Tractatus Logico-Philosophicus


The world is everything that is the case.


The world is the totality of facts, not of things.


The world is determined by the facts, and by these begin all the facts.


For the totality of facts determines both what is the case, and also all that is not the case.


The facts in logical space are the world.


The world divides into facts.


Any one can either be the case or not be the case, and everything else remain the same.


What is the case, the fact, is the existence of atomic facts.



An atomic fact is a combination of objects (entities, things)


It is essential to a thing that it can be a constituent part of an atomic fact.


In logic nothing is accidental: if a thing can occur in an atomic fact the possibility of that atomic fact must already be prejudged in the thing.


It would, so to speak, appear as an accident, when to a thing that could exist alone on its own account, subsequently a state of affairs could be made to fit.

If things can occur in atomic facts, this possibility must already lie in them.

(A logical entity cannot be merely possible. Logic treats of every possibility, and all possibilities are its facts.)

Just as we cannot think of spatial objects at all apart from space, or temporal objects apart from time, so we cannot think of any object apart from the possibility of its connection with other things.

If I can think of an object in the context of an atomic fact, I cannot think of it apart from the possibility of this context.


The thing is independent, in so far as it can occur in all possible circumstances, but this form of independence is a form of connection with the atomic fact, a form of dependence. (It is impossible for words to occur in two different ways, alone and in the proposition.)


If I know an object, then I also know all the possibilities of its occurrence in atomic facts.

(Every such possibility must lie in the nature of the object.)

A new possibility cannot subsequently be found.


In order to know an object, I must know not its external but all its internal qualities.


If all objects are given, then thereby are all possible atomic facts also given.


Every thing is, as it were, in a space of possible atomic facts. I can think of this space as empty, but not of the thing without the space.


A spatial object must lie in infinite space. (A point in space is an argument place.)

A speck in a visual field need not be red, but it must have a color; it has, so to speak, a color space round it. A tone must have a pitch, the object of the sense of touch a hardness, etc.


Objects contain the possibility of all states of affairs.


The possibility of its occurrence in atomic facts is the form of the object.


The object is simple.



Every statement about complexes can be analyzed into a statement about their constituent parts, and into those propositions which completely describe the complexes.


Objects form the substance of the world. Therefore they cannot be compound.


If the world had no substance, then whether a proposition had sense would depend on whether another proposition was true.


It would then be impossible to form a picture of the world (true or false).


It is clear that however different from the real one an imagined world may be, it must have something -- a form -- in common with the real world.


This fixed form consists of the objects.


The substance of the world can only determine a form and not any material properties. For these are first presented by the propositions -- first formed by the configuration of the objects.


Roughly speaking: objects are colorless.


Two objects of the same logical form are -- apart from their external properties -- only differentiated from one another in that they are different.


Either a thing has properties which no other has, and then one can distinguish it straight away from the others by a description and refer to it; or, on the other hand, there are several things which have the totality of their properties in common, and then it is quite impossible to point to any one of them.

For it a thing is not distinguished by anything, I cannot distinguish it -- for otherwise it would be distinguished.


Substance is what exists independently of what is the case.


It is form and content.


Space, time and color (color ness) are forms of objects.


Only if there are objects can there be a fixed form of the world.


The fixed, the existent and the object are one.


The object is the fixed, the existent; the configuration is the changing, the variable.


The configuration of the objects forms the atomic fact.


In the atomic fact objects hang one in another, like the links of a chain.


In the atomic fact the objects are combined in a definite way.


The way in which objects hang together in the atomic fact is the structure of the atomic fact.


The form is the possibility of the structure.


The structure of the fact consists of the structures of the atomic facts.


The totality of existent atomic facts is the world.


The totality of existent atomic facts also determines which atomic facts do not exist.


The existence and non-existence of atomic facts is the reality.

(The existence of atomic facts we also call a positive fact, their non-existence a negative fact.)


Atomic facts are independent of one another.


