5.2
The structures of propositions stand to one another in internal relations.
5.21
We can bring out these internal relations in our manner of expression, by presenting a proposition as the result of an operation which produces it from other propositions (the bases of the operation).
5.22
The operation is the expression of a relation between the structures of its result and its bases.
5.23
The operation is that which must happen to a proposition in order to make another out of it.
5.231
And that will naturally depend on their formal properties, on the internal similarity of their forms.
5.232
The internal relation which orders a series is equivalent to the operation by which one term arises from another.
5.233
The first place in which an operation can occur is where a proposition arises from another in a logically significant way; i.e. where the logical construction of the proposition begins.
5.234
The truth-functions of elementary proposition. are results of operations which have the elementary propositions as bases. (I call these operations, truth-operations.)
5.2341
The sense of a truth-function of p is a function of the sense of p.
Denial, logical addition, logical multiplication, etc., etc., are operations.
(Denial reverses the sense of a proposition.)
5.24
An operation shows itself in a variable; it shows how we can proceed from one form of proposition to another.
It gives expression to the difference between the forms.
(And that which is common the bases, and the result of an operation, is the bases themselves.)
5.241
The operation does not characterize a form but only the difference between forms.
5.242
The same operation which makes "q" from "p", makes "r" from "q", and so on. This can only be expressed by the fact that "p", "q", "r", etc., are variables which give general expression to certain formal relations.
5.25
The occurrence of an operation does not characterize the sense of a proposition.
For an operation does not assert anything; only its result does, and this depends on the bases of the operation.
(Operation and function must not be confused with one another.)
5.251
A function cannot be its own argument, but the result of an operation can be its own basis.
5.252
Only in this way is the progress from term to term in a formal series possible (from type to type in the hierarchy of Russell and Whitehead). (Russell and Whitehead have not admitted the possibility of this progress but have made use of it all the same.)
5.2521
The repeated application of an operation to its own result I call its successive application ("O' O' O' a" is the result of the threefold successive application of "O'" to "a").
In a similar sense I speak of the successive application of several operations to a number of propositions.
5.2522
The general term of the formal series a, O' a, O' O' a, . . . I write thus: "[a, x, O' x]". This expression in brackets is a variable. The first term of the expression is the beginning of the formal series, the second the form of an arbitrary term x of the series, and the third the form of that term of the series which immediately follows x.
5.2523
The concept of the success application of an operation is equivalent to the concept "and so on".
5.253
One operation can reverse the effect of another. Operations can cancel one another.
5.254
Operations can vanish (e.g. denial in "~~p". ~~p = p).
5.3
All propositions are results of truth-operations on the elementary propositions.
The truth-operation is the way in which a truth-function arises from elementary propositions.
According to the nature of truth-operations, in the same way as out of elementary propositions arise their truth-functions, from truth-functions arises a new one. Every truth-operation creates from truth-functions of elementary propositions, another truth-function of elementary propositions i.e. a proposition. The result of every truth-operation on the results of truth-operations on elementary propositions is also the result of one truth-operation on elementary propositions.
Every proposition is the result of truth-operations on elementary propositions.
5.31
The Schemata No. 4.31 is also significant, if "p", "q", "r", etc. are not elementary propositions.
And it is easy to see that the propositional sign in No. 4.42 expresses one truth-function of elementary propositions even when "p" and "q" are truth-functions of elementary propositions.
5.32
All truth-functions are results of the successive application of a finite number of truth-operations to elementary propositions.
5.4
Here it becomes clear that there are no such things as "logical objects" or "logical constants" (in the sense of Frege and Russell).
5.41
For all those results of truth-operations on truth-functions are identical, which are one and the same truth-function of elementary propositions.
5.42
That v, 
, etc., are not relations in the sense of right and left, etc., is obvious.
The possibility of crosswise definition of the logical "primitive signs" of Frege and Russell shows by itself that these are not primitive signs and that they signify no relations.
And it is obvious that the "" which we define by means of "~" and "v" is identical with that by which we define "v" with the help of "~", and that this "v" is the same as the first, and so on.
5.43
That from a fact p an infinite number of others should follow, namely, ~~p, ~~~~p, etc., is indeed hardly to be believed, and it is no less wonderful that the infinite number of propositions of logic (of mathematics) should follow from half a dozen "primitive propositions".
