Tractatus Logico-Philosophicus
by Ludwig Wittgenstein
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Introduction By Bertrand Russell
it is necessary to realize what is the problem with which he is concerned. In the part of his theory which deals with Symbolism he is concerned with the conditions which would have to be fulfilled by a logically perfect language. There are various problems as regards language. First, there is the problem what actually occurs in our minds when we use language with the intention of meaning something by it; this problem belongs to psychology. Secondly, there is the problem as to what is the relation subsisting between thoughts, words, or sentences, and that which they refer to or mean; this problem belongs to epistemology. Thirdly, there is the problem of using sentences so as to convey truth rather that falsehood; this belongs to the special sciences dealing with the subject-matter of the sentences in question. Fourthly, there is the question: what relation must one fact (such as a sentence) have to another in order to be capable of being a symbol for that other? Read more at location 25
In practice, language is always more or less vague, so that what we assert is never quite precise. Thus, logic has two problems to deal with in regard to Symbolism: (1) the conditions for sense rather than nonsense in combinations of words; (2) the conditions for uniqueness of meaning or reference in symbols or combinations of symbols. A logically perfect language has rules of syntax which prevent nonsense, and has single symbols which always have a definite and unique meaning. Mr Wittgenstein is concerned with the conditions for a logically perfect language Read more at location 33
**** (Note: godel-esque) In order that a certain sentence should assert a certain fact there must, however the language may be constructed, be something in common between the structure of the sentence and the structure of the fact. This is perhaps the most fundamental thesis of Mr Wittgenstein’s theory. That which has to be in common between the sentence and the fact cannot, he contends, be itself in turn said in language. It can, in his phraseology, only be shown, not said, for whatever we may say will still need to have the same structure. Read more at location 40
The first requisite of an ideal language would be that there should be one name for every simple, and never the same name for two different simples. Read more at location 44
He compares linguistic expression to projection in geometry. A geometrical figure may be projected in many ways: each of these ways corresponds to a different language, but the projective properties of the original figure remain unchanged whichever of these ways may be adopted. Read more at location 54
The world consists of facts: facts cannot strictly speaking be defined, but we can explain what we mean by saying that facts are what makes propositions true, or false. Read more at location 86
If an atomic fact is analyzed as fully as possibly (theoretical, not practical possibility is meant) the constituents finally reached may be called “simples” or “objects.” It is a logical necessity demanded by theory, like an electron. His ground for maintaining that there must be simples is that every complex presupposes a fact. Read more at location 91
(Note: focus on coherence. correlation (to reality) can never be established) the naming of complexes presupposes propositions, while propositions presupposes the naming of simples. In this way the naming of simples is shown to be what is logically first in logic. The world is fully described if all atomic facts are known, together with the fact that these are all of them. The world is not described by mearly naming all the objects in it; it is necessary also to know the atomic facts of which these objects are constituents. Given this total of atomic facts, every true proposition, however complex, can theoretically be inferred. A proposition (true or false) asserting an atomic fact is called an atomic proposition. All atomic propositions are logically independent of each other. No atomic proposition implies any other or is inconsistent with any other. Thus the whole business of logical inference is concerned with proposition which are not atomic. Such propositions may be called molecular. Wittgenstein’s theory of molecular propositions turns upon his theory of the construction of truth-functions. Read more at location 98
The symbol is intended to describe a process by the help of which, given the atomic propositions, all others can be manufactured. The process depends upon: (a) Sheffer’s proof that all truth-functions can be obtained out of simultaneous negation, i.e. out of “not-p and not-q”; (b) Mr Wittgenstein’s theory of the derivation of general propositions from conjunctions and disjunctions; (c) The assertion that a proposition can only occur in another proposition as argument to a truth-function. Given these three foundations, it follows that all propositions which are not atomic can be derived from such as are, buy a uniform process, and it is this process which is indicated by Mr Wittgenstein’s symbol. Read more at location 138
There is no way whatever, according to him, by which we can describe the totality of things that can be names, in other words, the totality of what there is in the world. In order to be able to do this we should have to know of some property which must belong to every thing by a logical necessity. It has been sought to find such a property in self-identity, but the conception of identity is subjected by Wittgenstein to a destructive criticism from which there seems no escape. The definition of identity by means of the identity of indiscernibles is rejected, because the identity of indiscernibles appears to be not a logically necessary principle. Read more at location 157
**** We here touch one instance of Wittgenstein’s fundamental thesis, that it is impossible to say anything about the world as a whole, and that whatever can be said has to be about bounded portions of the world. Read more at location 175
Logic, he says, fills the world. The boundaries of the world are also its boundaries. In logic, therefore, we cannot say, there is this and this in the world, but not that, for to say so would apparently presuppose that we exclude certain possibilities, and this cannot be the case, since it would require that logic should go beyond the boundaries of the world as if it could contemplate these boundaries from the other side also. What we cannot think we cannot think, therefore we also cannot say what we cannot think. This, he says, gives the key to solipsism. What Solipsism intends is quite correct, but this cannot be said, it can only be shown. That the world is my world appears in the fact that the boundaries of language (the only language I understand) indicate the boundaries of my world. Read more at location 186
**** (Note: Same idea as Schroedinger, subject and object are only One) The metaphysical subject does not belong to the world but is a boundary of the world. Read more at location 192
The real point is that in believing, desiring, etc., what is logically fundamental is the relation of a proposition considered as a fact to the fact which makes it true or false, and that this relation of two facts is reducible to a relation of their constituents. Thus the proposition does not occur at all in the same sense in which it occurs in a truth-function. Read more at location 223
