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Appendix from H.O. Mounce, Wittgenstein's Tractatus (Links to English text)

Facts 1-1.21
'The world is all that is the case.' The propositions that follow this are its elucidation. Thus 'all that is the case' is the totality of facts, not of things. The difference between 'facts' and 'things' is elucidated by the statement that it is the facts in logical space that are the world.

+-------- 1 ------- 1.1 ----- 1.11
|         |         |
|         |         +-------- 1.12
|         |         |
|         |         +-------- 1.13
|         |
|         +-------- 1.2 ----- 1.21
|

States of affairs 2-2.0141
'What is the case - a fact - is the existence of states of affairs.' The section beginning with this proposition is a further elucidation of the propositions falling under one. A fact is a state of affairs; it is something complex. The things that make up the complex appear in a certain combination; but they might have been combined differently. In logic, however, 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. A speck in the visual field need not be red, but it must have some colour; notes must have some pitch, objects of touch some degree of hardness, and so on. Objects exist in logical space, so that if I know an object I also know all its possible occurrences in states of affairs.

|
+-------- 2 ------- 2.0 ----- 2.01 ---- 2.011
|         |         |         |
|         |         |         +-------- 2.012 --- 2.0121
|         |         |         |         |
|         |         |         |         +-------- 2.0122
|         |         |         |         |
|         |         |         |         +-------- 2.0123 -- 2.01231
|         |         |         |         |
|         |         |         |         +-------- 2.0124
|         |         |         |
|         |         |         +-------- 2.013 --- 2.0131
|         |         |         |
|         |         |         +-------- 2.014 --- 2.0141
|         |         |

Objects 2.02-2.063
'Objects are simple.' A state of affairs is complex, so every statement about a state of affairs can be resolved into a statement about its constituents. But every statement about its constituents cannot itself be resolved into a further statement, otherwise there would be no contact between language and the world. At some point therefore words must just stand for objects, these objects being simple. Only when words stand for objects can something be said. Moreover whatever is said involves complexity, involves the combination of objects. To say that something is red, for example, is to represent a combination of objects, a state of affairs. (This is why an object in itself is, in a manner of speaking, colourless.) A combination of objects can be represented because it is possible for the objects so represented to appear in that combination. This is a matter of logic. What is the case, reality, the world depends on which among these possible states of affairs actually exist. This is a matter of fact. Logic determines only what is possible; it cannot determine what is the case.

|         |         |
|         |         +-------- 2.02 ---- 2.020 --- 2.0201
|         |         |         |
|         |         |         +-------- 2.021 --- 2.0211
|         |         |         |         |
|         |         |         |         +-------- 2.0212
|         |         |         |
|         |         |         +-------- 2.022
|         |         |         |
|         |         |         +-------- 2.023 --- 2.0231
|         |         |         |         |
|         |         |         |         +-------- 2.0232
|         |         |         |         |
|         |         |         |         +-------- 2.0233 -- 2.02331
|         |         |         |
|         |         |         +-------- 2.024
|         |         |         |
|         |         |         +-------- 2.025 --- 2.0251
|         |         |         |
|         |         |         +-------- 2.026
|         |         |         |
|         |         |         +-------- 2.027 --- 2.0271
|         |         |                   |
|         |         |                   +-------- 2.0272
|         |         |
|         |         +-------- 2.03 ---- 2.031
|         |         |         |
|         |         |         +-------- 2.032
|         |         |         |
|         |         |         +-------- 2.033
|         |         |         |
|         |         |         +-------- 2.034
|         |         |
|         |         +-------- 2.04
|         |         |
|         |         +-------- 2.05
|         |         |
|         |         +-------- 2.06 ---- 2.061
|         |                   |
|         |                   +-------- 2.062
|         |                   |
|         |                   +-------- 2.063
|         |

Pictures 2.1-2.225
'We picture facts to ourselves.' A proposition pictures; it is the representation of a possible state of affairs, of what might be so. The elements of a proposition stand for objects; they are their representatives. These elements are related to one another in a determinate form. The form that the elements take constitutes the picturing. The fact that the elements of the picture are related in a determinate form is a representation of how things are in the world. So there must be something in common between the form of the proposition and the form of the objects it represents. But what the proposition pictures is a possible state of affairs; it cannot picture its own form. Moreover, whether or not it is true is a different question from what it pictures. In order to know whether it is true one must first know what it pictures and then compare it with reality. It will picture the same thing whether or not it is true.

