Tractatus Logico-Philosophicus (English): Difference between revisions

1 The world is everything that is the case.

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

1.11 The world is determined by the facts, and by these being ''all'' the facts.

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

1.13 The facts in logical space are the world.

1.2 The world divides into facts.

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

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

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.

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.

2.0121 It would, so to speak, appear as an accident, when to a thing that could exist alone on its own account, subsequently a state of affairs could be made to fit.

2.0122 The thing is independent, in so far as it can occur in all ''possible'' circumstances, but this form of independence is a form of connexion with the atomic fact, a form of dependence. (It is impossible for words to occur in two different ways, alone and in the proposition.)

2.0123 If I know an object, then I also know all the possibilities of its occurrence in atomic facts.

2.01231 In order to know an object, I must know not its external but all its internal qualities.

2.0124 If all objects are given, then thereby are all ''possible'' atomic facts also given.

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.

2.0131 A spatial object must lie in infinite space. (A point in space is an argument place.)

2.014 Objects contain the possibility of all states of affairs.

2.0141 The possibility of its occurrence in atomic facts is the form of the object.

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.

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

2.0211 If the world had no substance, then whether a proposition had sense would depend on whether another proposition was true.

2.0212 It would then be impossible to form a picture of the world (true or false).

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.

2.023 This fixed form consists of the objects.

2.0231 The substance of the world ''can'' only determine a form and not any material properties. For these are first presented by the propositions—first formed by the configuration of the objects.

2.0232 Roughly speaking: objects are colourless.

2.0233 Two objects of the same logical form are—apart from their external properties—only differentiated from one another in that they are different.

2.02331 Either a thing has properties which no other has, and then one can distinguish it straight away from the others by a description and refer to it; or, on the other hand, there are several things which have the totality of their properties in common, and then it is quite impossible to point to any one of them.

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

2.025 It is form and content.

2.0251 Space, time and colour (colouredness) are forms of objects.

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

2.027 The fixed, the existent and the object are one.

2.0271 The object is the fixed, the existent; the configuration is the changing, the variable.

2.0272 The configuration of the objects forms the atomic fact.

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

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

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

2.033 The form is the possibility of the structure.

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

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.

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

2.061 Atomic facts are independent of one another.

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

2.063 The total reality is the world.

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.

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

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

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

2.141 The picture is a fact.

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.

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

2.1511 Thus the picture is linked with reality; it reaches up to it.

2.1512 It is like a scale applied to reality.

2.15121 Only the outermost points of the dividing lines ''touch'' the object to be measured.

2.1513 According to this view the representing relation which makes it a picture, also belongs to the picture.

2.1514 The representing relation consists of the co-ordinations of the elements of the picture and the things.

2.1515 These co-ordinations are as it were the feelers of its elements with which the picture touches reality.

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

2.161 In the picture and the pictured there must be something identical in order that the one can be a picture of the other at all.

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.

2.171 The picture can represent every reality whose form it has.

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

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

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

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.

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

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

2.19 The logical picture can depict the world.

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

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

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

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

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

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

2.221 What the picture represents is its sense.

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

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

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

2.225 There is no picture which is a priori true.

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.

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 co-ordinates a figure which contradicts the laws of space; or to give the co-ordinates of a point which does not exist.

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.

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.

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

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

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.

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.

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''.

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.

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.

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).

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.

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

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.

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

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.

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.

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.

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.

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.

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.

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

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''”).

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 co-ordinates: 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.

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

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.

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.

4.0031 All philosophy is “Critique of language” (but not at all in Mauthner’s sense). Russell’s merit is to have shown that the apparent logical form of the proposition need not be its real form.

4.01 The proposition is a picture of reality.

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).

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.

4.0141 In the fact that there is a general rule by which the musician is able to read the symphony out of the score, and that there is a rule by which one could reconstruct the symphony from the line on a gramophone record and from this again—by means of the first rule—construct the score, herein lies the internal similarity between these things which at first sight seem to be entirely different. And the rule is the law of projection which projects the symphony into the language of the musical score. It is the rule of translation of this language into the language of the gramophone record.

