Tractatus Logico-Philosophicus

Author:Ludwig Wittgenstein
Language: eng
Format: mobi
Published: 2010-07-30T00:20:39.174674+00:00

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6. The general form of a truth-function is [p, E, N(E)].

This is the general form of a proposition.

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.

6.01 Therefore the general form of an operation /'(n) is [E, N(E)]' (n) ( = [n, E, N(E)]). This is the most general form of transition from one proposition to another.

6.02 And this is how we arrive at numbers. I give the following definitions x = /0x Def., /'/v'x = /v+1'x Def. So, in accordance with these rules, which deal with signs, we write the series x, /'x, /'/'x, /'/'/'x,..., in the following way /0'x, /0+1'x, /0+1+1'x, /0+1+1+1'x,.... Therefore, instead of '[x, E, /'E]', I write '[/0'x, /v'x, /v+1'x]'. And I give the following definitions 0 + 1 = 1 Def., 0 + 1 + 1 = 2 Def., 0 + 1 + 1 +1 = 3 Def., (and so on).

6.021 A number is the exponent of an operation.

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.

6.03 The general form of an integer is [0, E, E +1].

6.031 The theory of classes is completely superfluous in mathematics. This is connected with the fact that the generality required in mathematics is not accidental generality.

6.1 The propositions of logic are tautologies.

6.11 Therefore the propositions of logic say nothing. (They are the analytic propositions.)

6.111 All theories that make a proposition of logic appear to have content are false. One might think, for example, that the words 'true' and 'false' signified two properties among other properties, and then it would seem to be a remarkable fact that every proposition possessed one of these properties. On this theory it seems to be anything but obvious, just as, for instance, the proposition, 'All roses are either yellow or red', would not sound obvious even if it were true. Indeed, the logical proposition acquires all the characteristics of a proposition of natural science and this is the sure sign that it has been construed wrongly.

6.112 The correct explanation of the propositions of logic must assign to them a unique status among all propositions.

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.

6.12 The fact that the propositions of logic are tautologies shows the formal--logical--properties of language and the world.

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