From the existence of non-existence of an atomic fact we cannot infer the existence of non-existence of another.


The total reality is the world.


We make to ourselves pictures of facts.


The picture presents the facts in logical space, the existence and non-existence of atomic facts.


The picture is a model of reality.


To the objects correspond in the picture the elements of the picture.


The elements of the picture stand, in the picture, for the objects.


The picture consists in the fact that its elements are combined with one another in a definite way.


The picture is a fact.


That the elements of the picture are combined with one another in a definite way, represents that the things are so combined with one another.

This connection of the elements of the picture is called its structure, and the possibility of this structure is called the form of representation of the picture.


The form of representation is the possibility that the things are combined with one another as are the elements of the picture.


Thus the picture is linked with reality; it reaches up to it.


The picture, however, cannot represent its form of representation; it shows it forth.


The picture represents its object from without (its standpoint is its form of representation), therefore the picture represents its object rightly or falsely.


But the picture cannot place itself outside of its form of representation.


What every picture, of whatever form, must have in common with reality in order to be able to represent it at all -- rightly or falsely -- is the logical form, that is, the form of reality.


If the form of representation is the logical form, then the picture is called a logical picture.


Every picture is also a logical picture. (On the other hand, for example, not every picture is spatial.)


The logical picture can depict the world.

The picture has the logical form of representation in common with what it pictures.



The picture depicts reality by representing a possibility of the existence and non-existence of atomic facts.


The picture represents a possible state of affairs in logical space.


The picture contains the possibility of the state of affairs which it represents.


The picture agrees with reality or not; it is right or wrong, true or false.


The picture represents what it represents, independently of its truth or falsehood, through the form of representation.


What the picture represents is its sense.


In the agreement or disagreement of its sense with reality, its truth or falsity consists.


In order to discover whether the picture is true or false we must compare it with reality.


It cannot be discovered from the picture alone whether it is true or false.


There is no picture which is a priori true.

The logical picture of the facts is the thought




"An atomic fact is thinkable" -- means: we can imagine it.


The totality of true thoughts is a picture of the world.


The though contains the possibility of the state of affairs which it thinks.

What is thinkable is also possible.


We cannot think anything unlogical, for otherwise we should have to think unlogically.


It used to be said that God could create everything, except what was contrary to the laws of logic. The truth is, we could not say of an "unlogical" world how it would look.


To present in language anything which "contradicts logic" is as impossible as in geometry to present by its co-ordinates a figure which contradicts the laws of space; or to give the co-ordinates of a point which does not exist.


We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry.


An a priori true thought would be one whose possibility guaranteed its truth.


Only if we could know a priori that a thought is true if its truth was to be recognized from the thought itself (without an object of comparison).


In the proposition the thought is expressed perceptibly through the senses.


We use the sensibly perceptible sign (sound or written sign, etc.) of the proposition as a projection of the possible state of affairs.

The method of projection is the thinking of the sense of the proposition.


The sign through which we express the though I call the proposition sign. And the proposition is the proposition sign in its projective relation to the world.


To the proposition belongs everything which belongs to the projection; but not what is projected.

Therefore the possibility of what is projected but not this itself.

In the proposition, therefore, its sense is not yet contained, but the possibility of expressing it.

("The content of the proposition" means the content of the signicant proposition.)

In the proposition the form of its sense is contained, but not its content.


The propositional sign consists in the fact that its elements, the words, are combined in it in a definite way.

The propositional sign is a fact.


The proposition is not a mixture of words (just as the musical theme is not a mixture of tones).

The proposition is articulate.


Only facts can express a sense, a class of names cannot.


That the propositional sign is a fact is concealed by the ordinary form of expression, written or printed.

(For in the printed proposition, for example, the sign of a proposition does not appear essentially different from a word. Thus it was possible for Frege to call the proposition a compounded name.)