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.

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

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

4.022 The proposition ''shows'' its sense.

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

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

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

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

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

4.03 A proposition must communicate a new sense with old words.

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

4.0311 One name stands for one thing, and another for another thing, and they are connected together. And so the whole, like a living picture, presents the atomic fact.

4.0312 The possibility of propositions is based upon the principle of the representation of objects by signs.

4.032 The proposition is a picture of its state of affairs, only in so far as it is logically articulated. (Even the proposition “ambulo” is composite, for its stem gives a different sense with another termination, or its termination with another stem.)

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

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

4.0411 If we tried, for example, to express what is expressed by “(''x'').''fx''” by putting an index before ''fx'', like: “Gen.''fx''”, it would not do, we should not know what was generalized. If we tried to show it by an index ''g'', like: “''f''(''x_g'')” it would not do—we should not know the scope of the generalization.

4.0412 For the same reason the idealist explanation of the seeing of spatial relations through “spatial spectacles” does not do, because it cannot explain the multiplicity of these relations.

4.05 Reality is compared with the proposition.

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.

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.

4.0621 That, however, the signs “''p''” and “~''p''” can say the same thing is important, for it shows that the sign “''~''” corresponds to nothing in reality.

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.

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.

4.0641 One could say, the denial is already related to the logical place determined by the proposition that is denied.

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

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

4.111 Philosophy is not one of the natural sciences.

4.112 The object of philosophy is the logical clarification of thoughts.

4.1121 Psychology is no nearer related to philosophy, than is any other natural science.

4.1122 The Darwinian theory has no more to do with philosophy than has any other hypothesis of natural science.

4.113 Philosophy limits the disputable sphere of natural science.

4.114 It should limit the thinkable and thereby the unthinkable.

4.115 It will mean the unspeakable by clearly displaying the speakable.

4.116 Everything that can be thought at all can be thought clearly. Everything that can be said can be said clearly.

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.

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

4.1211 Thus a proposition “''fa''” shows that in its sense the object ''a'' occurs, two propositions “''fa''” and “''ga''” that they are both about the same object.

4.1212 What ''can'' be shown ''cannot'' be said.

4.1213 Now we understand our feeling that we are in possession of the right logical conception, if only all is right in our symbolism.

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.

4.1221 An internal property of a fact we also call a feature of this fact. (In the sense in which we speak of facial features.)

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

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.

4.1241 One cannot distinguish forms from one another by saying that one has this property, the other that: for this assumes that there is a sense in asserting either property of either form.

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.

4.1251 Now this settles the disputed question “whether all relations are internal or external”.

4.1252 Series which are ordered by ''internal'' relations I call formal series.

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

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.

4.1272 So the variable name “''x''” is the proper sign of the pseudo-concept ''object''.

4.12721 The formal concept is already given with an object, which falls under it. One cannot, therefore, introduce both, the objects which fall under a formal concept ''and'' the formal concept itself, as primitive ideas. One cannot, therefore, ''e.g.'' introduce (as Russell does) the concept of function and also special functions as primitive ideas; or the concept of number and definite numbers.

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.)

4.1274 The question about the existence of a formal concept is senseless. For no proposition can answer such a question.

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.

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

4.211 It is a sign of an elementary proposition, that no elementary proposition can contradict it.

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

4.221 It is obvious that in the analysis of propositions we must come to elementary propositions, which consist of names in immediate combination.

4.2211 Even if the world is infinitely complex, so that every fact consists of an infinite number of atomic facts and every atomic fact is composed of an infinite number of objects, even then there must be objects and atomic facts.

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

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

4.241 If I use two signs with one and the same meaning, I express this by putting between them the sign “=”.

4.242 Expressions of the form “''a=b''” are therefore only expedients in presentation: They assert nothing about the meaning of the signs “''a''” and “''b''”.

4.243 Can we understand two names without knowing whether they signify the same thing or two different things? Can we understand a proposition in which two names occur, without knowing if they mean the same or different things?

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>K_n = \sum_{v=0}^n \binom{n}{v}</math> possibilities.

