When you see X ⊃ Y, you are to read "X implies Y" or "If X, then Y." Sometimes you may see the modern form, X —> Y, in my comments.
When you see X ≡ Y, you should read "X equivalent to Y" or "X strictly implies Y" or "If and only if X, then Y" or "If X, then Y and if Y, then X" (these forms are sometimes shortened to "iff"). "X ≡ Y" can also be read: "Y is necessary for X and X is sufficient for Y." Occasionally one also sees "X ≡ Y" interpreted as "Y holds just in case X holds" or "Y holds only in case X holds" and the converse, "X holds just in case Y holds." Sometimes my comments use the modern X <—> Y.
For the quantifier all, you will see ( ), as in (x)Fx. In my comments I sometimes use the modern symbol ∀, as in ∀xFx. The existential quantifer ∃, as in ∃xFx, is used in W's text and in my comments. P●Q should be read "P and Q." Often the dot is dropped, as in PQ. Occasionally dots are used instead of parentheses.
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.
3.03
We cannot think anything unlogical, for otherwise we should have to think unlogically.
3.031
It used to be said that God could create everything, except what was contrary to the laws of logic. The truth is, we could not say of an “unlogical” world how it would look.
3.032
To present in language anything which “contradicts logic” is as impossible as in geometry to present by its coordinates a figure which contradicts the laws of space; or to give the coordinates of a point which does not exist.
This claim appears to have been refuted by Goedel's undecidability theorem, as well as being denied by other paradoxes. Even in ordinary language we have the liar's paradox and Richard's paradox.
Here are three paradoxes, all based on the same idea, taken from a Cambridge web page
https://www.dpmms.cam.ac.uk/~wtg10/richardsparadox.html
3.0321
We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry.
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).
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.
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.
3.141
The proposition is not a mixture of words (just as the musical theme is not a mixture of tones).
The proposition is articulate.
3.142
Only facts can express a sense, a class of names cannot.
3.143
That the propositional sign is a fact is concealed by the ordinary form of expression, written or printed.
For in the printed proposition, for example, the sign of a proposition does not appear essentially different from a word.
(Thus it was possible for Frege to call the proposition a compounded name.)
3.1431
The essential nature of the propositional sign becomes very clear when we imagine it made up of spatial objects (such as tables, chairs, books) instead of written signs.
The mutual spatial position of these things then expresses the sense of the proposition.
3.1432
We must not say, “The complex sign ‘aRb’ says ‘a stands in relation R to b’ ”; but we must say, “That ‘a’ stands in a certain relation to ‘b’ says that aRb”.
3.144
States of affairs can be described but not named.
(Names resemble points; propositions resemble arrows, they have sense.)
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.
3.22
In the proposition the name represents the object.
3.221
Objects I can only name. Signs represent them. I can only speak of them. I cannot assert them. A proposition can only say how a thing is, not what it is.
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.
3.25
There is one and only one complete analysis of the proposition.
3.251
The proposition expresses what it expresses in a definite and clearly specifiable way: the proposition is articulate.
3.26
The name cannot be analysed further by any definition. It is a primitive sign.
3.261
Every defined sign signifies via those signs by which it is defined, and the definitions show the way.
Two signs, one a primitive sign, and one defined by primitive signs, cannot signify in the same way. Names cannot be taken to pieces by definition (nor any sign which alone and independently has a meaning).
3.262
What does not get expressed in the sign is shown by its application. What the signs conceal, their application declares.
3.263
The meanings of primitive signs can be explained by elucidations. Elucidations are propositions which contain the primitive signs. They can, therefore, only be understood when the meanings of these signs are already known.
3.3
Only the proposition has sense; only in the context of a proposition has a name meaning.
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 supposes 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.
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.
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.)
3.33
In logical syntax the meaning of a sign ought never to play a role; it must admit of being established without mention being thereby made of the meaning of a sign; it ought to presuppose only the description of the expressions.
3.331
From this observation we get a further view—into Russell’s “Theory of Types.” Russell’s error is shown by the fact that in drawing up his symbolic rules he has to speak about the things his signs mean.
3.332
No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the “whole theory of types”).
3.333
A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself.
If, for example, we suppose that the function F(fx) could be its own argument, then there would be a proposition “F(F(fx))”, and in this the outer 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(Fu)” we write “(∃φ):F(φu).φu=Fu”.
