Tractatus Logico-Philosophicus (tree-like view)

Tractatus Logico-Philosophicus

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Ludwig Wittgenstein

Tractatus Logico-Philosophicus (tree-like view)

Translated by F. P. Ramsey and edited by C. K. Ogden

The text of this interactive digital edition is based on Project Gutenberg's Tractatus Logico-Philosophicus, by Ludwig Wittgenstein, which was produced by Jana Srna, Norbert H. Langkau, and the Online Distributed Proofreading Team at pgdp.net; the Project Gutenberg's edition is, in turn, a reproduction of Ludwig Wittgenstein. Tractatus Logico-Philosophicus, translated by F. P. Ramsey and edited by C. K. Ogden, Kegan Paul, Trench, Trubner & Co., 1922. The original used a lower-case 'v' for the logical or operator; it has been replaced with the correct '∨' character. Every effort has been made to replicate the original text as faithfully as possible. The interactive presentation of this edition was designed by Michele Lavazza and reviewed by David Chandler for the Ludwig Wittgenstein Project. The original-language text is in the public domain in its country of origin and other countries and areas where the copyright term is the author's life plus 70 years or fewer. This translation is in the public domain in its country of origin and other countries and areas where the copyright term is the author's life plus 70 years or fewer. Additionally, both the original-language text and the translation are in the public domain in the United States, because they were published before 1 January 1929.

Ludwig Wittgenstein

Tractatus Logico-Philosophicus

Dedicated
to the memory of my friend
DAVID H. PINSENT

. . . und alles, was man weiss, nicht bloss rauschen und brausen gehört hat, lässt sich in drei Worten sagen.

Kürnberger.

Preface

This book will perhaps only be understood by those who have themselves already thought the thoughts which are expressed in it—or similar thoughts. It is therefore not a text-book. Its object would be attained if there were one person who read it with understanding and to whom it afforded pleasure.

The book deals with the problems of philosophy and shows, as I believe, that the method of formulating these problems rests on the misunderstanding of the logic of our language. Its whole meaning could be summed up somewhat as follows: What can be said at all can be said clearly; and whereof one cannot speak thereof one must be silent.

The book will, therefore, draw a limit to thinking, or rather—not to thinking, but to the expression of thoughts; for, in order to draw a limit to thinking we should have to be able to think both sides of this limit (we should therefore have to be able to think what cannot be thought).

The limit can, therefore, only be drawn in language and what lies on the other side of the limit will be simply nonsense.

How far my efforts agree with those of other philosophers I will not decide. Indeed what I have here written makes no claim to novelty in points of detail; and therefore I give no sources, because it is indifferent to me whether what I have thought has already been thought before me by another.

I will only mention that to the great works of Frege and the writings of my friend Bertrand Russell I owe in large measure the stimulation of my thoughts.

If this work has a value it consists in two things. First that in it thoughts are expressed, and this value will be the greater the better the thoughts are expressed. The more the nail has been hit on the head.—Here I am conscious that I have fallen far short of the possible. Simply because my powers are insufficient to cope with the task.—May others come and do it better.

On the other hand the truth of the thoughts communicated here seems to me unassailable and definitive. I am, therefore, of the opinion that the problems have in essentials been finally solved. And if I am not mistaken in this, then the value of this work secondly consists in the fact that it shows how little has been done when these problems have been solved.

L.W.

Vienna, 1918.

[Expand]1 The world is everything that is the case.^[1]

[Expand]2 What is the case, the fact, is the existence of atomic facts.

[Expand]2.01 An atomic fact is a combination of objects (entities, things).

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

[Expand]2.012 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.

[Expand]2.013 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.

[Expand]2.014 Objects contain the possibility of all states of affairs.

[Expand]2.02 The object is simple.

2.0201 Every statement about complexes can be analysed into a statement about their constituent parts, and into those propositions which completely describe the complexes.

[Expand]2.021 Objects form the substance of the world. Therefore they cannot be compound.

2.022 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.

[Expand]2.023 This fixed form consists of the objects.

2.024 Substance is what exists independently of what is the case.

[Expand]2.025 It is form and content.

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

[Expand]2.027 The fixed, the existent and the object are one.

[Expand]2.03 In the atomic fact objects hang one in another, like the links of a chain.

2.04 The totality of existent atomic facts is the world.

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

[Expand]2.06 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.)

[Expand]2.1 We make to ourselves pictures of facts.

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

2.12 The picture is a model of reality.

[Expand]2.13 To the objects correspond in the picture the elements of the picture.

[Expand]2.14 The picture consists in the fact that its elements are combined with one another in a definite way.

[Expand]2.15 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 connexion 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.

[Expand]2.16 In order to be a picture a fact must have something in common with what it pictures.

[Expand]2.17 What the picture must have in common with reality in order to be able to represent it after its manner—rightly or falsely—is its form of representation.

[Expand]2.18 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.

2.19 The logical picture can depict the world.

[Expand]2.2 The picture has the logical form of representation in common with what it pictures.

[Expand]3 The logical picture of the facts is the thought.

3.001 “An atomic fact is thinkable”—means: we can imagine it.

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

3.02 The thought contains the possibility of the state of affairs which it thinks.

What is thinkable is also possible.

[Expand]3.03 We cannot think anything unlogical, for otherwise we should have to think unlogically.

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

3.05 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).

[Expand]3.1 In the proposition the thought is expressed perceptibly through the senses.

3.11 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.

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

3.13 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 significant proposition.)

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

[Expand]3.14 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.

[Expand]3.2 In propositions thoughts can be so expressed that to the objects of the thoughts correspond the elements of the propositional sign.

3.201 These elements I call “simple signs” and the proposition “completely analysed”.

3.202 The simple signs employed in propositions are called names.

3.203 The name means the object. The object is its meaning. (“A” is the same sign as “A”.)

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

[Expand]3.22 In the proposition the name represents the object.

