Tractatus Logico-Philosophicus (English): Difference between revisions

4.0621 That, however, the signs “''p''” and “~''p''” can say the same thing is important, for it shows that the sign “''~''” corresponds to nothing in reality.

The propositions “''p''” and “''~p''” have opposite senses, but to them corresponds one and the same reality.

In a similar sense I speak of the successive application of ''several'' operations to a number of propositions.

5.2523 The concept of the successive application of an operation is equivalent to the concept "and so on".

5.253 One operation can reverse the effect of another. Operations can cancel one another.

5.3 All propositions are results of truth-operations on the elementary propositions.

And it is obvious that the "⊃" which we define by means of "~" and "∨" is identical with that by which we define "∨" with the help of "~", and that this "∨" is the same as the first, and so on.

But all propositions of logic say the same thing. That is, nothing.

If ''e.g.'' an affirmation can be produced by repeated denial, is the denial — in any sense — contained in the affirmation?

5.441 This disappearance of the apparent logical constants also occurs if "~(∃''x'') . ~''fx''" says the same as "(''x'') . ''fx''", or "(∃''x'') . ''fx'' . ''x'' = ''a''" the same as "''fa''".

5.452 The introduction of a new expedient in the symbolism of logic must always be an event full of consequences. No new symbol may be introduced in logic in brackets or in the margin — with, so to speak, an entirely innocent face. (Thus in the "Principia Mathematica" of Russell and Whitehead there occur definitions and primitive propositions in words. Why suddenly words here? This would need a justification. There was none, and can be none for the process is actually not allowed.)

5.453 All numbers in logic must be capable of justification.

Where there is composition, there is argument and function, and where these are, all logical constants already are.

That however is the general form of proposition.

Signs which serve ''one'' purpose are logically equivalent, signs which serve ''no'' purpose are logically meaningless.

(Even if we believe that we have done so.)

5.531 I write therefore not "f(''a'', ''b'') . ''a'' = ''b''" but "''f'' (''a'', ''a'')" (or "''f'' (''b'', ''b'')"). And not "''f'' (''a'', ''b'') . ~''a'' = b" but "''f'' (''a'', ''b'')"

(Therefore instead of Russell's "(∃ ''x'', ''y'') . ''f'' (''x'', ''y'')": "(∃ ''x'', ''y'') . ''f'' (''x'', ''y'') . ∨ . (∃''x'') . ''f'' (''x'', ''x'')".)

5.5321 Instead of "(''x'') : ''f'' ''x'' ⊃ ''x'' = ''a''" we therefore write ''e.g.'' "(∃''x'') . ''f'' ''x'' . ⊃ . ''f'' ''a'' : ~(∃ ''x'', ''y'') . ''f'' ''x'' . ''f'' ''y''".

5.533 The identity sign is therefore not an essential constituent of logical notation.

5.535 So all problems disappear which are connected with such pseudo-propositions.

It is before the How, not before the What.

(There is no pre-eminent number.)

5.554 The enumeration of any special forms would be entirely arbitrary.

5.5542 May we then ask this at all? Can we set out a sign form and not know whether anything can correspond to it?

<p style="text-align:center;">„<math>[ \Omega^{0 \prime} x, \Omega^{\nu \prime} x, \Omega^{\nu + 1 \prime} x ]</math>“.</p>

:<math>0 + 1 = 1 \text{ Def.}</math>

And the concept of equality of numbers is the general form of all special equalities of numbers.

6.031 The theory of classes is altogether superfluous in mathematics.

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6.1265 Logic can always be conceived to be such that every proposition is its own proof.

Every tautology itself shows that it is a tautology.

6.36 If there were a law of causality, it might run: "There are natural laws".

6.361 In the terminology of Hertz we might say: Only ''uniform'' connexions are ''thinkable''.

6.3751 For two colours, ''e.g.'' to be at one place in the visual field, is impossible, logically impossible, for it is excluded by the logical structure of colour.

(It is clear that the logical product of two elementary propositions can neither be a tautology nor a contradiction. The assertion that a point in the visual field has two different colours at the same time, is a contradiction.)