Herewith Russell’s paradox vanishes.
3.333 makes sense for the liar's paradox. But Russell's paradox arises from the naive definition of set. Perhaps W. is right about rules of definition, but we know that set theorists have preferred axioms that prevent the paradox from arising, such as the prohibition of any set being an element of itself.
3.334
The rules of logical syntax must follow of themselves, if we only know how every single sign signifies.
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.
3.341
The essential in a proposition is therefore that which is common to all propositions which can express the same sense.
And in the same way in general the essential in a symbol is that which all symbols which can fulfill the same purpose have in common.
3.3411
One could therefore say the real name is that which all symbols, which signify an object, have in common. It would then follow, step by step, that no sort of composition was essential for a name.
3.342
In our notations there is indeed something arbitrary, but this is not arbitrary, namely that if we have determined anything arbitrarily, then something else must be the case. (This results from the essence of the notation.)
3.3421
A particular method of symbolizing may be unimportant, but it is always important that this is a possible method of symbolizing. And this happens as a rule in philosophy: The single thing proves over and over again to be unimportant, but the possibility of every single thing reveals something about the nature of the world.
3.343
Definitions are rules for the translation of one language into another. Every correct symbolism must be translatable into every other according to such rules. It is this which all have in common.
3.344
What signifies in the symbol is what is common to all those symbols by which it can be replaced according to the rules of logical syntax.
3.3441
We can, for example, express what is common to all notations for the truth-functions as follows: It is common to them that they all, for example, can be replaced by the notations of “~p” (“not p”) and “p∨q” (“p or q”).
(Herewith is indicated the way in which a special possible notation can give us general information.)
3.3442
The sign of the complex is not arbitrarily resolved in the analysis, in such a way that its resolution would be different in every propositional structure.
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.
3.41
The propositional sign and the logical coordinates: that is the logical place.
3.411
The geometrical and the logical place agree in that each is the possibility of an existence.
3.42
Although a proposition may only determine one place in logical space, the whole logical space must already be given by it.
(Otherwise denial, the logical sum, the logical product, etc., would always introduce new elements—in coordination.)
(The logical scaffolding round the picture determines the logical space. The proposition reaches through the whole logical space.)
3.5
The applied, thought, propositional sign, is the thought.
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.
3.03
We cannot think anything unlogical, for otherwise we should have to think unlogically.
3.031
It used to be said that God could create everything, except what was contrary to the laws of logic. The truth is, we could not say of an “unlogical” world how it would look.
3.032
To present in language anything which “contradicts logic” is as impossible as in geometry to present by its coordinates a figure which contradicts the laws of space; or to give the coordinates of a point which does not exist.
This claim appears to have been refuted by Goedel's undecidability theorem, as well as being denied by other paradoxes. Even in ordinary language we have the liar's paradox and Richard's paradox.
Here are three paradoxes, all based on the same idea, taken from a Cambridge web page
https://www.dpmms.cam.ac.uk/~wtg10/richardsparadox.html
1. Let A be the set of all positive integers that can be defined in under 100 words. Since there are only finitely many of these, there must be a smallest positive integer n that does not belong to A. But haven't I just defined n in under 100 words?We should observe that when he wrote Tractatus, Wittgenstein did not have access to Principia Mathematica, in which the various paradoxes are laid out in detail, but only a detailed summary of that massive work. Yet the young war veteran was certainly highly focused on Russell's paradox.
2. Let B be the set of all reasonably interesting positive integers. Let n be the smallest integer not belonging to B. But surely the defining property of n makes it reasonably interesting.
3. Let X be the set of all definable real numbers. Since there are only countably many definitions, X is countable. Indeed, we can explicitly count X - just list the elements in alphabetical order of their definitions. Now apply to this list some explicit diagonal process, obtaining a number y that does not belong to X. But haven't I just defined y?
We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry.
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).
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.
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.
3.141
The proposition is not a mixture of words (just as the musical theme is not a mixture of tones).
The proposition is articulate.
3.142
Only facts can express a sense, a class of names cannot.
3.143
That the propositional sign is a fact is concealed by the ordinary form of expression, written or printed.
For in the printed proposition, for example, the sign of a proposition does not appear essentially different from a word.
(Thus it was possible for Frege to call the proposition a compounded name.)
3.1431
The essential nature of the propositional sign becomes very clear when we imagine it made up of spatial objects (such as tables, chairs, books) instead of written signs.