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

3.24 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 an 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.

[Expand]3.25 There is one and only one complete analysis of the proposition.

[Expand]3.26 The name cannot be analysed further by any definition. It is a primitive sign.

[Expand]3.3 Only the proposition has sense; only in the context of a proposition has a name meaning.

[Expand]3.31 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.

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

3.312 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.

3.313 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”.

3.314 An expression has meaning only in a proposition. Every variable can be conceived as a propositional variable. (Including the variable name.)

3.315 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.

3.316 What values the propositional variable can assume is determined.

The determination of the values is the variable.

3.317 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 it is only a description of symbols and asserts nothing about what is symbolized.

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

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

[Expand]3.32 The sign is the part of the symbol perceptible by the senses.

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

3.322 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.

3.323 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.)

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

3.325 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.)

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

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

3.328 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.)

[Expand]3.33 In logical syntax the meaning of a sign ought never to play a rôle; 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.

[Expand]3.34 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.

[Expand]3.4 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.

3.5 The applied, thought, propositional sign, is the thought.

[Expand]4 The thought is the significant proposition.

4.001 The totality of propositions is the language.

4.002 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 it is humanly impossible to gather immediately the logic 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.

[Expand]4.003 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 the 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.

[Expand]4.01 The proposition is a picture of reality.

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

4.011 At the first glance the proposition—say as it stands printed on paper—does not seem to be 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.

4.012 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.

4.013 And if we penetrate to the essence of this pictorial nature we see that this is not disturbed by apparent irregularities (like the use of ♯ and ♭ in the score).

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

[Expand]4.014 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.)

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

4.016 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.

[Expand]4.02 This we see from the fact that we understand the sense of the propositional sign, without having had it explained to us.

[Expand]4.03 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 connexion is, in fact, that it is its logical picture.

The proposition only asserts something, in so far as it is a picture.

[Expand]4.04 In the proposition there must be exactly as many thing 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).

4.05 Reality is compared with the proposition.

[Expand]4.06 Propositions can be true or false only by being pictures of the reality.

4.061 If one does not observe 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.

[Expand]4.062 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.

4.063 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 knowing 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.

[Expand]4.064 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.

[Expand]4.1 A proposition presents the existence and non-existence of atomic facts.

[Expand]4.11 The totality of true propositions is the total natural science (or the totality of the natural sciences).

[Expand]4.12 Propositions can represent the whole reality, but they cannot represent what they must have in common with reality in order to be able to represent it—the logical form.

To be able to represent the logical form, we should have to be able to put ourselves with the propositions outside logic, that is outside the world.

[Expand]4.121 Propositions cannot represent the logical form: this mirrors itself in the propositions.

That which mirrors itself in language, language cannot represent.

That which expresses itself in language, we cannot express by language.

The propositions show the logical form of reality.

They exhibit it.

[Expand]4.122 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.

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

(This blue colour and that stand in the internal relation of brighter 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”.)

[Expand]4.124 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 internal property of this proposition.

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

[Expand]4.125 The existence of an internal relation between possible states of affairs expresses itself in language by an internal relation between the propositions presenting them.

4.126 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.

[Expand]4.127 The propositional variable signifies the formal concept, and its values signify the objects which fall under this concept.

4.1271 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.

[Expand]4.1272 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 “(∃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 ℵ₀ 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.)

4.1273 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, (∃x) : aRx . xRb, (∃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.

4.1274 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 unanalysable subject-predicate propositions?”)

4.128 The logical forms are anumerical.

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

[Expand]4.2 The sense of a proposition is its agreement and disagreement with the possibilities of the existence and non-existence of the atomic facts.

[Expand]4.21 The simplest proposition, the elementary proposition, asserts the existence of an atomic fact.

[Expand]4.22 The elementary proposition consists of names. It is a connexion, a concatenation, of names.

4.23 The name occurs in the proposition only in the context of the elementary proposition.

[Expand]4.24 The names are the simple symbols, I indicate them by single letters (x, y, z).

The elementary proposition I write as function of the names, in the form “fx”, “ϕ(x, y)”, etc.

Or I indicate it by the letters p, q, r.

4.25 If the elementary proposition is true, the atomic fact exists; if it is false the atomic fact does not exist.

4.26 The specification of all true elementary propositions describes the world completely. The world is completely described by the specification of all elementary propositions plus the specification, which of them are true and which false.

4.27 With regard to the existence of n atomic facts there are [math]\displaystyle{ K_n = \sum_{\nu=0}^n \binom{n}{\nu} }[/math] possibilities.

It is possible for all combinations of atomic facts to exist, and the others not to exist.

4.28 To these combinations correspond the same number of possibilities of the truth—and falsehood—of n elementary propositions.

[Expand]4.3 The truth-possibilities of the elementary propositions mean the possibilities of the existence and non-existence of the atomic facts.

[Expand]4.4 A proposition is the expression of agreement and disagreement with the truth-possibilities of the elementary propositions.

[Expand]4.41 The truth-possibilities of the elementary propositions are the conditions of the truth and falsehood of the propositions.

4.42 With regard to the agreement and disagreement of a proposition with the truth-possibilities of n elementary propositions there are [math]\displaystyle{ \sum_{\kappa=0}^{K_n} \binom{K_n}{\kappa} = L_n }[/math] possibilities.

[Expand]4.43 Agreement with the truth-possibilities can be expressed by co-ordinating with them in the schema the mark “T” (true).

Absence of this mark means disagreement.

[Expand]4.44 The sign which arises from the co-ordination of that mark “T” with the truth-possibilities is a propositional sign.

4.45 For n elementary propositions there are L_n possible groups of truth-conditions.

The groups of truth-conditions which belong to the truth-possibilities of a number of elementary propositions can be ordered in a series.

