6

The general form of truth-function is: [, , N()].

This is the general form of proposition.

6.0

6.00

6.001

This says nothing else than that every proposition is the result of successive applications of the operation N'() 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 ' () is therefore:
[, N ()]'() (= [, , N()]).

This is the most general form of transition from one proposition to another.

6.02

And thus we come to numbers: I define

x = ⁰'x Def. and
'^v'x = ^v+1'x Def.

According, then, to these symbolic rules we write the series x, 'x, ''x, '''x . . . . . as: ⁰'x, ⁰⁺¹'x, ⁰⁺¹⁺¹'x, ^0+1+1+1'x . . . . .

Therefore I write in place of "[x, , ' ]",

"[⁰, ^v'x, ^v+1'x]",

And I define:

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 number is nothing else than that which is common to all numbers, the general form of a number.

6.03

The general form of the cardinal number is: [0, , +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. Once 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 seem 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 given 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.

6.120

6.1201

That e.g. the propositions "p" and "~p" in the connection "~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.:

and the co-ordination of the truth or falsity of the whole proposition with the truth-combinations of the truth-arguments by lines in the following way:

T (T/F)->F" v:shapes="_x0000_i1072" border="0" height="96" width="128">

This sign, for example, would therefore present the proposition p q. Now I will proceed to inquire whether such a proposition as ~(p . ~p) (The Law of Contradiction) is a tautology. The form "~" is written in our notation

T, (T)->F" v:shapes="_x0000_i1065" border="0" height="64" width="64">

the form " . " thus :--

F (T/T)->T" v:shapes="_x0000_i1068" border="0" height="96" width="128">

Hence the proposition ~(p . ~q) runs thus :--

T, T/F->F" v:shapes="_x0000_i1069" border="0" height="128" width="128">

If here we put "p" instead of "q" and examine the combination of the outermost T and F with the innermost, it is seen that the truth of the whole proposition is coordinated with all the truth-combinations of its argument, its falsity with none of the truth-combinations.