. Therefore we may use the WF scheme itself instead of the function. Now what the WF scheme does is, it correlates the letters W and F with each proposition. These two letters are the poles of atomic propositions. Then the scheme correlates another W and F to these poles. In this notation all that matters is the correlation of the outside poles to the pole of the atomic propositions. Therefore not-not-p is the same symbol as p. And therefore we shall never get two symbols for the same molecular function.

A3 The meaning of a proposition is the fact which actually corresponds to it.
As the ab functions of atomic propositions are bi-polar propositions again we can perform ab operations on them. We shall, by doing so, correlate two new outside poles via the old outside poles to the poles of the atomic propositions.

The symbolising fact in a-p-b is that, say2 a is on the left of p and b on the right of p; then the correlation of new poles is to be transitive, so that for instance if a new pole a in whatever way i.e. via whatever poles is correlated to the inside a, the symbol is not changed thereby. It is therefore possible to construct all possible ab functions by performing one ab operation repeatedly, and we can therefore talk of all ab functions as of all those functions which can be obtained by performing this ab operation repeatedly.

[Note by Bertrand Russell]

[NB. ab means the same as WF, which means true-false.]

Naming is like pointing. A function is like a line dividing points of a plane into right and left ones; then „p or not-p” has no meaning because it does not divide the plane.

But though a particular proposition „p or not-p” has no meaning, a general proposition „for all p's, p or not-p” has a meaning because this does not contain the nonsensical function „p or not-p” but the function „p or not-q” just as „for all x's xRx” contains the function „xRy”.

A4 A proposition is a standard to which all facts behave, with names it is otherwise; it is thus bi-polarity and sense comes in; just as one arrow behaves to another arrow by being in the same sense or the opposite, so a fact behaves to a proposition.
The form of a proposition has meaning in the following way. Consider a symbol „xRy”. To symbols of this form correspond couples of things whose names are respectively „x” and „y”. The things x y stand to one another in all sorts of relations, amongst others some stand in the relation R, and some not; just as I single out a particular thing by a particular name I single out all behaviours of the points x and y with respect to the relation R. I say that if an x stands in the relation R to a y the sign „x R y” is to be called true to the fact and otherwise false. This is a definition of sense.

In my theory p has the same meaning as not-p but opposite sense. The meaning is the fact. The proper theory of judgment must make it impossible to judge nonsense.

It is not strictly true to say that we understand a proposition p if we know that p is equivalent to „p is true” for this would be the case if accidentally both were true or false. What is wanted is the formal equivalence with respect to the forms of the proposition, i.e., all the general indefinables involved. The sense of an ab function of a proposition is a function of its sense. There are only unasserted propositions. ¦ Assertion is merely psychological. In not-p, p is exactly the same as if it stands alone; this point is absolutely fundamental. Among the facts which make „p or q” true there are also facts which make „p and q” true; if propositions have only meaning, we ought, in such a case, to say that these two propositions are identical, but in fact, their sense is different for we have introduced sense by talking of all p's and all q's. Consequently the molecular propositions will only be used in cases where there ab function stands under a generality sign or enters into another function such as „I believe that, etc.,” because then the sense enters.

A5 In „a judges p” p cannot be replaced by a proper name. This appears if we substitute „a judges that p is true and not p is false”. The proposition „a judges p” consists of the proper name a, the proposition p with its 2 poles, and a being related to both of these poles in a certain way. This is obviously not a relation in the ordinary sense.
The ab notation makes it clear that not and or are dependent on one another and we can therefore not use them as simultaneous indefinables. Same objections in the case of apparent variables to old indefinables, as in the case of molecular functions: The application of the ab notation to apparent-variable propositions becomes clear if we consider that, for instance, the proposition „for all x, x” is to be true when x is true for all x's and false when jx is false for some x's. We see that some and all occur simultaneously in the proper apparent variable notation.

A6 The Notation is:
for (x) x ; a - (x) - a x b - ( x) - b

and

for (x) x : a - (x) - a x b - (x) - b

Old definitions now become tautologous.

In aRb it is not the complex that symbolises but the fact that the symbol a stands in a certain relation to the symbol b. Thus facts are symbolised by facts, or more correctly: that a certain thing is the case in the symbol says that a certain thing is the case in the world.

Judgment, question and command are all on the same level. What interests logic in them is only the unasserted proposition. Facts cannot be named.

A proposition cannot occur in itself. This is the fundamental truth of the theory of types.

Every proposition that says something about one thing is a subject-predicate proposition, and so on.

Therefore we can recognize a subject-predicate proposition if we know it contains only one name and one form, etc. This gives the construction of types. Hence the type of a proposition can be recognized by its symbol alone.

A7 What is essential in a correct apparent-variable notation is this:- (1) it must mention a type of propositions; (2) it must show which components of a proposition of this type are constants.
[Components are forms and constituents.]

Take ().!x. Then if we describe the kind of symbols, for which ! stands and which, by the above, is enough to determine the type, then automatically „().!x” cannot be fitted by this description, because it contains „!x” and the description is to describe all that symbolizes in symbols of the ! - kind. If the description is thus complete vicious circles can just as little occur as for instance in ().(x) (where (x) is a subject-predicate proposition).

B1 First MS.
Indefinables are of two sorts: names, and forms. Propositions cannot consist of names alone; they cannot be classes of names. A name can not only occur in two different propositions, but can occur in the same way in both.

Propositions [which are symbols having reference to facts] are themselves facts: that this inkpot is on this table may express that I sit in this chair.

It can never express the common characteristic of two objects that we designate them by the same name but by two different ways of designation, for, since names are arbitrary, we might also choose different names, and where then would be the common element in the designations? Nevertheless one is always tempted, in a difficulty, to take refuge in different ways of designation.