[Expand]4.46 Among the possible groups of truth-conditions there are two extreme cases.

In the one case the proposition is true for all the truth-possibilities of the elementary propositions. We say that the truth-conditions are tautological.

In the second case the proposition is false for all the truth-possibilities. The truth-conditions are self-contradictory.

In the first case we call the proposition a tautology, in the second case a contradiction.

[Expand]4.461 The proposition shows what it says, the tautology and the contradiction that they say nothing.

The tautology has no truth-conditions, for it is unconditionally true; and the contradiction is on no condition true.

Tautology and contradiction are without sense.

(Like the point from which two arrows go out in opposite directions.)

(I know, e.g. nothing about the weather, when I know that it rains or does not rain.)

4.462 Tautology and contradiction are not pictures of the reality. They present no possible state of affairs. For the one allows every possible state of affairs, the other none.

In the tautology the conditions of agreement with the world—the presenting relations—cancel one another, so that it stands in no presenting relation to reality.

4.463 The truth-conditions determine the range, which is left to the facts by the proposition.

(The proposition, the picture, the model, are in a negative sense like a solid body, which restricts the free movement of another: in a positive sense, like the space limited by solid substance, in which a body may be placed.)

Tautology leaves to reality the whole infinite logical space; contradiction fills the whole logical space and leaves no point to reality. Neither of them, therefore, can in any way determine reality.

4.464 The truth of tautology is certain, of propositions possible, of contradiction impossible.

(Certain, possible, impossible: here we have an indication of that gradation which we need in the theory of probability.)

4.465 The logical product of a tautology and a proposition says the same as the proposition. Therefore that product is identical with the proposition. For the essence of the symbol cannot be altered without altering its sense.

[Expand]4.466 To a definite logical combination of signs corresponds a definite logical combination of their meanings; every arbitrary combination only corresponds to the unconnected signs.

That is, propositions which are true for every state of affairs cannot be combinations of signs at all, for otherwise there could only correspond to them definite combinations of objects.

(And to no logical combination corresponds no combination of the objects.)

Tautology and contradiction are the limiting cases of the combinations of symbols, namely their dissolution.

[Expand]4.5 Now it appears to be possible to give the most general form of proposition; i.e. to give a description of the propositions of some one sign language, so that every possible sense can be expressed by a symbol, which falls under the description, and so that every symbol which falls under the description can express a sense, if the meanings of the names are chosen accordingly.

It is clear that in the description of the most general form of proposition only what is essential to it may be described—otherwise it would not be the most general form.

That there is a general form is proved by the fact that there cannot be a proposition whose form could not have been foreseen (i.e. constructed). The general form of proposition is: Such and such is the case.

[Expand]5 Propositions are truth-functions of elementary propositions.

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

5.01 The elementary propositions are the truth-arguments of propositions.

5.02 It is natural to confuse the arguments of functions with the indices of names. For I recognize the meaning of the sign containing it from the argument just as much as from the index.

In Russell’s “+_c”, for example, “c” is an index which indicates that the whole sign is the addition sign for cardinal numbers. But this way of symbolizing depends on arbitrary agreement, and one could choose a simple sign instead of “+_c”: but in “~p” “p” is not an index but an argument; the sense of “~p” cannot be understood, unless the sense of “p” has previously been understood. (In the name Julius Caesar, Julius is an index. The index is always part of a description of the object to whose name we attach it, e.g. The Caesar of the Julian gens.)

The confusion of argument and index is, if I am not mistaken, at the root of Frege’s theory of the meaning of propositions and functions. For Frege the propositions of logic were names and their arguments the indices of these names.

[Expand]5.1 The truth-functions can be ordered in series.

That is the foundation of the theory of probability.

5.101 The truth-functions of every number of elementary propositions can be written in a schema of the following kind:

(TTTT)(p, q)	Tautology	(if p then p, and if q then q.) [p ⊃ p . q ⊃ q]
(FTTT)(p, q)	in words:	Not both p and q. [~(p . q)]
(TFTT)(p, q)	” ”	If q then p. [q ⊃ p]
(TTFT)(p, q)	” ”	If p then q. [p ⊃ q]
(TTTF)(p, q)	” ”	p or q. [p ∨ q]
(FFTT)(p, q)	” ”	Not q. ~q
(FTFT)(p, q)	” ”	Not p. ~p
(FTTF)(p, q)	” ”	p or q, but not both. [p . ~q : ∨ : q . ~p]
(TFFT)(p, q)	” ”	If p, then q; and if q, then p. [p ≡ q]
(TFTF)(p, q)	” ”	p
(TTFF)(p, q)	” ”	q
(FFFT)(p, q)	” ”	Neither p nor q. [~p . ~q or p \| q]
(FFTF)(p, q)	” ”	p and not q. [p . ~q]
(FTFF)(p, q)	” ”	q and not p. [q . ~p]
(TFFF)(p, q)	” ”	q and p. [q . p]
(FFFF)(p, q)	Contradiction	(p and not p; and q and not q.) [p . ~p . q . ~q]

Those truth-possibilities of its truth-arguments, which verify the proposition, I shall call its truth-grounds.

5.11 If the truth-grounds which are common to a number of propositions are all also truth-grounds of some one proposition, we say that the truth of this proposition follows from the truth of those propositions.

[Expand]5.12 In particular the truth of a proposition p follows from that of a proposition q, if all the truth-grounds of the second are truth-grounds of the first.

[Expand]5.13 That the truth of one proposition follows from the truth of other propositions, we perceive from the structure of the propositions.

[Expand]5.131 If the truth of one proposition follows from the truth of others, this expresses itself in relations in which the forms of these propositions stand to one another, and we do not need to put them in these relations first by connecting them with one another in a proposition; for these relations are internal, and exist as soon as, and by the very fact that, the propositions exist.

