THE PARADOXES OF ME. BUSSELL
W I T H A B R I E F ACCOUNT OF T H E I R HISTORY

BY

EDWIN R. GUTHRIE, JR.

A THESIS
PRESENTED TO THE FACULTY AND TRUSTEES OF THE UNIVERSITY
OF PENNSYLVANIA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE D E G R E E OP
DOCTOR OF PHILOSOPHY

OCTOBER, 1914

PRESS OF
THE NEW ERA PRINTING COMPANY
LANCASTER, PA.

1915

THE PARADOXES OF MR. RUSSELL
In his Principles of Mathematics published in 1903 Mr.
Bertrand Russell found himself involved in a difficulty in his
theory of classes for which he could at that time see no solution.
This difficulty lay in a series of paradoxes which were in one
form or another much discussed in Scholastic logic but which
had fallen into disrepute and had been little studied until the
recent development of symbolic logic brought them again to
notice. In the recent discussion the paradoxes have presented
themselves in a form somewhat different from the insoluhilia
which occupied so much space in Scholastic text-books, but
they are essentially the same and the Scholastic attempts at
solution bear many points of resemblance to the principal
solutions offered by the writers who have worked over the
difficulty since Russell's rediscovery of it.
The paradoxes mentioned by Russell are the first seven of
the following list, to which have been added a number of
others of the same general character.
1. The first is the classic example of the Cretan Epimenides
who says, according to St. Paul, that all Cretans are liars, as
well as other things that have nothing to do with the case,—
and by this statement makes himself a liar.
2. Represent the class of all classes that are not members
of themselves by w. Then (xew) < (xex)'.
Now let x
have the value w; we get: (wew) <
(wew)'}
3. Let T be the relation that exists between R and 8 whenever R does not have the relation R to 8. Then R TS = (RRS)'.
Making R and S both T, we have TTT=
(TTT)'.
1
(e) in Russell's notation indicates the membership of an individual in
a class.

1

2

4. The next contradiction is one given by Burali-Forti.
Any series of ordinals beginning with 0 has an ordinal number
one greater than the highest term. The series of all ordinals
would then have an ordinal number greater than the highest
ordinal.
5. "The least integer not namable in fewer than nineteen
syllables" (which Russell states to be 111,777) has been
named in eighteen.
6. All finite combinations of the letters of the alphabet
can be ranged1 so as to be in one to one correspondence with
the series of ordinals, so that the total number of definitions
is aleph, the first transfinite number. The number of transfinite ordinals exceeds this so that there are indefinable ordinals, and among these there is a least. This has, however,
been defined.
7. The paradox of Richard is: Let E be the class of all
decimals that can be defined in a finite number of words. Its
terms can be ordered as were the definitions in (6). Then
define N so that the nth figure of the nth decimal of N is one
greater than the nth figure of the nth decimal of E. N is then
defined in a finite number of words, but still is not a member
oiE.
For convenience of reference we may state here several
other paradoxes which have been mentioned in the discussion.
The next is due to Russell:
8. Every concept can be predicated of itself or it cannot.
If it can, let it be called predicable, if not, impredicable. Then
the concept impredicable, if predicable is impredicable, and if
impredicable is predicable.
9. This proposition is false.
10. All propositions are false.
11. I lie.
1

This arrangement Gorsch has shown in his Mengenlehre, Abhandlungen
der Fries'chen Schule, I : 508. The combinations one at a time, then two at a
time, then three, and so on, can be given in a definite order.

3

Mr. Alexander Riistow has collected a large number of
references to similar paradoxes in ancient writings, notably
in the works of Aristotle, Plato, and Diogenes Laertius.1
Among these ancient writers the paradox of the Liar was
the one to attract the most attention. As a rule the paradox
was accepted as final, and the fact that it existed was used in
support of an attack on the validity of human knowledge, as
Montaigne used it later on. What solutions were proposed
were crude attempts to place it under one of the Aristotelian
forms of fallacy. No analysis of the paradoxes was made,
nor were they recognized as a class. It was not until the time
of the Scholastic logicians that they were presented in a form
which offers interesting parallels to our present statement and
analysis.
During the Scholastic period the interest which the paradoxes or Insolubilia aroused was so great that many of the
text-books of logic written from the fourteenth to the sixteenth
centuries devoted lengthy chapters to them and there were a
number of separate treatises in which their solution was
attempted.
Anything like a careful analysis of the Scholastic doctrines
concerning the Insolubilia would require an extensive knowledge of medieval logic and might not be justified in a study of
the logical side of the problem. But it is of some interest to
see the form in which the difficulties arose and the types of
solution offered, and in particular the close analogies which
these bear to the solutions of more modern writers.
The first collections of Insolubilia contained a number of
1
Mr. Rustow's thesis consists of an extended examination of ancient literature for references to the paradox of the " L i a r " and a discussion of the uses
to which the fact of paradox was put. There is also a short list of sentences
from Scholastic writers giving the substance of their solutions. He has
stated the paradoxes in two type forms that are very enlightening, and with
these statements as a basis he offers a solution of his own which will come in
for a share of our consideration. Der Liigner, Leipsig, 1910.

