Against bounded-indefinite-extensibility

Absolute generality Paradox lost Paradox regained Against the third way
Against Bounded Indeﬁnite Extensibility
James Studd
Language, Truth & Logic Workshop
Princeton
March 23rd 2013

Outline
1 Absolute generality: an introduction
2 Paradox lost: is there a coherent case for relativism?
3 Paradox regained: a regimentation of the argument
4 Against bounded indeﬁnite extensibility

Absolute generality
Can we quantify over a domain comprising absolutely everything?
Absolutist about quantifiers: Yes
Some domain comprises absolutely everything.
Relativist about quantifiers: No
No domain comprises absolutely everything.
(First approximation: notoriously difficult to formulate.)
James Studd Against bounded indefinite extensibility 1/26

Absolutism
Some domain—M say—is absolutely comprehensive
Absolutism enjoys prima facie plausibility.
Often quantiﬁers are restricted.
(1) No donkey talks.
(2) Everyone turned up.
But unrestricted quantiﬁcation seems possible (and necessary).
(3) Nothing is God.
(4) Everything is self-identical.

Indefinite extensibility
But there is also a well known argument against absolutism.
‘Set’ seems to be ‘indefinitely extensible’
Our intuitive understanding of ‘set’ seems to preclude pinning
down an absolutely comprehensive extension for it.
Given any initial domain—including M—we appear able to
specify sets which demonstrably lie outside it.
Zermelo-Russell Argument
Consider r = {x in M | x x}. Assume r in M (for reductio).
Consequently: r ∈ r iff r r. Contradiction! Thus r is not in M.
Note: phenomenon by no means limited to set theory.

‘Interpretation’ seems indefinitely extensible.
Consider a language L with P as a predicate letter.
Interpreting L
Intuitively: an interpretation I specifies what P applies to.
Notation: App(P, a, I) abbreviates P applies to a under I.
Can give: e.g. Standard Tarskian semantics for L.
Px is true under I and σ iff App(P, σ(x), I)
¬φ is true under I and σ iff φ is not true under I and σ, etc.
Extensible: given M, we seem able to specify J not in M.
Williamson-Zermelo-Russell Argument
Let J interpret P to apply to interpretations I in M s.t ¬App(P, I, I).
Suppose J is in M. Then: App(P, J, J) iff ¬App(P, J, J). Contradiction!
Note: J and M aren’t assumed to be sets.

Apparent indeﬁnite extensibility tends to evoke two responses:
Focus on the language of set theory with ‘urelements’: Lß,∈.
(urelement = non-set)
Absolutist: no such set as r
Lß,∈ has a single intended interpretation:
M, S, E
M is the comprehensive domain, S and E the extensions of ß and ∈.
(The absolutist won’t construe this interpretation as a set-model.)
Relativist: ‘forming’ r leads to a wider interpretation
Lß,∈ has an unbounded sequence of ever-more liberal interpretations.
M0, S0, E0 , M1, S1, E1 , M2, S2, E2 , . . .
with M0 ⊂ M1 ⊂ . . . and S0 ⊂ S1 ⊂ . . . and E0 ⊂ E1 ⊂ . . .

End point: trade off.
The two views face a trade off between two sorts of generality.
Relativist: Illiberal about expressive generality
(3) fails to capture seeming intended generality of atheism.
(3) Nothing is God.
Response: draw on non-quantificational generality
Relativist can capture absolute generality schematically.
(3i) Nothingi is God.
‘Nothing0’ ranges over M0, ‘Nothing1’ ranges over M1, ...
Problem: schemas don’t embed. (5i) fails to capture theism.
(5i) It is not the case that nothingi is God.

Absolutist: Illiberal about applicative generality
Cannot apply set theory to theorise about any domain
Consider e.g. Barwise-Cooper style semantics for quantiﬁers:
No donkey talks = T iﬀ donkey , talks ∈ No
= { A, B : A ⊆ M − B}
Absolutist: when M = M there is no such set as { A, B : A ⊆ M − B}
Response: draw on non-set-theoretic resources.
Capture the semantics in higher-order logic (HOL).
donkey ∼ the denotation of a second-order variable.
No ∼ the denotation of a third-order variable, etc.
Problems:
Ideological cost: HOL must be taken seriously
Error theory: semanticists don’t employ HOL.

End point: trade oﬀ.
Expressive generality Applicative generality
Absolutism Liberal Illiberal
Can quantify over
absolutely every set
Can’t apply set theory on
top of any domain
Relativism Illiberal Liberal
Can’t quantify over
Can apply set theory on
top of any domain
Aim: not to settle this trade oﬀ but to reach it.

