Tractatus Logico-Philosophicus (tree-like view)
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Ludwig Wittgenstein
Tractatus Logico-Philosophicus (tree-like view)
Translated by F. P. Ramsey
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, edited by C. K. Ogden and F. P. Ramsey, 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 1927.
Ludwig Wittgenstein
Tractatus Logico-Philosophicus
Dedicated
to the memory of my friend
DAVID H. PINSENT
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.
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 connexion 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.
(Every such possibility must lie in the nature of the object.)
A new possibility cannot subsequently be found.
A speck in a visual field need not be red, but it must have a colour; it has, so to speak, a colour space round it. A tone must have a pitch, the object of the sense of touch a hardness, etc.
For if a thing is not distinguished by anything, I cannot distinguish it—for otherwise it would be distinguished.
(The existence of atomic facts we also call a positive fact, their non-existence a negative fact.)
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.
The spatial picture, everything spatial, the coloured, everything coloured, etc.
What is thinkable is also possible.
The method of projection is the thinking of the sense of the proposition.
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.
The propositional sign is a fact.
The proposition is articulate.
(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 mutual spatial position of these things then expresses the sense of the proposition.
(Names resemble points; propositions resemble arrows, they have sense.)
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.
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).
(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.
And in this form the expression is constant and everything else variable.
(In the limiting case the variable becomes constant, the expression a proposition.)
I call such a variable a “propositional variable”.
The determination of the values is the variable.
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.
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.)
(The logical symbolism of Frege and Russell is such a language, which, however, does still not exclude all errors.)
(If everything in the symbolism works as though a sign had meaning, then it has meaning.)
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 function F and the inner function F must have different meanings; for the inner has the form ϕ(fx), the outer the form ψ(ϕ(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 “(∃ϕ):F(ϕu).ϕu=Fu”.
Herewith Russell’s paradox vanishes.
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.
And in the same way in general the essential in a symbol is that which all symbols which can fulfil the same purpose have in common.
(Herewith is indicated the way in which a special possible notation can give us general information.)
(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.)
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.
(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.
The proposition is a model of the reality as we think it is.
For these irregularities also picture what they are to express; only in another way.
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.)
And from it came the alphabet without the essence of the representation being lost.
The proposition shows how things stand, if it is true. And it says, that they do so stand.
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.
(One can therefore understand it without knowing whether it is true or not.)
One understands it if one understands it constituent parts.
(And the dictionary does not only translate substantives but also adverbs and conjunctions, etc., and it treats them all alike.)
By means of propositions we explain ourselves.
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.
One can say, instead of, This proposition has such and such a sense, This proposition represents such and such a state of affairs.
My fundamental thought is that the “logical constants” do not represent. That the logic of the facts cannot be represented.
They must both possess the same logical (mathematical) multiplicity (cf. Hertz’s Mechanics, on Dynamic Models).
If we were to try it by introducing a mark in the argument places, like “(G, G).F(G, G)”, it would not do—we could not determine the identity of the variables, etc.
All these ways of symbolizing are inadequate because they have not the necessary mathematical multiplicity.
One could, then, for example, say that “p” signifies in the true way what “~p” signifies in the false way, etc.
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.
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.
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.
(The word “philosophy” must mean something which stands above or below, but not beside the natural sciences.)
Philosophy is not a theory but an activity.
A philosophical work consists essentially of elucidations.
The result of philosophy is not a number of “philosophical propositions”, but to make propositions clear.
Philosophy should make clear and delimit sharply the thoughts which otherwise are, as it were, opaque and blurred.
The theory of knowledge is the philosophy of psychology.
Does not my study of sign-language correspond to the study of thought processes which philosophers held to be so essential to the philosophy of logic? Only they got entangled for the most part in unessential psychological investigations, and there is an analogous danger for my method.
It should limit the unthinkable from within through the thinkable.
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.
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.
If two propositions contradict one another, this is shown by their structure; similarly if one follows from another, etc.
(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.
(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”.)
It would be as senseless to ascribe a formal property to a proposition as to deny it the formal property.
The series of numbers is ordered not by an external, but by an internal relation.