The essential nature of the propositional sign becomes very clean when we imagine it made up of spatial objects (such as tables, chairs, books) instead of written signs.

The mutual spatial position of these things then expresses the sense of the proposition.


We must not say, "The complex sign `aRb' says `a stands in relation R to b'"; but we must say, "That `a' stands in a certain relation to `b' says that aRb".


States of affairs can be described but not named.

(Names resemble points; propositions resemble arrows, they have senses.)


In propositions thoughts can be so expressed that to the objects of the thoughts correspond the elements of the propositional sign.



These elements I call "simple signs" and the proposition "completely analyzed".


The simple signs employed in propositions are called names.


The name means the object. The object is its meaning. ("A" is the same sign as "A".)


To the configuration of the simple signs in the propositional sign corresponds the configuration of the objects in the state of affairs.


In the proposition the name represents the object.


Objects I can only name. Signs represent them. I can only speak of them. I cannot assert them. A proposition can only say how a thing is, not what it is.


The postulate of the possibility of the simple signs is the postulate of the determinateness of the sense.


A proposition about a complex stands in internal relation to the proposition about its constituent part.

A complex can only be given by its description, and this will either be right or wrong. The proposition in which there is mention of a complex, if this does not exist, becomes not nonsense but simply false.

That a propositional element signifies a complex can be seen from indeterminateness in the propositions in which it occurs. We know that everything is not yet determined by this proposition. (The notation for generality contains a prototype.)

The combination of the symbols of a complex in a simple symbol can be expressed by a definition.


There is one and only one complete analysis of the proposition.


The proposition expresses what it expresses in a definite and clearly specifiable way: the proposition is articulate.


The name cannot be analyzed further by any definition. It is a primitive sign.


Every defined sign signifies via those signs by which it is defined, and the definitions show the way.

Two signs, one a primitive sign, and one defined by primitive signs, cannot signify in the same way. Names cannot be taken to pieces by definition (nor any sign which alone and independently has a meaning).


What does not get expressed in the sign is shown by its application. What the signs conceal, their application declares.


The meanings of primitive signs can be explained by elucidations. Elucidations are propositions which contain the primitive signs. They can, therefore, only be under stood when the meanings of these signs are already known.


Only the proposition has sense; only in the context of a proposition has a name meaning.


Every part of a proposition which characterizes its sense I call an expression (a symbol).

(The proposition itself is an expression.)

Expressions are everything -- essential for the sense of the proposition -- that propositions can have in common with one another.

An expression characterizes a form and a content.


An expression presupposes the forms of all propositions in which it can occur. It is the common characteristic mark of a class of propositions.


It is therefore represented by the general form of the propositions which it characterizes.

And in this form the expression is constant and everything else variable.


An expression is thus presented by a variable, whose values are the propositions which contain the expression.

(In the limiting case the variable becomes constant, the expression a proposition.)

I call such a variable a "propositional variable".


An expression has meaning only in a proposition. Every variable can be conceived as a propositional variable.

(Including the variable name.)


If we change a constituent part of a proposition into a variable, there is a class of propositions which are all the values of the resulting variable proposition. This class in general still depends on what, by arbitrary agreement, we mean by parts of that proposition. But if we change all those signs, whose meaning was arbitrarily determined, into variables, there always remains such a class. But this is now no longer dependent on any agreement; it depends only on the nature of the proposition. It corresponds to a logical form, to a logical prototype.


What values the propositional variable can assume is determined.

The determination of the values is the variable.


The determination of the values of the propositional variable is done by indicating the propositions whose common mark the variable is.

The determination is a description of these propositions.

The determination will therefore deal only with symbols not with their meaning.

And only this is essential to the determination that is only a description of symbols and asserts nothing about what is symbolized.

The way in which we describe the propositions is not essential.


I conceive the proposition -- like Frege and Russell -- as a function of the expressions contained in it.


The sign is the part of the symbol perceptible by the senses.