4
paradoxes which had in common only the characteristic that
they were difficult of solution. It came gradually to be
recognized that there was a class of these that contained a
peculiar difficulty and were not to be attacked by ordinary
methods. These were the propositions which purported to
apply to themselves and to state directly or indirectly their
own falsity.
One of the longest of these lists, one compiled after this
distinction had been recognized, was that of Albertus de
Saxonia who wrote in Vienna before 1390. Four of his
examples are typical of the rest. These are: 1
1. Ego dico falsum.
2. Haec propositio est falsa.
3. Ponatur, quod Socrates dicat illam, "Plato dicit falsum,"
et Plato dicat illam, "Socrates dicit verum."
4. Si deus est, aliqua conditionalis est falsa, et sit nulla alia
conditionalis.
The first of these reduce to the type of "This proposition is
false/' and the last makes its own truth imply the falsity of a
class of propositions to which it itself belongs, as does "All
propositions are false." The rest of his long list for the most
part reduce to these two types, intrigued by the use of terms
whose meaning is not always definite; for example, "posito
quod in mente Socratis sit ista, 'Socrates decipitur,' et nulla alia,
et Socrates credat illam esse wram, quaeritur, an Socrates eredendo earn esse wram, decipiatur."
Of the solutions proposed for these two forms of the paradoxes Prantl has mentioned the most important and these
have been compiled by Riistow. Among the whole number
some are obscure and are dependent on distinctions and rules
which belong to the complicated apparatus of medieval logic
1
For this period Prantl has collected a great number of passages
with the Insolubilia. The only important omissions that I have been
find are Wy cliffe's Solution in his Tractatus de Logica of which Prantl
no mention, and an incomplete representation of the solution of
Venetus.

dealing
able to
makes
Paulus

5
which modern logic has discarded. But the majority make
use of comparatively simple devices.
The first solution seems to have been that of Buridan,
written in the fourteenth century. One of the premises which
leads to the paradox is the law of excluded middle. In the
Insolubilia we have examples of propositions which are both
true and false. Therefore the law must be rejected. 1
Another attempt to deal with the situation is made by
Hentisberus and mentioned by Buridan. 2 This is based on
the doctrine of restrictio which was worked out in great detail
by the Scholastics. This doctrine states that the denotation
of a term, which as far as the explicit statement goes might
be taken to be general, is often limited by the context,—an
approximation to the more modern universe of discourse.
In the case of the Insolubilia this unexpressed restriction
limits the denotation of a term in a proposition whose verb is
in the present tense to time immediately preceding the present
instant, with the idea that the time indicated is that at which
the proposition is begun, not that in which it is expressed.
This solution has, like the next, a basis in a psychological
analysis. We feel that no thought can really be "of itself."
The restriction which prevents this can be made in another
way. Peter von Ailly in his tractate on the Insolubilia 3
argues that these Insolubilia all lie in the field of verbal or
written propositions as distinguished from mental propositions.
These written and spoken propositions are not really propositions at all, and are merely the expression of mental propo1
Grelling-Nelson note this possibility, and mention that it is this axiom
that is used to define paradox. Abhandlungen der Fries'chen Schule: II.
2 Prantl, IV: 89.
3
Prantl, IV: 104. "Nulla propositio mentalis proprie dicta potest significare
se ipsam esse falsam. . . . Impossibile est intellectui primo formare propositionem
universalem mentalem proprie dictam significantem omnem propositionem mentalem esse falsam. . . . Nulla propositio mentalis proprie dicta potest significare,
se ipsam esse veram, . . . nee potest habere reflexionem supra se ipsam.1'
Cf. also Occam, quoted by Johannis Majoris (Pr. IV: 250). A simpliciter
insolubile is not possible.

