Mathematics Appendix B Paradox, is a contradiction that arises in the
logical treatment of classes and "propositions", where "propositions"
are understood as mind-independent and language-independent logical
objects. If propositions are treated as objectively existing objects,
then they can be members of classes. But propositions can also be
about classes, including classes of propositions. Indeed, for each
class of propositions, there is a proposition stating that all
propositions in that class are true. Propositions of this form are
said to "assert the logical product" of their associated classes. Some
such propositions are themselves in the class whose logical product
they assert. For example, the proposition asserting that
all-propositions-in-the-class-of-all-propositions-are-true is itself a
proposition, and therefore it itself is in the class whose logical
product it asserts. However, the proposition stating that
all-propositions-in-the-null-class-are-true is not itself in the null
class. Now consider the class w, consisting of all propositions that
state the logical product of some class m in which they are not
included. This w is itself a class of propositions, and so there is a
proposition r, stating its logical product. The contradiction arises
from asking the question of whether r is in the class w. It seems that
r is in w just in case it is not.
This antinomy was discovered by Bertrand Russell in 1902, a year after
discovering a simpler paradox usually called "Russell's paradox." It
was discussed informally in Appendix B of his 1903 Principles of
Mathematics. In 1958, the antinomy was independently rediscovered by
John Myhill, who found it to plague the "Logic of Sense and
Denotation" developed by Alonzo Church.
1. History and Historical Importance
In his early work (prior to 1907) Russell held an ontology of
propositions understood as being mind independent entities
corresponding to possible states of affairs. The proposition
corresponding to the English sentence "Socrates is wise" would be
thought to contain both Socrates the person and wisdom (understood as
a Platonic universal) as constituent entities. These entities are the
meanings of declarative sentences.
After discovering "Russell's paradox" in 1901 while working on his
Principles of Mathematics, Russell began searching for a solution. He
soon came upon the Theory of Types, which he describes in Appendix B
of the Principles. This early form of the theory of types was a
version of what has later come to be known as the "simple theory of
types" (as opposed to ramified type theory). The simple theory of
types was successful in solving the simpler paradox. However, Russell
soon asked himself whether there were other contradictions similar to
Russell's paradox that the simple theory of types could not solve. In
1902, he discovered such a contradiction. Like the simpler paradox,
Russell discovered this paradox by considering Cantor's power class
theorem: the mathematical result that the number of classes of
entities in a certain domain is always greater than the number in the
domain itself. However, there seems to be a 1-1 correspondence between
the number of classes of propositions and the number of propositions
themselves. A different proposition can seemingly be generated for
each class of propositions, for instance, the proposition stating that
all propositions in the class are true. This would mean that the
number of propositions is as great as the number of classes of
propositions, in violation of Cantor's theorem.
Unlike Russell's paradox, this paradox cannot be blocked by the simple
theory of types. The simple theory of types divides entities into
individuals, properties of individuals, properties of properties of
individuals, and so forth. The question of whether a certain property
applies to itself does not arise, because properties never apply to
entities of their own type. Thus there is no question as to whether
the property that a property has just in case it does not apply to
itself applies to itself. Classes can only have entities of a certain
type: the type to which the property defining the class applies. There
can be classes of individuals, classes of classes of individuals, and
classes of classes of classes of individuals, etc., but never classes
that contain members of different types. Thus, there is no such thing
as the class of all classes that are not in themselves. However, on
the simple theory of types, propositions are not properties of
anything, and thus, they are all in the type of individuals. However,
they can include classes or properties as constituents. But consider
the property a proposition has just in case it states the logical
product of a class it is not in. This property defines a class. This
class will be a class of individuals; for any individual, the question
arises whether that individual is in the class. However, the
proposition stating the logical product of this class is also an
individual. Thus, the problematic question is not avoided by the
simple theory of types.
Some authors have speculated that this antinomy was the first hint
Russell found that what was needed to solve the paradoxes was
something more than the simple theory of types. If so, then this
antinomy is of considerable importance, as it might represent the
first motivation for the ramified theory of types adopted by Russell
and Whitehead in Principia Mathematica.
2. Formulation and Derivation
In 1902, when he discovered this paradox, Russell's logical notation
was borrowed mostly from Peano. However, translating into more
contemporary notation, the class w of all propositions stating the
logical product of a class they are not in, and r, the proposition
stating its logical product, are written as follows:
w = {p: (∃m)[(p = (∀q)(q ∈m →q)) & ~(p ∈m)]}
r = (∀q)(q ∈w →q)
Because propositions are entities, variables for them in Russell's
logic can be bound by quantifiers and can flank the identity sign.