Outline
1. Absolute generality: an introduction
2. Paradox lost: is there a coherent case for relativism?
3. Paradox regained: a regimentation of the argument
4. Against bounded indeﬁnite extensibility

Dummett’s Argument
Let’s focus on a line of argument due to Dummett.
(Note: primary aim in this section is not exegetic.)
If there is some definite totality over which the variable ‘x’
ranges, and if F(ξ) is any specific predicate which is well defined
over that totality, then of course there will be some definite
subset of objects of the totality that satisfy the predicate
‘F(ξ)’...What there is no warrant for is the assumption that the
objects so denoted must belong to the totality with which we
started...It is of no use to say that we assumed that that totality
comprised all objects whatever, because we have no ground for
supposing that there is any totality closed under the operations of
mapping arbitrary predicates defined over it on to classes...in
fact...Russell’s paradox shows that there can be no such totality.
(Frege: Philosophy of Mathematics, p. 530.)

Dummett’s argument may be regimented as follows.
(P) Dummettian Separation
For any definite totality T, and predicate φ(x), there is a set
whose elements comprise the members of T that satisfy φ(x).
Given this, the Zermelo/Russell argument shows that
{x in T : x x} is not in T.
(C) No Comprehensive Totality
No definite totality T comprises everything.
But what are we to make of this premiss and conclusion?
It all depends on how we understand: ‘definite totality’
Option 1: definite totality = setlike object
Option 2: definite totality = ‘plurality’

Option 1a: deﬁnite totality = set.
(P1a) Separation
For any set s and predicate φ(x), there is a set whose elements
comprise the elements of s that satisfy φ(x)
∀s∃t∀x(x ∈ t ↔ x ∈ s ∧ φ)
First-order reasoning allows us to conclude:
(C1a) No Comprehensive Set
No set comprises everything.
¬∃s∀x(x ∈ s)
Premiss (P1a): seems ﬁne.
Conclusion (C1a): seems harmless to absolutism

(C1a) No Comprehensive Set: trouble only if we accept All-in-One.
All-in-One Principle ...rejected by absolutists
To quantify over some things presupposes that there be some one
set-like collection comprising those things.
Consider what it implies: that we cannot speak of the cookies in
the jar unless they constitute a set; ...I do not mean to imply that
there is no set the members of which are the cookies in the jar,
.. .The point is rather that the needs of quantiﬁcation are already
served by there being simply the cookies in the jar,...no
additional objects are required.
(Cartwright, ‘Speaking of Everything’, p. 8)
Instead absolutists adopt: No domain theory of domains
‘Domain’ talk may be paraphrased away in plural terms.
e.g. ‘The domain of ∀x comprises the non-self membered collections’
⇒ ‘∀x ranges over the non-self-membered collections, severally’.

Option 1a fails.
Option 1a: definite totality = set
(C1a) No comprehensive set: no threat to absolutism.
Some things may comprise absolutely everything even if
no set has these things as its elements.
Taking totalities to be proper classes fares no better.
Option 1b: definite totality = class.
(C1b) No comprehensive class: no threat to absolutism.
Some things may comprise absolutely everything even if
no class has these things as its elements.
Other variants of option 1 face similar difficulties.

‘Totality’ talk might be paraphrased away like ‘domain’ talk.
Formally: add to Lß,∈
Plural quantiﬁers: ∀vv (‘for any zero or more things vv’)
Member-‘plurality’ predicate: v vv (‘v is one of vv’)
Dummett’s premiss and conclusion become:
(P2) Plural Separation
For any zero or more things, and predicate φ(x), there is a set whose
elements are those of them that satisfy φ(x).
∀xx∃s∀x(x ∈ s ↔ x xx ∧ φ)
(C2) No Comprehensive Domain
No zero or more things comprise everything.
¬∃xx∀x(x xx)

This time Dummett’s premiss (P2) is problematic.
Plural setting: standard to add comprehension axioms:
Plural Comprehension
Any predicate φ(x) has zero or more satisﬁers.
∃xx∀x(x xx ↔ φ)
(P2) Plural Separation is inconsistent in plural logic.
Dummett’s argument is not sound.
(Note for cognoscenti: Plural Comprehension + Separation
remain inconsistent even if we restrict Comprehension to
predicative φ.)
Similar problem faces:
e.g. deﬁnite totality = ‘second-order term denotation’

Paradox lost?
Dummett’s argument faces a dilemma:
Option 1: deﬁnite totality = setlike collection
The conclusion is no threat to absolutism.
The premiss of the argument is inconsistent.
Can the relativist make a better case from the paradoxes?