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

4.3 The truth-possibilities of the elementary propositions mean the possibilities of the existence and non-existence of the atomic facts. 4.31 The truth-possibilities can be presented by schemata of the following kind ("T" means "true", "F" "false". The rows of T's and F's under the row of the elementary propositions mean their truth-possibilities in an easily intelligible symbolism).

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

4.411 It seems probable even at first sight that the introduction of the elementary propositions is fundamental for the comprehension of the other kinds of propositions. Indeed the comprehension of the general propositions depends ''palpably'' on that of the elementary propositions.

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

4.431 The expression of the agreement and disagreement with the truth-possibilities of the elementary-propositions expresses the truth-conditions of the proposition. The proposition is the expression of its truth-conditions.

4.44 The sign which arises from the co-ordination of that mark "T" with the truth-possibilities is a propositional sign.

4.441 It is clear that to the complex of the signs "F" and "T" no object (or complex of objects) corresponds; any more than to horizontal and vertical lines or to brackets. There are no "logical objects".

4.442 Thus e.g.

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

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

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

4.4611 Tautology and contradiction are, however, not nonsensical; they are part of the symbolism, in the same way that "0" is part of the symbolism of Arithmetic.

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''.

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

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

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.

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.

4.4661 Of course the signs are also combined with one another in the tautology and contradiction, ''i.e.'' they stand in relations to one another, but these relations are meaningless, unessential to the ''symbol''.

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.

4.51 Suppose ''all'' elementary propositions were given me: then we can simply ask: what propositions I can build out of them. And these are ''all'' propositions and ''so'' are they limited.

4.52 The propositions are everything which follows from the totality of all elementary propositions (of course also from the fact that it is the ''totality of them all''). (So, in some sense, one could say, that ''all'' propositions are generalizations of the elementary propositions.)

4.53 The general propositional form is a variable.

5 Propositions are truth-functions of elementary propositions.

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.

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:

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.

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.

5.121 The truth-grounds of ''q'' are contained in those of ''p''; ''p'' follows from ''q''.

5.122 If ''p'' follows from ''q'', the sense of "''p''" is contained in that of "''q''".

5.123 If a god creates a world in which certain propositions are true, he creates thereby also a world in which all propositions consequent on them are true. And similarly he could not create a world in which the proposition "''p''" is true without creating all its objects.

5.124 A proposition asserts every proposition which follows from it.

5.1241 "''p.q''" is one of the propositions which assert "''p''" and at the same time one of the propositions which assert "''q''".

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

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.

5.1311 When we conclude from ''p''v''q'' and ~''p'' to ''q'' the relation between the forms of the propositions "''p''v''q''" and "~''p''" is here concealed by the method of symbolizing. But if we write, ''e.g.'' instead of "''p''v''q''" "''p''|''q'' .|. ''p''|''q''" and instead of "~''p''" "''p''|''p''" (''p''|''q'' = neither ''p'' nor ''q''), then the inner connexion becomes obvious.

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

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.

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

5.1361 The events of the future ''cannot'' be inferred from those of the present.

5.1362 The freedom of the will consists in the fact that future actions cannot be known now. We could only know them if causality were an ''inner'' necessity, like that of logical deduction.—The connexion of knowledge and what is known is that of logical necessity.

5.1363 If from the fact that a proposition is obvious to us it does not ''follow'' that it is true, then obviousness is no justification for our belief in its truth.

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

5.141 If ''p'' follows from ''q'' and ''q'' from ''p'' then they are one and the same proposition.

5.142 A tautology follows from all propositions: it says nothing.

5.143 Contradiction is something shared by propositions, which ''no'' proposition has in common with another. Tautology is that which is shared by all propositions, which have nothing in common with one another.

5.15 If ''T_r'' is the number of the truth-grounds of the proposition "''r''", ''T_r'' 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 {{nowrap|''T_rs : T_r''}} the measure of the ''probability'' which the proposition "''r''" gives to the proposition "''s''".

5.151 Suppose in a schema like that above in No. [[#5.101|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 {{nowrap|''T_rs : T_r''}}.

5.1511 There is no special object peculiar to probability propositions.

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

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.