Two different symbols can therefore have the sign (the written sign or the sound sign) in common -- they then signify in different ways.


It can never indicate the common characteristic of two objects that we symbolize them with the same signs but by different methods of symbolizing. For the sign is arbitrary. We could therefore equally well choose two different signs and where then would be what was common in the symbolization.


In the language of everyday life it very often happens that the same word signifies in two different ways -- and therefore belongs to two different symbols -- or that two words, which signify in different ways, are apparently applied in the same way in the proposition.

Thus the word "is" appears as the copula, as the sign of equality, and as the expression of existence; "to exist" as an intransitive verb like "to go"; "identical" as an adjective; we speak of something but also of the fact of something happening.

(In the proposition "Green is green" -- where the first word is a proper name as the last an adjective -- these words have not merely different meanings but they are different symbols.)


Thus there easily arise the most fundamental confusions (of which the whole of philosophy is full).


In order to avoid these errors, we must employ a symbolism which excludes them, by not applying the same sign in different symbols and by not applying signs in the same way which signify in different ways. A symbolism, that is to say, which obeys the rules of logical grammar -- of logical syntax.

(The logical symbolism of Frege and Russell is such a language, which, however, does still not exclude all errors.)


In order to recognize the symbol in the sign we must consider the significant use.


The sign determines a logical form only together with its logical syntactic application.


If a sign is not necessary then it is meaningless. That is the meaning of Occam's razor.

(If everything in the symbolism works as though a sign had meaning, then it has meaning.)


In logical syntax the meaning of a sign ought never to play a role; it must admit of being established without mention being thereby made of the meaning of a sign; it ought to presuppose only the description of the expressions.


From this observation we get a further view -- into Russell's Theory of Types. Russell's error is shown by the fact that in drawing up his symbolic rules he has to speak about the things his signs mean.


No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the "whole theory of types").


A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself.

If, for example, we suppose that the function F(fx) could be its own argument, then there would be a proposition "F(F(fx))", and in this the outer functions F and the inner function F must have different meanings; for the inner has the form �psi�(fx), the outer the form �psi�( �phi�(fx)). Common to both functions is only the letter "F", which by itself signifies nothing.

This is at once clear, if instead of "F(F(u ))" we write

( �EXISTS��phi�):F( �phi�u). �phi�u=Fu".*

Herewith Russell's paradox vanishes.


The rules of logical syntax must follow of themselves, if we only know how every single sign signifies.


A proposition possesses essential and accidental features.

Accidental are the features which are due to a particular way of producing the propositional sign. Essential are those which alone enable the proposition to express its sense.


The essential in a proposition is therefore that which is common to all propositions which can express the same sense.

And in the same way in general the essential in a symbol is that which all symbols which can fulfill the same purpose have in common.


One could therefore say the real name is that which all symbols, which signify an object, have in common. It would then follow, step by step, that no sort of composition was essential for a name.


In our notations there is indeed something arbitrary, but this is not arbitrary, namely that if we have determined anything arbitrarily, then something else must be the case. (This results from the essence of the notation.)


A particular method of symbolizing may be unimportant, but it is always important that this is a possible method of symbolizing. And this happens as a rule in philosophy: The single thing proves over and over again to be unimportant, but the possibility of every single thing reveals something about the nature of the world.


Definitions are rules for the translation of one language into another. Every correct symbolism must be translatable into every other according to such rules. It is this which all has in common.


What signifies in the symbol is what is common to all those symbols by which it can be replaced according to the rules of logical syntax.


We can, for example, express what is common to all notations for the truth-functions as follows: It is common to them that they all, for example, can be replaced by the notations of "~p" ("not p") and "p v q" ("p or q").

(Herewith is indicated the way in which a special possible notation can give us general information.)


The sign of the complex is not arbitrarily resolved in the analysis, in such a way that its resolution would be different in every propositional structure.


The proposition determines a place in logical space: the existence of this logical place is guaranteed by the existence of the constituent parts alone, by the existence of the significant proposition.