**** detail is Mr Wittgenstein’s attitude towards the mystical. His attitude upon this grows naturally out of his doctrine in pure logic, according to which the logical proposition is a picture (true or false) of the fact, and has in common with the fact a certain structure. It is this common structure which makes it capable of being a picture of the fact, but the structure cannot itself be put into words, since it is a structure of words, as well as of the fact to which they refer. Everything, therefore, which is involved in the very idea of the expressiveness of language must remain incapable of being expressed in language, and is, therefore, inexpressible in a perfectly precise sense. This inexpressible contains, according to Mr Wittgenstein, the whole of logic and philosophy, he says, would be to confine oneself to propositions of the sciences, stated with all possible clearness and exactness, leaving philosophical assertions to the learner, and proving to him, whenever he made them, that they are meaningless. Read more at location 230
What causes hesitation is the fact that, after all, Mr Wittgenstein manages to say a good deal about what cannot be said, thus suggesting to the sceptical reader that possibly there may be some loophole through a hierarchy of languages, or by some other exit. The whole subject of ethics, for example, is placed by Mr Wittgenstein in the mystical, inexpressible region. Nevertheless he is capable of conveying his ethical opinions. His defence would be that what he calls the mystical can be shown, although it cannot be said. It may be that this defence is adequate, but, for my part, I confess that it leaves me with acertain sense of intellectual discomfort. Read more at location 239
These difficulties suggest to my mind some such possibility as this: that every language has, as Mr Wittgenstein says, a structure concerning which in the language, nothing can be said, but that there may be another language dealing with the structure of the first language, and having itself a new structure, and that to this hierarchy of languages there may be no limit. Mr Wittgenstein would of course reply that his whole theory is applicable unchanged to the totality of such languages. The only retort would be to deny that there is any such totality. The totalities concerning which Mr Wittgenstein holds that it is impossible to speak logically are nevertheless thought by him to exist, and are the subject-matter of his mysticism. The totality resulting from our hierarchy would be not merely logically inexpressible, but a fiction, a mere delusion, and in this way the supposed sphere of the mystical would be abolished. Read more at location 251
to have constructed a theory of logic which is not at any point obviously wrong is to have achieved a work of extraordinary difficulty and importance. This merit, in my opinion, belongs to Mr Wittgenstein’s book, and makes it one which no serious philosopher can afford to neglect. Bertrand Russell May 1922 Read more at location 261
Preface
**** 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. Read more at location 271
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 thnk 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. Read more at location 273
On the other hand the truth of the thoughts communicated here seems to me unassailable and definitive. I am, therefore, of the opinion that the problems have in essentials been finally solved. And if I am not mistaken in this, then the value of this work secondly consists in the fact that it shows how little has been done when these problems have been solved. L. W. Vienna, 1918 Read more at location 283
Tractatus Logico-Philosophicus
**** 1 The world is all that is the case. Read more at location 289
1.1 The world is the totality of facts, not of things.
1.11 The world is determined by the facts, and by their being all the facts.
1.12 For the totality of facts determines what is the case, and also whatever is not the case.
1.13 The facts in logical space are the world.
1.2 The world divides into facts.
1.21 Each item can be the case or not the case while everything else remains the same. Read more at location 289
**** 2 What is the case—a fact—is the existence of states of affairs. Read more at location 294
2.01 A state of affairs (a state of things) is a combination of objects (things).
2.011 It is essential to things that they should be possible constituents of states of affairs.
2.012 In logic nothing is accidental: if a thing can occur in a state of affairs, the possibility of the state of affairs must be written into the thing itself. Read more at location 295
2.021 Objects make up the substance of the world. That is why they cannot be composite.
2.0211 If they world had no substance, then whether a proposition had sense would depend on whether another proposition was true.
2.0212 In that case we could not sketch any picture of the world (true or false).
2.022 It is obvious that an imagined world, however difference it may be from the real one, must have something—a form—in common with it.
2.023 Objects are just what constitute this unalterable form.
2.0231 The substance of the world can only determine a form, and not any material properties. For it is only by means of propositions that material properties are represented—only by the configuration of objects that they are produced.
2.0232 In a manner of speaking, objects are colourless. Read more at location 317
2.024 The substance is what subsists independently of what is the case.
2.025 It is form and content.
2.0251 Space, time, colour (being coloured) are forms of objects.
2.026 There must be objects, if the world is to have unalterable form.
2.027 Objects, the unalterable, and the subsistent are one and the same.
2.0271 Objects are what is unalterable and subsistent; their configuration is what is changing and unstable. 2.0272 The configuration of objects produces states of affairs.
2.03 In a state of affairs objects fit into one another like the links of a chain.
2.031 In a state of affairs objects stand in a determinate relation to one another. Read more at location 330
**** 2.06 The existence and non-existence of states of affairs is reality. (We call the existence of states of affairs a positive fact, and their a negative fact.) Read more at location 343
(Note: for LW, a "fact" can be true OR false. It is, really, a proposition)
2.1 We picture facts to ourselves.
2.11 A picture presents a situation in logical space, the existence and of states of affairs.
2.12 A picture is a model of reality.
2.13 In a picture objects have the elements of the picture corresponding to them.
2.131 In a picture the elements of the picture are the representatives of objects.
2.14 What constitutes a picture is that its elements are related to one another in a determinate way.
2.141 A picture is a fact. Read more at location 347
2.18 What any picture, of whatever form, must have in common with reality, in order to be able to depict it—correctly or incorrectly—in any way at all, is logical form, i.e. the form of reality.
2.181 A picture whose pictorial form is logical form is called a logical picture.
2.182 Every picture is at the same time a logical one. (On the other hand, not every picture is, for example, a spatial one.)
2.19 Logical pictures can depict the world.
2.2 A picture has logico-pictorial form in common with what it depicts. Read more at location 370
2.22 What a picture represents it represents independently of its truth or falsity, by means of its pictorial form.
2.221 What a picture represents is its sense.
2.222 The agreement or disagreement or its sense with reality constitutes its truth or falsity.
2.223 In order to tell whether a picture is true or false we must compare it with reality.
2.224 It is impossible to tell from the picture alone whether it is true or false. Read more at location 379
***** 2.225 There are no pictures that are true a priori. Read more at location 383
**** 3 A logical picture of facts is a thought. Read more at location 384
3.001 ‘A state of affairs is thinkable’: what this means is that we can picture it to ourselves.
