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Tractatus Logico-Philosophicus (English): Difference between revisions
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The philosophical I is not the man, not the human body or the human soul of which psychology treats, but the metaphysical subject, the limit—not a part of the world. | The philosophical I is not the man, not the human body or the human soul of which psychology treats, but the metaphysical subject, the limit—not a part of the world. | ||
6 The general form of truth function is: <math>[ \bar{p}, \bar{\xi}, N (\bar{\xi}) ]</math>. | |||
This is the general form of proposition. | |||
6.001 This says nothing else than that every proposition is the result of successive applications of the operation <math>N (\bar{\xi})</math> to the elementary propositions. | |||
6.002 If we are given the general form of the way in which a proposition is constructed, then thereby we are also given the general form of the way in which by an operation out of one proposition another can be created. | |||
6.01 The general form of the operation <math>\Omega ' (\bar{\eta})</math> is therefore: <math>[\bar{\xi}, N(\bar{\xi})]' (\bar{\eta}) (= [ \bar{\eta}, \bar{\xi}, N (\bar{\xi}) ])</math>. | |||
This is the most general form of transition from one proposition to another. | |||
6.02 And thus we come to numbers: I define | |||
<p style="text-align:center;"><math>x = \Omega^{0 \prime} x \text{ Def.}</math> und<br> | |||
<math>\Omega^{\prime} \Omega^{\nu \prime} x = \Omega^{\nu + 1 \prime} x \text{ Def.}</math></p> | |||
According, then, to these symbolic rules we write the series <math>x, \Omega ' x, \Omega ' \Omega ' x, \Omega ' \Omega ' \Omega ' x, .....</math> | |||
<p style="text-align:center;">as: <math>\Omega^{0 \prime} x, \Omega^{0+1 \prime} x, \Omega^{0 + 1 + 1 \prime} x, \Omega^{0 + 1 + 1 + 1 \prime} x, .....</math></p> | |||
Therefore I write in place of "<math>[ x, \xi, \Omega ' \xi ]</math>", | |||
<p style="text-align:center;">„<math>[ \Omega^{0 \prime} x, \Omega^{\nu \prime} x, \Omega^{\nu + 1 \prime} x ]</math>“.</p> | |||
And I define : | |||
:<math>0 + 1 = 1 \text{ Def.}</math> | |||
:<math>0 + 1 + 1 = 2 \text{ Def.}</math> | |||
:<math>0 + 1 + 1 + 1 = 3 \text{ Def.}</math> | |||
:<math>\text{(and so on.)}</math> | |||
6.021 A number is the exponent of an operation. | |||
6.022 The concept number is nothing else than that which is common to all numbers, the general form of number. | |||
The concept number is the variable number. | |||
And the concept of equality of numbers is the general form of all special equalities of numbers. | |||
6.03 The general form of the cardinal number is : [0, ''ξ'', ''ξ'' + 1]. | |||
6.031 The theory of classes is altogether superfluous in mathematics. | |||
This is connected with the fact that the generality which we need in mathematics is not the ''accidental'' one. | |||
6.1 The propositions of logic are tautologies. | |||
6.11 The propositions of logic therefore say nothing. (They are the analytical propositions.) | |||
6.111 Theories which make a proposition of logic appear substantial are always false. One could ''e.g.'' believe that the words "true" and "false" signify two properties among other properties, and then it would appear as a remarkable fact that every proposition possesses one of these properties. This now by no means appears self-evident, no more so than the proposition "All roses are either yellow or red" would sound even if it were true. Indeed our proposition now gets quite the character of a proposition of natural science and this is a certain symptom of its being falsely understood. | |||
6.112 The correct explanation of logical propositions must give them a peculiar position among all propositions. | |||
6.113 It is the characteristic mark of logical propositions that one can perceive in the symbol alone that they are true; and this fact contains in itself the whole philosophy of logic. And so also it is one of the most important facts that the truth or falsehood of non-logical propositions can ''not'' be recognized from the propositions alone. | |||
6.12 The fact that the propositions of logic are tautologies ''shows'' the formal—logical—properties of language, of the world. | |||
That its constituent parts connected together ''in this way'' give a tautology characterizes the logic of its constituent parts. | |||
In order that propositions connected together in a definite way may give a tautology they must have definite properties of structure. That they give a tautology when ''so'' connected shows therefore that they possess these properties of structure. | |||
6.1201 That ''e.g.'' the propositions "''p''" and "~''p''" in the connexion "~(''p.~p'')" give a tautology shows that they contradict one another. That the propositions "''p⊃q''", "''p''" and "''q''" connected together in the form "''(p⊃q).(p):⊃:(q)''" give a tautology shows that "''q''" follows from "''p''" and "''p⊃q''". That "(''x'').''fx'':⊃:''fa''" is a tautology shows that ''fa'' follows from (''x'') ,''fx'', etc. etc. | |||
6.1202 It is clear that we could have used for this purpose contradictions instead of tautologies. | |||
6.1203 In order to recognize a tautology as such, we can, in cases in which no sign of generality occurs in the tautology, make use of the following intuitive method: I write instead of "''p", "q", "r''", etc., "''TpF", "TqF", "TrF''", etc. The truth-combinations I express by brackets, ''e.g.'': | |||