6
sitions. It is to the last only that the predicates true and false
apply. Furthermore, no mental proposition can really denote
itself or have itself as one of its terms, nor can any term represent itself "formally." This would represent an impossible state of mind.
The rule for avoiding the paradoxes becomes, "Pars
propositionis non potest supponere pro toto."
This method of dealing with the problem appealed to a
number of later writers, Johannis Majoris Scotus, Olkot, and
Rosetus among them. It is much the same as Russell's
device of the theory of types which depends on the principle
of the vicious circle, namely that no term in a proposition can
presuppose the proposition or have it as one of its possible
values.
The only Scholastic to make a serious criticism of this view
was Wycliffe1 and the difficulty which he points out is much the
same as that contained in the theory of types.2 If we are
not to allow a proposition to refer to itself we make a general
proposition like "AH propositions are true or false" exceptive.
It becomes, "All propositions are true or false except this
proposition." We would seem to do away with all general
propositions about propositions and there are some of these
which we do not wish to reject. Wycliffe's criticism takes for
consideration the proposition, "A is A." This would not be
true universally, for there is one value of A, namely the proposition itself, to which it could not apply.
Wycliffe's own solution lies in denying that the paradoxical
statement, "This proposition is false," is true or false in one
sense of truth and falsehood, and so denying that it is a
proposition at all. The criterion of truth is correct representation of a situation independent of the proposition itself,
and in the given instance there is no such independent situation
represented. He does not go on to consider what the effect of
1
2

Tractatus de logica, p. 197.
This criticism parallels that of Russell by Grelling-Nelson.

7
this theory would be on "All propositions are true or false"
where there is both independent reference and self reference.1
Wycliffe's theory is really of the type of another theory
termed by an unknown author of a Paris manuscript2 cassatio
which involved denying that the propositions in question were
propositions at all. Since this theory was not accompanied
by a definition of proposition which would exclude these
propositions only it was only a beginning of a solution.
Besides the theory of restriction the theory which found the
most support was that of Paulus Venetus, of a type which we
shall have occasion to refer to later as null-class solution.
This pointed out that every proposition implies its own truth,
and that the propositions in question assert also their own
falsity. Any proposition thus implying contradictories is
false, and these propositions are then false. Paulus states
this concisely.3 "Hoc est falsum significant quod hoc estfalsum
et quod hoc est verum, sed quod hoc sit f ahum et hoc sit verum
est impossibile simpliciter." This is like the comment of
ManselFs Aldrich, that these sentences are merely contradictory and so say nothing.4
I have chosen only a few of the Scholastic doctrines for
mention for there were many which did not represent anything
more than a rhetorical treatment like that of the Scot David
1
This is like the solution proposed in Mansell's Aldrich. " Incipiat Socrates
sic loqui, 'Socrates dicit falsum1 et nihil amplius loquatur: turn interrogat aliquis
utrum vera an falsa sit haec propositio. Respondeo, nee veram nee falsam esse,
sed nihil significare, nisi aliquid aliud respiciat, quod a Socrate ante dictum supponitur. Qui enim profert haec verba, ' Socrates dicit falsum,'' fert judicium de
dicto Socratis: quique fert judicium, necessario praesupponit aliquid de quo
judicet: unde cum sententia praesupponat objectum suum, clarum est eandem
numero propositionem, et sententiam et eius objectum esse non posse. Quare et
Scholarum subtilitas hie nihil profecit; nihilque opus est plura dicere de Insolubilibus.
2 Prantl, IV: 41.
3
Logica Pauli Veneti, Venetiis, 1654, p. 84.
4
"Sed qui ut verum simul dicat et mentiatur dicit unum aliquid, cuius partes
sibi invicem contradicunt, is nee verum, nee falsum, sed omnino nihil dicat; quando
enim sententiae pars una evertit alteram, tota nihil prorsus significat sed inaniter
strepit."

8
Cranston who dismissed them with the rule that no proposition
really falsified itself when the contrary could be rationally
maintained.
After the decline of the Scholastic logic the problem of the
Insolubilia fell into disrepute. The only mention of it is the
expression of a wonder that men could have been inclined to
discuss so subtle and vain a question. This attitude found
occasional expression until modern times along with a contempt for all purely formal difficulties. Even in the last
century Lotze dismisses the problem with the remark that
when an assertion involves something in regard to the fact of
its assertion which makes the assertion impossible or untrue,
it is formally insoluble, and he does not seem to feel any
desire to change this state of affairs. He seems to think that
the essential difficulty is overcome in pointing out that we
can say " I lied" instead of " I lie" and there will be no contradiction.
In 1869 Mr. C. S. Peirce published an article on " T h e
Validity of the Laws of Logic" in which he considered the
paradox of the proposition, "This proposition is false." 1
Although Mr. Peirce was one of the leaders in opening the
field of symbolic logic his discussion of the Insolubilia has more
in common with the Scholastic logicians than with the modern.
He himself calls attention to the fact that his solution is identical with that of Paulus Venetus except that he offers a proof
for a principle which Paulus assumes to be true.
Mr. Peirce shows that the proposition, "This proposition is
false" whether it be assumed false or true can be proved both
false and true by regular procedure. But an examination of
the reasoning which leads from the assumption that the
proposition is false to the conclusion that it is true shows that
it is based on a false premise, namely that all that the proposition says is that it is false, whereas every proposition
1

Journal of Speculative Philosophy, I I : no. 4.

