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

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.

6.1221

If for example two propositions "p" and "q" give a tautology in the connection "p q", then it is clear that q follows from p.

E.g. that "q" follows from "p q . p" we see from these two propositions themselves, but we can also show it by combining them to "p q . p :: q" and then showing that this is a tautology.

6.1222

This throws light on the question why logical propositions can no more be empirically confirmed than they can be empirically refuted. not only must a proposition of logic be incapable of being contradicted by any possible experience, but it must also be incapable of being confirmed by any such.

6.1223

It now becomes clear why we often feel as though "logical truths" must be "postulated" by us. We can in fact postulate them in so far as we can postulate an adequate notation.

6.1224

It also becomes clear why logic has been called the theory of forms and of inference.

6.123

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.)

6.1231

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.

6.1232

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.

6.1233

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.

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 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.

6.125

It is possible, also with the old conception of logic, to give at the outset a description of all "true" logical propositions.

6.1251

Hence there can never be surprises in logic.

6.126

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.

6.1261

In logic process and result are equivalent. (Therefore no surprises.)

6.1262

Proof in logic is only a mechanical expedient to facilitate the recognition of tautology, where it is complicated.

6.1263

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.

6.1264

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 present in signs. (And the modus ponens can not be expressed by a proposition.)

6.1265

Logic can always be conceived to be such that every proposition is its own proof.

6.127

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.

6.1271

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.)

6.13

Logic is not a theory but a reflexion of the world.

Logic is transcendental.

6.2

Mathematics is a logical method.

The propositions of mathematics are equations, and therefore pseudo-propositions.

6.21

Mathematical propositions express no thoughts.

6.211

In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics.

In philosophy the question is "Why do we really use that word, that proposition?" constantly leads to valuable results.)

6.22

The logic of the world which the propositions of logic show in tautologies, mathematics shows in equations.

6.23

If two expressions are connected by the sign of equality, this means that they can be substituted for one another. But whether this is the case must show itself in the two expressions themselves.

It characterizes the logical form of two expressions, that they can be substituted for one another.

6.231

It is a property of affirmation that it can be conceived as double denial.

It is a property of "1+1+1=1" that it can be conceived as "(1+1)+(1+1)".

6.232

Frege says that these expressions have the same meaning but different senses.

But what is essential about equation is that it is not necessary in order to show that both expressions, which are connected by the sign of equality, have the same meaning: for this can be perceived from the two expressions themselves.

6.2321

And, that the propositions of mathematics can be proved means nothing else than that their correctness can be seen without our having to compare what they express with the facts as regards correctness.

6.2322

The identity of the meaning of two expressions cannot be asserted. For in order to be able to assert anything about their meaning, I must know their meaning, and if I know their meaning, I know whether they mean the same or something different.

6.2323

The equation characterizes only the standpoint from which I consider the two expressions, that is to say from the standpoint of their equality of meaning.

6.233

To the question whether we need intuition for the solution of mathematical problems it must be answered that language itself here supplies the necessary intuition.

6.2331

The process of calculation brings about just this intuition.

Calculation is not an experiment.

6.234

Mathematics is a method of logic.

6.2341

The essential of mathematical method is working with equations. On this method depends the fact that every proposition of mathematics must be self-evident.

6.24

The method by which mathematics arrives at its equations is the method of substitution.

For equations express the substitutability of two expressions, and we proceed from a number of equations to new equations, replacing expressions by others in accordance with the equations.

6.241

Thus the proof of the proposition 2×2=4 runs:
(^v)^µ'x=^v×µ'x Def.
^2×2'x = (²)²'x = (²)¹⁺¹'x = ²'²'x = ¹⁺¹'¹⁺¹'x
= (')'(')'x = ''''x = ^1+1+1+1'x = ⁴'x.