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.

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

<math>x = \Omega^{0 \prime} x \text{ Def.}</math> und

<math>\Omega^{\prime} \Omega^{\nu \prime} x = \Omega^{\nu + 1 \prime} x \text{ Def.}</math>

According, then, to these symbolic rules we write the series <math>x, \Omega ' x, \Omega ' \Omega ' x, \Omega ' \Omega ' \Omega ' x, .....</math>

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>

Therefore I write in place of "<math>[ x, \xi, \Omega ' \xi ]</math>",

„<math>[ \Omega^{0 \prime} x, \Omega^{\nu \prime} x, \Omega^{\nu + 1 \prime} x ]</math>“.

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.'':

@@ Line 1,521: / Line 1,521: @@
 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 general form of truth function is: <math>[ \bar{p}, \bar{\xi}, N (\bar{\xi}) ]</math>.
+This is the general form of proposition.
+.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.
+.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.
+.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.
+.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>
+.021 A number is the exponent of an operation.
+.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.
+.03 The general form of the cardinal number is : [0, ''ξ'', ''ξ'' + 1].
+.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.
+.1 The propositions of logic are tautologies.
+.11 The propositions of logic therefore say nothing. (They are the analytical propositions.)
+.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.
+.112 The correct explanation of logical propositions must give them a peculiar position among all propositions.
+.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.
+.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.
+.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.
+.1202 It is clear that we could have used for this purpose contradictions instead of tautologies.
+.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.'':