5.132 If p follows from q, I can conclude from q to p; infer p from q.

The method of inference is to be understood from the two propositions alone.

Only they themselves can justify the inference.

Laws of inference, which—as in Frege and Russell—are to justify the conclusions, are senseless and would be superfluous.

5.133 All inference takes place a priori.

5.134 From an elementary proposition no other can be inferred.

5.135 In no way can an inference be made from the existence of one state of affairs to the existence of another entirely different from it.

[Expand]5.136 There is no causal nexus which justifies such an inference.

[Expand]5.14 If a proposition follows from another, then the latter says more than the former, the former less than the latter.

[Expand]5.15 If T_r is the number of the truth-grounds of the proposition “r”, T_rs the number of those truth-grounds of the proposition “s” which are at the same time truth-grounds of “r”, then we call the ratio T_rs : T_r the measure of the probability which the proposition “r” gives to the proposition “s”.

[Expand]5.151 Suppose in a schema like that above in No. 5.101 T_r is the number of the “T”’s in the proposition r, T_rs the number of those “T”’s in the proposition s, which stand in the same columns as “T”’s of the proposition r; then the proposition r gives to the proposition s the probability T_rs : T_r.

5.152 Propositions which have no truth-arguments in common with one another we call independent.

Two elementary propositions give to one another the probability ½.

If p follows from q, the proposition q gives to the proposition p the probability 1. The certainty of logical conclusion is a limiting case of probability.

(Application to tautology and contradiction.)

5.153 A proposition is in itself neither probable nor improbable. An event occurs or does not occur, there is no middle course.

5.154 In an urn there are equal numbers of white and black balls (and no others). I draw one ball after another and put them back in the urn. Then I can determine by the experiment that the numbers of the black and white balls which are drawn approximate as the drawing continues.

So this is not a mathematical fact.

If then, I say, It is equally probable that I should draw a white and a black ball, this means, All the circumstances known to me (including the natural laws hypothetically assumed) give to the occurrence of the one event no more probability than to the occurrence of the other. That is they give—as can easily be understood from the above explanations—to each the probability ½.

What I can verify by the experiment is that the occurrence of the two events is independent of the circumstances with which I have no closer acquaintance.

5.155 The unit of the probability proposition is: The circumstances—with which I am not further acquainted—give to the occurrence of a definite event such and such a degree of probability.

5.156 Probability is a generalization.

It involves a general description of a propositional form. Only in default of certainty do we need probability.

If we are not completely acquainted with a fact, but know something about its form.

(A proposition can, indeed, be an incomplete picture of a certain state of affairs, but it is always a complete picture.)

The probability proposition is, as it were, an extract from other propositions.

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

5.21 We can bring out these internal relations in our manner of expression, by presenting a proposition as the result of an operation which produces it from other propositions (the bases of the operation).

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

[Expand]5.23 The operation is that which must happen to a proposition in order to make another out of it.

[Expand]5.24 An operation shows itself in a variable; it shows how we can proceed from one form of proposition to another.

It gives expression to the difference between the forms.

(And that which is common to the bases, and the result of an operation, is the bases themselves.)

[Expand]5.25 The occurrence of an operation does not characterize the sense of a proposition.

For an operation does not assert anything; only its result does, and this depends on the bases of the operation.

(Operation and function must not be confused with one another.)

[Expand]5.3 All propositions are results of truth-operations on the elementary propositions.

The truth-operation is the way in which a truth-function arises from elementary propositions.

According to the nature of truth-operations, in the same way as out of elementary propositions arise their truth-functions, from truth-functions arises a new one. Every truth-operation creates from truth-functions of elementary propositions another truth-function of elementary propositions, i.e., a proposition. The result of every truth-operation on the results of truth-operations on elementary propositions is also the result of one truth-operation on elementary propositions.

Every proposition is the result of truth-operations on elementary propositions.

[Expand]5.4 Here it becomes clear that there are no such things as “logical objects” or “logical constants” (in the sense of Frege and Russell).

5.41 For all those results of truth-operations on truth-functions are identical, which are one and the same truth-function of elementary propositions.

5.42 That ∨, ⊃ etc., are not relations in the sense of right and left, etc., is obvious.

The possibility of crosswise definition of the logical “primitive signs” of Frege and Russell shows by itself that these are not primitive signs and that they signify no relations.

And it is obvious that the “⊃” which we define by means of “~” and “∨” is identical with that by which we define “∨” with the help of “~”, and that this “∨” is the same as the first, and so on.

5.43 That from a fact p an infinite number of others should follow, namely ~~p, ~~~~p etc., is indeed hardly to be believed, and it is no less wonderful that the infinite number of propositions of logic (of mathematics) should follow from half a dozen “primitive propositions”.

But all propositions of logic say the same thing. That is, nothing.

[Expand]5.44 Truth-functions are not material functions.

If e.g. an affirmation can be produced by repeated denial, is the denial—in any sense—contained in the affirmation?

Does “~~p” deny ~p, or does it affirm p; or both?

The proposition “~~p” does not treat of denial as an object, but the possibility of denial is already prejudged in affirmation.

And if there was an object called “~”, then “~~p” would have to say something other than “p”. For the one proposition would then treat of ~, the other would not.

[Expand]5.45 If there are logical primitive signs a correct logic must make clear their position relative to one another and justify their existence. The construction of logic out of its primitive signs must become clear.

5.451 If logic has primitive ideas these must be independent of one another. If a primitive idea is introduced it must be introduced in all contexts in which it occurs at all. One cannot therefore introduce it for one context and then again for another. For example, if denial is introduced, we must understand it in propositions of the form “~p” just as in propositions like “~(p ∨ q)”, “(∃x) . ~fx” and others. We may not first introduce it for one class of cases and then for another, for it would then remain doubtful whether its meaning in the two cases was the same, and there would be no reason to use the same way of symbolizing in the two cases.