Indeed, Russell also allows complete sentences or formulae to flank
the identity sign. If α is some complex formula, then "p = α" is to be
understood as asserting that p is the proposition that "α". Thus, w is
defined as the class of propositions p such that there is a class of m
for which p is the proposition that all propositions q in m are true,
and such that p is not in m. The proposition r is then defined as the
proposition stating that all propositions in w are true.
The derivation of the contradiction requires certain principles
involving the identity conditions of propositions understood as
entities. These principles were never explicitly formulated by
Russell, but are informally stated in his discussion of the antinomy
in the Principles. However, other writers have sought to make these
principles explicit, and even to develop a fully formulated
intensional logic of propositions based on Russell's views. The
principles relevant for the derivation of the contradiction are the
following:
Principle 1: (∀p)(∀q)(∀r)(∀s)[((p → q) = (r → s)) →((p = r) & (q = s))]
Principle 2: [(∀x)A(x) = (∀x)B(x)] →(∀y)[A(y) = B(y)]
The first principle states that identical conditional propositions
have identical antecedent and consequent component propositions. The
second states that if the universal proposition that everything
satisfies open formula A(x) is the same as the universal proposition
that everything satisfies open formula B(x), then for any particular
entity y, the proposition that A(y) is identical to the proposition
that B(y).
Then, from either the assumption that r ∈w or the assumption ~(r ∈w),
the opposite follows.
Assume:
1. r ∈w
From (1), by class abstraction and the definition of w:
2. (∃m)[(r = (∀q)(q ∈m →q)) & ~(r ∈m)]
(2) allows us to consider some m such that:
3. (r = (∀q)(q ∈m →q)) & ~(r ∈m)
From the first conjunct of (3) definition of r we arrive at:
4. (∀q)(q ∈w →q) = (∀q)(q ∈m →q)
By (4) and principle 2, then:
5. (∀q)[(q ∈w →q) = (q ∈m →q)]
Instantiating (5) to r, we conclude:
6. (r ∈w →r) = (r ∈m →r)
By (6), and principle 1, then:
7. (r ∈w) = (r ∈m)
This, with the second disjunct of (3), yields:
8. ~(r ∈m)
By (7) and (8) and substitution of identicals, we get:
9. ~(r ∈w)
This contradicts our assumption. However, assume instead:
10. ~(r ∈w)
By (10) and class abstraction:
11. ~(∃m)[(r = (∀q)(q ∈m →q)) & ~(r ∈m)]
By the rules of the quantifiers and propositional logic, (11) becomes:
12. (∀m)[(r = (∀q)(q ∈m →q)) → (r ∈m)]
Instantiating (12) to w:
13. (r = (∀q)(q ∈w →q)) → (r ∈w)
By (13), the definition of r, and modus ponens:
14. r ∈w
Thus, from either assumption the opposite follows.
3. Frege's Response
Soon after discovering this antinomy, in September of 1902, Russell
related his discovery to Gottlob Frege. Although Frege was clearly
devastated by the simpler "Russell's paradox", which Russell had
related to Frege three months prior, Frege was not similarly impressed
by the Russell-Myhill antinomy. Russell had formulated the antinomy in
Peano's logical notation, and Frege charged that the apparent paradox
derived from defects of Peano's symbolism.
In Frege's own way of speaking, a "proposition" is understood simply
as a declarative sentence, a bit of language. Frege certainly did not
ascribe to propositions the sort of ontology Russell did. However, he
thought propositions had both senses and references (see
sense/reference distinction ). He called the senses of propositions
"thoughts" and believed that their references were truth-values,
either the True or the False. An expression written in his logical
language was thought to stand for its reference (though express a
thought). When propositions flank the identity sign, e.g. "p = q" this
is taken as expressing that the two propositions have the same
truth-value, not that they express the same thought.
Thus, Frege was unsatisfied with Russell's formulation of the
antinomy. In Russell's definition "w = {p: (∃m)[(p = (∀q)(q ∈m →q)) &
~(p ∈m)]}", the part "p = (∀q)(q ∈m →q)" seems to mean not an identity
of truth-values, but thoughts. However, if this is the case, then
"(∀q)(q ∈m →q)" must be understood as referring to, rather than simply
expressing, a thought. However, on Frege's view, this would mean that
the expressions that occur in it have indirect reference, i.e. they
refer to the thoughts they customarily express. However, in indirect
reference, the variable "m" in that context must be understood not as
standing for a class, but as standing for a sense picking out a class.