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.

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.

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

5.231 And that will naturally depend on their formal properties, on the internal similarity of their forms.

5.232 The internal relation which orders a series is equivalent to the operation by which one term arises from another.

5.233 The first place in which an operation can occur is where a proposition arises from another in a logically significant way; ''i.e.'' where the logical construction of the proposition begins.

5.234 The truth-functions of elementary proposition, are results of operations which have the elementary propositions as bases. (I call these operations, truth-operations.)

5.2341 The sense of a truth-function of ''p'' is a function of the sense of ''p''.

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

5.241 The operation does not characterize a form but only the difference between forms.

5.242 The same operation which makes "''q''" from "''p''", makes "''r''" from "''q''", and so on. This can only be expressed by the fact that "''p''", "''q''", "''r''", etc., are variables which give general expression to certain formal relations.

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

5.251 A function cannot be its own argument, but the result of an operation can be its own basis.

5.252 Only in this way is the progress from term to term in a formal series possible (from type to type in the hierarchy of Russell and Whitehead). (Russell and Whitehead have not admitted the possibility of this progress but have made use of it all the same.)

5.2521 The repeated application of an operation to its own result I call its successive application ("''O' O' O' a''" is the result of the threefold successive application of "''O' ξ''" to "''a''").

5.2522 The general term of the formal series ''a, O', a, O' O' a,'' ... I write thus: "[''a, x, O' x'']". This expression in brackets is a variable. The first term of the expression is the beginning of the formal series, the second the form of an arbitrary term ''x'' of the series, and the third the form of that term of the series which immediately follows ''x''.

5.2523 The concept of the successive application of an operation is equivalent to the concept "and so on".

5.253 One operation can reverse the effect of another. Operations can cancel one another.

5.254 Operations can vanish (''e.g.'' denial in "~~''p''", ~~''p'' = ''p'').

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

5.31 The Schemata No. [[#4.31|4.31]] are also significant, if "''p''", "''q''", "''r''", etc. are not elementary propositions.

5.32 All truth-functions are results of the successive application of a finite number of truth-operations to elementary propositions.

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.

5.43 That from a fact ''p'' an infinite number of ''others'' should follow, namely ~~''p'', <nowiki>~~~~</nowiki>''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".

5.44 Truth-functions are not material functions.

5.441 This disappearance of the apparent logical constants also occurs if "~(∃''x'') . ~''fx''" says the same as "(''x'') . ''fx''", or "(∃''x'') . ''fx'' . ''x'' = ''a''" the same as "''fa''".

5.442 If a proposition is given to us then the results of all truth-operations which have it as their basis are given ''with'' it.

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.

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.)

5.453 All numbers in logic must be capable of justification.

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

5.4541 The solution of logical problems must be neat for they set the standard of neatness.

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.

5.461 The apparently unimportant fact that the apparent relations like ∨ and ⊃ need brackets—unlike real relations—is of great importance.

5.4611 Logical operation signs are punctuations.

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''.

5.471 The general form of proposition is the essence of proposition.

5.4711 To give the essence of proposition means to give the essence of all description, therefore the essence of the world.

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

5.473 Logic must take care of itself.

5.4731 Self-evidence, of which Russell has said so much, can only be discarded in logic by language itself preventing every logical mistake. That logic is a priori consists in the fact that we ''cannot'' think illogically.

5.4732 We cannot give a sign the wrong sense.

5.47321 Occam's razor is, of course, not an arbitrary rule nor one justified by its practical success. It simply says that ''unnecessary'' elements in a symbolism mean nothing.

5.4733 Frege says: Every legitimately constructed proposition must have a sense; and I say: Every possible proposition is legitimately constructed, and if it has no sense this can only be because we have given no ''meaning'' to some of its constituent parts.

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.

5.5 Every truth-function is a result of the successive application of the operation (- - - - T)(''ξ'', . . . .) to elementary 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>( \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>( \bar{\xi} )</math> = (P, Q, R).)

5.502 Therefore I write instead of "(- - - - T)(''ξ'', . . . .)", "<math>N ( \bar{\xi} )</math>".