3.01 The totality of true thoughts is a picture of the world.
3.02 A thought contains the possibility of the situation of which it is the thought. What is thinkable is possible too. Read more at location 385
3.032 It is as impossible to represent in language anything that ‘contradicts logic’ as it is in geometry to represent by its coordinates a figure that contradicts the laws of space, or to give the coordinates of a point that does not exist.
3.0321 Though a state of affairs that would contravene the laws of physics can be represented by us spatially, one that would contravene the laws of geometry cannot. Read more at location 391
3.1 In a proposition a thought finds an expression that can be perceived by the senses.
3.11 We use the perceptible sign of a proposition (spoken or written, etc.) as a projection of a possible situation. The method of projection is to think of the sense of the proposition.
3.12 I call the sign with which we express a thought a propositional sign.And a proposition is a propositional sign in its projective relation to the world.
3.13 A proposition, therefore, does not actually contain its sense, but does contain the possibility of expressing it. (‘The content of a proposition’ means the content of a proposition that has sense.) A proposition contains the form, but not the content, of its sense.
3.14 What constitutes a propositional sign is that in its elements (the words) stand in a determinate relation to one another. A propositional sign is a fact. Read more at location 397
3.2 In a proposition a thought can be expressed in such a way that elements of the propositional sign correspond to the objects of the thought. 3.201 I call such elements ‘simple signs’, and such a proposition ‘complete analysed’. 3.202 The simple signs employed in propositions are called names. 3.203 A name means an object. The object is its meaning. (‘A’ is the same sign as ‘A’.) Read more at location 413
3.262 What signs fail to express, their application shows. What signs slur over, their application says clearly.
3.263 The meanings of primitive signs can be explained by means of elucidations. Elucidations are propositions that stood if the meanings of those signs are already known.
3.3 Only propositions have sense; only in the nexus of a proposition does a name have meaning. Read more at location 430
3.314 An expression has meaning only in a proposition. All variables can be construed as propositional variables. (Even variable names.) Read more at location 441
3.32 A sign is what can be perceived of a symbol.
3.321 So one and the same sign (written or spoken, etc.) can be common to two different symbols—in which case they will signify in different ways.
3.322 Our use of the same sign to signify two different objects can never indicate a common characteristic of the two, if we use it with two different modes of signification. For the sign, of course, is arbitrary. So we could choose two different signs instead, and then what would be left in common on the signifying side? Read more at location 453
3.324 In this way the most fundamental confusions are easily produced (the whole of philosophy is full of them).
3.325 In order to avoid such errors we must make use of a sign-language that excludes them by not using the same sign for different symbols and by not using in a superficially similar way signs that have different modes of signification: that is to say, a sign-language that is governed by logical grammar—by logical syntax. (The conceptual notation of Frege and Russell is such a language, though, it is true, it fails to exclude all mistakes.)
3.326 In order to recognize a symbol by its sign we must observe how it is used with a sense.
3.327 A sign does not determine a logical form unless it is taken together with its logico-syntactical employment. Read more at location 463
3.331 From this observation we turn to Russell’s ‘theory of types’. It can be seen that Russell must be wrong, because he had to mention the meaning of signs when establishing the rules for them.
3.332 No proposition can make a statement about itself, because a propositional sign cannot be contained in itself (that is the whole of the ‘theory of types’).
3.333 The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. Read more at location 472
3.4 A proposition determines a place in logical space. The existence of this logical place is guaranteed by the mere existence of the constituents—by the existence of the proposition with a sense. Read more at location 500
3.5 A propositional sign, applied and thought out, is a thought. Read more at location 506
**** 4 A thought is a proposition with a sense. Read more at location 507
4.001 The totality of propositions is language.
4.022 Man possesses the ability to construct languages capable of expressing every sense, without having any idea how each word has meaning or what its meaning is—just as people speak without knowing how the individual sounds are produced. Everyday language is a part of the human organism and is no less complicated than it. It is not humanly possible to gather immediately from it what the logic of language is. Language disguises thought. So much so, that from the outward form of the clothing it is impossible to infer the form of the thought beneath it, because the outward form of the clothing is not designed to reveal the form of the body, but for entirely different purposes. The tacit conventions on which the understanding of everyday language depends are enormously complicated.
4.003 Most of the propositions and questions to be found in philosophical works are not false but nonsensical. Consequently we cannot give any answer to questions of this kind, but can only point out that they are nonsensical. Most of the propositions and questions of philosophers arise from our failure to understand the logic of our language. (They belong to the same class as the question whether the good is more or less identical than the beautiful.) And it is not surprising that the deepest problems are in fact not problems at all.
4.0031 All philosophy is a ‘critique of language’ Read more at location 508
4.022 A proposition shows its sense. A proposition shows how things stand if it is true. And it says that they do so stand.
4.023 A proposition must restrict reality to two alternatives: yes or no. In order to do that, it must describe reality completely. A proposition is a description of a state of affairs. Just as a description of an object describes it by giving its external properties, so a proposition describes reality by its internal properties. A proposition constructs a world with the help of a logical scaffolding, so that one can actually see from the proposition how everything stands logically if it is true. One can draw inferences from a false proposition.
4.024 To understand a proposition means to know what is the case if it is true. (One can understand it, therefore, without knowing whether it is true.) It is understood by anyone who understands its constituents. Read more at location 541
4.03 A proposition must use old expressions to communicate a new sense. A proposition communicates a situation to us, and so it must be essentially connected with the situation. And the connexion is precisely that it is its logical picture. A proposition states something only in so far as it is a picture.