9
implies in addition to this that it is true. Hence, though
the proposition is false, that does not make it true or make it
assert the truth, for it asserts that it is both false and true.
This solution we may call a nul-class solution since it lies in
showing that the proposition implies contradictories and so is
false.
Besides these notices and a comment by the editor of
WyclifiVs Latin Works in which he suggests that the paradoxical propositions are not really propositions at all, I have
been able to find no serious consideration of the paradoxes
until the development of symbolic logic brought them to the
light in a form which could not be ignored. It was Mr. Russell
who first noticed these difficulties in connection with the theory
of classes and considered several devices for overcoming them.
The one to which he gave preference was the no-class theory
and it was from this that his perfected theory of types grew.
This theory of types will require an extended statement.

To deal with these paradoxes Russell has invented a theory
of types that Mr. Harold Chapman Brown has very sagely
characterized as a disguised universe of discourse, but it is
rather more than that,—it is a theory of the universe of discourse.
Russell's logic differs from the traditional logic in admitting
expressions that he calls propositional functions, or simply
functions. These are not propositions but are derived from
propositions by making one or more of the terms indefinite.
The proposition "Socrates is a man" of the text books can be
expressed "#is a man" in which x means that a term is to be
supplied. These propositional functions are of two kinds,
those in which the truth value of the function depends on the
value given the variable and those in which the truth value is
the same for all values of the variable. The function "x is a
man" is of the first type; "(x is a man' implies (x is mortal'"

10
is of the second. Only functions of the second type can be
called true or false and treated as propositions in logic. The
variable that occurs in functions of this type Russell calls
an apparent variable. Examination of the paradoxes has
led him to conclude that all of them contain an apparent
variable that makes reference to " a l l " of a totality that is
really illegitimate.
The apparency of the variable is deceitful.
The understanding "for all values" is never quite justified,
for there is sometimes possible one value that will result in
paradox. This troublesome value is that of the function
itself. In the case of " All propositions are false " we may make
the statement, "For all values of tiuy tC i s a false proposition,"
and then insert as a value of x this statement itself and the
result is paradox. This leads to the conclusion that such variables must have ranges of values that result in significant
propositions and that the function itself must be excluded
from this range in each case. "Whatever contains an apparent variable must not be one of the possible values of that
variable."
This exclusion of a function from the range of significance
of a variable occurring in it is accomplished by the theory of
types which is really a theory of ranges of significance, or,
to use the more familiar term, of universes of discourse. A
logical type is defined as the range of significance of a propositional function. The purpose of the theory of types is to rule
the function itself out of the range of significance of its
variables. A proposition is not one of the things to which it
can refer. This exclusion is accomplished by establishing a
series of ranges of significance. The first range is defined as
including all "individuals" and these are known by the character of lack of complexity, which only means that they are
not analyzable into term and predicate as are propositions.
These are the first logical type, and the functions in which they
occur are first order functions. These first order functions
constitute the second logical type, and the functions into which

11
they enter, the second order of functions, and the process may
be extended indefinitely. The purpose of the theory is accomplished by defining functions whose terms are of a type
below their own as "predicative functions" and ruling that
only predicative functions are to be admitted.
The effect of this on the paradoxes is evident. Where "all
propositions" occurs in a proposition of order n the proposition,
to be legitimate, must have, understood or expressed, the addition "of order less than w." " I lie" becomes "There is a
proposition of order n which I affirm and which is false," and
this is of order one greater than n; hence no proposition of
order n has been uttered and the statement is simply false.
This is the theory in its essentials and as far as the paradoxes
are concerned is enough to meet the situation. When the
theory is incorporated into the logic Russell finds an addition
necessary, and with this addition we are led into a number of
difficulties of serious nature. Incidentally we find ourselves
unable to make any statement concerning truth or meaning,
or logic and have the statement itself true or meaningful or
logical within the meaning of its own terms. But what is of
more concern to the mathematical logician is that at this
point we find ourselves unable to speak of all the properties
of a term, and yet such reference is very necessary at times. It
is through such reference that mathematical logic has been in
the habit of defining identity. The usual definition is: For
all values of <p, <p(x) implies <p(y), or y has all the properties
of x. But this statement itself is a function of # or a property
of x, and so is a possible value of x and the definition is not
predicative.7 To make it possible to refer to all the properties
of a term the axiom of reducibility is invented, the assumption
that there exists for any given function a predicative function
that is true when it is true and false when it is false. Applied
to the non-predicative function "All propositions are true or
false" this would assert that for every set of propositions there
will be found a proposition of higher order which will assert

