It also becomes clear why logic has been called the theory of forms and of inference.
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.)
The mark of logical propositions is not their general validity.
To be general is only to be accidentally valid for all things. An ungeneralized proposition can be tautologous just as well as a generalized one.
Logical general validity, we could call essential as opposed to accidental general validity, e.g. of the proposition “all men are mortal.” Propositions like Russell’s “axiom of reducibility” are not logical propositions, and this explains our feeling that, if true, they can only be true by a happy chance.
We can imagine a world in which the axiom of reducibility is not valid. But it is clear that logic has nothing to do with the question whether our world is really of this kind or not.
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 connection 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.
It is possible, also with the old conception of logic, to give at the outset a description of all “true” logical propositions.
Hence there can never be surprises in logic.
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.
In logic process and result are equivalent. (Therefore no surprises.)
Proof in logic is only a mechanical expedient to facilitate the recognition of tautology, where it is complicated.
It would be too remarkable, if one could prove a significant proposition logically from another, and a logical proposition also. It is clear from the beginning that the logical proof of a significant proposition and the proof in logic must be two quite different things.
The significant proposition asserts something, and its proof shows that it is so; in logic every proposition is the form of a proof.
Every proposition of logic is a modus ponens presented in signs. (And the modus ponens can not be expressed by a proposition.)
Logic can always be conceived to be such that every proposition is its own proof.
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.
It is clear that the number of “primitive propositions of logic” is arbitrary, for we could deduce logic from one primitive proposition by simply forming, for example, the logical produce of Frege’s primitive propositions. (Frege would perhaps say that this would no longer be immediately self-evident. But it is remarkable that so exact a thinker as Frege should have appealed to the degree of self-evidence as the criterion of a logical proposition.)
Logic is not a theory but a reflection of the world.
Logic is transcendental.