4.031 In a proposition a situation is, as it were, constructed by way of experiment. Instead of, ‘This proposition has such and such a sense, we can simply say, ‘This proposition represents such and such a situation’.
4.0311 One name stands for one thing, another for another thing, and they are combined with one another. In this way the whole group—like a tableau vivant—presents a state of affairs.
4.0312 The possibility of propositions is based on the principle that objects have signs as their representatives. My fundamental idea is that the ‘logical constants’ are not representatives; that there can be no representatives of the logic of facts. Read more at location 554
(Note: but language grossly oversimplifies in ways that can never be extracted from the language itself. From "forest", trees, branches, leaves, etc can be inferred, but not the rusty metal rod on the forest floor. And it is part of the picture of "forest" in some scenario.) 4.04 In a proposition there must be exactly as many distinguishable parts as in the situation that it represents. The two must possess the same logical (mathematical) multiplicity. (Compare Hertz’s Mechanics on dynamical models.) Read more at location 564
(Note: true or false, as here described, is strictly (internal) coherence, not correlation (to reality)) 4.05 Reality is compared with propositions.
4.06 A proposition can be true or false only in virtue of being a picture of reality.
4.061 It must not be overlooked that a proposition has a sense that is independent of the facts: otherwise one can easily suppose that true and false are relations of equal status between signs and what they signify. Read more at location 574
4.1 Propositions represent the existence and non-existence of states of affairs.
4.11 The totality of true propositions is the whole of natural science (or the whole corpus of the natural sciences). Read more at location 599
4.1212 What can be shown, cannot be said. Read more at location 622
4.125 The existence of an internal relation between possible situations expresses itself in language by means of an internal relation between the propositions representing them.
4.1251 Here we have the answer to the vexed question ‘whether all relations are internal or external’.
4.1252 I call a series that is ordered by an internal relation a series of forms. The order of the number-series is not governed by an external relation but by an internal relation. Read more at location 639
4.127 The propositional variable signifies the formal concept, and its values signify the objects that fall under the concept.
4.1271 Every variable is the sign for a formal concept. For every variable represents a constant form that all its values possess, and this can be regarded as a formal property of those values.
4.1272 Thus the variable name ‘x’ is the proper sign for the pseudo-concept object. Wherever the word ‘object’ (‘thing’, etc.) is correctly used, it is expressed in conceptual notation by a variable name. For example, in the proposition, ‘There are 2 objects which. . .’, it is expressed by ‘ (dx,y) ... ‘. Wherever it is used in a different way, that is as a proper , nonsensical pseudo-propositions are the result. So one cannot say, for example, ‘There are objects’, as one might say, ‘There are books’. And it is just as impossible to say, ‘There are 100 objects’, or, ‘There are !0 objects’. And it is nonsensical to speak of the total number of objects. The same applies to the words ‘complex’, ‘fact’, ‘function’, ‘number’, etc. They all signify formal concepts, and are represented in conceptual notation by variables, not by functions or classes (as Frege and Russell believed). ‘1 is a number’, ‘There is only one zero’, and all similar expressions are nonsensical. (It is just as nonsensical to say, ‘There is only one 1', as it would be to say, ‘2 + 2 at 3 o’clock equals 4'.) Read more at location 652
4.1274 To ask whether a formal concept exists is nonsensical. For no proposition can be the answer to such a question. (So, for example, the question, ‘Are there unanalysable subject-predicate propositions?’ cannot be asked.)
4.128 Logical forms are without number. Hence there are no preeminent numbers in logic, and hence there is no possibility of philosophical monism or dualism, etc.
4.2 The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs.
4.21 The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs. Read more at location 672
4.2211 Even if the world is infinitely complex, so that every fact consists of infinitely many states of affairs and every state of affairs is composed of infinitely many objects, there would still have to be objects and states of affairs.
4.23 It is only in the nexus of an elementary proposition that a name occurs in a proposition. Read more at location 681
4.25 If an elementary proposition is true, the state of affairs exists: if an elementary proposition is false, the state of affairs does not exist.
4.26 If all true elementary propositions are given, the result is a complete description of the world. The world is completely described by giving all elementary propositions, and adding which of them are true and which false. For n states of affairs, there are possibilities of existence and non-existence. Of these states of affairs any combination can exist and the remainder not exist. Read more at location 696
4.3 Truth-possibilities of elementary propositions mean Possibilities of existence and non-existence of states of affairs.
4.31 We can represent truth-possibilities by schemata of the following kind (‘T’ means ‘true’, ‘F’ means ‘false’; the rows of ‘T’s’ and ‘F’s’ under the row of elementary propositions symbolize their truth-possibilities in a way that can easily be understood):
4.4 A proposition is an expression of agreement and disagreement with of elementary propositions.
4.41 Truth-possibilities of elementary propositions are the conditions of the truth and falsity of propositions. Read more at location 702
4.44 The sign that results from correlating the mark ‘I” with is a propositional sign. Read more at location 717
4.461 Propositions show what they say; tautologies and contradictions show that they say nothing. A tautology has no truth-conditions, since it is unconditionally true: and a contradiction is true on no condition. Tautologies and contradictions lack sense. (Like a point from which two arrows go out in opposite directions to one another.) (For example, I know nothing about the weather when I know that it is either raining or not raining.)
4.46211 Tautologies and contradictions are not, however, nonsensical. They are part of the symbolism, much as ‘0' is part of the symbolism of arithmetic.