|         |
|         +-------- 2.1 ----- 2.11
|         |         |
|         |         +-------- 2.12
|         |         |
|         |         +-------- 2.13 ---- 2.131
|         |         |
|         |         +-------- 2.14 ---- 2.141
|         |         |
|         |         +-------- 2.15 ---- 2.151 --- 2.1511
|         |         |                   |
|         |         |                   +-------- 2.1512 -- 2.15121
|         |         |                   |
|         |         |                   +-------- 2.1513
|         |         |                   |
|         |         |                   +-------- 2.1514
|         |         |                   |
|         |         |                   +-------- 2.1515
|         |         |
|         |         +-------- 2.16 ---- 2.161
|         |         |
|         |         +-------- 2.17 ---- 2.171
|         |         |         |
|         |         |         +-------- 2.172
|         |         |         |
|         |         |         +-------- 2.173
|         |         |         |
|         |         |         +-------- 2.174
|         |         |
|         |         +-------- 2.18 ---- 2.181
|         |         |         |
|         |         |         +-------- 2.182
|         |         |
|         |         +-------- 2.19
|         |
|         +-------- 2.2 ----- 2.20 ---- 2.201
|                   |         |
|                   |         +-------- 2.202
|                   |         |
|                   |         +-------- 2.203
|                   |
|                   +-------- 2.21
|                   |
|                   +-------- 2.22 ---- 2.221
|                             |
|                             +-------- 2.222
|                             |
|                             +-------- 2.223
|                             |
|                             +-------- 2.224
|                             |
|                             +-------- 2.225
|

Thought 3-3.13
'A logical picture of facts is a thought.' This can be read the other way round: a thought is a logical picture of facts. In other words a thought is a thought only when it has the logical structure of a proposition or picture. (But this interpretation is controversial. The student is advised to look at pages 30-3 of this book where the matter is discussed in detail.)

|
+-------- 3 ------- 3.0 ----- 3.00 ---- 3.001
|         |         |
|         |         +-------- 3.01
|         |         |
|         |         +-------- 3.02
|         |         |
|         |         +-------- 3.03 ---- 3.031
|         |         |         |
|         |         |         +-------- 3.032 --- 3.0321
|         |         |
|         |         +-------- 3.04
|         |         |
|         |         +-------- 3.05
|         |
|         +-------- 3.1 ----- 3.11
|         |         |
|         |         +-------- 3.12
|         |         |
|         |         +-------- 3.13
|         |         |

Proposition and name 3.14-3.261
'What constitutes a propositional sign is that in it its elements (the words) stand in a determinate relation to one another.' The sense of a proposition lies in its structure. In this it is to be contrasted with a name. A name does not posses pictorial form; it just stands for an object in the world, which is its meaning. But the meaning, or better, the sense of a proposition is not something for which it stands in the world. The sense of a proposition is not external to that proposition as the meaning of a name is external to that name. This is why a proposition has the same sense whether or not it is true, whether or not it corresponds to something in the world.

|         |         |
|         |         +-------- 3.14 ---- 3.141
|         |                   |
|         |                   +-------- 3.142
|         |                   |
|         |                   +-------- 3.143 --- 3.1431
|         |                   |         |
|         |                   |         +-------- 3.1432
|         |                   |
|         |                   +-------- 3.144
|         |
|         +-------- 3.2 ----- 3.20 ---- 3.201
|         |         |         |
|         |         |         +-------- 3.202
|         |         |         |
|         |         |         +-------- 3.203
|         |         |
|         |         +-------- 3.21
|         |         |
|         |         +-------- 3.22 ---- 3.221
|         |         |
|         |         +-------- 3.23
|         |         |
|         |         +-------- 3.24
|         |         |
|         |         +-------- 3.25 ---- 3.251
|         |         |
|         |         +-------- 3.26 ---- 3.261
|         |                   |