The propositional sign and the logical co-ordinates: that is the logical place.

The ge

ometrical and the logical place agree in that each is the possibility of an existence.


Although a proposition may only determine one place in logical space, the whole logical space must already be given by it.

(Otherwise denial, the logical sum, the logical product, etc., would always introduce new elements -- in co-ordination.)

(The logical scaffolding round the picture determines the logical space. The proposition reaches through the whole logical space.


The applied, thought, propositional sign is the thought.


The thought is the significant proposition.



The totality of propositions is the language.


Man possesses the capacity of constructing languages, in which every sense can be expressed, without having an idea how and what each word means -- just as one speaks without knowing how the single sounds are produced.

Colloquial language is a part of the human organism and is not less complicated than it.

From it is humanly impossible to gather immediately the logical of language.

Language disguises the thought; so that from the external form of the clothes one cannot infer the form of the thought they clothe, because the external form of the clothes is constructed with quite another object than to let the form of the body be recognized.

The silent adjustments to understand colloquial language are enormously complicated.


Most propositions and questions that have been written about philosophical matters are not false, but senseless. We cannot, therefore, answer questions of this kind at all, but only state their senselessness. Most questions and propositions of t he philosophers result from the fact that we do not understand the logic of our language.

(They are of the same kind as the question whether the Good is more or less identical than the Beautiful.)

And so it is not to be wondered at that the deepest problems are really no problems.


All philosophy is "Critique of language" (but not at all in Mathner's sense). Russell's merit is to have shown that the apparent logical form of the proposition need not be its real form.


The proposition is a picture of reality.

The proposition is a model of the reality as we think it is.


At the first glance the proposition -- say as it stands printed on paper -- does not seem to be a picture of the reality of which it treats. But nor does the musical score appear at first sight to be a picture of a musical piece; nor does our phonetic spelling (letters) seem to be a picture of our spoken language. And yet these symbolisms prove to be pictures -- even in the ordinary sense of the word -- of what they represent.


It is obvious that we perceive a proposition of the form aRb as a picture. Here the sign is obviously a likeness of the signified.


And if we penetrate to the essence of this pictorial nature we see that his is not disturbed by apparent irregularities (like the use of �musical sharp�and �musical-flat�in the score).

For these irregularities also picture what they are to express; only in another way.


The gramophone record, the musical thought, the score, the waves of sound, all stand to one another in that pictorial internal relation, which holds between language and the world.

To all of them the logical structure is common.

(Like the two youths, their two horses and their lilies in the story. They are all in a certain sense one.)


In the fact that there is a general rule by which the musician is able to read the symphony out of the score, and that there is a rule by which one could reconstruct the symphony from the line on a gramophone record and from this again -- by means of the first rule -- construct the score, herein lies the internal similarity between these things which at first sight seem to be entirely different. And the rule is the law of projection which projects the symphony into the language of the musical score. It is the rule of translation of this language into the language of the gramophone record.


The possibility of all similes, of all the images of our language, rests on the logic of representation.


In order to understand the essence of the proposition, consider hieroglyphic writing, which pictures the facts it describes.

And from it came the alphabet without the essence of the representation being lost.


This we see from the fact that we understand the sense of the propositional sign, without having had it explained to us.


The proposition is a picture of reality, for I know the state of affaires presented by it, if I understand the proposition. And I understand the proposition, without its sense having been explained to me.


The proposition shows its sense.

The proposition shows how things stand, if it is true. And it says, that they do so stand.


The proposition determines reality to this extent that one only needs to say "Yes" or "No" to it to make it agree with reality.

Reality must therefore be completely described by the proposition.

A proposition is the description of a fact.

As the description of an object describes it by its external properties so propositions describe reality by its internal properties.

The proposition constructs a world with the help of a logical scaffolding, and therefore one can actually see in the proposition all the logical features possessed by reality if it is true. One can draw conclusions from a false proposition.


