I. THEY EYTHYPHRONIC DILEMMA
II. SEMANTIC DOUBLING AND THE SCHELLINGIAN MOMENT
III. MEILLASSOUX’SARGUMENT FOR CONTINGENCY
IV. KAPLAN’S PARADOX
V. WHAT MEILLASSOUX SHOULD SAY ABOUT RUSSELL’S PARADOX
explain why being careful as possible
Va. Russell’s Paradox as the Canonical Inclosure Paradox
Graham Priest’s great discovery is that an astonishing number of traditional paradoxes fit the
following schema:
Priest’s Inclosure Schema:
(1)
Ω = {y; φ(y)} exists and ψ(Ω)
Existence
(2)
if x  Ω and ψ(x)
(a)
δ(x)  x
Transcendence
(b)
δ(x)  Ω
Closure
In Beyond the Limits of Thought he gives values for φ, ψ, and δ to derive the paradoxes of
Russell, Burali-Forti, Mirimanoff, Kant (according to Priest’s reconstructed “fifth antinomy”),
König, Berry, Richard, Berkeley, Weyl, Montague, as well as the traditional liar paradox. In
addition, he is able to present a guerilla retelling of the history of philosophy (including Plato,
Aristotle, Cusanus, Aquinas, Anselm, Leibniz, Berkeley, Kant, Hegel, Cantor, Russell, Frege,
Quine, Davidson, Derrida, Wittgenstein, Heidegger, and Nāgārjuna) in terms of the schema.
For Priest, Russell’s paradox has a kind of pride of place, in part because it is “the heart
of Cantor’s Paradox” and “undoubtedly the simplest of all the set-theoretic paradoxes” (Priest
2006, 129). We will explain the connection with Cantor below, but Priest’s key insight is that the
“Russell set,” the set of all sets that are not members of themselves, is arrived at by applying
Cantor’s own diagonalization function to the identity function of the set of all pure sets. 1 First,
we define diag in the usual manner. Where fα is a one-place function defined over the set α,
equals the set of x in α that are such that x is not a member of fα(x):
diag(fα) = {x  α | x  fα(x)}
This is the key function that Cantor constructed to show that the cardinality of any set is less than
the cardinality of the set of all of that set’s subsets. Let the id be the function that takes every
object to itself.
diag(idα) = {x  α | x  idα(x)}
But since by definition idα(x) = x, we can replace idα(x) with x in the above, giving us:
diag(idα) = {x  α | x  x}
Now assume that α is V, the set of all pure sets. Since V contains everything the set of x in V is
the same as the set of all sets.
diag(idV) = {x  V | x  x} = {x | x  x}
1
This is pretty insightful. The way Russell’s Paradox is usually presented is by noting that if the
set of all sets exists then the Russell set would also exist as a subset of it. Priest shows that this
way of presentation is also an instance of the inclusure schema (Priest 2006, 130).
The Russell set.
And now we can consider Priest’s inclosure schema for Russell’s paradox. We should
note that part of what makes Russell’s paradox the simplest such one is that it is in the family of
paradoxes (including the Burali-Forti, Mirimanoff, and Kan’ts 5th Antinomy) where ψ can be a
trivial property such as being self-identical. Thus we don’t need to represent it in the inclosure
schema. Also, since Ω = {y; y  V} is equal to V, we can simplify the inclosure schema thusly:
Priest’s Inclosure Schema:
(1)
V exists
(2)
if x  V
Existence
(a)
diag(idx)  x Transcendence
(b)
diag(idx)  V Closure
The contradiction is arrived at by the characteristic self-application of (a) Transcendence and (b)
Closure to V itself, which yields the contradiction (a) diag(idV)  V and (b) diag(idV)  V. Note
that this is licensed because any set is a subset of itself. From the nature of the contradiction we
can see why Priest uses the phrases he does. The Transcendence lemma shows that the
diagonalizing function yields a value external to the postulated totality (the set of all sets), while
the Closure lemma shows that the same function yields a value inside the postulated totality.
Clearly the self-application yields a contradiction. The interesting work then is always
demonstrating Transcendence and Closure with respect to all of the subsets of the totality in
question. In the case of Russell’s Paradox, Closure follows immediately. Since V is the set of all
pure sets, any pure set is an element of it by definition. Transcendence is a bit trickier, but in the
case of Russell’s Paradox, easily formalizable in a Fitch style system, if we make one
assumption:
1. z(z = {y  z | y  y}  {y  z | y  y})
FACT
This says that any set is the union of two of its subsets: (1) the one composed of elements of that
set that are members of themselves, and (2) the one composed of elements of that set that are not
members of themselves. Since, by the law of bivalence a set either is a member of itself or is not,
this exhausts the possibilities and so hence the union of the two subsets is equal to the superset.
