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.
6.121 The propositions of logic demonstrate the logical properties of propositions, by combining them into propositions which say nothing.
This method could be called a zero-method. In a logical proposition propositions are brought into equilibrium with one another, and the state of equilibrium then shows how these propositions must be logically constructed.
6.122 (4) Whence it follows that we can get on without logical propositions, for we can recognize in an adequate notation the formal properties of the propositions by mere inspection.
6.123 (3) It is clear that the laws of logic cannot themselves obey further logical laws.
(There is not, as Russell supposed, for every "type" a special law of contradiction; but one is sufficient, since it is not applied to itself.)
6.124 The logical propositions describe the scaffolding of the world, or rather they present it. They "treat" of nothing. They presuppose that names have meaning, and that elementary propositions have sense. And this is their connexion with the world. It is clear that it must show something about the world that certain combinations of symbols - which essentially have a definite character - are tautologies. Herein lies the decisive point. We said that in the symbols which we use something is arbitrary, something not. In logic only this expresses: but this means that in logic it is not we who express, by means of signs, what we want, but in logic the nature of the essentially necessary signs itself asserts. That is to say, if we know the logical syntax of any sign language, then all the propositions of logic are already given.
6.125 (1) It is possible, also with the old conception of logic, to give at the outset a description of all "true" logical propositions.
6.126 (5) Whether a proposition belongs to logic can be calculated by calculating the logical properties of the symbol.
And this we do when we prove a logical proposition. For without troubling ourselves about a sense and a meaning, we form the logical propositions out of others by mere symbolic rules.
We prove a logical proposition by creating it out of other logical propositions by applying in succession certain operations, which again generate tautologies out of the first. (And from a tautology only tautologies follow.)
Naturally this way of showing that its propositions are tautologies is quite unessential to logic. Because the propositions, from which the proof starts, must show without proof that they are tautologies.
6.127 (1) All propositions of logic are of equal rank; there are not some which are essentially primitive and others deduced from there.
Every tautology itself shows that it is a tautology.