12
of these that they are true or false, though this proposition
will not apply to itself and if it is to be included must be taken
to be a part of a new set consisting of the propositions to which
it referred, and itself.
The intended effect of the axiom of reducibility is to make
the theory of types less "drastic," and drastic it surely is,
as its elaborate introduction into Russell's logic shows, but
whether it accomplishes this result is not so clear. Through
the theory of types we have declared that non-predicative
functions are not to be allowed and that they cannot be called
true or false. If the predicative function which the axiom
assures us accompanies each non-predicative function is to be
true only when the non-predicative function is true, then we
must have had a true non-predicative function and there
could have been no need for the other. All that the axiom can
mean is that for every function there exists a predicative function that is true when the given function is meant to be true,
and false when it is meant to be false, and this stands in great
need of explanation that would have to include among other
things a definition of "meant to be true."
The theory of types avoids the paradoxes by declaring them
illegitimate propositions and we might accept this outcome as
satisfactory if it were not that the theory does not confine its
effect to the propositions that give trouble but extends to
others in which no contradiction appears, and in some of
these the doubtful recourse of the axiom of reducibility is not
available as it is for "All propositions are true or false." In
"This proposition contains five words" and "This proposition
is t r u e " we have sentences that seem to have meaning and
yet are declared to be non-predicative. We are forbidden to
use them along with the propositions for which the theory was
invented.
1
That the axiom must avail itself of its own protection in order not to be
rejected by the principle of the vicious circle presents another curious paradox
like the one pointed out in the latter principle by Grelling-Nelson quoted below.

13
One of the criticisms of Russell's theory brings up an
interesting question. K. Grelling and L. Nelson 1 call attention
to the fact that the principle of the vicious circle, that no
function may make reference to itself in one of its terms,
makes a statement concerning all functions and hence concerning itself. They might have gone farther and pointed out
that since the antecedent of every implication contains understood all the axioms and principles of our logic, on Russell's
view every proposition asserts " I make no reference to
myself." When we ask whether our principle that "only
predicative functions are to be allowed" is itself predicative,
we find ourselves in as serious a difficulty as any of the paradoxes could offer, but it is doubtful whether we can ever require
a principle to stand its own test. Many principles of logistic
cannot be expressed in the logistic for which they legislate.
We cannot express in symbolic language the convention to use
Roman type to represent propositions; nor can the right to
substitute an expression for its equivalent be symbolically
expressed without assuming the right in the expression*
With every logic there must be a body of extra-symbolic
principles that govern the use of the symbolism.
There has been one other solution for the paradoxes worked
out with the care of Mr. Russell's. In 1910 Mr. Alexander
^ustow published a monograph on " T h e Liar" 2 in which he
has collected a number of references to the paradox in ancient
writings and made a brief summary of the scholastic solutions
given in Prantl's history of logic. His own theory is a development from the analysis of Grelling-Nelson who had stated the
paradoxes in a type form in the language of theory of classes.3
Riistow finds that all the paradoxes can be stated in one of
two forms, one of them for paradoxes of the type of "This
proposition is false" and the other for those akin to "All
propositions are false." The first of these is:
1

Abhandlungen der Fries'chen Schule, II.
Der Liigner, Theorie, Geschichte und Auflosung, Leipzig, 1910.
3
Abhandlungen der Fries'chen Schule, vol. II, p. 306.
2

14
I
z < x + x' logic
II
(z < x) < (z < a),
(a < a;') < (» < a')
definition
III
a < # • a + x' - a' logic
IV
x s a'
V
a < a • a' + a' • a
VI
(z < a) < (z < a'),
(z < a') < (z < a).
Applied to the paradox, "This proposition is false" this
gives the following analysis:
I. This proposition is what its predicate says or it is not.
II. That this proposition is what its predicate says implies
that it is true; and that it is not what its predicate says implies
that it is false.
III. Then it is either what its predicate says and true, or
not what its predicate says and false.
IV. Let its predicate say it is false.
V. Then it is either false and true, or true and false.
VI. Stating the conclusion in another way we have: That
this proposition is false implies that it is true, and that it is
true implies that it is false.
With this form as a starting point Rustow constructs a
theory of the paradoxes that has close resemblance to RusselPs
theory of types, though the only form of RusselPs solution
with which he was acquainted was the earlier " no-class theory"
offered in the Principles of Mathematics. The result of the
symbolic statement of the paradoxes has been to show a
definitional element into which a variable term enters, the
element I I of the form given above. The paradox does not
result until for this term there has been substituted a value
that "contradicts one of the constants in the definition." A
definition that contains a variable Rustow calls mehrdeutig,
and finds the remedy for paradox in a restriction of such definitions just as Russell finds it in a restriction of propositional
functions. This restriction consists in declaring any definition
containing a variable illegitimate if it is possible for the
variable to take on a value that "contradicts one of the