We continue:
2. | x  V
assumption for  introduction
3. | | diag(idx)  x
assumption for  introduction
4. | | x = {y  x | y  y}  {y  x | y  y}
1  elimination
5. | | diag(idx)  {y  x | y  y}  {y  x | y  y}
3,4 = elimination
6. | | | diag(idx)  {y  x | y  y}
assumption for  elimination
7. | | | diag(idx)  diag(idx)
6 definition 
8. | | | diag(idx)  {y  x | y  y}
6 definition of diag(idx)
9. | | | diag(idx)  diag(idx)
8 definition 
10.| | | 
7, 9  elimination
11.| | | diag(idx)  {y  x | y  y}
assumption for  elimination
12.| | | diag(idx)  diag(idx)
11 definition 
13.| | | diag(idx)  diag(idx)
11 definition diag(idx)
14.| | | 
12, 13  elimination
15.| | 
5, 6-10, 11-14  elimination
16 | diag(idx)  x
3-15  introduction
17. x(x  V → diag(idx)  x)
2-16  introduction
We (2) start with an arbitrary x which is a subset of V, and (3) assume for reductio that the
diagonalization on the identity function applied to that x is itself a member of x. Then, (4) note
that the x in question will be equal to the union set of two subsets, one consisting of members of
x that are members of themselves and one consisting of members that aren’t. Since we’ve
assumed that the diagonalization set is a member of x it follows that (5) it is a member of this
union. But then if we reduce to absurdity the assumption that it is in either member of the union
(6-10, 11-14) we will have reduced to absurdity the assumption that it is a member of x (15), and
hence (16, 17) proved that for any subset x of V the diagonalization of the identity function on x
is not a member of x. The two subproofs (6-10, 11-14) both straightforwardly hinge on the
definition of diagonalization and set membership.
The typical response to such paradoxes is to refuse to assert the Existence of the totality
in question. Kant’s original response to his mathematical antinomies is the model of how to
attempt such a thing, and Zermelo-Fränkel ZF set theory is a modern day version of Kant’s
gambit. But there are other possibilities. The problematic open sentences in ZF set theory are in
von Neumann’s set theory expressible as classes, which are like sets but not prohibited from
being members of anything. Thus Closure is blocked, since it cannot apply to the proper class V.
And the proof to Transcendence can be blocked by adopting weaker logics, such the De Morgan
logic developed by Hartry Field in Saving Truth From Paradox.
Priest notoriously accepts Existence, Closure, and Transcendence, holding that inclosure
paradoxes are true paradoxes. When critiquing alternative approaches he appeals to two
principles: (1) the Principle of Uniformed Solutions (PUS) which states that similar paradoxes
should have similar solutions, and (2) the Domain Principle, which in its simplest formulation
states that “For every potential infinity there is a corresponding actual infinity (Priest 2006,
124).”
Jon- how Domain principle works w.r.t. ZF
Vb. Meillassoux, Russell’s Paradox, and the Domain Principle
Joshuaif step ten is valid then he should accept domain principle wrt Russell’s paradox, but wait on as
applied to modality until VIIb
for step seven to do the work the domain principle must be an enthymeme
VII. WHAT MEILLASSOUX SHOULD SAY ABOUT KAPLAN’S PARADOX
VIIa. The Principle of Uniform Solution
Jon- point out that don’t need identity function because proof that set is smaller than powerset
doesn’t assume that they are equal in the premise. So Kaplan’s Paradox is quasi Priestian
VIIb. The Domain Principle Revisited
Jon- show how PUS concerns aren’t relevant anyhow, because the Domain Principle on its own
entails
Joshua- if he can be a Kantian wrt the domain principle applied to modality, then there is no
reason at all to think that step seven overturns correlationism. Give damning quasi Kantian quote
here.
VI. MEILLASSOUX’S CIRCLE
CONCLUSION
Garcia and Gabriel how issues of totality are again at the forefront of continental philosophy.
Hartry Field in Saving Truth From Paradox.
Priest, Graham. Beyond the Limits of Thought.