Frege said „propositions are names”; Russell said „propositions correspond to complexes”. Both are false; and especially false is the statement „propositions are names of complexes”.

It is easy to suppose that only such symbols are complex as contain names of objects, and that accordingly „(x,).x” or „(x,R,y).xRy” must be simple. It is then natural to call the first of these the name of a form, the second the name of a relation. But in that case what is the meaning of (e.g.) „˜(x,y).xRy”? Can we put „not” before a name?

B2 The reason why „˜Socrates” means nothing is that „˜x” does not express a property of x.
There are positive and negative facts: if the proposition „this rose is not red” is true, then what it signifies is negative. But the occurrence of the word „not” does not indicate this unless we know that the signification of the proposition „this rose is red” (when it is true) is positive. It is only from both, the negation and the negated proposition, that we can conclude to a characteristic of the significance of the whole proposition. (We are not here speaking of negations of general propositions, i.e. of such as contain apparent variables. Negative facts only justify the negations of atomic propositions.)

Positive and negative facts there are, but not true and false facts.

If we overlook the fact that propositions have a sense which is independent of their truth or falsehood, it easily seems as if true and false were two equally justified relations between the sign and what is signified. (We might then say e.g. that „q” signifies in the true way what „not-q” signifies in the false way). But are not true and false in fact equally justified? Could we not express ourselves by means of false propositions just as well as hitherto with true ones, so long as we know that they are meant falsely? ¦ No! For a proposition is then true when it is as we assert in this proposition; and accordingly if by „q” we mean „not-q”, and it is as we mean to assert, then in the new interpretation „q” is actually true and not false. But it is important that we can mean the same by „q” as by „not-q”, for it shows that neither to the symbol „not” nor to the manner of its combination with „q” does a characteristic of the denotation of „q” correspond.

B4 2nd MS.
We must be able to understand propositions which we have never heard before. But every proposition is a new symbol. Hence we must have general indefinable symbols; these are unavoidable if propositions are not all indefinable.

Whatever corresponds in reality to compound propositions must not be more than what corresponds to their several atomic propositions.

Not only must logic not deal with [particular] things, but just as little with relations and predicates.

There are no propositions containing real variables.

What corresponds in reality to a proposition depends upon whether it is true or false. But we must be able to understand a proposition without knowing if it is true or false.

What we know when we understand a proposition is this: We know what is the case if the proposition is true, and what is the case if it is false. But we do not know [necessarily] whether it is true or false.

Propositions are not names.

We can never distinguish one logical type from another by attributing a property to members of the one which we deny to members of the other.

Symbols are not what they seem to be. In „aRb”, „R” looks like a substantive, but is not one. What symbolizes in „aRb” is that R occurs between a and b. Hence „R” is not the indefinable in „aRb”. Similarly in „x”, „” looks like a substantive but is not one; in „˜p”, „˜” looks like „” but is not like it. This is the first thing that indicates that there may not be logical constants. A reason against them is the generality of logic: logic cannot treat a special set of things.

B5 Molecular propositions contain nothing beyond what is contained in their atoms; they add no material information above that contained in their atoms.
All that is essential about molecular functions is their T-F schema [i.e. the statement of the cases when they are true and the cases when they are false].

Alternative indefinability shows that the indefinables have not been reached.

Every proposition is essentially true-false: to understand it, we must know both what must be the case if it is true, and what must be the case if it is false. Thus a proposition has two poles, corresponding to the case of its truth and the case of its falsehood. We call this the sense of a proposition.

In regard to notation, it is important to note that not every feature of a symbol symbolizes. In two molecular functions which have the same T-F schema, what symbolizes must be the same. In „not-not-p”, „not-p” does not occur; for „not-not-p” is the same as ”p”, and therefore, if „not-p” occurred in „not-not-p”, it would occur in „p”.

Logical indefinables cannot be predicates or relations, because propositions, owing to sense, cannot have predicates or relations. Nor are „not” and „or”, like judgment, analogous to predicates or relations, because they do not introduce anything new.

Propositions are always complex even if they contain no names.

B6 A proposition must be understood when all its indefinables are understood. The indefinables in „aRb” are introduced as follows:
„a” is indefinable;

„b” is indefinable;

Whatever „x” and „y” may mean, „xRy” says something indefinable about their meanings.

A complex symbol must never be introduced as a single indefinable. (Thus e.g. no proposition is indefinable.) For if one of its parts occurs also in another connection, it must there be re-introduced. And would it then mean the same?

The ways by which we introduce our indefinables must permit us to construct all propositions that have sense from these indefinables alone. It is easy to introduce „all” and „some” in a way that will make the construction of (say) „(x,y).xRy” possible from „all” and „xRy” as introduced before.

B7 3rd MS.
An analogy for the theory of truth: Consider a black patch on white paper; then we can describe the form of the patch by mentioning, for each point of the surface, whether it is white or black. To the fact that a point is black corresponds a positive fact, to the fact that a point is white (not black) corresponds a negative fact. If I designate a point of the surface (one of Frege's „truth-values”), this is as if I set up an assumption to be decided upon. But in order to be able to say of a point that it is black or that it is white, I must first know when a point is to be called black and when it is to be called white. In order to be able to say that „p” is true (or false), I must first have determined under what circumstances I call a proposition true, and thereby I determine the sense of a proposition. The point in which the analogy fails is this: I can indicate a point of the paper that is white and black, but to a proposition without sense nothing corresponds, for it does not designate a thing (truth-value), whose properties might be called „false” or „true”; the verb of a proposition is not „is true” or „is false”, as Frege believes, but what is true must already contain the verb.

The comparison of language and reality is like that of retinal image and visual image: to the blind spot nothing in the visual image seems to correspond, and thereby the boundaries of the blind spot determine the visual image - as true negations of atomic propositions determine reality.

