The schemata No. 4.31 are also significant, if “p”, “q”, “r”, etc. are not elementary propositions.
And it is easy to see that the propositional sign in No. 4.442 expresses one truth-function of elementary propositions even when “p” and “q” are truth-functions of elementary propositions.
All truth-functions are results of the successive application of a finite number of truth-operations to elementary propositions.
Here it becomes clear that there are no such things as “logical objects” or “logical constants” (in the sense of Frege and Russell).
For all those results of truth-operations on truth-functions are identical, which are one and the same truth-function of elementary propositions.
That ∨, ⊃, etc., are not relations in the sense of right and left, etc., is obvious.
The possibility of crosswise definition of the logical “primitive signs” of Frege and Russell shows by itself that these are not primitive signs and that they signify no relations.
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.
That from a fact p an infinite number of others should follow, namely, ~~p, ~~~~p, etc., is indeed hardly to be believed, and it is no less wonderful that the infinite number of propositions of logic (of mathematics) should follow from half a dozen “primitive propositions.”
But the propositions of logic say the same thing. That is, nothing.
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.
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”.
If a proposition is given to us then the results of all truth-operations which have it as their basis are given with it.
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.
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∨q)”, “(∃x).~fx” and others. We may not first introduce it for one 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.)
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.
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 preeminent numbers.