(In short, what Frege (“Grundgesetze der Arithmetik”) has said about the introduction of signs by definitions holds, mutatis mutandis, for the introduction of primitive signs also.)

5.452 The introduction of a new expedient in the symbolism of logic must always be an event full of consequences. No new symbol may be introduced in logic in brackets or in the margin—with, so to speak, an entirely innocent face.

(Thus in the “Principia Mathematica” of Russell and Whitehead there occur definitions and primitive propositions in words. Why suddenly words here? This would need a justification. There was none, and can be none for the process is actually not allowed.)

But if the introduction of a new expedient has proved necessary in one place, we must immediately ask: Where is this expedient always to be used? Its position in logic must be made clear.

5.453 All numbers in logic must be capable of justification.

Or rather it must become plain that there are no numbers in logic.

There are no pre-eminent numbers.

[Expand]5.454 In logic there is no side by side, there can be no classification.

In logic there cannot be a more general and a more special.

[Expand]5.46 When we have rightly introduced the logical signs, the sense of all their combinations has been already introduced with them: therefore not only “p ∨ q” but also “~(p ∨ ~q)”, etc. etc. We should then already have introduced the effect of all possible combinations of brackets; and it would then have become clear that the proper general primitive signs are not “p ∨ q”, “(∃x) . fx”, etc., but the most general form of their combinations.

[Expand]5.47 It is clear that everything which can be said beforehand about the form of all propositions at all can be said on one occasion.

For all logical operations are already contained in the elementary proposition. For “fa” says the same as “(∃x) . fx . x = a”.

Where there is composition, there is argument and function, and where these are, all logical constants already are.

One could say: the one logical constant is that which all propositions, according to their nature, have in common with one another.

That however is the general form of proposition.

[Expand]5.471 The general form of proposition is the essence of proposition.

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

[Expand]5.473 Logic must take care of itself.

A possible sign must also be able to signify. Everything which is possible in logic is also permitted. (“Socrates is identical” means nothing because there is no property which is called “identical”. The proposition is senseless because we have not made some arbitrary determination, not because the symbol is in itself unpermissible.)

In a certain sense we cannot make mistakes in logic.

5.474 The number of necessary fundamental operations depends only on our notation.

5.475 It is only a question of constructing a system of signs of a definite number of dimensions—of a definite mathematical multiplicity.

5.476 It is clear that we are not concerned here with a number of primitive ideas which must be signified but with the expression of a rule.

[Expand]5.5 Every truth-function is a result of the successive application of the operation (– – – – – T)(ξ, ....) to elementary propositions.

This operation denies all the propositions in the right-hand bracket and I call it the negation of these propositions.

5.501 An expression in brackets whose terms are propositions I indicate—if the order of the terms in the bracket is indifferent—by a sign of the form “[math]\displaystyle{ ( \bar{\xi} ) }[/math]”. “ξ” is a variable whose values are the terms of the expression in brackets, and the line over the variable indicates that it stands for all its values in the bracket.

(Thus if ξ has the 3 values P, Q, R, then [math]\displaystyle{ ( \bar{\xi} ) }[/math] = (P, Q, R).)

The values of the variables must be determined.

The determination is the description of the propositions which the variable stands for.

How the description of the terms of the expression in brackets takes place is unessential.

We may distinguish 3 kinds of description: Direct enumeration. In this case we can place simply its constant values instead of the variable. Giving a function fx whose values for all values of x are the propositions to be described. Giving a formal law, according to which those propositions are constructed. In this case the terms of the expression in brackets are all the terms of a formal series.

5.502 Therefore I write instead of “(– – – – – T)(ξ, ....)”, “[math]\displaystyle{ N ( \bar{\xi} ) }[/math]”.

[math]\displaystyle{ N ( \bar{\xi} ) }[/math] is the negation of all the values of the propositional variable ξ.

5.503 As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.

[Expand]5.51 If ξ has only one value, then [math]\displaystyle{ N ( \bar{\xi} ) }[/math] = ~p (not p), if it has two values then [math]\displaystyle{ N ( \bar{\xi} ) }[/math] = ~p . ~q (neither p nor q).

5.511 How can the all-embracing logic which mirrors the world use such special catches and manipulations? Only because all these are connected into an infinitely fine network, to the great mirror.

5.512 “~p” is true if “p” is false. Therefore in the true proposition “~p” “p” is a false proposition. How then can the stroke “~” bring it into agreement with reality?

That which denies in “~p” is however not “~” but that which all signs of this notation, which deny p, have in common.

Hence the common rule according to which “~p”, “~~~p”, “~p ∨ ~p”, “~p . ~p”, etc. etc. (to infinity) are constructed. And this which is common to them all mirrors denial.

5.513 We could say: What is common to all symbols, which assert both p and q, is the proposition “p . q”. What is common to all symbols, which assert either p or q, is the proposition “p ∨ q”.

And similarly we can say: Two propositions are opposed to one another when they have nothing in common with one another; and every proposition has only one negative, because there is only one proposition which lies altogether outside it.

Thus in Russell’s notation also it appears evident that “q : p ∨ ~p” says the same as “q”; that “p ∨ ~p” says nothing.

5.514 If a notation is fixed, there is in it a rule according to which all the propositions denying p are constructed, a rule according to which all the propositions asserting p are constructed, a rule according to which all the propositions asserting p or q are constructed, and so on. These rules are equivalent to the symbols and in them their sense is mirrored.

[Expand]5.515 It must be recognized in our symbols that what is connected by “∨”, “.”, etc., must be propositions.