B8 Logical inferences can, it is true, be made in accordance with Frege's or Russell's laws of deduction, but this cannot justify the inference; and therefore they are not primitive propositions of logic. If p follows from q, it can also be inferred from q, and the „manner of deduction” is indifferent.
Those symbols which are called propositions in which „variables occur” are in reality not propositions at all, but only schemes of propositions, which only become propositions when we replace the variables by constants. There is no proposition which is expressed by „x = x”, for „x” has no signification; but there is a proposition „(x).x = x” and propositions such as „Socrates = Socrates” etc.

In books on logic, no variables ought to occur, but only the general propositions which justify the use of variables. It follows that the so-called definitions of logic are not definitions, but only schemes of definitions, and instead of these we ought to put general propositions; and similarly the so-called primitive ideas (Urzeichen) of logic are not primitive ideas, but the schemes of them. The mistaken idea that there are things called facts or complexes and relations easily leads to the opinion that there must be a relation of questioning to the facts, and then the question arises whether a relation can hold between an arbitrary number of things, since a fact can follow from arbitrary cases. It is a fact that the proposition which e.g. expresses that q follows from p and pq is this: p.pq.p.q.q.

B9 At a pinch, one is tempted to interpret „not-p” as „everything else, only not p”. That from a single fact p an infinity of others, not-not-p etc., follow, is hardly credible. Man possesses an innate capacity for constructing symbols with which some sense can be expressed, without having the slightest idea what each word signifies. The best example of this is mathematics, for man has until lately used the symbols for numbers without knowing what they signify or that they signify nothing.
Russell's „complexes” were to have the useful property of being compounded, and were to combine with this the agreeable property that they could be treated like „simples”. But this alone made them unserviceable as logical types, since there would have been significance in asserting, of a simple, that it was complex. But a property cannot be a logical type.

Every statement about apparent complexes can be resolved into the logical sum of a statement about the constituents and a statement about the proposition which describes the complex completely. How, in each case, the resolution is to be made, is an important question, but its answer is not unconditionally necessary for the construction of logic.

B10 That „or” and „not” etc. are not relations in the same sense as „right” and „left” etc., is obvious to the plain man. The possibility of cross-definitions in the old logical indefinables shows, of itself, that these are not the right indefinables, and, even more conclusively, that they do not denote relations.
If we change a constituent a of a proposition (a) into a variable, then there is a class

˜p {(x).(x) = p}.

This class in general still depends upon what, by an arbitrary convention, we mean by „(x)”. But if we change into variables all those symbols whose significance was arbitrarily determined, there is still such a class. But this is now not dependent upon any convention, but only upon the nature of the symbol „(x)”. It corresponds to a logical type.

Types can never be distinguished from each other by saying (as is often done) that one has these but the other has that properties, for this presupposes that there is a meaning in asserting all these properties of both types. But from this it follows that, at best, these properties may be types, but certainly not the objects of which they are asserted.

B11 At a pinch, we are always inclined to explanations of logical functions of propositions which aim at introducing into the function either only the constituents of these propositions, or only their forms, etc. etc.; and we overlook that ordinary language would not contain the whole propositions if it did not need them: However, e.g., „not-p” may be explained, there must always be a meaning given to the question „what is denied?”
The very possibility of Frege's explanations of „not-p” and „if p then q”, from which it follows that not-not-p denotes the same as p, makes it probable that there is some method of designation in which „not-not-p” corresponds to the same symbol as „p”. But if this method of designation suffices for logic, it must be the right one.

Names are points, propositions arrows - they have sense. The sense of a proposition is determined by the two poles true and false. The form of a proposition is like a straight line, which divides all points of a plane into right and left. The line does this automatically, the form of proposition only by convention.

B12 Just as little as we are concerned, in logic, with the relation of a name to its meaning, just so little are we concerned with the relation of a proposition to reality, but we want to know the meaning of names and the sense of propositions - as we introduce an indefinable concept „A” by saying: „‚A’ denotes something indefinable”, so we introduce e.g. the form of propositions aRb by saying: „For all meanings of „x” and „y”, „xRy” expresses something indefinable about x and y”.
In place of every proposition „p”, let us write „abp”. Let every correlation of propositions to each other or of names to propositions be effected by a correlation of their poles „a” and „b”. Let this correlation be transitive. Then accordingly „a-ab-bp” is the same symbol as „abp”. Let n propositions be given. I then call a „class of poles” of these propositions every class of n members, of which each is a pole of one of the n propositions, so that one member corresponds to each proposition. I then correlate with each class of poles one of two poles (a and b). The sense of the symbolizing fact thus constructed I cannot define, but I know it.

If p = not-not-p etc., this shows that the traditional method of symbolism is wrong, since it allows a plurality of symbols with the same sense; and thence it follows that, in analyzing such propositions, we must not be guided by Russell's method of symbolizing.

B13 It is to be remembered that names are not things, but classes: „A” is the same letter as „A”. This has the most important consequences for every symbolic language.
Neither the sense nor the meaning of a proposition is a thing. These words are incomplete symbols.

It is impossible to dispense with propositions in which the same argument occurs in different positions. It is obviously useless to replace (a,a) by (a,b).a = b.

Since the ab-functions of p are again bi-polar propositions, we can form ab-functions of them, and so on. In this way a series of propositions will arise, in which in general the symbolizing facts will be the same in several members. If now we find an ab-function of such a kind that by repeated application of it every ab-function can be generated, then we can introduce the totality of ab-functions as the totality of those that are generated by application of this function. Such a function is ˜pv˜q.