Similarly the series of propositions “aRb”,
- “(∃x):aRx.xRb”,
- “(∃x,y):aRx.xRy.yRb”, etc.
(If b stands in one of these relations to a, I call b a successor of a.)
(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.
For every variable presents a constant form, which all its values possess, and which can be conceived as a formal property of these values.
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 ℵ_{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.)
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.
(For example, one cannot ask: “Are there unanalysable subject-predicate propositions?”)
Therefore there are in logic no pre-eminent numbers, and therefore there is no philosophical monism or dualism, etc.
The question arises here, how the propositional connexion comes to be.
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.
“a=b” means then, that the sign “a” is replaceable by the sign “b”.
(If I introduce by an equation a new sign “b”, by determining that it shall replace a previously known sign “a”, I write the equation—definition—(like Russell) in the form “a=b Def.”. A definition is a symbolic rule.)
If I know the meaning of an English and a synonymous German word, it is impossible for me not to know that they are synonymous, it is impossible for me not to be able to translate them into one another.
Expressions like “a=a”, or expressions deduced from these are neither elementary propositions nor otherwise significant signs. (This will be shown later.)
It is possible for all combinations of atomic facts to exist, and the others not to exist.
p | q | r |
---|---|---|
T | T | T |
F | T | T |
T | F | T |
T | T | F |
F | F | T |
F | T | F |
T | F | F |
F | F | F |
p | q |
---|---|
T | T |
F | T |
T | F |
F | F |
p |
---|
T |
F |
Absence of this mark means disagreement.
(Frege has therefore quite rightly put them at the beginning, as explaining the signs of his logical symbolism. Only Frege's explanation of the truth-concept is false: if "the true" and "the false" were real objects and the arguments in ~p etc., then the sense of ~p would by no means be determined by Frege's determination.)
Something analogous holds of course for all signs, which express the same as the schemata of "T"and "F".
“
p | q | |
---|---|---|
T | T | T |
F | T | T |
T | F | |
F | F | T |
”
is a propositional sign.
(Frege's assertion sign "$\vdash$ " is logically altogether meaningless; in Frege (and Russell) it only shows that these authors hold as true the propositions marked in this way.
"$\vdash$ " belongs therefore to the propositions no more than does the number of the proposition. A proposition cannot possibly assert of itself that it is true.)
If the sequence of the truth-possibilities in the schema is once for all determined by a rule of combination, then the last column is by itself an expression of the truth-conditions. If we write this column as a row the propositional sign becomes: "(T—T) (p,q)" or more plainly: "(T T F T) (p,q)".
(The number of places in the left-hand bracket is determined by the number of terms in the right-hand bracket.)
The groups of truth-conditions which belong to the truth-possibilities of a number of elementary propositions can be ordered in a series.
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.
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.)
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.
(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.
(Certain, possible, impossible: here we have an indication of that gradation which we need in the theory of probability.)
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.
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.
(An elementary proposition is a truth-function of itself.)
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.
(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] |
Two propositions are opposed to one another if there is no significant proposition which asserts them both.
Every proposition which contradicts another, denies it.
(The fact that we can infer fa from (x)fx shows that generality is present also in the symbol "(x).fx".)
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.
Superstition is the belief in the causal nexus.
("A knows that p is the case" is senseless if p is a tautology.)
Contradiction vanishes so to speak outside, tautology inside all propositions.
Contradiction is the external limit of the propositions, tautology their substanceless centre.
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.)
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.
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.
Denial, logical addition, logical multiplication, etc. etc., are operations.
(Denial reverses the sense of a proposition.)
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.)
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.)
In a similar sense I speak of the successive application of several operations to a number of 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.
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.
But all propositions of logic say the same thing. That is, nothing.
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.
(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.)
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.
Or rather it must become plain that there are no numbers in logic.
There are no pre-eminent numbers.
In logic there cannot be a more general and a more special.
Men have always thought that there must be a sphere of questions whose answers—a priori—are symmetrical and united into a closed regular structure.
A sphere in which the proposition, simplex sigillum veri, is valid.
The use of brackets with these apparent primitive signs shows that these are not the real primitive signs; and nobody of course would believe that the brackets have meaning by themselves.
