Logical objects or logical constants in Frege's and Russell's sense are superfluous.

We now see that 'logical objects' or 'logical constants' in Frege's and Russell's sense are superfluous. This is because truth operations on truth functions that are one and the same truth function of elemental propositions will yield identical results.

It is self-evident that ∨, ⊃, etc. are relations in a different sense than that in which right and left etc. are. The interdefinability of Frege's and Russell's 'primitive signs' of logic is enough to show that they are not primitive signs, still less signs for relations. And it is obvious that the '⊃' which we define by means of '~' and '∨' is identical with the one by which we define '∨' using '~'. It is also obvious that the latter '∨' is identical with the former; and so on.

On the other hand, it seems implausible one fact p should generate infinitely many others, namely (~~p), (~~~~p), etc. And it is no more plausible that the infinite number of propositions of (mathematical) logic follow from half a dozen 'fundamental laws'. All the propositions of logic say the same thing; namely nothing.

Truth-functions are not material functions. When one can, for example, produce an affirmation by double negation, is then negation - in some sense - part of affirmation? Does (~~p) negate (~p), or does it affirm (p) - or both?

The proposition (~~p) does not treat negation as an object; but the possibility of negation is nevertheless already hinted at in affirmation. And if there were an object called '~', it would follow that (~~p) said something different from what 'p' said, just because the one proposition would then be about '~' and the other would not.

The apparent logical constants also disappear in the case of (~(∃x).~fx) that says the same as ((x).fx), or when((∃x).fx.x = a), says the same as (fa).

If we are given a proposition, then with it we are already given the results of all truth operations that have it as their base as well.