And this is the case, for the symbols “p” and “q” presuppose “∨”, “~”, etc. If the sign “p” in “p ∨ q” does not stand for a complex sign, then by itself it cannot have sense; but then also the signs “p ∨ p”, “p . p” etc. which have the same sense as “p” have no sense. If, however, “p ∨ p” has no sense, then also “p ∨ q” can have no sense.

[Expand]5.52 If the values of ξ are the total values of a function fx for all values of x, then [math]\displaystyle{ N ( \bar{\xi} ) }[/math] = ~(∃x) . fx.

5.521 I separate the concept all from the truth-function.

Frege and Russell have introduced generality in connexion with the logical product or the logical sum. Then it would be difficult to understand the propositions “(∃x) . fx” and “(x) . fx” in which both ideas lie concealed.

5.522 That which is peculiar to the “symbolism of generality” is firstly, that it refers to a logical prototype, and secondly, that it makes constants prominent.

5.523 The generality symbol occurs as an argument.

5.524 If the objects are given, therewith are all objects also given.

If the elementary propositions are given, then therewith all elementary propositions are also given.

5.525 It is not correct to render the proposition “(∃x) . fx”—as Russell does—in words “fx is possible”.

Certainty, possibility or impossibility of a state of affairs are not expressed by a proposition but by the fact that an expression is a tautology, a significant proposition or a contradiction.

That precedent to which one would always appeal, must be present in the symbol itself.

[Expand]5.526 One can describe the world completely by completely generalized propositions, i.e., without from the outset co-ordinating any name with a definite object.

In order then to arrive at the customary way of expression we need simply say after an expression “there is one and only one x, which ....”: and this x is a.

[Expand]5.53 Identity of the object I express by identity of the sign and not by means of a sign of identity. Difference of the objects by difference of the signs.

5.5301 That identity is not a relation between objects is obvious. This becomes very clear if, for example, one considers the proposition “(x) : fx . ⊃ . x = a”. What this proposition says is simply that only a satisfies the function f, and not that only such things satisfy the function f which have a certain relation to a.

One could of course say that in fact only a has this relation to a but in order to express this we should need the sign of identity itself.

5.5302 Russell’s definition of “=” won’t do; because according to it one cannot say that two objects have all their properties in common. (Even if this proposition is never true, it is nevertheless significant.)

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

5.531 I write therefore not “f(a, b) . a = b” but “f(a, a)” (or “f(b, b)”). And not “f(a, b) . ~a = b” but “f(a, b)”.

[Expand]5.532 And analogously: not “(∃x, y) . f(x, y) . x = y”, but “(∃x) . f(x, x)”; and not “(∃x, y) . f(x, y) . ~x = y” but “(∃x, y) . f(x, y)”.

(Therefore instead of Russell’s “(∃x, y) . f(x, y)”: “(∃x, y) . f(x, y) . ∨ . (∃x) . f(x, x)”.)

5.533 The identity sign is therefore not an essential constituent of logical notation.

5.534 And we see that apparent propositions like: “a = a”, “a = b . b = c . ⊃ a = c”, “(x) . x = x”, “(∃x) . x = a”, etc. cannot be written in a correct logical notation at all.

[Expand]5.535 So all problems disappear which are connected with such pseudo-propositions.

This is the place to solve all the problems which arise through Russell’s “Axiom of Infinity”.

What the axiom of infinity is meant to say would be expressed in language by the fact that there is an infinite number of names with different meanings.

[Expand]5.54 In the general propositional form, propositions occur in a proposition only as bases of the truth-operations.

[Expand]5.55 We must now answer a priori the question as to all possible forms of the elementary propositions.

The elementary proposition consists of names. Since we cannot give the number of names with different meanings, we cannot give the composition of the elementary proposition.

5.551 Our fundamental principle is that every question which can be decided at all by logic can be decided off-hand.

(And if we get into a situation where we need to answer such a problem by looking at the world, this shows that we are on a fundamentally wrong track.)

[Expand]5.552 The “experience” which we need to understand logic is not that such and such is the case, but that something is; but that is no experience.

Logic precedes every experience—that something is so.

It is before the How, not before the What.

5.553 Russell said that there were simple relations between different numbers of things (individuals). But between what numbers? And how should this be decided—by experience?

(There is no pre-eminent number.)

[Expand]5.554 The enumeration of any special forms would be entirely arbitrary.

5.555 It is clear that we have a concept of the elementary proposition apart from its special logical form.

Where, however, we can build symbols according to a system, there this system is the logically important thing and not the single symbols.

And how would it be possible that I should have to deal with forms in logic which I can invent: but I must have to deal with that which makes it possible for me to invent them.

[Expand]5.556 There cannot be a hierarchy of the forms of the elementary propositions. Only that which we ourselves construct can we foresee.

[Expand]5.557 The application of logic decides what elementary propositions there are.

What lies in its application logic cannot anticipate.

It is clear that logic may not conflict with its application.

But logic must have contact with its application.

Therefore logic and its application may not overlap one another.

[Expand]5.6 The limits of my language mean the limits of my world.

5.61 Logic fills the world: the limits of the world are also its limits.

We cannot therefore say in logic: This and this there is in the world, that there is not.

For that would apparently presuppose that we exclude certain possibilities, and this cannot be the case since otherwise logic must get outside the limits of the world: that is, if it could consider these limits from the other side also.

What we cannot think, that we cannot think: we cannot therefore say what we cannot think.

[Expand]5.62 This remark provides a key to the question, to what extent solipsism is a truth.

In fact what solipsism means, is quite correct, only it cannot be said, but it shows itself.

That the world is my world, shows itself in the fact that the limits of the language (the language which I understand) mean the limits of my world.

[Expand]5.63 I am my world. (The microcosm.)

[Expand]5.64 Here we see that solipsism strictly carried out coincides with pure realism. The I in solipsism shrinks to an extensionless point and there remains the reality co-ordinated with it.