4.462 Tautologies and contradictions are not pictures of reality. They do not represent any possible situations. For the former admit all possible situations, and latter none . In a tautology the conditions of agreement with the world—the representational relations—cancel one another, so that it does not stand in any representational relation to reality. Read more at location 732
4.5 It now seems possible to give the most general propositional form: that is, to give a description of the propositions of any sign-language whatsoever in such a way that every possible sense can be expressed by a symbol satisfying the description, and every symbol satisfying the description can express a sense, provided that the meanings of the names are suitably chosen. It is clear that only what is essential to the most general propositional form may be included in its description—for otherwise it would not be the most general form. The existence of a general propositional 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 a proposition is: This is how things stand. Read more at location 753
**** 5 A proposition is a truth-function of elementary propositions. (An elementary proposition is a truth-function of itself.) Read more at location 764
5.01 Elementary propositions are the truth-arguments of propositions.
5.02 The arguments of functions are readily confused with the affixes of names. For both arguments and affixes enable me to recognize the meaning of the signs containing them. Read more at location 765
5.1 Truth-functions can be arranged in series. That is the foundation of the theory of probability. 5.101 The truth-functions of a given number of elementary propositions can always be set out in a schema of the following kind: (TTTT) (p, q) Tautology (If p then p, and if q then q.) (p z p . q z q) (FTTT) (p, q) In words : Not both p and q. (P(p . q)) (TFTT) (p, q) “ : If q then p. (q z p) (TTFT) (p, q) “ : If p then q. (p z q) (TTTF) (p, q) “ : p or q. (p C q) (FFTT) (p, q) “ : Not g. (Pq) (FTFT) (p, q) “ : Not p. (Pp) (FTTF) (p, q) “ : p or q, but not both. (p . Pq : C : q . Pp) (TFFT) (p, q) “ : If p then p, and if q then p. (p + q) (TFTF) (p, q) “ : p (TTFF) (p, q) “ : q (FFFT) (p, q) “ : Neither p nor q. (Pp . Pq or p | q) (FFTF) (p, q) “ : p and not q. (p . Pq) (FTFF) (p, q) “ : q and not p. (q . Pp) (TFFF) (p,q) “ : q and p. (q . p) (FFFF) (p, q) Contradiction (p and not p, and q and not q.) (p . Pp . q . Pq) I will give the name truth-grounds of a proposition to those truth-possibilities of its truth-arguments that make it true. Read more at location 773
5.132 If p follows from q, I can make an inference from q to p, deduce p from q. The nature of the inference can be gathered only from the two propositions. They themselves are the only possible justification of the inference. ‘Laws of inference’, which are supposed to justify inferences, as in the works of Frege and Russell, have no sense, and would be superfluous. Read more at location 800
5.133 All deductions are made a priori.
5.134 One elementary proposition cannot be deduced form another.
5.135 There is no possible way of making an inference form the existence of one situation to the existence of another, entirely different situation.
5.136 There is no causal nexus to justify such an inference.
5.1361 We cannot infer the events of the future from those of the present. Belief in the causal nexus is superstition.
5.1362 The freedom of the will consists in the impossibility of knowing actions that still lie in the future. We could know them only if causality were an inner necessity like that of logical inference.—The connexion between knowledge and what is known is that of logical necessity. (‘A knows that p is the case’, has no sense if p is a tautology.) Read more at location 803
5.143 Contradiction is that common factor of propositions which no proposition has in common with another. Tautology is the common factor of all propositions that have nothing in common with one another. Contradiction, one might say, vanishes outside all propositions: tautology vanishes inside them. Contradiction is the outer limit of propositions: tautology is the unsubstantial point at their centre. Read more at location 815
5.2 The structures of propositions stand in internal relations to one another. Read more at location 840
5.232 The internal relation by which a series is ordered is equivalent to the operation that produces one term from another.
5.233 Operations cannot make their appearance before the point at which one proposition is generated out of another in a logically meaningful way; i.e. the point at which the logical construction of propositions begins.
5.234 Truth-functions of elementary propositions are results of operations with elementary propositions as bases. (These operations I call .) Read more at location 846
5.3 All propositions are results of truth-operations on elementary propositions. A truth-operation is the way in which a truth-function is produced out of elementary propositions. It is of the essence of that, just as elementary propositions yield a truth-function of themselves, so too in the same way truth-functions yield a further . When a truth-operation is applied to truth-functions of elementary propositions, it always generates another truth-function of elementary propositions, another proposition. When a truth-operation is applied to the results of truth-operations on elementary propositions, there is always a single operation on elementary propositions that has the same result. Every proposition is the result of truth-operations on elementary propositions. Read more at location 872
5.32 All truth-functions are results of successive applications to elementary propositions of a finite number of truth-operations.
5.4 At this point it becomes manifest that there are no ‘logical objects’ or ‘logical constants’ (in Frege’s and Russell’s sense).
5.41 The reason is that the results of truth-operations on truth-functions are always identical whenever they are one and the same truth-function of elementary propositions. Read more at location 880
5.453 All numbers in logic stand in need of justification. Or rather, it must become evident that there are no numbers in logic. There are no numbers.
5.454 In logic there is no co-ordinate status, and there can be no classification. In logic there can be no distinction between the general and the specific.
5.4541 The solutions of the problems of logic must be simple, since they set the standard of simplicity. Men have always had a presentiment that there must be a realm in which the answers to questions are symmetrically combined—a priori—to form a self-contained system. A realm subject to the law: Simplex sigillum veri. Read more at location 912
5.47 It is clear that whatever we can say in advance about the form of all propositions, we must be able to say all at once . An elementary proposition really contains all logical operations in itself. For ‘fa’ says the same thing as ‘(dx) . fx . x = a’ Wherever there is compositeness, argument and function are present, and where these are present, we already have all the logical constants. One could say that the sole logical constant was what all propositions, by their very nature, had in common with one another. But that is the general propositional form.
5.471 The general propositional form is the essence of a proposition.
5.4711 To give the essence of a proposition means to give the essence of all description, and thus 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. Read more at location 924
5.4731 Self-evidence, which Russell talked about so much, can become dispensable in logic, only because language itself prevents every logical mistake.—What makes logic a priori is the impossibility of illogical thought.