B14 It is easy to suppose a contradiction in the fact that on the one hand every possible complex proposition is a simple ab-function of simple propositions, and that on the other hand the repeated application of one ab-function suffices to generate all these propositions. If e.g. an affirmation can be generated by double negation, is negation in any sense contained in affirmation? Does „p” deny „not-p” or assert „p”, or both? And how do matters stand with the definition of „” by „v” and „·”, or of „v” by „·” and „”? And how e.g. shall we introduce p|q (i.e. ˜pv˜q), if not by saying that this expression says something indefinable about all arguments p and q? But the ab-functions must be introduced as follows: The function p|q is merely a mechanical instrument for constructing all possible symbols of ab-functions. The symbols arising by repeated application of the symbol „¦” do not contain the symbol „p¦q”. We need a rule according to which we can form all symbols of ab-functions, in order to be able to speak of the class of them; and we now speak of them e.g. as those symbols of functions which can be generated by repeated application of the operation „¦”. And we say now: For all p's and q's, „p¦q” says something indefinable about the sense of those simple propositions which are contained in p and q.
B15 The assertion-sign is logically quite without significance. It only shows, in Frege and Whitehead and Russell, that these authors hold the propositions so indicated to be true. „” therefore belongs as little to the proposition as (say) the number of the proposition. A proposition cannot possibly assert of itself that it is true.
Every right theory of judgment must make it impossible for me to judge that this table penholders the book. Russell's theory does not satisfy this requirement.

It is clear that we understand propositions without knowing whether they are true or false. But we can only know the meaning of a proposition when we know if it is true or false. What we understand is the sense of the proposition.

The assumption of the existence of logical objects makes it appear remarkable that in the sciences propositions of the form „pvq”, „pq”, etc. are only then not provisional when „v” and ”” stand within the scope of a generality-sign [apparent variable].

B16 4th MS.
If we formed all possible atomic propositions, the world would be completely described if we declared the truth or falsehood of each. [I doubt this.]

The chief characteristic of my theory is that, in it, p has the same meaning as not-p.

A false theory of relations makes it easily seem as if the relation of fact and constituent were the same as that of fact and fact which follows from it. But the similarity of the two may be expressed thus: a..,a a = a.

If a word creates a world so that in it the principles of logic are true, it thereby creates a world in which the whole of mathematics holds; and similarly it could not create a world in which a proposition was true, without creating its constituents.

Signs of the form „pv˜p” are senseless, but not the proposition „(p).p v ˜p”. If I know that this rose is either red or not red, I know nothing. The same holds of all ab-functions.

To understand a proposition means to know what is the case if it is true. Hence we can understand it without knowing if it is true. We understand it when we understand its constituents and forms. If we know the meaning of „a” and „b”, and if we know what „xRy” means for all x's and y's, then we also understand „aRb”.

I understand the proposition „aRb” when I know that either the fact that aRb or the fact that not aRb corresponds to it; but this is not to be confused with the false opinion that I understand „aRb” when I know that „aRb or not-aRb” is the case.

B17 But the form of a proposition symbolizes in the following way: Let us consider symbols of the form „xRy”; to these correspond primarily pairs of objects, of which one has the name „x”, the other the name „y”. The x's and y's stand in various relations to each other, among others the relation R holds between some, but not between others. I now determine the sense of „xRy” by laying down: when the facts behave in regard to „xRy” so that the meaning of „x” stands in the relation R to the meaning of „y”, then I say that they [the facts] are „of like sense” [„gleichsinnig”] with the proposition „xRy”; otherwise, „of opposite sense” [„entgegengesetzt”]; I correlate the facts to the symbol „xRy” by thus dividing them into those of like sense and those of opposite sense. To this correlation corresponds the correlation of name and meaning. Both are psychological. Thus I understand the form „xRy” when I know that it discriminates the behaviour of x and y according as these stand in the relation R or not. In this way I extract from all possible relations the relation R, as, by a name, I extract its meaning from among all possible things.
Strictly speaking, it is incorrect to say: We understand the proposition p when we know that ‚„p” is true’ p; for this would naturally always be the case if accidentally the propositions to right and left of the symbol „” were both true or both false. We require not only an equivalence, but a formal equivalence, which is bound up with the introduction of the form of p.

The sense of an ab-function of p is a function of the sense of p.

B18 The ab-functions use the discrimination of facts, which their arguments bring forth, in order to generate new discriminations.
Only facts can express sense, a class of names cannot. This is easily shown.

There is no thing which is the form of a proposition, and no name which is the name of a form. Accordingly we can also not say that a relation which in certain cases holds between things holds sometimes between forms and things. This goes against Russell's theory of judgment.

It is very easy to forget that, tho. the propositions of a form can be either true or false, each one of these propositions can only be either true or false, not both.

Among the facts which make „p or q” true, there are some which make „p and q” true; but the class which makes „p or q” true is different from the class which makes „p and q” true; and only this is what matters. For we introduce this class, as it were, when we introduce ab-functions.

A very natural objection to the way in which I have introduced e.g. propositions of the form xRy is that by it propositions such as (x,y).xRy and similar ones are not explained, which yet obviously have in common with aRb what cRd has in common with aRb. But when we introduced propositions of the form xRy we mentioned no one particular proposition of this form; and we only need to introduce (x,y).(x,y) for all 's in any way which makes the sense of these propositions dependent on the sense of all propositions of the form (a,b), and thereby the justification of our procedure is proved.

B19 The indefinables of logic must be independent of each other. If an indefinable is introduced, it must be introduced in all combinations in which it can occur. We cannot therefore introduce it first for one combination, then for another; e.g., if the form xRy has been introduced, it must henceforth be understood in propositions of the form aRb just in the same way as in propositions such as (x,y). xRy and others. We must not introduce it first for one class of cases, then for the other; for it would remain doubtful if its meaning was the same in both cases, and there would be no ground for using the same manner of combining symbols in both cases. In short, for the introduction of indefinable symbols and combinations of symbols the same holds, mutatis mutandis, that Frege has said for the introduction of symbols by definitions.
It is a priori likely that the introduction of atomic propositions is fundamental for the understanding of all other kinds of propositions. In fact the understanding of general propositions obviously depends on that of atomic propositions.