[Expand]6 The general form of truth function is: [math]\displaystyle{ [ \bar{p}, \bar{\xi}, N (\bar{\xi}) ] }[/math].

This is the general form of proposition.

6.001 This says nothing else than that every proposition is the result of successive applications of the operation [math]\displaystyle{ N (\bar{\xi}) }[/math] to the elementary propositions.

6.002 If we are given the general form of the way in which a proposition is constructed, then thereby we are also given the general form of the way in which by an operation out of one proposition another can be created.

6.01 The general form of the operation [math]\displaystyle{ \Omega ' (\bar{\eta}) }[/math] is therefore: [math]\displaystyle{ [\bar{\xi}, N(\bar{\xi})]' (\bar{\eta}) (= [ \bar{\eta}, \bar{\xi}, N (\bar{\xi}) ]) }[/math].

This is the most general form of transition from one proposition to another.

[Expand]6.02 And thus we come to numbers: I define

[math]\displaystyle{ x = \Omega^{0 \prime} x \text{ Def.} }[/math] and
[math]\displaystyle{ \Omega^{\prime} \Omega^{ \nu \prime} x = \Omega^{ \nu + 1 \prime} x \text{ Def.} }[/math]

According, then, to these symbolic rules we write the series [math]\displaystyle{ x, \Omega ' x, \Omega ' \Omega ' x, \Omega ' \Omega ' \Omega ' x, ..... }[/math]

as: [math]\displaystyle{ \Omega^{0 \prime} x, \Omega^{0+1 \prime} x, \Omega^{0 + 1 + 1 \prime} x, \Omega^{0 + 1 + 1 + 1 \prime} x, ..... }[/math]

Therefore I write in place of “[math]\displaystyle{ [ x, \xi, \Omega ' \xi ] }[/math]”,

“[math]\displaystyle{ [ \Omega^{0 \prime} x, \Omega^{ \nu \prime} x, \Omega^{ \nu + 1 \prime} x ] }[/math]”.

And I define:

[math]\displaystyle{ 0 + 1 = 1 \text{ Def.} }[/math]

[math]\displaystyle{ 0 + 1 + 1 = 2 \text{ Def.} }[/math]

[math]\displaystyle{ 0 + 1 + 1 + 1 = 3 \text{ Def.} }[/math]

(and so on.)

[Expand]6.03 The general form of the cardinal number is: [0, ξ, ξ + 1].

[Expand]6.1 The propositions of logic are tautologies.

[Expand]6.11 The propositions of logic therefore say nothing. (They are the analytical propositions.)

[Expand]6.12 The fact that the propositions of logic are tautologies shows the formal—logical—properties of language, of the world.

That its constituent parts connected together in this way give a tautology characterizes the logic of its constituent parts.

In order that propositions connected together in a definite way may give a tautology they must have definite properties of structure. That they give a tautology when so connected shows therefore that they possess these properties of structure.

6.1201 That e.g. the propositions “p” and “~p” in the connexion “~(p . ~p)” give a tautology shows that they contradict one another. That the propositions “p ⊃ q”, “p” and “q” connected together in the form “(p ⊃ q) . (p) : ⊃ : (q)” give a tautology shows that “q” follows from “p” and “p ⊃ q”. That “(x) . fx : ⊃ : fa” is a tautology shows that fa follows from (x) . fx, etc. etc.

6.1202 It is clear that we could have used for this purpose contradictions instead of tautologies.

6.1203 In order to recognize a tautology as such, we can, in cases in which no sign of generality occurs in the tautology, make use of the following intuitive method: I write instead of “p”, “q”, “r”, etc., “T p F”, “T q F”, “T r F”, etc. The truth-combinations I express by brackets, e.g.:

and the co-ordination of the truth or falsity of the whole proposition with the truth-combinations of the truth-arguments by lines in the following way:

This sign, for example, would therefore present the proposition “p ⊃ q”. Now I will proceed to inquire whether such a proposition as ~(p . ~p) (The Law of Contradiction) is a tautology. The form “~ ξ” is written in our notation

the form “ξ . η” thus:—

Hence the proposition ~(p . ~q) runs thus:—

If here we put “p” instead of “q” and examine the combination of the outermost T and F with the innermost, it is seen that the truth of the whole proposition is co-ordinated with all the truth-combinations of its argument, its falsity with none of the truth-combinations.

6.121 The propositions of logic demonstrate the logical properties of propositions, by combining them into propositions which say nothing.

This method could be called a zero-method. In a logical proposition propositions are brought into equilibrium with one another, and the state of equilibrium then shows how these propositions must be logically constructed.

[Expand]6.122 Whence it follows that we can get on without logical propositions, for we can recognize in an adequate notation the formal properties of the propositions by mere inspection.

[Expand]6.123 It is clear that the laws of logic cannot themselves obey further logical laws.

(There is not, as Russell supposed, for every “type” a special law of contradiction; but one is sufficient, since it is not applied to itself.)

6.124 The logical propositions describe the scaffolding of the world, or rather they present it. They “treat” of nothing. They presuppose that names have meaning, and that elementary propositions have sense. And this is their connexion with the world. It is clear that it must show something about the world that certain combinations of symbols—which essentially have a definite character—are tautologies. Herein lies the decisive point. We said that in the symbols which we use something is arbitrary, something not. In logic only this expresses: but this means that in logic it is not we who express, by means of signs, what we want, but in logic the nature of the essentially necessary signs itself asserts. That is to say, if we know the logical syntax of any sign language, then all the propositions of logic are already given.

[Expand]6.125 It is possible, also with the old conception of logic, to give at the outset a description of all “true” logical propositions.

[Expand]6.126 Whether a proposition belongs to logic can be calculated by calculating the logical properties of the symbol.

And this we do when we prove a logical proposition. For without troubling ourselves about a sense and a meaning, we form the logical propositions out of others by mere symbolic rules.

