5.4
Truth-functions are not material functions.
If e.g. an affirmation can be produced by repeated denial, is the denial -- in any sense -- contained in the affirmation?
Does "~~p" deny "~p", or does it affirm p; or both?
The proposition "~~p" does not treat of denial as an object, but the possibility of denial is already prejudged in affirmation.
And if there was an object called "~", then "~~p" would have to say something other than "p". For the one proposition would then treat of ~, the other would not.
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".
5.442
If a proposition is given to us then the results of all truth-operations which have it as their basis are given with it.
5.45
If there are logical primitive signs a correct logic must make clear their position relative to one another and justify their existence. The construction of logic out of its primitive signs must become clear.
5.451
If logic has primitive ideas these must be independent of one another. If a primitive idea is introduced it must be introduced in all contexts in which it occurs at all. One cannot therefore introduce it for one context and then again for another. For example, if denial is introduced, we must understand it in propositions of the form "~p", just as in propositions like "~(p v q)", "(x) . ~fx" and others. We may not first introduce it for oone class of cases and then for another, for it would then remain doubtful whether its meaning in the two cases was the same, and there would be no reason to use the same way of symbolizing in the two cases.
(In short, what Frege ("Grundgesetze der Arithmetik") has said about the introduction of signs by definitions holds, mutatis mutandis, for the introduction of primitive signs also.)
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.)
But if the introduction of a new expedient has proved necessary in one place, we must immediately ask: Where is this expedient always to be used? Its position in logic must be made clear.
5.453
All numbers in logic must be capable of justification.
Or rather it must become plain that there are no numbers in logic.
There are no pre-eminent numbers.
5.454
In logic there is no side by side, there can be no classification.
In logic there cannot be a more general and a more special.
5.4541
The solution of logical problems must be neat for they set the standard of neatness.
Men have always thought that there must be a sphere of questions whose answers -- a priori -- are symmetrical and united into a closed regular structure.
A sphere in which the proposition, simplex sigillum veri, is valid.
5.46
When we have rightly introduced the logical signs, the sense of all their combinations has been already introduced with them: therefore not only "p v q" but also "~(p v ~q)", etc. etc. We should then already have introduced the effect of all possible combinations of brackets; and it would then have become clear that the proper general primitive signs are not "p v q", "(x) . fx", etc., but the most general form of their combinations.
5.461
The apparently unimportant fact that the apparent relations like v and 
need brackets -- unlike real relations -- is of great importance.
The use of brackets with these apparent primitive signs shows that these are not the real primitive signs; and nobody of course would believe that the brackets have meaning by themselves.
5.4611
Logical operation signs are punctuations.
5.47
It is clear that everything which can be said beforehand about the form of all propositions at all can be said on one occasion.
For all logical operations are already contained in the elementary proposition. For "fa" says the same as "(x) . fx . x=a".
Where there is composition, there is argument and function, and where these are, all logical constants already are.
One could say: the one logical constant is that which all propositions, according to their nature, have in common with one another.
That however is the general form of proposition.
5.471
The general form of proposition is the essence of proposition.
5.4711
To give the essence of proposition means to give the essence of all description, therefore the essence of the world.
5.472
The description of the most general propositional form is the description of the one and only general primitive sign in logic.
5.473
Logic must take care of itself.
A possible sign must also be able to signify. Everything which is possible in logic is also permitted. ("Socrates is identical" means nothing because there is no property which is called "identical". The proposition is senseless because we have not made some arbitrary determination, not because the symbol is in itself impermissible.)
In a certain sense we cannot make mistakes in logic.
5.4731
Self-evidence, of which Russell has said so much, can only be discard in logic by language itself preventing every logical mistake. That logic is a priori consists in the fact that we cannot think illogically.
5.4732
We cannot give a sign the wrong sense.
5.47321
Occam's razor is, of course, not an arbitrary rule nor one justified by its practical success. It simply says that unnecessary elements in a symbolism mean nothing.
Signs which serve one purpose are logically equivalent; signs which serve no purpose are logically meaningless.
5.4733
Frege says: Every legitimately constructed proposition msut have a sense; and I say: Every possible proposition is legitimately constructed, and if it has no sense this can only be because we have given no meaning to some of its constituent parts.
(Even if we believe that we have done so.)
Thus "Socrates is identical" says nothing, because we have given no meaning to the word "identical" as adjective. For when it occurs as the sign of equality it symbolizes in an entirely different way -- the symbolizing relation is another -- therefore the symbol is in the two cases entirely different; the two symbols have the sign in common with one another only by accident.
5.474
The number of necessary fundamental operations depends only on our notation.
5.475
It is only a question of constructing a system of signs of a definite number of dimensions -- of a definite mathematical multiplicity.
5.476
It is clear that we are not concerned here with a number of primitive ideas which must be signified but with the expression of a rule.
5.5
Every truth-function is a result of the successive application of the operation
(- - - - -T) (
, . . . .) to elementary propositions.
This operation denies all the propositions in the right-hand bracket and I call it the negation of these propositions.
5.50
5.501
An expression in brackets whose terms are propositions I indicate -- if the order of the terms in the bracket is indifferent -- by a sign of the form "(
)". "" is a variable whose values are the terms of the expression in brackets, and the line over the variable indicates that it stands for all its values in the bracket.
(Thus if has the 3 values P, Q, R, then () = (P, Q, R).)
The values of the variables must be determined.
The determination is the description of the propositions which the variable stands for.
How the description of the terms of the expression in brackets takes place is unessential.
We may distinguish 3 kinds of description: 1. direct enumeration. In this case we can place simply its constant values instead of the variable. 2. Giving a function fx, whose values for all values of x are the propositions to be described. 3. Giving a formal law, according to which those propositions are constructed. In this case the terms of the expression in brackets are all the terms of a formal series.
5.502
Therefore I write instead of "(- - - - - T)(, . . . .)", "N()".
N() is the negation of all the values of the propositional variable .
5.503
As it is obviously easy to express how propositions can be constructioned by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.
5.51
has only one value, then N()=~p (not p), if it has two values then N()=~p . ~q (neither p nor q).