For all logical operations are already contained in the elementary proposition. For "f a" says the same as "(∃x) . f x . 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.
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.
Signs which serve one purpose are logically equivalent, signs which serve no purpose are logically meaningless.
(Even if we believe that we have done so.)
Thus "Socrates is identical" says nothing, because we have given no meaning to the word "identical" as adjective. For when it occurs as the sign of equality it symbolizes in an entirely different way—the symbolizing relation is another—therefore the symbol is in the two cases entirely different; the two symbols have the sign in common with one another only by accident.
This operation denies all the propositions in the right-hand bracket and I call it the negation of these propositions.
(Thus if ξ has the 3 values P, Q, R, then $({\bar {\xi }})$ = (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 f x 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.
$N({\bar {\xi }})$ is the negation of all the values of the propositional variable ξ.
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.
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.
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.
But here also the negative proposition is indirectly constructed with the positive.
The positive proposition must presuppose the existence of the negative proposition and conversely.
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) . f x" and "(x) . f x" in which both ideas lie concealed.
If the elementary propositions are given, then therewith all elementary propositions are also given.
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.
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,
A characteristic of a composite symbol: it has something: in common with other symbols.
(If an elementary proposition is true, then, at any rate, there is one more elementary proposition true.)
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.
(Therefore instead of Russell's "(∃ x, y) . f (x, y)": "(∃ x, y) . f (x, y) . ∨ . (∃x) . f (x, x)".)
And the proposition "only one x satisfies f( )" reads: "(∃ x) . f x : ~(∃ x, y) . f x . f y".
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.
(It is nonsense to place the hypothesis p ⊃ p before a proposition in order to ensure that its arguments have the right form, because the hypothesis for a non-proposition as argument becomes not false but meaningless, and because the proposition itself becomes senseless for arguments of the wrong kind, and therefore it survives the wrong arguments no better and no worse than the senseless hypothesis attached for this purpose.)
Especially in certain propositional forms of psychology, like "A thinks, that p is the case", or "A thinks p", etc.
Here it appears superficially as if the proposition p stood to the object A in a kind of relation.
(And in modern epistemology (Russell, Moore, etc.) those propositions have been conceived in this way.)
A composite soul would not be a soul any longer.
This perhaps explains that the figure
can be seen in two ways as a cube; and all similar phenomena. For we really see two different facts.
(If I fix my eyes first on the corners a and only glance at b, a appears in front and b behind, and vice versa.)
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.
(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.)
Logic precedes every experience—that something is so.
It is before the How, not before the What.
(There is no pre-eminent number.)
Has the question sense: what must there be in order that anything can be the case?
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.
(Our problems are not abstract but perhaps the most concrete that 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.
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.
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.
If I wrote a book "The world as I found it", I should also have therein to report on my body and say which members obey my will and which do not, etc. This then would be a method of isolating the subject or rather of showing that in an important sense there is no subject: that is to say, of it alone in this book mention could not be made.
You say that this case is altogether like that of the eye and the field of sight. But you do not really see the eye.
And from nothing in the field of sight can it be concluded that it is seen from an eye.
Everything we see could also be otherwise.
Everything we can describe at all could also be otherwise.
There is no order of things a priori.
The I occurs in philosophy through the fact that the "world is my world".
The philosophical I is not the man, not the human body or the human soul of which psychology treats, but the metaphysical subject, the limit—not a part of the world.
This is the general form of proposition.
This is the most general form of transition from one proposition to another.
$x=\Omega ^{0\prime }x{\text{ Def.}}$ and
$\Omega ^{\prime }\Omega ^{\nu \prime }x=\Omega ^{\nu +1\prime }x{\text{ Def.}}$
According, then, to these symbolic rules we write the series $x,\Omega 'x,\Omega '\Omega 'x,\Omega '\Omega '\Omega 'x,.....$
as: $\Omega ^{0\prime }x,\Omega ^{0+1\prime }x,\Omega ^{0+1+1\prime }x,\Omega ^{0+1+1+1\prime }x,.....$
Therefore I write in place of "$[x,\xi ,\Omega '\xi ]$ ",
"$[\Omega ^{0\prime }x,\Omega ^{\nu \prime }x,\Omega ^{\nu +1\prime }x]$ ".