We prove a logical proposition by creating it out of other logical propositions by applying in succession certain operations, which again generate tautologies out of the first. (And from a tautology only tautologies follow.)

Naturally this way of showing that its propositions are tautologies is quite unessential to logic. Because the propositions, from which the proof starts, must show without proof that they are tautologies.

[Expand]6.127 All propositions of logic are of equal rank; there are not some which are essentially primitive and others deduced from these.

Every tautology itself shows that it is a tautology.

6.13 Logic is not a theory but a reflexion of the world.

Logic is transcendental.

[Expand]6.2 Mathematics is a logical method.

The propositions of mathematics are equations, and therefore pseudo-propositions.

[Expand]6.21 Mathematical propositions express no thoughts.

6.22 The logic of the world which the propositions of logic show in tautologies, mathematics shows in equations.

[Expand]6.23 If two expressions are connected by the sign of equality, this means that they can be substituted for one another. But whether this is the case must show itself in the two expressions themselves.

It characterizes the logical form of two expressions, that they can be substituted for one another.

[Expand]6.24 The method by which mathematics arrives at its equations is the method of substitution.

For equations express the substitutability of two expressions, and we proceed from a number of equations to new equations, replacing expressions by others in accordance with the equations.

[Expand]6.3 Logical research means the investigation of all regularity. And outside logic all is accident.

6.31 The so-called law of induction cannot in any case be a logical law, for it is obviously a significant proposition.—And therefore it cannot be a law a priori either.

[Expand]6.32 The law of causality is not a law but the form of a law.

6.33 We do not believe a priori in a law of conservation, but we know a priori the possibility of a logical form.

[Expand]6.34 All propositions, such as the law of causation, the law of continuity in nature, the law of least expenditure in nature, etc. etc., all these are a priori intuitions of possible forms of the propositions of science.

6.341 Newtonian mechanics, for example, brings the description of the universe to a unified form. Let us imagine a white surface with irregular black spots. We now say: Whatever kind of picture these make I can always get as near as I like to its description, if I cover the surface with a sufficiently fine square network and now say of every square that it is white or black. In this way I shall have brought the description of the surface to a unified form. This form is arbitrary, because I could have applied with equal success a net with a triangular or hexagonal mesh. It can happen that the description would have been simpler with the aid of a triangular mesh; that is to say we might have described the surface more accurately with a triangular, and coarser, than with the finer square mesh, or vice versa, and so on. To the different networks correspond different systems of describing the world. Mechanics determine a form of description by saying: All propositions in the description of the world must be obtained in a given way from a number of given propositions—the mechanical axioms. It thus provides the bricks for building the edifice of science, and says: Whatever building thou wouldst erect, thou shalt construct it in some manner with these bricks and these alone.

(As with the system of numbers one must be able to write down any arbitrary number, so with the system of mechanics one must be able to write down any arbitrary physical proposition.)

6.342 And now we see the relative position of logic and mechanics. (We could construct the network out of figures of different kinds, as out of triangles and hexagons together.) That a picture like that instanced above can be described by a network of a given form asserts nothing about the picture. (For this holds of every picture of this kind.) But this does characterize the picture, the fact, namely, that it can be completely described by a definite net of definite fineness.

So too the fact that it can be described by Newtonian mechanics asserts nothing about the world; but this asserts something, namely, that it can be described in that particular way in which as a matter of fact it is described. The fact, too, that it can be described more simply by one system of mechanics than by another says something about the world.

[Expand]6.343 Mechanics is an attempt to construct according to a single plan all true propositions which we need for the description of the world.

6.35 Although the spots in our picture are geometrical figures, geometry can obviously say nothing about their actual form and position. But the network is purely geometrical, and all its properties can be given a priori.

Laws, like the law of causation, etc., treat of the network and not of what the network describes.

[Expand]6.36 If there were a law of causality, it might run: “There are natural laws”.

But that can clearly not be said: it shows itself.

[Expand]6.37 A necessity for one thing to happen because another has happened does not exist. There is only logical necessity.

[Expand]6.4 All propositions are of equal value.

6.41 The sense of the world must lie outside the world. In the world everything is as it is and happens as it does happen. In it there is no value—and if there were, it would be of no value.

If there is a value which is of value, it must lie outside all happening and being-so. For all happening and being-so is accidental.

What makes it non-accidental cannot lie in the world, for otherwise this would again be accidental.

It must lie outside the world.

[Expand]6.42 Hence also there can be no ethical propositions.

Propositions cannot express anything higher.

[Expand]6.43 If good or bad willing changes the world, it can only change the limits of the world, not the facts; not the things that can be expressed in language.

In brief, the world must thereby become quite another. It must so to speak wax or wane as a whole.

The world of the happy is quite another than that of the unhappy.

6.44 Not how the world is, is the mystical, but that it is.

6.45 The contemplation of the world sub specie aeterni is its contemplation as a limited whole.

The feeling of the world as a limited whole is the mystical feeling.

[Expand]6.5 For an answer which cannot be expressed the question too cannot be expressed.

The riddle does not exist.

If a question can be put at all, then it can also be answered.

7 Whereof one cannot speak, thereof one must be silent.

↑ The decimal figures as numbers of the separate propositions indicate the logical importance of the propositions, the emphasis laid upon them in my exposition. The propositions n.1, n.2, n.3, etc., are comments on proposition No. n; the propositions n.m1, n.m2, etc., are comments on the proposition No. n.m; and so on.

[tlp-note-1-1] The decimal figures as numbers of the separate propositions indicate the logical importance of the propositions, the emphasis laid upon them in my exposition. The propositions n.1, n.2, n.3, etc., are comments on proposition No. n; the propositions n.m1, n.m2, etc., are comments on the proposition No. n.m; and so on.

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