The mutual spatial position of these things then expresses the sense of the proposition.
3.1432
We must not say, “The complex sign ‘aRb’ says ‘a stands in relation R to b’ ”; but we must say, “That ‘a’ stands in a certain relation to ‘b’ says that aRb”.
3.144
States of affairs can be described but not named.
(Names resemble points; propositions resemble arrows, they have sense.)
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.
3.22
In the proposition the name represents the object.
3.221
Objects I can only name. Signs represent them. I can only speak of them. I cannot assert them. A proposition can only say how a thing is, not what it is.
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.
3.25
There is one and only one complete analysis of the proposition.
3.251
The proposition expresses what it expresses in a definite and clearly specifiable way: the proposition is articulate.
3.26
The name cannot be analysed further by any definition. It is a primitive sign.
3.261
Every defined sign signifies via those signs by which it is defined, and the definitions show the way.
Two signs, one a primitive sign, and one defined by primitive signs, cannot signify in the same way. Names cannot be taken to pieces by definition (nor any sign which alone and independently has a meaning).
3.262
What does not get expressed in the sign is shown by its application. What the signs conceal, their application declares.
3.263
The meanings of primitive signs can be explained by elucidations. Elucidations are propositions which contain the primitive signs. They can, therefore, only be understood when the meanings of these signs are already known.
3.3
Only the proposition has sense; only in the context of a proposition has a name meaning.
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 supposes 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.
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.
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.)
3.33
In logical syntax the meaning of a sign ought never to play a role; it must admit of being established without mention being thereby made of the meaning of a sign; it ought to presuppose only the description of the expressions.
3.331
From this observation we get a further view—into Russell’s “Theory of Types.” Russell’s error is shown by the fact that in drawing up his symbolic rules he has to speak about the things his signs mean.
3.332
No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the “whole theory of types”).
3.333
A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself.
If, for example, we suppose that the function F(fx) could be its own argument, then there would be a proposition “F(F(fx))”, and in this the outer 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(Fu)” we write “(∃φ):F(φu).φu=Fu”.
Herewith Russell’s paradox vanishes.
3.333 makes sense for the liar's paradox. But Russell's paradox arises from the naive definition of set. Perhaps W. is right about rules of definition, but we know that set theorists have preferred axioms that prevent the paradox from arising, such as the prohibition of any set being an element of itself.
The rules of logical syntax must follow of themselves, if we only know how every single sign signifies.
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.
3.341
The essential in a proposition is therefore that which is common to all propositions which can express the same sense.
And in the same way in general the essential in a symbol is that which all symbols which can fulfill the same purpose have in common.
3.3411
One could therefore say the real name is that which all symbols, which signify an object, have in common. It would then follow, step by step, that no sort of composition was essential for a name.
3.342
In our notations there is indeed something arbitrary, but this is not arbitrary, namely that if we have determined anything arbitrarily, then something else must be the case. (This results from the essence of the notation.)
3.3421
A particular method of symbolizing may be unimportant, but it is always important that this is a possible method of symbolizing. And this happens as a rule in philosophy: The single thing proves over and over again to be unimportant, but the possibility of every single thing reveals something about the nature of the world.
3.343
Definitions are rules for the translation of one language into another. Every correct symbolism must be translatable into every other according to such rules. It is this which all have in common.
3.344
What signifies in the symbol is what is common to all those symbols by which it can be replaced according to the rules of logical syntax.
3.3441
We can, for example, express what is common to all notations for the truth-functions as follows: It is common to them that they all, for example, can be replaced by the notations of “~p” (“not p”) and “p∨q” (“p or q”).
(Herewith is indicated the way in which a special possible notation can give us general information.)
3.3442
The sign of the complex is not arbitrarily resolved in the analysis, in such a way that its resolution would be different in every propositional structure.
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.
3.41
The propositional sign and the logical coordinates: that is the logical place.
3.411
The geometrical and the logical place agree in that each is the possibility of an existence.
3.42
Although a proposition may only determine one place in logical space, the whole logical space must already be given by it.
(Otherwise denial, the logical sum, the logical product, etc., would always introduce new elements—in coordination.)
(The logical scaffolding round the picture determines the logical space. The proposition reaches through the whole logical space.)
3.5
The applied, thought, propositional sign, is the thought.
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