Logic and convention 3.262-3.5
'What signs fail to express, their application shows. What signs slur over, their application says clearly.' It is only because a proposition is a collection of signs having logical structure that it has a sense and it is only within such a structure that name has meaning. But logical structure is not always clearly revealed by signs. For example, one and the same sign, as written or spoken, may have different uses, as when the word 'is' figures sometimes as the copula, sometimes as a sign of identity, and sometimes as an expression of existence. Here the word 'is' stands really stands for three different symbols and this is evident in its application, in there being three quite different rules for its use. Thus logical form or structure is revealed not by the way signs look or sound, not by what is conventional, but by their application. In this way logic distinguishes itself from what is arbitrary or conventional. For although it is an arbitrary matter that the word 'is' should be used at all, it is not an arbitrary matter that certain things should follow when it is given one use and not when it is given another, that one can say some things when it is used as the copula but not when it is used as a sign of identity. The advantage of a formal notation or symbolism is that it makes that clear. In an adequate symbolism the difference in the application of signs would be marked by differences in the signs themselves, so that logical form would be adequately displayed. In this way what is essential to a proposition would be clearly distinguishable from what is conventional or arbitrary. (Propositions 3.33-3.333 concern Russell's theory of types. For a discussion of this, see pages 55-7 of this book.)

|         |                   |
|         |                   +-------- 3.262
|         |                   |
|         |                   +-------- 3.263
|         |
|         +-------- 3.3 ----- 3.31 ---- 3.311
|         |         |         |
|         |         |         +-------- 3.312
|         |         |         |
|         |         |         +-------- 3.313
|         |         |         |
|         |         |         +-------- 3.314
|         |         |         |
|         |         |         +-------- 3.315
|         |         |         |
|         |         |         +-------- 3.316
|         |         |         |
|         |         |         +-------- 3.317
|         |         |         |
|         |         |         +-------- 3.318
|         |         |
|         |         +-------- 3.32 ---- 3.321
|         |         |         |
|         |         |         +-------- 3.322
|         |         |         |
|         |         |         +-------- 3.323
|         |         |         |
|         |         |         +-------- 3.324
|         |         |         |
|         |         |         +-------- 3.325
|         |         |         |
|         |         |         +-------- 3.326
|         |         |         |
|         |         |         +-------- 3.327
|         |         |         |
|         |         |         +-------- 3.328
|         |         |
|         |         +-------- 3.33 ---- 3.331
|         |         |         |
|         |         |         +-------- 3.332
|         |         |         |
|         |         |         +-------- 3.333
|         |         |         |
|         |         |         +-------- 3.334
|         |         |
|         |         +-------- 3.34 ---- 3.341 --- 3.3411
|         |                   |
|         |                   +-------- 3.342 --- 3.3421
|         |                   |
|         |                   +-------- 3.343
|         |                   |
|         |                   +-------- 3.344 --- 3.3441
|         |                             |
|         |                             +-------- 3.3442
|         |
|         +-------- 3.4 ----- 3.41 ---- 3.411
|         |         |
|         |         +-------- 3.42
|         |
|         +-------- 3.5
|

Philosophy 4-4.0031
'A thought is a proposition with a sense.' What does not have sense is not a proposition and cannot be thought. But, as we have seen, the sense of a proposition can be disguised; grammar, convention, can mislead one as to logical form. So misled, one may utter words which have only the appearance of constituting a proposition. Words may be uttered, in short, which have no clear application, no clear logic. Much of philosophy consists of such utterances. It springs from a failure to understand the logic of our language. Philosophy, properly understood, is therefore in a special way 'a critique of language'; it is a restoring of words to their proper sense.

|
+-------- 4 ------- 4.0 ----- 4.00 ---- 4.001
|         |         |         |
|         |         |         +-------- 4.002
|         |         |         |
|         |         |         +-------- 4.003 --- 4.0031
|         |         |

True and false 4.01-4.0641
'A proposition is a picture of reality.' If I understand a proposition I know the situation it represents. To understand a proposition is to know what is the case if it is true and one can indicate the meaning of a proposition by indicating what would make it true as opposed to what would make it false. So a proposition's being either true or false is not a consequence of its having a meaning. Rather, to understand what would make it true and what false is to understand its meaning. It follows that the negation sign does not introduce a new discrimination of fact. If one understands a proposition one knows what would make it false and so far as the facts are concerned one has nothing further to grasp in order to understand the negation of that proposition. (See pages 38-9 for a further discussion.)