To understand a proposition means to know what is the case, if it is true.

(One can therefore understand it without knowing whether it is true or not.)

One understands it if one understands it constituent parts.


The translation of one language into another is not a process of translating each proposition of the one into a proposition of the other, but only the constituent parts of propositions are translated.

(And the dictionary does not only translate substantives but also adverbs and conjunctions, etc., and it treats them all alike.)


The meanings of the simple signs (the words) must be explained to us, if we are to understand them.

By means of propositions we explain ourselves.


It is essential to propositions, that they can communicate a new sense to us.


A proposition must communicate a new sense with old words.

The proposition communicates to us a state of affairs; therefore it must be essentially connected with the state of affairs.

And the connection is, in fact, that it is its logical picture.


In the proposition a state of affairs is, as it were, put together for the sake of experiment.

One can say, instead of, this proposition has such and such a sense, this proposition represents such and such a state of affairs.


The proposition is a picture of its state of affairs, only in so far as it is logically articulated.

(Even the proposition "ambulo" is composite, for its stem gives a different sense with another termination, or its termination with another stem.)


In the proposition there must be exactly as many things distinguishable as there are in the state of affairs, which it represents.

They must both possess the same logical (mathematical) multiplicity (cf. Hertz's Mechanics, on Dynamic Models).


This mathematical multiplicity naturally cannot in its turn be represented. One cannot get outside it in the representation.


Reality is compared with the proposition.


Propositions can be true or false only by being pictures of the reality.


If one does not observer that propositions have a sense independent of the facts, one can easily believe that true and false are two relations between signs and things signified with equal rights.

One could, then, for example, say that "p" signifies in the true way what "~p" signifies in the false way, etc.


Can we not make ourselves understood by means of false propositions as hitherto with true ones, so long as we know that they are meant to be false? No! For a proposition is true, if what we assert by means of it is the case; and if by "p" we mean ~p, and what we mean is the case, then "p" in the new conception is true and not false.


That, however, the signs "p" and "~p" can say the same thing is important, for it shows that the sign "~" corresponds to nothing in reality.

That negation occurs in a proposition, is no characteristic of its sense (~~p?=?p).

The propositions "p" and "~p" have opposite senses, but to them corresponds one and the same reality.


An illustration to explain the concept of truth. A black spot on white paper; the form of the spot can be described by saying of each point of the plane 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), a negative fact. If I indicate a point of the plane (a truth-value in Frege's terminology), this corresponds to the assumption proposed for judgment, etc. etc.

But to be able to say that a point is black or white, I must first know under what conditions a point is called white or black; in order to be able to say "p" is true (or false) I must have determined under what conditions I call "p" true, and thereby I determine the sense of the proposition.

The point at which the simile breaks down is this: we can indicate a point on the paper, without know what white and black are; but to a proposition without a sense corresponds nothing at all, for it signifies no thing (truth-value) whose properties are called "false" or "true"; the verb of the proposition is not "is true" or "is false" -- as Frege thought -- but that which "is true" must already contain the verb.


Every proposition must already have a sense; assertion cannot give it a sense, for what it asserts is the sense itself. And the same holds of denial, etc.


One could say, the denial is already related to the logical place determined by the proposition that is denied.

The denying proposition determines a logical place other than does the proposition denied.

The denying proposition determines a logical place, with the help of the logical place of the proposition denied, by saying that it lies outside the latter place.

That one can deny again the denied proposition shows that what is denied is already a proposition and not merely the preliminary to a proposition.

What can be shown cannot be said.

Now we understand our feeling that we are in possession of the right logical conception, if only all is right in our symbolism.

We can speak in a certain sense of formal properties of objects and atomic facts, or of properties of the structure of facts, and in the same sense of formal relations and relations of structures.

(Instead of property of the structure I also say "internal property"; instead of relation of structures "internal relation".