Cross-definability in the realm of general propositions leads to the quite similar questions to those in the realm of ab-functions.

B20 When we say „A believes p”, this sounds, it is true, as if here we could substitute a proper name for „p”; but we can see that here a sense, not a meaning, is concerned, if we say „A believes that ‚p’ is true”; and in order to make the direction of p even more explicit, we might say „A believes that ‚p’ is true and ‚not-p’ is false”. Here the bi-polarity of p is expressed, and it seems that we shall only be able to express the proposition „A believes p” correctly by the ab-notation; say by making „A” have a relation to the poles „a” and „b” of a-p-b.

The epistemological questions concerning the nature of judgment and belief cannot be solved without a correct apprehension of the form of the proposition.

The ab-notation shows the dependence of or and not, and thereby that they are not to be employed as simultaneous indefinables.

Not: „The complex sign ‚aRb’” says that a stands in the relation R to b; but that ‚a’ stands in a certain relation to ‚b’ says that aRb.

In philosophy there are no deductions: it is purely descriptive.

Philosophy gives no pictures of reality.

Philosophy can neither confirm nor confute scientific investigation.

B21 Philosophy consists of logic and metaphysics: logic is its basis.
Epistemology is the philosophy of psychology.

Distrust of grammar is the first requisite for philosophizing.

Propositions can never be indefinables, for they are always complex. That also words like „ambulo” are complex appears in the fact that their root with a different termination gives a different sense.

Only the doctrine of general indefinables permits us to understand the nature of functions. Neglect of this doctrine leads to an impenetrable thicket.

Philosophy is the doctrine of the logical form of scientific propositions (not only of primitive propositions).

The word „philosophy” ought always to designate something over or under, but not beside, the natural sciences.

Judgment, command and question all stand on the same level; but all have in common the propositional form, which does interests us.

The structure of the proposition must be recognized, the rest comes of itself. But ordinary language conceals the structure of the proposition: in it, relations look like predicates, predicates like names, etc.

Facts cannot be named.

B22 It is easy to suppose that „individual”, „particular”, „complex” etc. are primitive ideas of logic. Russell e.g. says „individual” and „matrix” are „primitive ideas”. This error presumably is to be explained by the fact that, by employment of variables instead of the generality-sign, it comes to seem as if logic dealt with things which have been deprived of all properties except thing-hood, and with propositions deprived of all properties except complexity. We forget that the indefinables of symbols [Urbilder von Zeichen] only occur under the generality-sign, never outside it.
Just as people used to struggle to bring all propositions into the subject-predicate form, so now it is natural to conceive every proposition as expressing a relation, which is just as incorrect. What is justified in this desire is fully satisfied by Russell's theory of manufactured relations.

One of the most natural attempts at solution consists in regarding „not-p” as „the opposite of p”, where then „opposite” would be the indefinable relation. But it is easy to see that every such attempt to replace the ab-functions by descriptions must fail.

B23 The false assumption that propositions are names leads us to believe that there must be logical objects: for the meanings of logical propositions will have to be such things.
A correct explanation of logical propositions must give them a unique position as against all other propositions.

No proposition can say anything about itself, because the symbol of the proposition cannot be contained in itself; this must be the basis of the theory of logical types.

Every proposition which says something indefinable about a thing is a subject-predicate proposition; every proposition which says something indefinable about two things expresses a dual relation between these things, and so on. Thus every proposition which contains only one name and one indefinable form is a subject-predicate proposition, and so on. An indefinable simple symbol can only be a name, and therefore we can know, by the symbol of an atomic proposition, whether it is a subject-predicate proposition.

B24 I. Bi-polarity of propositions: sense and meaning, truth and falsehood.
II. Analysis of atomic propositions: general indefinables, predicates, etc.

III. Analysis of molecular propositions: ab-functions.

IV. Analysis of general propositions.

V. Principles of symbolism: what symbolizes in a symbol. Facts for facts.

VI. Types.

B25

This is the symbol for

˜p v ˜q

B26

1 you for instance imagine every fact as a spatial complex.
2 This is quite arbitrary but if we such have fixed on which sides the poles have to stand we must of course stick to our convention. If for instance „apb” says p then bpa says nothing. (It does not say ˜p.) But a-apb-b is the same symbol as apb the ab function vanishes automatically for here the new poles are related to the same side of p as the old ones. The question is always: how are the new poles correlated to p compared with the way the old poles are correlated to p.

-- Tuomas (tuomas.alh...), July 8th, 2003.
That's not a summary of the Bible, is it? Wait, let me read that again.

-- Sommermute (wintermute_v0.3...), July 8th, 2003.
Summary
One reason for thinking the old notation wrong is that it is very unlikely that from every proposition p an infinite number of other propositions not-not-p, not-not-not-not-p, etc., should follow.
If only those signs which contain proper names were complex then propositions containing nothing but apparent variables would be simple. Then what about their denials?

The verb of a proposition cannot be „is true” or „is false”, but whatever is true or false must already contain the verb.

Deductions only proceed according to the laws of deduction, but these laws cannot justify the deduction.

One reason for supposing that not all propositions which have more than one argument are relational propositions is that if they were, the relations of judgement and inference would have to hold between an arbitrary number of things.

Every proposition which seems to be about a complex can be analysed into a proposition about its constituents and about the proposition which describes the complex perfectly; i.e., that proposition which is equivalent to saying the complex exists.

The idea that propositions are names of complexes suggests that whatever is not a proper name is a sign for a relation. Because spatial complexes1 consist of Things and Relations only and the idea of a complex is taken from space.