|         |         |
|         |         +-------- 4.01 ---- 4.011
|         |         |         |
|         |         |         +-------- 4.012
|         |         |         |
|         |         |         +-------- 4.013
|         |         |         |
|         |         |         +-------- 4.014 --- 4.0141
|         |         |         |
|         |         |         +-------- 4.015
|         |         |         |
|         |         |         +-------- 4.016
|         |         |
|         |         +-------- 4.02 ---- 4.021
|         |         |         |
|         |         |         +-------- 4.022
|         |         |         |
|         |         |         +-------- 4.023
|         |         |         |
|         |         |         +-------- 4.024
|         |         |         |
|         |         |         +-------- 4.025
|         |         |         |
|         |         |         +-------- 4.026
|         |         |         |
|         |         |         +-------- 4.027
|         |         |
|         |         +-------- 4.03 ---- 4.031 --- 4.0311
|         |         |         |         |
|         |         |         |         +-------- 4.0312
|         |         |         |
|         |         |         +-------- 4.032
|         |         |
|         |         +-------- 4.04 ---- 4.041 --- 4.0411
|         |         |                   |
|         |         |                   +-------- 4.0412
|         |         |
|         |         +-------- 4.05
|         |         |
|         |         +-------- 4.06 ---- 4.061
|         |                   |
|         |                   +-------- 4.062 --- 4.0621
|         |                   |
|         |                   +-------- 4.063
|         |                   |
|         |                   +-------- 4.064 --- 4.0641
|         |

Philosophy and science 4.1-4.116
'Propositions represent the existence and non-existence of states of affairs.' The totality of true propositions constitutes the natural sciences. Philosophy is not one of the natural sciences. It is not a body of doctrine but an activity. Its task is the clarification of thought. So far as it takes itself to be a doctrine it is confused. It confuses what can be said with what can only be shown.

|         |
|         +-------- 4.1 ----- 4.11 ---- 4.111
|         |         |         |
|         |         |         +-------- 4.112 --- 4.1121
|         |         |         |         |
|         |         |         |         +-------- 4.1122
|         |         |         |
|         |         |         +-------- 4.113
|         |         |         |
|         |         |         +-------- 4.114
|         |         |         |
|         |         |         +-------- 4.115
|         |         |         |
|         |         |         +-------- 4.116
|         |         |

Formal concepts 4.12-4.2
Propositions can represent the whole of reality, but they cannot represent what they must have in common with reality in order to be able to represent it - logical form.' Logical relations are formal properties. The attempt to state the format properties of a concept is confused. These cannot be stated but show themselves in the application of the symbol. Thus to assert '"It is raining" is a proposition.', or 'Red is a colour' or 'One is a number' is nonsensical. 'It is raining', for example, shows that it is a proposition, that it is intelligible, in what is says. Nothing further is added in trying to state that it is. (Wittgenstein in this section introduces the important notion of a formal series. But see propositions 5.2-5.541 for a more detailed treatment.)

|         |         |
|         |         +-------- 4.12 ---- 4.121 --- 4.1211
|         |                   |         |
|         |                   |         +-------- 4.1212
|         |                   |         |
|         |                   |         +-------- 4.1213
|         |                   |
|         |                   +-------- 4.122 --- 4.1221
|         |                   |
|         |                   +-------- 4.123
|         |                   |
|         |                   +-------- 4.124 --- 4.1241
|         |                   |
|         |                   +-------- 4.125 --- 4.1251
|         |                   |         |
|         |                   |         +-------- 4.1252
|         |                   |
|         |                   +-------- 4.126
|         |                   |
|         |                   +-------- 4.127 --- 4.1271
|         |                   |         |
|         |                   |         +-------- 4.1272 -- 4.12721
|         |                   |         |
|         |                   |         +-------- 4.1273
|         |                   |         |
|         |                   |         +-------- 4.1274
|         |                   |
|         |                   +-------- 4.128
|         |
|         +-------- 4.2 ----- (4.21 ---- 4.211)
|         |