I introduce these expressions in order to show the reason for the confusion, very widespread among philosophers, between internal relations and proper (external) relations.)

The holding of such internal properties and relations cannot, however, be asserted by propositions, but it shows itself in the propositions, which present the facts and treat of the objects in question.

An internal property of a fact we also call a feature of this fact. (In the sense in which we speak of facial features.)

A property is internal if it is unthinkable that its object does not possess it.

(This bright blue color and that stand in the internal relation of bright and darker eo ipso. It is unthinkable that these two objects should not stand in this relation.)

(Here to the shifting use of the words "property" and "relation" there corresponds the shifting use of the word "object".)

The existence of an internal property of a possible state of affairs is not expressed by a proposition, but it expresses itself in the proposition which presents that state of affairs, by an intern al property of this proposition.

It would be as senseless to ascribe a formal property to a proposition as to deny it the formal property.

One cannot distinguish forms from one another by saying that one has this property, the other that: for this assumes that there is a sense in asserting either property of either form

The existence of an internal relation between possible states of affairs expresses itself in language by an internal relation between the propositions presenting them

In the sense in which we speak of formal properties we can now speak also of formal concepts

(I introduce this expression in order to make clear the confusion of formal concepts with proper concepts which runs through the whole of the old logic.)

That anything falls under a formal concept as an object belonging to it, cannot be expressed by a proposition. But it is shown in the symbol for the object itself. (The name shows that it signifies an object, the numerical sign that it signifies a number, etc.)

Formal concepts, cannot, like proper concepts, be presented by a function.

For their characteristics, the formal properties are not expressed by the functions.

The expression of a formal property is a feature of certain symbols.

The sign that signifies the characteristics of a formal concept is, therefore, a characteristic feature of all symbols, whose meanings fall under the concept.

The expression of the formal concept is therefore a propositional variable in which only this characteristic feature is constant.

The propositional variable signifies the formal concept, and its values signify the objects which fall under this concept

Every variable is the sign of a formal concept

For every variable presents a constant form, which all its values possess, and which can be conceived as a formal property of these values.

So the variable name "x" is the proper sign of the pseudo-concept object

Wherever the word "object" ("thing", "entity", etc.) is rightly used, it is expressed in logical symbolism by the variable name.

For example in the proposition "there are two objects which ?.?.?.", by "( �EXISTS�x,?y)?.?.?.".

Wherever it is used otherwise, i.e. as a proper concept word, there arise senseless pseudo-propositions.

So one cannot, e.g. say "There are objects" as one says "There are books". Nor "There are 100 objects" or "There are �ALEPH�0 objects".

And it is senseless to speak of the number of all objects.

The same holds of the words "Complex", "Fact", "Function", "Number", etc.

They all signify formal concepts and are presented in logical symbolism by variables, not by functions or classes (as Frege and Russell thought).

Expressions like "1 is a number", "there is only one number nought", and all like them are senseless.

(It is as senseless to say, "there is only one 1" as it would be to say: 2?+?2 is at 3 o'clock equal to 4.)

If we want to express in logical symbolism the general proposition "b is a successor of a" we need for this an expression for the general term of the formal series: aRb,
( �EXISTS�x)?:?aRx?.?xRb, ( �EXISTS�x,?y)?:?aRx?.?xRy?.?yRb,?.?.?. The general term of a formal series can only be expressed by a variable, for the concept symbolized by "term of this formal series" is a formal concept. (This Frege and Russell overlooked; the way in which they express general propositions like the above is, therefore, false; it contains a vicious circle.)

We can determine the general term of the formal series by giving its first term and the general form of the operation, which generates the following term out of the preceding proposition.

The question about the existence of a formal concept is senseless. For no proposition can answer such a question

(For example, one cannot ask: "Are there unanalyzable subject-predicate propositions?")

The logical forms are anumerical

Therefore there are in logic no pre-eminent numbers, and therefore there is no philosophical monism or dualism, etc.