In a proposition convert all its indefinables into variables; there then remains a class of propositions which is not all propositions but a type.

A2 There are thus two ways in which signs are similar. The names Socrates and Plato are similar: they are both names. But whatever they have in common must not be introduced before Socrates and Plato are introduced. The same applies to subject-predicate form etc. Therefore, thing, proposition, subject-predicate form, etc., are not indefinables, i.e., types are not indefinables.
When we say A judges that etc., then we have to mention a whole proposition which A judges. It will not do either to mention only its constituents, or its constituents and form, but not in the proper order. This shows that a proposition itself must occur in the statement that it is judged; however, for instance, „not-p” may be explained. The question „What is negated” must have a meaning.

To understand a proposition p it is not enough to know that p implies ‚„p” is true’, but we must also know that ~p implies „p is false”. This shows the bipolarity of the proposition.

W-F = Wahr-Falsch

To every molecular function a WF scheme corresponds. Therefore we may use the WF scheme itself instead of the function. Now what the WF scheme does is, it correlates the letters W and F with each proposition. These two letters are the poles of atomic propositions. Then the scheme correlates another W and F to these poles. In this notation all that matters is the correlation of the outside poles to the pole of the atomic propositions. Therefore not-not-p is the same symbol as p. And therefore we shall never get two symbols for the same molecular function.

A3 The meaning of a proposition is the fact which actually corresponds to it.
As the ab functions of atomic propositions are bi-polar propositions again we can perform ab operations on them. We shall, by doing so, correlate two new outside poles via the old outside poles to the poles of the atomic propositions.

The symbolising fact in a-p-b is that, say2 a is on the left of p and b on the right of p; then the correlation of new poles is to be transitive, so that for instance if a new pole a in whatever way i.e. via whatever poles is correlated to the inside a, the symbol is not changed thereby. It is therefore possible to construct all possible ab functions by performing one ab operation repeatedly, and we can therefore talk of all ab functions as of all those functions which can be obtained by performing this ab operation repeatedly.

[Note by Bertrand Russell]

[NB. ab means the same as WF, which means true-false.]

Naming is like pointing. A function is like a line dividing points of a plane into right and left ones; then „p or not-p” has no meaning because it does not divide the plane.

But though a particular proposition „p or not-p” has no meaning, a general proposition „for all p's, p or not-p” has a meaning because this does not contain the nonsensical function „p or not-p” but the function „p or not-q” just as „for all x's xRx” contains the function „xRy”.

A4 A proposition is a standard to which all facts behave, with names it is otherwise; it is thus bi-polarity and sense comes in; just as one arrow behaves to another arrow by being in the same sense or the opposite, so a fact behaves to a proposition.
The form of a proposition has meaning in the following way. Consider a symbol „xRy”. To symbols of this form correspond couples of things whose names are respectively „x” and „y”. The things x y stand to one another in all sorts of relations, amongst others some stand in the relation R, and some not; just as I single out a particular thing by a particular name I single out all behaviours of the points x and y with respect to the relation R. I say that if an x stands in the relation R to a y the sign „x R y” is to be called true to the fact and otherwise false. This is a definition of sense.

In my theory p has the same meaning as not-p but opposite sense. The meaning is the fact. The proper theory of judgment must make it impossible to judge nonsense.

It is not strictly true to say that we understand a proposition p if we know that p is equivalent to „p is true” for this would be the case if accidentally both were true or false. What is wanted is the formal equivalence with respect to the forms of the proposition, i.e., all the general indefinables involved. The sense of an ab function of a proposition is a function of its sense. There are only unasserted propositions. ¦ Assertion is merely psychological. In not-p, p is exactly the same as if it stands alone; this point is absolutely fundamental. Among the facts which make „p or q” true there are also facts which make „p and q” true; if propositions have only meaning, we ought, in such a case, to say that these two propositions are identical, but in fact, their sense is different for we have introduced sense by talking of all p's and all q's. Consequently the molecular propositions will only be used in cases where there ab function stands under a generality sign or enters into another function such as „I believe that, etc.,” because then the sense enters.

A5 In „a judges p” p cannot be replaced by a proper name. This appears if we substitute „a judges that p is true and not p is false”. The proposition „a judges p” consists of the proper name a, the proposition p with its 2 poles, and a being related to both of these poles in a certain way. This is obviously not a relation in the ordinary sense.
The ab notation makes it clear that not and or are dependent on one another and we can therefore not use them as simultaneous indefinables. Same objections in the case of apparent variables to old indefinables, as in the case of molecular functions: The application of the ab notation to apparent-variable propositions becomes clear if we consider that, for instance, the proposition „for all x, x” is to be true when x is true for all x's and false when jx is false for some x's. We see that some and all occur simultaneously in the proper apparent variable notation.

A6 The Notation is:
for (x) x ; a - (x) - a x b - ( x) - b

and

for (x) x : a - (x) - a x b - (x) - b

Old definitions now become tautologous.

In aRb it is not the complex that symbolises but the fact that the symbol a stands in a certain relation to the symbol b. Thus facts are symbolised by facts, or more correctly: that a certain thing is the case in the symbol says that a certain thing is the case in the world.

Judgment, question and command are all on the same level. What interests logic in them is only the unasserted proposition. Facts cannot be named.

A proposition cannot occur in itself. This is the fundamental truth of the theory of types.

Every proposition that says something about one thing is a subject-predicate proposition, and so on.

Therefore we can recognize a subject-predicate proposition if we know it contains only one name and one form, etc. This gives the construction of types. Hence the type of a proposition can be recognized by its symbol alone.

A7 What is essential in a correct apparent-variable notation is this:- (1) it must mention a type of propositions; (2) it must show which components of a proposition of this type are constants.
[Components are forms and constituents.]