Truth function 4.21-4.45
'The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs.' The propositions of ordinary language are complex; they are made up of elementary propositions. A complex proposition is a truth function of elementary propositions, i.e. the truth or falsity of the proposition as a whole will depend on the truth or falsity of its elementary constituents. The ways in which the truth or falsity of the proposition as a whole may be determined by the truth or falsity of its constituents can be set out in the form of a truth table. A truth table is a propositional sign. For example, the same propositional sign can be written either as 'p v q' or as '(TTTF)(p,q)'.

|         |
|         +-------- (4.2) ----- 4.21 ---- 4.211
|         |         |
|         |         +-------- 4.22 ---- 4.221 --- 4.2211
|         |         |
|         |         +-------- 4.23
|         |         |
|         |         +-------- 4.24 ---- 4.241
|         |         |         |
|         |         |         +-------- 4.242
|         |         |         |
|         |         |         +-------- 4.243
|         |         |
|         |         +-------- 4.25
|         |         |
|         |         +-------- 4.26
|         |         |
|         |         +-------- 4.27
|         |         |
|         |         +-------- 4.28
|         |
|         +-------- 4.3 ----- 4.31
|         |
|         +-------- 4.4 ----- 4.41 ---- 4.411
|         |         |
|         |         +-------- 4.42
|         |         |
|         |         +-------- 4.43 ---- 4.431
|         |         |
|         |         +-------- 4.44 ---- 4.441
|         |         |         |
|         |         |         +-------- 4.442
|         |         |
|         |         +-------- 4.45
|         |

Tautology 4.46-5.101
Among the possible groups of truth-conditions there are two extreme cases.' We can construct propositions which are false whatever the truth possibilities of their constituent propositions and others which are true whatever these possibilities. We can construct contradictions and tautologies. Tautologies say nothing. One knows nothing about the weather if one knows that it is either raining or not raining. But tautologies are not nonsensical. They are part of the symbolism. Unlike gibberish, they show something about logical form. The propositions of logic are tautologies.

|         |         |
|         |         +-------- 4.46 ---- 4.461 --- 4.4611
|         |                   |
|         |                   +-------- 4.462
|         |                   |
|         |                   +-------- 4.463
|         |                   |
|         |                   +-------- 4.464
|         |                   |
|         |                   +-------- 4.465
|         |                   |
|         |                   +-------- 4.466 --- 4.4661
|         |
|         +-------- 4.5 ----- 4.51
|                   |
|                   +-------- 4.52
|                   |
|                   +-------- 4.53
|
+-------- 5 ------- 5.0 ----- 5.01
|         |         |
|         |         +-------- 5.02
|         |
|         +-------- 5.1 ----- 5.10 ---- 5.101
|         |         |

Inference 5.11-5.156
If all the truth-grounds that are common to a number of propositions are at the smae time truth-grounds of a certain proposition, then we say that the truth of that proposition follows from the truth of the other.' Logical inference rests entirely on the internal relation between propositions. If 'p' follows from 'q' in logic, they are themselves the only possible justification for the inference. The 'laws of inference' which are supposed to justify inference are superfluous. There is no hierarchy amongst the propositions of logic. All are on the same level and all say the same thing, namely, nothing. In developing a logical system one is merely elaborating the internal connections between propositions, showing how their senses are interrelated. (In this section and briefly in the last Wittgenstein discusses probability. See pages 80 and 81 of this book for a further discussion).

