the modus ponens by means of a proposition.)

6.1265 It is always possible to construe logic in such a way that every
proposition is its own proof.

6.127 All the propositions of logic are of equal status: it is not
the case that some of them are essentially derived propositions. Every
tautology itself shows that it is a tautology.

6.1271 It is clear that the number of the 'primitive propositions
of logic' is arbitrary, since one could derive logic from a single
primitive proposition, e.g. by simply constructing the logical product
of Frege's primitive propositions. (Frege would perhaps say that
we should then no longer have an immediately self-evident primitive
proposition. But it is remarkable that a thinker as rigorous as Frege
appealed to the degree of self-evidence as the criterion of a logical
proposition.)

6.13 Logic is not a body of doctrine, but a mirror-image 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 A proposition of mathematics does not express a thought.

6.211 Indeed in real life a mathematical proposition is never what
we want. Rather, we make use of mathematical propositions only in
inferences from propositions that do not belong to mathematics to others
that likewise do not belong to mathematics. (In philosophy the question,
'What do we actually use this word or this proposition for?' repeatedly
leads to valuable insights.)

6.22 The logic of the world, which is shown in tautologies by the
propositions of logic, is shown in equations by mathematics.

6.23 If two expressions are combined by means of the sign of equality,
that means that they can be substituted for one another. But it must be
manifest in the two expressions themselves whether this is the case
or not. When two expressions can be substituted for one another, that
characterizes their logical form.

6.231 It is a property of affirmation that it can be construed as double
negation. It is a property of '1 + 1 + 1 + 1' that it can be construed
as '(1 + 1) + (1 + 1)'.

6.232 Frege says that the two expressions have the same meaning but
different senses. But the essential point about an equation is that it
is not necessary in order to show that the two expressions connected by
the sign of equality have the same meaning, since this can be seen from
the two expressions themselves.

6.2321