5.4732 We cannot give a sign the wrong sense.
5.47321 Occam’s maxim is, of course, not an arbitrary rule, nor one that is justified by its success in practice: its point is that unnecessary units in a sign-language mean nothing. Signs that serve one purpose are logically equivalent, and signs that serve none are logically meaningless. Read more at location 935
5.5 Every truth-function is a result of successive applications to elementary propositions of the operation ‘(——-T)(E, ....)’. This operation negates all the propositions in the right-hand pair of brackets, and I call it the negation of those propositions. Read more at location 949
**** (Note: attempt to gain correlation? but quantum physics seems to negate logical laws.) 5.511 How can logic—all-embracing logic, which mirrors the world—use such peculiar crotchets and contrivances? Only because they are all connected with one another in an infinitely fine network, the great mirror. Read more at location 963
(Note: law of identity) 5.5302 Russell’s definition of ‘=’ is inadequate, because according to it we cannot say that two objects have all their properties in common. (Even if this proposition is never correct, it still has sense .)
***********. (Mirror Neitzsche) 5.5303 Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all. Read more at location 1006
5.55 We now have to answer a priori the question about all the possible forms of elementary propositions. Elementary propositions consist of names. Since, however, we are unable to give the number of names with different meanings, we are also unable to give the composition of elementary propositions. Read more at location 1042
(Note: interesting. anti Hume. I disagree with Wittgenstein. I think logic is inductively constructed from experience before it is ever employed deductively. Experience provides the framework of sequential and causal relation in thought.) 5.551 Our fundamental principle is that whenever a question can be decided by logic at all it must be possible to decide it without more ado. (And if we get into a position where we have to look at the world for an answer to such a problem, that shows that we are on a completely wrong track.)
5.552 The ‘experience’ that we need in order to understand logic is not that something or other is the state of things, but that something is : that, however, is not an experience. Logic is prior to every experience—that something is so . It is prior to the question ‘How?’ not prior to the question ‘What?’ Read more at location 1044
5.61 Logic pervades the world: the limits of the world are also its limits. So we cannot say in logic, ‘The world has this in it, and this, but not that.’ For that would appear to presuppose that we were excluding certain possibilities, and this cannot be the case, since it would require that logic should go beyond the limits of the world; for only in that way could it view those limits from the other side as well. We cannot think what we cannot think; so what we cannot think we cannot say either.
5.62 This remark provides the key to the problem, how much truth there is in solipsism. For what the solipsist means is quite correct; only it cannot be said , but makes itself manifest. The world is my world: this is manifest in the fact that the limits of language (of that language which alone I understand) mean the limits of my world. Read more at location 1071
****** (Note: subject and object as one, immersive nonduality) 5.621 The world and life are one.
5.63 I am my world. (The microcosm.)
5.631 There is no such thing as the subject that thinks or entertains ideas. If I wrote a book called The World as l found it , I should have to include a report on my body, and should have to say which parts were subordinate to my will, and which were not, etc., this being a method of isolating the subject, or rather of showing that in an important sense there is no subject; for it alone could not be mentioned in that book.—
5.632 The subject does not belong to the world: rather, it is a limit of the world.
5.633 Where in the world is a metaphysical subject to be found? You will say that this is exactly like the case of the eye and the visual field. But really you do not see the eye. And nothing in the visual field allows you to infer that it is seen by an eye. Read more at location 1078
5.64 Here it can be seen that solipsism, when its implications are followed out strictly, coincides with pure realism. The self of solipsism shrinks to a point without extension, and there remains the reality co-ordinated with it. Read more at location 1088
**** 5.641 Thus there really is a sense in which philosophy can talk about the self in a non-psychological way. What brings the self into philosophy is the fact that ‘the world is my world’. The philosophical self is not the human being, not the human body, or the human soul, with which psychology deals, but rather the metaphysical subject, the limit of the world—not a part of it. Read more at location 1090
**** 6 The general form of a truth-function is [p, E, N(E)]. This is the general form of a proposition. Read more at location 1093
6.001 What this says is just that every proposition is a result of successive applications to elementary propositions of the operation N(E) 6.002 If we are given the general form according to which propositions are constructed, then with it we are also given the general form according to which one proposition can be generated out of another by means of an operation. Read more at location 1094
6.022 The concept of number is simply what is common to all numbers, the general form of a number. The concept of number is the variable number. And the concept of numerical equality is the general form of all particular cases of numerical equality. Read more at location 1103
******* 6.1 The propositions of logic are tautologies.
6.11 Therefore the propositions of logic say nothing. (They are the analytic propositions.) Read more at location 1108
6.113 It is the peculiar mark of logical propositions that one can recognize that they are true from the symbol alone, and this fact contains in itself the whole philosophy of logic. And so too it is a very important fact that the truth or falsity of non-logical propositions cannot be recognized from the propositions alone. Read more at location 1115
6.121 The propositions of logic demonstrate the logical properties of propositions by combining them so as to form propositions that say nothing. This method could also be called a zero-method. In a logical proposition, propositions are brought into equilibrium with one another, and the state of equilibrium then indicates what the logical constitution of these propositions must be.
6.122 It follows from this that we can actually do without logical propositions; for in a suitable notation we can in fact recognize the formal properties of propositions by mere inspection of the propositions themselves. Read more at location 1134
6.1222 This throws some light on the question why logical propositions cannot be confirmed by experience any more than they can be refuted by it. Not only must a proposition of logic be irrefutable by any possible experience, but it must also be unconfirmable by any possible experience.
6.1223 Now it becomes clear why people have often felt as if it were for us to ‘postulate ‘ the ‘truths of logic’. The reason is that we can postulate them in so far as we can postulate an adequate notation.
6.1224 It also becomes clear now why logic was called the theory of forms and of inference.
6.123 Clearly the laws of logic cannot in their turn be subject to laws of logic. (There is not, as Russell thought, a special law of contradiction for each ‘type’; one law is enough, since it is not applied to itself.) Read more at location 1141
6.1233 It is possible to imagine a world in which the axiom of reducibility is not valid. It is clear, however, that logic has nothing to do with the question whether our world really is like that or not.