And I define:
- $0+1=1{\text{ Def.}}$
- $0+1+1=2{\text{ Def.}}$
- $0+1+1+1=3{\text{ Def.}}$
- (and so on.)
This is connected with the fact that the generality which we need in mathematics is not the accidental one.
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.
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.
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.
E.g. that "q" follows from "p ⊃ q . p" we see from these two propositions themselves, but we can also show it by combining them to "p ⊃ q . p : ⊃ : q" and then showing that this is a tautology.
(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.)
To be general is only to be accidentally valid for all things. An ungeneralized proposition can be tautologous just as well as a generalized one.
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.
Every proposition of logic is a modus ponens presented in signs. (And the modus ponens can not be expressed by a proposition.)
Every tautology itself shows that it is a tautology.
Logic is transcendental.
The propositions of mathematics are equations, and therefore pseudo-propositions.
(In philosophy the question "Why do we really use that word, that proposition?" constantly leads to valuable results.)
It characterizes the logical form of two expressions, that they can be substituted for one another.
It is a property of "1 + 1 + 1 + 1" that it can be conceived as "(1 + 1) + (1 + 1)".
But what is essential about equation is that it is not necessary in order to show that both expressions, which are connected by the sign of equality, have the same meaning: for this can be perceived from the two expressions themselves.
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.
$(\Omega ^{\nu })^{\mu \prime }x=\Omega ^{\nu \times \mu \prime }x{\text{ Def.}}$
$(\Omega ^{2\times 2\prime }x=(\Omega ^{2})^{2\prime }x=(\Omega ^{2})^{1+1\prime }x=\Omega ^{2\prime }\Omega ^{2\prime }x=\Omega ^{1+1\prime }\Omega ^{1+1\prime }x$
$(\Omega '\Omega )^{\prime }(\Omega '\Omega )^{\prime }x=\Omega '\Omega '\Omega '\Omega 'x=\Omega ^{1+1+1+1\prime }x=\Omega ^{4\prime }x$
(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.)
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.
Laws, like the law of causation, etc., treat of the network and not of what the network describes.
But that can clearly not be said: it shows itself.
Hence the description of the temporal sequence of events is only possible if we support ourselves on another process.
It is exactly analogous for space. When, for example, we say that neither of two events (which mutually exclude one another) can occur, because there is no cause why the one should occur rather than the other, it is really a matter of our being unable to describe one of the two events unless there is some sort of asymmetry. And if there is such an asymmetry, we can regard this as the cause of the occurrence of the one and of the non-occurrence of the other.
A right-hand glove could be put on a left hand if it could be turned round in four-dimensional space.
It is clear that there are no grounds for believing that the simplest course of events will really happen.
And they both are right and wrong. But the ancients were clearer, in so far as they recognized one clear terminus, whereas the modern system makes it appear as though everything were explained.
Let us consider how this contradiction presents itself in physics. Somewhat as follows: That a particle cannot at the same time have two velocities, i.e. that at the same time it cannot be in two places, i.e. that particles in different places at the same time cannot be identical.
(It is clear that the logical product of two elementary propositions can neither be a tautology nor a contradiction. The assertion that a point in the visual field has two different colours at the same time, is a contradiction.)
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.
Ethics is transcendental.
(Ethics and aesthetics are one.)
(And this is clear also that the reward must be something acceptable, and the punishment something unacceptable.)
And the will as a phenomenon is only of interest to psychology.
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.
If by eternity is understood not endless temporal duration but timelessness, then he lives eternally who lives in the present.
Our life is endless in the way that our visual field is without limit.
(It is not problems of natural science which have to be solved.)
The feeling of the world as a limited whole is the mystical feeling.
The riddle does not exist.
If a question can be put at all, then it can also be answered.
For doubt can only exist where there is a question; a question only where there is an answer, and this only where something can be said.
(Is not this the reason why men to whom after long doubting the sense of life became clear, could not then say wherein this sense consisted?)
He must surmount these propositions; then he sees the world rightly.
- ↑ 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.