15
constants in the definition/' Riistow is not so definite as
Russell as to the way in which his solution is to be carried out.
If we take what has been given as final the result will be
merely that all ambiguous definitions are illegitimate and it is
doubtful whether we have made much progress or not; for
consider the proposition "This proposition is true." Here we
have the same definitional element as in "This proposition is
false." Is this proposition to be declared illegitimate as was
the other? If it is, just as in the case of the theory of types,
we have excluded propositions that offer no paradox. If we
interpret Riistow as meaning his restriction of ambiguous
definitions to be a principle that forms part of every logical
premise in the form, " N o definition containing a variable that
can take on values contradicting a constant in the definition
is legitimate" we have a situation exactly like the outcome of
Russell's theory. On the other hand, if we take it to mean that
This definition is valid for all values of the variable except the
one that contradicts one of its constants we have thrown no more
light on the situation than to say "Paradoxical propositions
are paradoxical."
Another writer who has devoted some attention to the theory
of types is the mathematician Poincare who carried on an
extended discussion with Russell in the Revue de Metaphysique
et de Morale. The substitute which Poincare offered was not
a logical theory but rather a practical rule for avoiding
paradox, namely the avoidance of non-predicative definitions.
Logic, he holds, is a study of the properties of classification,
but a classification is not amenable to the purposes of logic
unless it is a completed one. The paradoxical classes are not
complete since they are always open to the addition of new
members which presuppose the classification made. The
definition of E as the totality of decimals definable in a finite
number of words is not legitimate since the totality can never
be completed. New decimals can always be added which
presuppose the completion of the totality. All definitions

16
are classifications, and this definition is non-predicative, or
this classification cannot be completed. Poincare proposes
for this particular case the amended definition of Richard
which defines E as the totality of decimals definable in a finite
number of words without mention of the totality E itself.
Russell points out that this amended definition contains a
vicious circle in a very flagrant form.1
Another suggestion has been made by Zermelo.2 As
Poincare excludes definitions that are not predicative, so
Zermelo would exclude those that are not definit. This is
defined as follows: "Eine Frage oder Aussage E uber deren
Gultigkeit oder Ungultigkeit die Grundbezeiungen des Bereiches,
vermoge der Axiome und der allgemeingultigen logischen Gesetze
ohne wilkiir unterscheiden, heisst definit/' This amounts to
making the starting point the class rather than the individual,
and would necessitate the definition of an individual by some
class to which it belonged, and of a class by some higher class
of which it could be shown a part. 3 The class of all classes,
the ordinal of all ordinals, the totality of all decimals definable
in a finite number of words are not definit and so are not to
be allowed.4
This theory of Zermelo's is really only the starting point for a
theory since it does not go on to give a criterion by which
we could determine when a question was decided by the
fundamental relations of the range. In Russell's device we
1

La logique de Tinfini, Rev. de M., 1909: 461.
Untersuchungen uber die Grundlagen der Mengen-lehre, I, Math. Ann.,
65: 261-281.
3
D. Hilbert, Foundations of Logic and Arithmetic, Monist: XV. This is
like Herbart's solution in which he uses functions such as A(z (0) ), and A(x°),
which last is Russell's (x); f(x). In these functions the range of the "arbitraries" or variables is confined to "thought things that have been previously
defined," so that the function itself could not occur among the possible values.
Moreover he defines an aggregate as a thought thing m of which the combinations mx are elements, so that the class of all classes or the ordinal of all ordinals
is impossible.
4
This last paradox, Jourdairi has pointed out, rests on an ambiguity in the
word definable.
2