6.124 The propositions of logic describe the scaffolding of the world, or rather they represent it. They have no ‘subject-matter’. They presuppose that names have meaning and elementary propositions sense; and that is their connexion with the world. It is clear that something about the world must be indicated by the fact that certain combinations of symbols—whose essence involves the possession of a determinate character—are tautologies. This contains the decisive point. We have said that some things are arbitrary in the symbols that we use and that some things are not. In logic it is only the latter that express: but that means that logic is not a field in which we express what we wish with the help of signs, but rather one in which the nature of the absolutely necessary signs speaks for itself. If we know the logical syntax of any sign-language, then we have already been given all the propositions of logic. Read more at location 1153
6.126 One can calculate whether a proposition belongs to logic, by calculating the logical properties of the symbol. And this is what we do when we ‘prove’ a logical proposition. For, without bothering about sense or meaning, we construct the logical proposition out of others using only rules that deal with signs . The proof of logical propositions consists in the following process: we produce them out of other logical propositions by successively applying certain operations that always generate further tautologies out of the initial ones. (And in fact only tautologies follow from a tautology.) Of course this way of showing that the propositions of logic are tautologies is not at all essential to logic, if only because the propositions from which the proof starts must show without any proof that they are tautologies.
6.1261 In logic process and result are equivalent. (Hence the absence of surprise.)
6.1262 Proof in logic is merely a mechanical expedient to facilitate the recognition of tautologies in complicated cases. Read more at location 1164
6.127 All the propositions of logic are of equal status: it is not the case that some of them are essentially derived propositions. Every tautology itself shows that it is a tautology.
6.1271 It is clear that the number of the ‘primitive propositions of logic’ is arbitrary, since one could derive logic from a single primitive proposition, e.g. by simply constructing the logical product of Frege’s primitive propositions. (Frege would perhaps say that we should then no longer have an immediately self-evident primitive proposition. But it is remarkable that a thinker as rigorous as Frege appealed to the degree of self-evidence as the criterion of a logical proposition.) Read more at location 1177
**** 6.13 Logic is not a body of doctrine, but a mirror-image of the world. Logic is transcendental.
6.2 Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-propositions. Read more at location 1183
6.22 The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics. 6.23 If two expressions are combined by means of the sign of equality, that means that they can be substituted for one another. But it must be manifest in the two expressions themselves whether this is the case or not. When two expressions can be substituted for one another, that characterizes their logical form. Read more at location 1189
6.2331 The process of calculating serves to bring about that intuition. Calculation is not an experiment. 6.234 Mathematics is a method of logic. Read more at location 1204
6.3 The exploration of logic means the exploration of everything that is subject to law . And outside logic everything is accidental. 6.31 The so-called law of induction cannot possibly be a law of logic, since it is obviously a proposition with sense.—-Nor, therefore, can it be an a priori law. 6.32 The law of causality is not a law but the form of a law. 6.321 ‘Law of causality’—that is a general name. And just as in mechanics, for example, there are ‘minimum-principles’, such as the law of least action, so too in physics there are causal laws, laws of the causal form. Read more at location 1211
6.34 All such propositions, including the principle of sufficient reason, tile laws of continuity in nature and of least effort in nature, etc. etc.—all these are a priori insights about the forms in which the propositions of science can be cast.
6.341 Newtonian mechanics, for example, imposes a unified form on the description of the world. Let us imagine a white surface with irregular black spots on it. We then say that whatever kind of picture these make, I can always approximate as closely as I wish to the description of it by covering the surface with a sufficiently fine square mesh, and then saying of every square whether it is black or white. In this way I shall have imposed a unified form on the description of the surface. The form is optional, since I could have achieved the same result by using a net with a triangular or hexagonal mesh. Possibly the use of a triangular mesh would have made the description simpler: that is to say, it might be that we could describe the surface more accurately with a coarse triangular mesh than with a fine square mesh (or conversely), and so on. The different nets correspond to different systems for describing the world. Mechanics determines one form of description of the world by saying that all propositions used in the description of the world must be obtained in a given way from a given set of propositions—the axioms of mechanics. It thus supplies the bricks for building the edifice of science, and it says, ‘Any building that you want to erect, whatever it may be, must somehow be constructed with these bricks, and with these alone.’ (Just as with the number-system we must be able to write down any number we wish, so with the system of mechanics we must be able to write down any proposition of physics that we wish.)
6.342 And now we can see the relative position of logic and mechanics. (The net might also consist of more than one kind of mesh: e.g. we could use both triangles and hexagons.) The possibility of describing a picture like the one mentioned above with a net of a given form tells us nothing about the picture. (For that is true of all such pictures.) But what does characterize the picture is that it can be described completely by a particular net with a particular size of mesh. Similarly the possibility of describing the world by means of Newtonian mechanics tells us nothing about the world: but what does tell us something about it is the precise way in which it is possible to describe it by these means. We are also told something about the world by the fact that it can be described more simply with one system of mechanics than with another.
6.343 Mechanics is an attempt to construct according to a single plan all the true propositions that we need for the description of the world. Read more at location 1233
**** 6.35 Although the spots in our picture are geometrical figures, nevertheless geometry can obviously say nothing at all about their actual form and position. The network, however, is purely geometrical; all its properties can be given a priori. Laws like the principle of sufficient reason, etc. are about the net and not about what the net describes. Read more at location 1244
6.36 If there were a law of causality, it might be put in the following way: There are laws of nature. But of course that cannot be said: it makes itself manifest.
6.361 One might say, using Hertt:’s terminology, that only connexions that are subject to law are thinkable. Read more at location 1247
6.363 The procedure of induction consists in accepting as true the simplest law that can be reconciled with our experiences.