However, the second occurrence of "m" later on in the definition of w
must be understood as referring to a class, not a sense picking out a
class. However, if the two occurrences of "m" do not refer to the same
thing, it is extremely problematic that they be bound by the same
quantifier. Moreover, Russell's derivation of the contradiction
requires treating the two occurrences of "m" as referring to the same
thing. Thus, Frege himself concluded that the antinomy was due to
unclarities in the symbolism Russell used to formulate the paradox. He
suggests that the antinomy can only be derived in a system that
conflates or assimilates sense and reference.
However, it is not clear that Frege's response is adequate. Frege
criticizes only the syntactic formulation of the antinomy in a logical
language, not the violation of Cantor's theorem lying behind the
paradox. Frege does not have an ontology of propositions, but he does
have an ontology of thoughts. Thoughts, as objectively existing
entities, can be members of classes. Moreover, it seems that there
will be as many thoughts as there are classes of thoughts. One can
generate a different thought for every class, i.e. the thought that
everything is in the class or that all thoughts in the class are true.
We now consider the class of all thoughts that state the logical
product of a class they are not in, and a thought stating the logical
product of this class, and arrive at the same contradiction. Frege's
metaphysics seems to have similar difficulties.
It is true that the antinomy cannot be formulated in Frege's own
logical systems. However, this is only because those systems are
entirely extensional. In them, it is impossible to refer to thoughts
(as opposed to simply express them) and assert their identity–one can
only refer to truth-values and assert their identity. However, it
appears that if Frege's logical systems were expanded to include
commitment to the realm of sense, to make it possible to refer not
only to truth-values and classes, but thoughts and other senses, a
version of the antinomy would be provable. In 1951, Alonzo Church
developed an expanded logical system based loosely on Frege's views,
which he called "the Logic of Sense and Denotation". In 1958, John
Myhill discovered that the antinomy considered here was formulable in
Church's system. Myhill seems to have rediscovered the paradox
independently of Russell. Hence the term, "Russell-Myhill Antinomy."
4. Possible Solutions
The antinomy results from the following commitments
(A) The commitment to classes, defined for every property,
(B) The commitment to propositions as intensional entities (or to
similar entities, such as Frege's thoughts),
(C) An understanding of propositions such that there must exist as
many propositions as there are classes of propositions; i.e. a
different proposition can be generated for every class,
(D) An understanding of propositions and classes such that for
every proposition and every class of propositions, the question arises
as to whether the proposition falls in the class.
One might hope to solve the antinomy by abandoning any one of these
commitments. Let us examine them in turn.
Abandoning (A), the commitment to classes, is very tempting,
especially given the other paradoxes of class theory. However, in this
context, this option may be not be as fruitful as it might appear.
Russell himself worked on a "no classes" theory from 1905 though 1907.
However, he soon discovered a classless version of the same paradox.
Here, rather than considering a class w consisting of propositions, we
consider a property W that a proposition p has just in case there is
some property F for which p states that all propositions with F are
true but which p does not itself have. Thus:
(∀p)[Wp ↔ (∃F)[(p = (∀q)(Fq →q)) & ~Fp]]
We then define proposition r as the proposition that all propositions
with property W are true:
r = (∀q)(Wq →q)
Then, via a similar deduction to that given above, from the assumption
of Wr one can prove ~Wr and vice versa. Thus it does not do to simply
abandon classes. One would also have to abandon a robust ontology of
properties; perhaps eschewing all of higher-order logic.
One might simply want to abandon (B), the commitment to propositions
or Fregean thoughts understood as logical entities. The commitment to
logical entities in a Platonic realm has grown less and less popular,
especially given the widespread view that logic ought to be without
ontological commitment. The challenge would be to abandon such
intensional entities while maintaining a plausible account of meaning
and intentionality.
However, one might hope to maintain commitment to propositions or
thoughts, but attempt to reduce the number posited. This would likely
involve denying (C). The Cantorian construction lying at the heart of
the antinomy involves the claim that one can generate a different
proposition for every class. In the construction given above, this
claim is justified by showing that for each class, one can generate a
proposition stating its logical product, and showing that, for each
class, the class so generated is different. To deny this, one could
either deny that one can generate such a proposition for each class,
or instead, deny that the proposition so generated is different for
every class. The first strategy is difficult to justify if one
understands propositions and classes as objectively existing entities,
independent of mind and language. If a proposition exists for every
possible state of affairs, then one such proposition will exist for
every class.
However, if one adopts looser identity conditions for propositions or
thoughts, one might attempt to take the second approach to denying
(C). That is, one would allow that the proposition stating the logical
product of one class might be the same proposition as the proposition
stating the logical product of a different class. This is perhaps not
an easy approach to justify. In the Russellian deduction given above,
principles 1 and 2 guarantee that the proposition stating the logical
product of one class is always different from the proposition stating
the logical product of another class. These principles seem justified
by the understanding of propositions as composite entities with a
certain fixed structure. Consider principle 1. It states that
identical conditional propositions have identical propositions in
their antecedent and consequent positions. However, this might be
denied if one were adopt looser identity conditions for propositions.