Take ().!x. Then if we describe the kind of symbols, for which ! stands and which, by the above, is enough to determine the type, then automatically „().!x” cannot be fitted by this description, because it contains „!x” and the description is to describe all that symbolizes in symbols of the ! - kind. If the description is thus complete vicious circles can just as little occur as for instance in ().(x) (where (x) is a subject-predicate proposition).

B1 First MS.
Indefinables are of two sorts: names, and forms. Propositions cannot consist of names alone; they cannot be classes of names. A name can not only occur in two different propositions, but can occur in the same way in both.

Propositions [which are symbols having reference to facts] are themselves facts: that this inkpot is on this table may express that I sit in this chair.

It can never express the common characteristic of two objects that we designate them by the same name but by two different ways of designation, for, since names are arbitrary, we might also choose different names, and where then would be the common element in the designations? Nevertheless one is always tempted, in a difficulty, to take refuge in different ways of designation.

Frege said „propositions are names”; Russell said „propositions correspond to complexes”. Both are false; and especially false is the statement „propositions are names of complexes”.

It is easy to suppose that only such symbols are complex as contain names of objects, and that accordingly „(x,).x” or „(x,R,y).xRy” must be simple. It is then natural to call the first of these the name of a form, the second the name of a relation. But in that case what is the meaning of (e.g.) „˜(x,y).xRy”? Can we put „not” before a name?

B2 The reason why „˜Socrates” means nothing is that „˜x” does not express a property of x.
There are positive and negative facts: if the proposition „this rose is not red” is true, then what it signifies is negative. But the occurrence of the word „not” does not indicate this unless we know that the signification of the proposition „this rose is red” (when it is true) is positive. It is only from both, the negation and the negated proposition, that we can conclude to a characteristic of the significance of the whole proposition. (We are not here speaking of negations of general propositions, i.e. of such as contain apparent variables. Negative facts only justify the negations of atomic propositions.)

Positive and negative facts there are, but not true and false facts.

If we overlook the fact that propositions have a sense which is independent of their truth or falsehood, it easily seems as if true and false were two equally justified relations between the sign and what is signified. (We might then say e.g. that „q” signifies in the true way what „not-q” signifies in the false way). But are not true and false in fact equally justified? Could we not express ourselves by means of false propositions just as well as hitherto with true ones, so long as we know that they are meant falsely? ¦ No! For a proposition is then true when it is as we assert in this proposition; and accordingly if by „q” we mean „not-q”, and it is as we mean to assert, then in the new interpretation „q” is actually true and not false. But it is important that we can mean the same by „q” as by „not-q”, for it shows that neither to the symbol „not” nor to the manner of its combination with „q” does a characteristic of the denotation of „q” correspond.

B4 2nd MS.
We must be able to understand propositions which we have never heard before. But every proposition is a new symbol. Hence we must have general indefinable symbols; these are unavoidable if propositions are not all indefinable.

Whatever corresponds in reality to compound propositions must not be more than what corresponds to their several atomic propositions.

Not only must logic not deal with [particular] things, but just as little with relations and predicates.

There are no propositions containing real variables.

What corresponds in reality to a proposition depends upon whether it is true or false. But we must be able to understand a proposition without knowing if it is true or false.

What we know when we understand a proposition is this: We know what is the case if the proposition is true, and what is the case if it is false. But we do not know [necessarily] whether it is true or false.

Propositions are not names.

We can never distinguish one logical type from another by attributing a property to members of the one which we deny to members of the other.

Symbols are not what they seem to be. In „aRb”, „R” looks like a substantive, but is not one. What symbolizes in „aRb” is that R occurs between a and b. Hence „R” is not the indefinable in „aRb”. Similarly in „x”, „” looks like a substantive but is not one; in „˜p”, „˜” looks like „” but is not like it. This is the first thing that indicates that there may not be logical constants. A reason against them is the generality of logic: logic cannot treat a special set of things.

B5 Molecular propositions contain nothing beyond what is contained in their atoms; they add no material information above that contained in their atoms.
All that is essential about molecular functions is their T-F schema [i.e. the statement of the cases when they are true and the cases when they are false].

Alternative indefinability shows that the indefinables have not been reached.

Every proposition is essentially true-false: to understand it, we must know both what must be the case if it is true, and what must be the case if it is false. Thus a proposition has two poles, corresponding to the case of its truth and the case of its falsehood. We call this the sense of a proposition.

In regard to notation, it is important to note that not every feature of a symbol symbolizes. In two molecular functions which have the same T-F schema, what symbolizes must be the same. In „not-not-p”, „not-p” does not occur; for „not-not-p” is the same as ”p”, and therefore, if „not-p” occurred in „not-not-p”, it would occur in „p”.

Logical indefinables cannot be predicates or relations, because propositions, owing to sense, cannot have predicates or relations. Nor are „not” and „or”, like judgment, analogous to predicates or relations, because they do not introduce anything new.

Propositions are always complex even if they contain no names.

B6 A proposition must be understood when all its indefinables are understood. The indefinables in „aRb” are introduced as follows:
„a” is indefinable;

„b” is indefinable;

Whatever „x” and „y” may mean, „xRy” says something indefinable about their meanings.

A complex symbol must never be introduced as a single indefinable. (Thus e.g. no proposition is indefinable.) For if one of its parts occurs also in another connection, it must there be re-introduced. And would it then mean the same?

The ways by which we introduce our indefinables must permit us to construct all propositions that have sense from these indefinables alone. It is easy to introduce „all” and „some” in a way that will make the construction of (say) „(x,y).xRy” possible from „all” and „xRy” as introduced before.