|         |         |
|         |         +-------- 5.11
|         |         |
|         |         +-------- 5.12 ---- 5.121
|         |         |         |
|         |         |         +-------- 5.122
|         |         |         |
|         |         |         +-------- 5.123
|         |         |         |
|         |         |         +-------- 5.124 --- 5.1241
|         |         |
|         |         +-------- 5.13 ---- 5.131 --- 5.1311
|         |         |         |
|         |         |         +-------- 5.132
|         |         |         |
|         |         |         +-------- 5.133
|         |         |         |
|         |         |         +-------- 5.134
|         |         |         |
|         |         |         +-------- 5.135
|         |         |         |
|         |         |         +-------- 5.136 --- 5.1361
|         |         |                   |
|         |         |                   +-------- 5.1362
|         |         |                   |
|         |         |                   +-------- 5.1363
|         |         |
|         |         +-------- 5.14 ---- 5.141
|         |         |         |
|         |         |         +-------- 5.142
|         |         |         |
|         |         |         +-------- 5.143
|         |         |
|         |         +-------- 5.15 ---- 5.151 --- 5.1511
|         |                   |
|         |                   +-------- 5.152
|         |                   |
|         |                   +-------- 5.153
|         |                   |
|         |                   +-------- 5.154
|         |                   |
|         |                   +-------- 5.155
|         |                   |
|         |                   +-------- 5.156
|         |

Formal operation 5.2-5.54

This is a complex section in which Wittgenstein's account of formal operations, the general form of the proposition, the significance of logical symbolism, and generality are intertwined. For an adequate account of the material the student is best advised to turn to chapters 4 and 6 of this book.

Wittgenstein has now largely completed his account of the proposition and of logic. In the remaining sections he is mainly concerned with those propositions which at first sight seemnot to fit conveniently into his account. Often the discussion in these sections is too complex to admit of useful summary. Where this is so, I shall adopt the procedure I have adopted in the case of section 5.2-5.54. I shall indicate the topic and then refer to the chapter in this book in which it is discussed in detail.


|         |
|         +-------- 5.2 ----- 5.21
|         |         |
|         |         +-------- 5.22
|         |         |
|         |         +-------- 5.23 ---- 5.231
|         |         |         |
|         |         |         +-------- 5.232
|         |         |         |
|         |         |         +-------- 5.233
|         |         |         |
|         |         |         +-------- 5.234 --- 5.2341
|         |         |
|         |         +-------- 5.24 ---- 5.241
|         |         |         |
|         |         |         +-------- 5.242
|         |         |
|         |         +-------- 5.25 ---- 5.251
|         |                   |
|         |                   +-------- 5.252 --- 5.2521
|         |                   |         |
|         |                   |         +-------- 5.2522
|         |                   |         |
|         |                   |         +-------- 5.2523
|         |                   |
|         |                   +-------- 5.253
|         |                   |
|         |                   +-------- 5.254
|         |
|         +-------- 5.3 ----- 5.31
|         |         |
|         |         +-------- 5.32
|         |
|         +-------- 5.4 ----- 5.41
|         |         |
|         |         +-------- 5.42
|         |         |
|         |         +-------- 5.43
|         |         |
|         |         +-------- 5.44 ---- 5.441
|         |         |         |
|         |         |         +-------- 5.442
|         |         |
|         |         +-------- 5.45 ---- 5.451
|         |         |         |
|         |         |         +-------- 5.452
|         |         |         |
|         |         |         +-------- 5.453
|         |         |         |
|         |         |         +-------- 5.454 --- 5.4541
|         |         |
|         |         +-------- 5.46 ---- 5.461 --- 5.4611
|         |         |
|         |         +-------- 5.47 ---- 5.471 --- 5.4711
|         |                   |
|         |                   +-------- 5.472
|         |                   |
|         |                   +-------- 5.473 --- 5.4731
|         |                   |         |
|         |                   |         +-------- 5.4732 -- 5.47321
|         |                   |         |
|         |                   |         +-------- 5.4733
|         |                   |
|         |                   +-------- 5.474
|         |                   |
|         |                   +-------- 5.475
|         |                   |
|         |                   +-------- 5.476
|         |
|         +-------- 5.5 ----- 5.50 ---- 5.501
|         |         |         |
|         |         |         +-------- 5.502
|         |         |         |
|         |         |         +-------- 5.503
|         |         |
|         |         +-------- 5.51 ---- 5.511
|         |         |         |
|         |         |         +-------- 5.512
|         |         |         |
|         |         |         +-------- 5.513
|         |         |         |
|         |         |         +-------- 5.514
|         |         |         |
|         |         |         +-------- 5.515 --- 5.5151
|         |         |
|         |         +-------- 5.52 ---- 5.521
|         |         |         |
|         |         |         +-------- 5.522
|         |         |         |
|         |         |         +-------- 5.523
|         |         |         |
|         |         |         +-------- 5.524
|         |         |         |
|         |         |         +-------- 5.525
|         |         |         |
|         |         |         +-------- 5.526 --- 5.5261
|         |         |                   |
|         |         |                   +-------- 5.5262
|         |         |
|         |         +-------- 5.53 ---- 5.530 --- 5.5301
|         |         |         |         |
|         |         |         |         +-------- 5.5302
|         |         |         |         |
|         |         |         |         +-------- 5.5303
|         |         |         |
|         |         |         +-------- 5.531
|         |         |         |
|         |         |         +-------- 5.532 --- 5.5321
|         |         |         |
|         |         |         +-------- 5.533
|         |         |         |
|         |         |         +-------- 5.534
|         |         |         |
|         |         |         +-------- 5.535 --- 5.5351
|         |         |                   |
|         |         |                   +-------- 5.5352
|         |         |
|         |         +-------- 5.54 ---- (5.541)
|         |         |         |

