Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

VICIOUS-CIRCLE FALLACIES

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appellations not involving any description. (b) First-order names will be such
as involve a description by means of a first-order function; that is to say, if
4! 5 is a first-order function, "the term which satisfies 4! 5" will be a firstorder name, though there will not always be an object named by this name.
(c) Second-order names will be such as involve a description by means of
a second-order function; among such names will be those involving a
reference to the totality of first-order names. And so we can proceed through
a whole hierarchy. But at no stage can we give a meaning to the word
"nameable " unless we specify the order of names to be employed; and any
name in which the phrase "nameable by names of order n" occurs is necessarily of a higher order than the nth. Thus the paradox disappears.
The solutions of the paradox about the least indefinable ordinal and
of Richard's paradox are closely analogous to the above. The notion of
"definable," which occurs in both, is nearly the same as " nameable," which
occurs in our fifth paradox: "definable" is what "nameable" becomes
when elementary names are excluded, i.e. "definable" means "nameable
by a name which is not elementary." But here there is the same ambiguity
as to type as there was before, and the same need for the addition of words
which specify the type to which the definition is to belong. And however
the type may be specified, " the least ordinal not definable by definitions of
this type" is a definition of a higher type; and in Richard's paradox, when
we confine ourselves, as we must, to decimals that have a definition of a
given type, the number N, which causes the paradox, is found to have a
definition which belongs to a higher type, and thus not to come within the
scope of our previous definitions.
An indefinite number of other contradictions, of similar nature to the
above seven, can easily be manufactured. In all of them, the solution is
of the same kind. In all of them, the appearance of contradiction is produced by the presence of some word which has systematic ambiguity of
type, such as truth, falsehood, function, property, class, relation, cardinal,
ordinal, name, definition. Any such word, if its typical ambiguity is overlooked, will apparently generate a totality containing members defined in
terms of itself, and will thus give rise to vicious-circle fallacies. In most
cases, the conclusions of arguments which involve vicious-circle fallacies
will not be self-contradictory, but wherever we have an illegitimate totality,
a little ingenuity will enable us to construct a vicious-circle fallacy leading to a contradiction, which disappears as soon as the typically ambiguous
words are rendered typically definite, i.e. are determined as belonging to
this or that type.
Thus the appearance of contradiction is always due to the presence of
words embodying a concealed typical ambiguity, and the solution of the
apparent contradiction lies in bringing the concealed ambiguity to light.
5-2