One might, for example, adopt logical equivalence as being a
sufficient condition for propositions to be identical. If so, then
principle 1 would be unjustified. For example p →q and ~q → ~p are
logically equivalent, however, they obviously need not have the same
antecedent propositions. However, this approach may lead to other
difficulties. Often, part of the motivation for intensional entities
such as propositions or Fregean thoughts is in order to view them as
relata in belief and other intentional states. If one adopts logical
equivalence as sufficient for propositions to be identical, this is
extremely problematic. The simple proposition p is logically
equivalent to the proposition ~(p & ~q) → ~(q → ~p). If we take these
two be the same proposition, then if propositions are relata in belief
states, we seemingly must conclude that anyone who believes p also
believes ~(p & ~q) → ~(q → ~p). This does not seem to be true.
W. V. Quine is famous for suggesting that intensional entities are
"creatures of darkness", having obscure identity conditions. Here it
appears that if the identity conditions of intensions are taken to be
too loose, then intensions cannot do many of the things we want of
them. If the identity conditions of intensions are too stringent,
however, it is difficult to avoid positing so many of them that
inconsistency with Cantor's theorem is a genuine threat.
Lastly, one could maintain commitment to a great number of
propositions or thoughts as entities, but block the paradox by
suggesting that these entities fall into different logical types. That
is, one could deny (D), and suggest instead that the question does not
always arise for every proposition and class of propositions whether
that proposition is in that class. This is in effect the approach
taken with ramified type-theory. In ramified type theory, the type of
a formula α depends not only on whether α stands for an individual, a
property of an individual, or a property of a property of an
individual, etc., but also on what sort of quantification α involves.
The core notion is that α cannot involve quantification over, or
classes including, entities within a domain that includes the thing
that α itself stands for. Consider the proposition r from the
antinomy. Recall that r was defined as (∀q)(q ∈m →q). Thus, r involves
quantification over propositions. In ramified type theory, we would
disallow r to fall within the range of the quantifier involved in the
definition of r. If a certain proposition involves quantification over
a range of propositions, it cannot be included in that range. Thus, we
divide the type of propositions into orders. Propositions of the
lowest order include mundane propositions such as the proposition that
Socrates is bald or the proposition that Hypatia is wise. Propositions
of the next highest order involve quantification over, or classes of,
propositions of this order, such as the proposition that all such
propositions are true, or the proposition that if such a proposition
is true, then God believes it, etc. Here, the challenge is to justify
the ramified hierarchy as something more than a simple ad hoc dodge of
the antinomies, to provide it with solid philosophical foundations.
Poincaré's Vicious Circle Principle is perhaps one way of providing
such justification.
Antinomies such as the Russell-Myhill antinomy must be a concern for
anyone with a robust ontology of intensional entities. Nevertheless,
there may be solutions to the antinomy short of eschewing intensions
altogether.
5. References and Further Reading
* Anderson, C. A. "Semantic Antinomies in the Logic of Sense and
Denotation." Notre Dame Journal of Formal Logic 28 (1987): 99-114.
* Anderson, C. A.. "Some New Axioms for the Logic of Sense and
Denotation: Alternative (0)." Noûs 14 (1980): 217-34.
* Church, Alonzo. "A Formulation of the Logic of Sense and
Denotation." In Structure, Method and Meaning: Essays in Honor of
Henry M. Sheffer, edited by P. Henle, H. Kallen and S. Langer. New
York: Liberal Arts Press, 1951.
* Church, Alonzo. "Russell's Theory of Identity of Propositions."
Philosophia Naturalis 21 (1984): 513-22.
* Frege, Gottlob. Correspondence with Russell. In Philosophical
and Mathematical Correspondence. Translated by Hans Kaal. Chicago:
University of Chicago Press, 1980.
* Klement, Kevin C. Frege and the Logic of Sense and Reference,
New York: Routledge, 2002.
* Myhill, John. "Problems Arising in the Formalization of
Intensional Logic." Logique et Analyse 1 (1958): 78-83.
* Russell, Bertrand. Correspondence with Frege. In Philosophical
and Mathematical Correspondence, by Gottlob Frege. Translated by Hans
Kaal. Chicago: University of Chicago Press, 1980.
* Russell, Bertrand. The Principles of Mathematics. 1902. 2d. ed.
Reprint, New York: W. W. Norton & Company, 1996, especially §500.
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