6.3631 This procedure, however, has no logical justification but only a psychological one. It is clear that there are no grounds for believing that the simplest eventuality will in fact be realized. Read more at location 1259
**** 6.371 The whole modern conception of the world is founded on the illusion that the so-called laws of nature are the explanations of natural phenomena.
6.372 Thus people today stop at the laws of nature, treating them as something inviolable, just as God and Fate were treated in past ages. And in fact both are right and both wrong: though the view of the ancients is clearer in so far as they have a clear and acknowledged terminus, while the modern system tries to make it look as if everything were explained. Read more at location 1265
6.373 The world is independent of my will. 6.374 Even if all that we wish for were to happen, still this would only be a favour granted by fate, so to speak: for there is no logical connexion between the will and the world, which would guarantee it, and the supposed physical connexion itself is surely not something that we could will. Read more at location 1269
(Note: fascinating example given quantum superposition) 6.375 Just as the only necessity that exists is logical necessity, so too the only impossibility that exists is logical impossibility.
6.3751 For example, the simultaneous presence of two colours at the same place in the visual field is impossible, in fact logically impossible, since it is ruled out by the logical structure of colour. Let us think how this contradiction appears in physics: more or less as follows—a particle cannot have two velocities at the same time; that is to say, it cannot be in two places at the same time; that is to say, particles that are 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 statement that a point in the visual field has two different colours at the same time is a contradiction.) Read more at location 1272
**** 6.4 All propositions are of equal value.
6.41 The sense of the world must lie outside the world. In the world everything is as it is, and everything happens as it does happen: in it no value exists—and if it did exist, it would have no value. If there is any value that does have value, it must lie outside the whole sphere of what happens and is the case. For all that happens and is the case is accidental. What makes it non-accidental cannot lie within the world, since if it did it would itself be accidental. It must lie outside the world.
6.42 So too it is impossible for there to be propositions of ethics. Propositions can express nothing that is higher.
6.421 It is clear that ethics cannot be put into words. Ethics is transcendental. (Ethics and aesthetics are one and the same.) Read more at location 1278
**** (Note: the reward for ethics must be self-contained, not external or transcendent) 6.422 When an ethical law of the form, ‘Thou shalt ...’ is laid down, one’s first thought is, ‘And what if I do, not do it?’ It is clear, however, that ethics has nothing to do with punishment and reward in the usual sense of the terms. So our question about the consequences of an action must be unimportant.—At least those consequences should not be events. For there must be something right about the question we posed. There must indeed be some kind of ethical reward and ethical punishment, but they must reside in the action itself. (And it is also clear that the reward must be something pleasant and the punishment something unpleasant.) Read more at location 1285
6.423 It is impossible to speak about the will in so far as it is the subject of ethical attributes. And the will as a phenomenon is of interest only to psychology. 6.43 If the good or bad exercise of the will does alter the world, it can alter only the limits of the world, not the facts—not what can be expressed by means of language. In short the effect must be that it becomes an altogether different world. It must, so to speak, wax and wane as a whole. The world of the happy man is a different one from that of the unhappy man. Read more at location 1290
**** 6.431 So too at death the world does not alter, but comes to an end.
*********** 6.4311 Death is not an event in life: we do not live to experience death. If we take eternity to mean not infinite temporal duration but timelessness, then eternal life belongs to those who live in the present. Our life has no end in just the way in which our visual field has no limits. Read more at location 1294
6.432 How things are in the world is a matter of complete indifference for what is higher. God does not reveal himself in the world.
6.4321 The facts all contribute only to setting the problem, not to its solution. Read more at location 1302
********* 6.44 It is not how things are in the world that is mystical, but that it exists.
6.45 To view the world sub specie aeterni is to view it as a whole—a limited whole. Feeling the world as a limited whole—it is this that is mystical.
6.5 When the answer cannot be put into words, neither can the question be put into words. The riddle does not exist. If a question can be framed at all, it is also possible to answer it. Read more at location 1304
6.51 Scepticism is not irrefutable, but obviously nonsensical, when it tries to raise doubts where no questions can be asked. For doubt can exist only where a question exists, a question only where an answer exists, and an answer only where something can be said. Read more at location 1308
**** 6.52 We feel that even when all possible scientific questions have been answered, the problems of life remain completely untouched. Of course there are then no questions left, and this itself is the answer.
6.521 The solution of the problem of life is seen in the vanishing of the problem. (Is not this the reason why those who have found after a long period of doubt that the sense of life became clear to them have then been unable to say what constituted that sense?)
6.522 There are, indeed, things that cannot be put into words. They make themselves manifest. They are what is mystical. Read more at location 1310
6.53 The correct method in philosophy would really be the following: to say nothing except what can be said, i.e. propositions of natural science—i.e. something that has nothing to do with philosophy—and then, whenever someone else wanted to say something metaphysical, to demonstrate to him that he had failed to give a meaning to certain signs in his propositions. Although it would not be satisfying to the other person—he would not have the feeling that we were teaching him philosophy—this method would be the only strictly correct one.
************ 6.54 My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.) Read more at location 1315
**** (Note: everything that is meaningful is outside of logic (which does not relate the "objective" reality anyway), and thus cannot be spoken of meaningfully or propositionally or logically. It is strictly private and personal. In the end, even the ladder of logic that we use to reach this conclusion must be discarded as an untrustworthy guide to higher truth.)
********** 7 What we cannot speak about we must pass over in silence. Read more at location 1321
SUMMARY:
1 The world is all that is the case.
2 What is the case—a fact—is the existence of states of affairs.
3 A logical picture of facts is a thought.
4 A thought is a proposition with a sense.
5 A proposition is a truth-function of elementary propositions. (An elementary proposition is a truth-function of itself.)
6 The general form of a truth-function is [p, E, N(E)]. This is the general form of a proposition.
7 What we cannot speak about we must pass over in silence.