B7 3rd MS.
An analogy for the theory of truth: Consider a black patch on white paper; then we can describe the form of the patch by mentioning, for each point of the surface, whether it is white or black. To the fact that a point is black corresponds a positive fact, to the fact that a point is white (not black) corresponds a negative fact. If I designate a point of the surface (one of Frege's „truth-values”), this is as if I set up an assumption to be decided upon. But in order to be able to say of a point that it is black or that it is white, I must first know when a point is to be called black and when it is to be called white. In order to be able to say that „p” is true (or false), I must first have determined under what circumstances I call a proposition true, and thereby I determine the sense of a proposition. The point in which the analogy fails is this: I can indicate a point of the paper that is white and black, but to a proposition without sense nothing corresponds, for it does not designate a thing (truth-value), whose properties might be called „false” or „true”; the verb of a proposition is not „is true” or „is false”, as Frege believes, but what is true must already contain the verb.

The comparison of language and reality is like that of retinal image and visual image: to the blind spot nothing in the visual image seems to correspond, and thereby the boundaries of the blind spot determine the visual image - as true negations of atomic propositions determine reality.

B8 Logical inferences can, it is true, be made in accordance with Frege's or Russell's laws of deduction, but this cannot justify the inference; and therefore they are not primitive propositions of logic. If p follows from q, it can also be inferred from q, and the „manner of deduction” is indifferent.
Those symbols which are called propositions in which „variables occur” are in reality not propositions at all, but only schemes of propositions, which only become propositions when we replace the variables by constants. There is no proposition which is expressed by „x = x”, for „x” has no signification; but there is a proposition „(x).x = x” and propositions such as „Socrates = Socrates” etc.

In books on logic, no variables ought to occur, but only the general propositions which justify the use of variables. It follows that the so-called definitions of logic are not definitions, but only schemes of definitions, and instead of these we ought to put general propositions; and similarly the so-called primitive ideas (Urzeichen) of logic are not primitive ideas, but the schemes of them. The mistaken idea that there are things called facts or complexes and relations easily leads to the opinion that there must be a relation of questioning to the facts, and then the question arises whether a relation can hold between an arbitrary number of things, since a fact can follow from arbitrary cases. It is a fact that the proposition which e.g. expresses that q follows from p and pq is this: p.pq.p.q.q.

B9 At a pinch, one is tempted to interpret „not-p” as „everything else, only not p”. That from a single fact p an infinity of others, not-not-p etc., follow, is hardly credible. Man possesses an innate capacity for constructing symbols with which some sense can be expressed, without having the slightest idea what each word signifies. The best example of this is mathematics, for man has until lately used the symbols for numbers without knowing what they signify or that they signify nothing.
Russell's „complexes” were to have the useful property of being compounded, and were to combine with this the agreeable property that they could be treated like „simples”. But this alone made them unserviceable as logical types, since there would have been significance in asserting, of a simple, that it was complex. But a property cannot be a logical type.

Every statement about apparent complexes can be resolved into the logical sum of a statement about the constituents and a statement about the proposition which describes the complex completely. How, in each case, the resolution is to be made, is an important question, but its answer is not unconditionally necessary for the construction of logic.

B10 That „or” and „not” etc. are not relations in the same sense as „right” and „left” etc., is obvious to the plain man. The possibility of cross-definitions in the old logical indefinables shows, of itself, that these are not the right indefinables, and, even more conclusively, that they do not denote relations.
If we change a constituent a of a proposition (a) into a variable, then there is a class

˜p {(x).(x) = p}.

This class in general still depends upon what, by an arbitrary convention, we mean by „(x)”. But if we change into variables all those symbols whose significance was arbitrarily determined, there is still such a class. But this is now not dependent upon any convention, but only upon the nature of the symbol „(x)”. It corresponds to a logical type.

Types can never be distinguished from each other by saying (as is often done) that one has these but the other has that properties, for this presupposes that there is a meaning in asserting all these properties of both types. But from this it follows that, at best, these properties may be types, but certainly not the objects of which they are asserted.

B11 At a pinch, we are always inclined to explanations of logical functions of propositions which aim at introducing into the function either only the constituents of these propositions, or only their forms, etc. etc.; and we overlook that ordinary language would not contain the whole propositions if it did not need them: However, e.g., „not-p” may be explained, there must always be a meaning given to the question „what is denied?”
The very possibility of Frege's explanations of „not-p” and „if p then q”, from which it follows that not-not-p denotes the same as p, makes it probable that there is some method of designation in which „not-not-p” corresponds to the same symbol as „p”. But if this method of designation suffices for logic, it must be the right one.

Names are points, propositions arrows - they have sense. The sense of a proposition is determined by the two poles true and false. The form of a proposition is like a straight line, which divides all points of a plane into right and left. The line does this automatically, the form of proposition only by convention.

B12 Just as little as we are concerned, in logic, with the relation of a name to its meaning, just so little are we concerned with the relation of a proposition to reality, but we want to know the meaning of names and the sense of propositions - as we introduce an indefinable concept „A” by saying: „‚A’ denotes something indefinable”, so we introduce e.g. the form of propositions aRb by saying: „For all meanings of „x” and „y”, „xRy” expresses something indefinable about x and y”.
In place of every proposition „p”, let us write

John 3:16

http://www.tvdewereld.be/gotv/images/thread.jpg
http://ritchie.jikos.cz/images/killed.gif

http://www.claretex.com/images/thread.jpg
http://www.stentorian.com/2ndamend/interact/killer.jpg

BUT!!! Give him his glory, he did win, in that he successfully killed the thread without locking it.

*hears someone whispering in her ear*

Er, wait a second. I'm not the threadkiller supreme. Sorry, bad me, I'm the invisible poster supreme. Someone else is the threadkiller supreme.

btw, it's way cool seeing something I did two months ago.

Dammit!

Curtis, you cheater!! I'll get you, you'll see!

There. That's better.

Dada, wanna post some more pictures?