17
have such a criterion, and the fact that it is necessary Russell
has expressed in much the same way as Zermelo. " W h a t is
necessary,1 is not that the values (of a variable in a function)
should be given individually and extensionally, but that the
totality of the values should be given intensionally, so that,
concerning any assigned object it is at least theoretically
determinate whether or not the said object is a value of the
function."
There are two proposed methods of dealing with the paradoxes through the universe of discourse which were both
developed in criticism of Russell. Mr. Shearman 2 in comment
on RusselFs chapter in the original work on the Contradiction
takes up the contradiction of the class of classes not members
of themselves. He holds that the subject term in any proposition can only denote "things." The sentence "Happy is
happy" can have one of three meanings; it can be "Happy
things are happy," " T h e concept happy is happy," or noise
merely. The first of these is a truism, the second is false. In
the proposition which states that a class is a member of itself
the mistake arises in taking a simple quality from the predicate
as subject and then assuming that it is complex. The attributes determining a class cannot themselves be members
of that class. The class of classes not members of themselves
cannot be said to be or not to be a member of itself. The
reason is that the class in its connotation is different from the
class as denotation.
This is practically what is suggested by Mr. Harold Chapman
Brown.3 He maintains that the propositional functions containing an apparent variable are what the traditional logic
held to be verbal propositions, and a verbal proposition must
be in a different universe of discourse from the connotation
of the subject term. It cannot be both a connotation and a
1

Principia Mathematica, I: 42.
Russell's Principles of Mathematics, Mind, XVI, 261.
3
Jour, Ph. Ps. Sci. Meth., VII, 85.

2

18
denotation in the same respects by the principle of contradiction.
Both of these suggestions along with Russell's apply as well
to "This proposition is true," "This proposition has subject
and predicate/' and "All propositions are true or false" as
to the paradoxical propositions, since all of these pretend to
be at the same time connotation and denotation in that they
denote themselves. If they denote themselves at all surely
they denote themselves as connotation. But none of these
propositions lead to paradox. The effect of the analysis
that leads to the conclusion that the propositions cannot be
at once connotation and denotation is substantially that of
the Schoolmen's "Pars propositionis non potest supponere
pro toto."
Mr. Brown is right in thinking that the theory of types
involves the familiar conception of the universe of discourse,
but it is not merely a rediscovery of the universe, it is rather
a theory of the universe of discourse. Without some regulative principle the universe of discourse is arbitrary and lawless. Mr. Brown says that two things are in different universes,
while the theory of types makes clear when two things are in
different universes.
The significant fact about the paradoxes is that they do not
exist until conditions amounting to definitions of the terms
are introduced. Russell and Riistow both notice that there
is one value of the variable, which means that there is one
definition, that results in paradox. Russell avoids the
paradoxes by a principle that rules out any premise that
refers to itself in one of its terms. Riistow's more obscure
provision rules out any premise that "contains a variable
unless there is excluded from the range of the variable any
value contradicting one of the constants of the definition."
It is clear what is meant by "contradicting a constant" in
the type forms given, but what this would be in general he

19
does not say. It cannot mean the contradictory of any constant in the definition, for definitions can be framed wherein
that is done with no harm resulting. His choice of this
phrase is probably due to the form in which he chose to state
the paradoxes, and the paradoxes can be given symbolic
statement in which no such substitution as x = a' is made and
the provision against it then becomes meaningless.
Russell and Riistow both find in the paradoxes propositions
that do not hold for all values of the terms. Both meet this
difficulty with a general principle that legislates for all values
of the terms, Russell with the principle of the vicious circle
that excludes a class of values that make reference to the
proposition itself, Riistow with a principle that excludes
"values that contradict a constant in the definition." Of
these two Russell's restriction excludes more values than are
necessary for the purpose of the restriction and makes illegitimate propositions in which was made not the troublesome
substitution but a very harmless one. Riistow avoids this
disadvantage at the expense of solving a special case that the
paradoxes need not resemble. We can so state any of the
paradoxes that the contradiction is not introduced by giving
a variable a value that is the contradictory of one of the
constants in the proposition. One of the paradoxes will be
stated in such a form below.

It is possible to approach the solution in another way that
does not involve any departure from traditional logic. In all
of the examples that have been given we find nothing paradoxical in the discovery that a proposition implies its contradictory. For this we are to some extent indebted to the
logistic. That p < pf is merely the criterion for the falseness
of p. We may express this by: p < 0. If nothing were true
except that certain propositions imply their own contradictories there would be no difficulty. But this is not the