Statements of belief 5.541-5.5423
The difficulty with 'A believes that p' is that it seems not to be truth-functional. For a discussion of how Wittgenstein resolves the difficulty see chapter 8.

|         |         |         |
|         |         +-------- (5.54) ---- 5.541
|         |         |         |
|         |         |         +-------- 5.542 --- 5.5421
|         |         |                   |
|         |         |                   +-------- 5.5422
|         |         |                   |
|         |         |                   +-------- 5.5423
|         |         |

Logic, the world and the self 5.55-5.641
This section includes Wittgenstein's discussion of solipsism. See chapter 9. It is important for the student to note that Wittgenstein is not putting forward a version of solipsism; rather, he is giving solipsism as an example of a philosophical confusion which has arisen through not seeing the difference between what can be said and what can only be shown.

|         |         |
|         |         +-------- 5.55 ---- 5.551
|         |                   |
|         |                   +-------- 5.552 --- 5.5521
|         |                   |
|         |                   +-------- 5.553
|         |                   |
|         |                   +-------- 5.554 --- 5.5541
|         |                   |         |
|         |                   |         +-------- 5.5542
|         |                   |
|         |                   +-------- 5.555
|         |                   |
|         |                   +-------- 5.556 --- 5.5561
|         |                   |         |
|         |                   |         +-------- 5.5562
|         |                   |         |
|         |                   |         +-------- 5.5563
|         |                   |
|         |                   +-------- 5.557 --- 5.5571
|         |
|         +-------- 5.6 ----- 5.61
|                   |
|                   +-------- 5.62 ---- 5.621
|                   |
|                   +-------- 5.63 ---- 5.631
|                   |         |
|                   |         +-------- 5.632
|                   |         |
|                   |         +-------- 5.633 --- 5.6331
|                   |         |
|                   |         +-------- 5.634
|                   |
|                   +-------- 5.64 ---- 5.641
|

Logic and mathematics 6-6.241
What is important in this section is to see the precise way in which mathematics and logic are related. Mathematics is a method of logic. It is not derived from a set of logical principles. Rather, it is an aspect of the fundamental logical operation by which any proposition is derived from another. For a detailed discussion see chapter 5.

