252 It is only in this way that the step from one term of a series of
forms to another is possible (from one type to another in the hierarchies
of Russell and Whitehead). (Russell and Whitehead did not admit the
possibility of such steps, but repeatedly availed themselves of it.)
5.2521 If an operation is applied repeatedly to its own results, I speak of
successive applications of it. ('O'O'O'a' is the result of three successive
applications of the operation 'O'E' to 'a'.) In a similar sense I speak of
successive applications of more than one operation to a number of
propositions.
5.2522 Accordingly I use the sign '[a, x, O'x]' for the general term of the
series of forms a, O'a, O'O'a, ... . This bracketed expression is a
variable: the first term of the bracketed expression is the beginning of
the series of forms, the second is the form of a term x arbitrarily
selected from the series, and the third is the form of the term that
immediately follows x in the series.
5.2523 The concept of successive applications of an operation is equivalent
to the concept 'and so on'.
5.253 One operation can counteract the effect of another. Operations can
cancel one another.
5.254 An operation can vanish (e.g. negation in 'PPp' : PPp = p).
5.3 All propositions are results of truth-operations on elementary
propositions. A truth-operation is the way in which a truth-function is
produced out of elementary propositions.
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