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.
The truth-functions of elementary proposition, are results of operations which have the elementary propositions as bases. (I call these operations, truth-operations.)
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.)
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 to the bases, and the result of an operation, is the bases themselves.)
The operation does not characterize a form but only the difference between forms.
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.
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.)
A function cannot be its own argument, but the result of an operation can be its own basis.
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.)
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′ξ” to “a”).
In a similar sense I speak of the successive application of several operations to a number of propositions.
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.
The concept of the successive application of an operation is equivalent to the concept “and so on.”
One operation can reverse the effect of another. Operations can cancel one another.
Operations can vanish (e.g. denial in “~~p”. ~~p=p).
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.