|
+-------- 6 ------- 6.0 ----- 6.00 ---- 6.001
|         |         |         |
|         |         |         +-------- 6.002
|         |         |
|         |         +-------- 6.01
|         |         |
|         |         +-------- 6.02 ---- 6.021
|         |         |         |
|         |         |         +-------- 6.022
|         |         |
|         |         +-------- 6.03 ---- 6.031
|         |
|         +-------- 6.1 ----- 6.11 ---- 6.111
|         |         |         |
|         |         |         +-------- 6.112
|         |         |         |
|         |         |         +-------- 6.113
|         |         |
|         |         +-------- 6.12 ---- 6.120 --- 6.1201
|         |         |         |         |
|         |         |         |         +-------- 6.1202
|         |         |         |         |
|         |         |         |         +-------- 6.1203
|         |         |         |
|         |         |         +-------- 6.121
|         |         |         |
|         |         |         +-------- 6.122 --- 6.1221
|         |         |         |         |
|         |         |         |         +-------- 6.1222
|         |         |         |         |
|         |         |         |         +-------- 6.1223
|         |         |         |         |
|         |         |         |         +-------- 6.1224
|         |         |         |
|         |         |         +-------- 6.123 --- 6.1231
|         |         |         |         |
|         |         |         |         +-------- 6.1232
|         |         |         |         |
|         |         |         |         +-------- 6.1233
|         |         |         |
|         |         |         +-------- 6.124
|         |         |         |
|         |         |         +-------- 6.125 --- 6.1251
|         |         |         |
|         |         |         +-------- 6.126 --- 6.1261
|         |         |         |         |
|         |         |         |         +-------- 6.1262
|         |         |         |         |
|         |         |         |         +-------- 6.1263
|         |         |         |         |
|         |         |         |         +-------- 6.1264
|         |         |         |         |
|         |         |         |         +-------- 6.1265
|         |         |         |
|         |         |         +-------- 6.127 --- 6.1271
|         |         |
|         |         +-------- 6.13
|         |
|         +-------- 6.2 ----- 6.21 ---- 6.211
|         |         |
|         |         +-------- 6.22
|         |         |
|         |         +-------- 6.23 ---- 6.231
|         |         |         |
|         |         |         +-------- 6.232 --- 6.2321
|         |         |         |         |
|         |         |         |         +-------- 6.2322
|         |         |         |         |
|         |         |         |         +-------- 6.2323
|         |         |         |
|         |         |         +-------- 6.233 --- 6.2331
|         |         |         |
|         |         |         +-------- 6.234 --- 6.2341
|         |         |
|         |         +-------- 6.24 ---- 6.241
|         |

Natural science 6.3-6.372
Here Wittgenstein further elucidates the difference between the generality of logic and accidental generality by considering the nature of scientific laws. See chapter 7.

|         |
|         +-------- 6.3 ----- 6.31
|         |         |
|         |         +-------- 6.32 ---- 6.321 --- 6.3211
|         |         |
|         |         +-------- 6.33
|         |         |
|         |         +-------- 6.34 ---- 6.341
|         |         |         |
|         |         |         +-------- 6.342
|         |         |         |
|         |         |         +-------- 6.343 --- 6.3431
|         |         |                   |
|         |         |                   +-------- 6.3432
|         |         |
|         |         +-------- 6.35
|         |         |
|         |         +-------- 6.36 ---- 6.361 --- 6.3611 -- 6.36111
|         |         |         |
|         |         |         +-------- 6.362
|         |         |         |
|         |         |         +-------- 6.363 --- 6.3631 -- 6.36311
|         |         |
|         |         +-------- 6.37 ---- 6.371
|         |                   |
|         |                   +-------- 6.372
|         |                   |

Value 6.373-6.522
An expression of value is not a statement of fact. All propositions are of equal value because all propositions merely say what is the case. But what is the case, what happens to be so, is not the same as what ought to be so, what is valuable. For a further discussion see chapter 10.

|         |                   |
|         |                   +-------- 6.373
|         |                   |
|         |                   +-------- 6.374
|         |                   |
|         |                   +-------- 6.375 --- 6.3751
|         |
|         +-------- 6.4 ----- 6.41
|         |         |
|         |         +-------- 6.42 ---- 6.421
|         |         |         |
|         |         |         +-------- 6.422
|         |         |         |
|         |         |         +-------- 6.423
|         |         |
|         |         +-------- 6.43 ---- 6.431 --- 6.4311
|         |         |         |         |
|         |         |         |         +-------- 6.4312
|         |         |         |
|         |         |         +-------- 6.432 --- 6.4321
|         |         |
|         |         +-------- 6.44
|         |         |
|         |         +-------- 6.45
|         |
|         +-------- 6.5 ----- 6.51
|                   |
|                   +-------- 6.52 ---- 6.521
|                   |         |
|                   |         +-------- 6.522
|                   |

What can be said and what can only be shown 6.53-7
For a discussion of the complex issues raised by these last propositions see chapter 11.

|                   |
|                   +-------- 6.53
|                   |
|                   +-------- 6.54
|
+-------- 7


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