20
whole case. The propositions under discussion seem to be
implied by their own contradictories as well as to imply them.
This is a true paradox for it makes p < pr and pf < p and
results in the ordinary algebra in making 1 equivalent to 0.
That this condition exists has been left to verbal statement.
To make it analytically true the proposition must be put in
symbolic form and the implication shown to follow, not merely
stated to follow. This involves supplying premises that are
implicit in the definitions of the terms used until we have an
implication that is true for all values of the terms. In other
words it is not analytically true that p < p'. This is a mere
statement.
That (this proposition is false) < (this proposition is false)' or, (a < b) < (a < h)' can be made logically
explicit by giving a and b definitions. It is perfectly clear that
it does not hold for all values of a and b as it now stands. The
introduction of these definitions will make the implication
explicit.
Let us introduce these necessary premises. The implication
may be so stated:
{[(a <b)<b]<(a<

b)f} • {(a < 6 ) < a] • {a < b}
< (a < b)f

Three premises have been necessary. The first is contained
in the definition of "false," the second in the definition of
"this proposition," and the third is the proposition which we
desire to imply its contradictory. As the whole implication
stands it is true for all values of the terms. Using " / " for
"false" we may express the fact that the first premise is a
definition of "false" so:
{[(a<b)<b]<(a<by}

= (&=/)

If we substitute / for b this premise becomes unity. That is,
it is true for the value / as defined. This may not be a complete definition of "false" but it is an element in a complete
definition and may be regarded as the part of a complete
definition that we need. The test of its correctness is that

21
there is no term save "false" that satisfies it, leaving out of
consideration the values 0 and 1, since the use of either of
these makes the complete proposition analytically true
directly.
If we regard this first premise as the definition of / then
the substitution of / for b makes it unity and it can be left
unexpressed. The complete statement then becomes:

[<P<f)<a]-[a<f\<[a<fy
The term a need not be completely defined in order to make
the factor " ( a < f) < a " unity and so reduce the expression
to the form, " ( a < b) < (a < 6)'." We need only introduce
the restriction:

(p<a)<[(a<f)<a]
This is true whether p is taken to represent "this proposition"
or "all propositions." The second premise in the original
proposition we may take to be an element common to the
definition of "this proposition" and "all propositions." For
the solution of the paradox it is not necessary to complete
the definition of either.
With the substitution of p for a we may now drop the factor
"(a < / ) < a " as a unit factor (to any implication such as
" 1 - 1 • x < y" the first two factors are unessential) and there
remains the proposition in the desired form:

All this goes only to make the implication of its contradictory analytically sound for "This proposition is false."
This makes of the proposition merely a null-proposition like
many others with which we are familiar, for any false proposition implies its contradictory. If there is to be paradox this
must be accompanied by the proof that (p < / ) ' < (p < / ) ,
but in order to reach this result it is necessary to use the term
p or "this proposition" in an ambiguous sense as referring
both to "this proposition is false," or (p < / ) and to " ' t h i s

22

proposition is false' is false," or (p < / ) ' . In other words the
second necessary part of the paradox is fallacious.
A like confusion is responsible for the paradox of the class of
classes that are not members of themselves. This Russell
has stated so: (x < w) < (x < x)f which leads to the result
(w < w) < (w < w)f on substitution of w for x, and the
symmetric proposition which Russell evidently takes for
granted, (x < x)f < (x < x) which would lead to (w < w)'
< (w < w). If this is to stand as a case in which a proposition
implies its contradictory and the contradictory implies the
proposition w must be substituted for x in the proposition in
each place that it occurs, but it is essential to the meaning of
the epsilon relation that its antecedent be an individual and
its consequent a class, and that if they are the same term that
term is first considered as individual and then as class. This
understood restriction should be expressed by using Greek
letter or capitals for the class. The statement xex really
means, when the nature of the relation in question is taken into
account, "x, the individual, is a member of x, the class and
the symbolic expression should take this into reckoning. As
now stated there is involved the substitution of w, the class,
for x, the individual, as well as x, the class. Only a definition
of the epsilon relation could avoid this and then it might be
possible to show the fallacy of one part of the paradox. So
long as the relation remains "indefinable" we have no right
to paradox or solution.
The proposition "All propositions are false" sometimes
taken to be a paradox of the same type as the others fails of
this in so far as it is merely a case of a proposition that implies
its contradictory without its contradictory implying the
proposition. If all propositions are false it is false that all
propositions are false; but if not all propositions are false it
does not follow that all propositions are false.
We may expect to find in all the paradoxes a pretended
equivalence or mutual implication between proposition and

23

contradictory, but we may be prepared also to find that one
side of the relation is fallacious. A proper symbolic statement
of each fallacy would require a complete logic and it must be
sufficient here to indicate a method by which the problem can
be attacked when such a logic is available. There is not
ground for the belief that this logic will require for the solution
of the paradoxes a special principle that is strange to the
traditional formal logic. Nor need we find in these paradoxes,
or in propositions that are implied by their contradictories
evidence of an Absolute in our thought as Mr. Royce suggests
in his essay in the "Problem of Truth" 1 but things amenable
to definitions and axioms in which we need find no last word.
1

William James and other Essays, p. 244.