of logic are tautologies shows the
formal--logical--properties of language and the world. The fact that
a tautology is yielded by this particular way of connecting its
constituents characterizes the logic of its constituents. If
propositions are to yield a tautology when they are connected in a
certain way, they must have certain structural properties. So their
yielding a tautology when combined in this shows that they possess these
structural properties.

6.1201 For example, the fact that the propositions 'p' and 'Pp' in the
combination '(p. Pp)' yield a tautology shows that they contradict one
another. The fact that the propositions 'p z q', 'p', and 'q', combined
with one another in the form '(p z q). (p):z: (q)', yield a tautology
shows that q follows from p and p z q. The fact that '(x). fxx:z: fa' is
a tautology shows that fa follows from (x). fx. Etc. etc.

6.1202 It is clear that one could achieve the same purpose by using
contradictions instead of tautologies.

6.1203 In order to recognize an expression as a tautology, in cases
where no generality-sign occurs in it, one can employ the following
intuitive method: instead of 'p', 'q', 'r', etc. I write 'TpF', 'TqF',
'TrF', etc. Truth-combinations I express by means of brackets, e.g. and
I use lines to express the correlation of the truth or falsity of the
whole proposition with the truth-combinations of its truth-arguments,
in the following way So this sign, for instance, would represent
the proposition p z q. Now, by way of example, I wish to examine the
proposition P(p.Pp) (the law of contradiction) in order to determine
whether it is a tautology. In our notation the form 'PE' is written as
and the form 'E. n' as Hence the proposition P(p. Pp). reads as follows
If we here substitute 'p' for 'q' and examine how the outermost T and F
are connected with the innermost ones, the result will be that the truth
of the whole proposition is correlated with all the truth-combinations
of its argument, and its falsity with none of the truth-combinations.

6.121 The propositions of logic demonstrate the logical properties
of propositions by combining them so as to form propositions that say
nothing. This method could also be called a zero-method. In a logical
proposition, propositions are brought into equilibrium with one
another, and the state of equilibrium then indicates what the logical
constitution of these propositions must be.

6.122 It follows from thi