Prior problem: stating relativism
David Lewis oﬀers a two-sentence repudiation of relativism.
Maybe [the relativist] replies that some mystical censor stops us
from quantifying over absolutely everything without restriction.
Lo, he violates his own stricture in the very act of proclaiming it.
(Parts of Classes, p. 68)
Vann McGee is almost as quick in his dismissal.
The reason [relativism] is not a serious worry is that the thesis
that, for any discussion, there are things that lie outside the
universe of discourse of that discussion is a position that cannot
be coherently maintained. Consider the discussion we are having
right now. We cannot coherently claim that there are things that
lie outside the universe of our discussion, for any witness to the
truth of that claim would have to lie outside the claim’s universe
of discourse.
(‘Everything’, p. 55)

Comprehensive Domain trivially true
Some zero or more things comprise everything.
∃xx∀z(z xx)
No Comprehensive Domain trivially false
No zero or more things comprise everything.
¬∃xx∀z(z xx)
Should we conclude that absolutism is trivially true?
Relativist: Comprehensive Domain fails to capture absolutism.
Relativism may be open to an inﬁnite, schematic axiomatisation.
No Comprehensivei+ Domaini
No zero or more members of Mi comprise every member of Mi+ .
¬∃xxi∀zi+ (zi+ xxi)

Stage setting
Suppose we have a sequence of interpretations of the language:
M0, S0, E0 , M1, S1, E1 , M2, S2, E2 , . . .
(Absolutist adds: M0 = M1 = · · · = M, etc.)
Regiment the argument in a sorted language
xi, yi, . . . range over Mi (‘thingsi’); si, ti, . . . over Si.
ßi and ∈i interpreted by Si and Ei (informally: ‘seti’, etc.).
(Logical predicates, = and , are left unsorted.)
Cross-sortal predication: e.g. ß1x0 =df ∃x1(x1 = x0 ∧ ß1x1).
Work in (full) sorted plural logic together with the following.
Auxiliary assumption: Pluralities0 Don’t Increase
Everything1 that is one of some things0 is a thing0.
∀xx0(∀y1 xx0)∃x0(x0 = y1)

Paradox regimented
(I) Some Domain0 is Comprehensive1
Some zero or more things0 comprise everything1.
∃xx0∀x1(x1 xx0)
(II) Pluralities0 Collapse into Sets1
For any sets0, some set1 has them as its elements1.
∀ss0∃s1(∀x1(x1 ∈1 s1 ↔ x1 ss0))
(III) Proper Sets1 are Proper Objects1
Everything1 that is a set1 but not a set0 is not a thing0.
∀x1(ß1x1 ∧ ¬ß0x1 → ¬∃y0(x1 = y0))
(I), (II) and (III) are jointly inconsistent but pairwise consistent.

Three package options
Absolutism about quantiﬁers (and predicates)
(I) Some Domain0 is Comprehensive1.
(II) Pluralities0 Collapse into Sets1.
(III) Proper Sets1 are Proper Objects1.
M, S, E
Relativism about quantiﬁers (and predicates)
M0, S0, E0 , M1, S1, E1 , M2, S2, E2 , . . .

Bounded indefinite extensibility (Bounded IE)
Combines quantifier absolutism with predicate relativism.
M, S0, E0 , M, S1, E1 , M, S2, E2 , . . .
with S0 ⊂ S1 ⊂ . . . M and E0 ⊂ E1 ⊂ . . . M
Prima facie appeal
Expressive generality? Quantification over M is available.
Applicative generality? Collapse is available.

Trouble with expressive generality
Bounded IE: we can’t generalise over absolutely every set.
The Axiom of Foundation
Every set that is non-empty has an ∈-minimal element
∀x ß(x) → ∃z(z ∈ x) → (∃z ∈ x)(∀w ∈ z)(w x)
Rules out ‘inﬁnite descending ∈-chains’ e.g.
a a a · · ·
a b c · · ·
Bounded IE: Foundationi
Predicate ß has Si as its extension.
Foundationi fails to rule out inﬁnite descending ∈-chains in Si+1

Trouble with applicative generality
Sets of non-sets are necessary for applied set theory: ZFCU.
McGee’s Urelement Set Axiom
Some set contains every urelement as an element. ∃s∀x(¬ßx → x ∈ s)
Natural addition to ZFCU: ensures any urelements form a set.
Bounded IE: proper sets1 are urelements0.
Shapiro: consequently the Urelement Set Axiom fails.
Cardinality considerations (after a fashion) entail that most
of the ordinals1 cannot be pure sets0....It follows
that...Vann McGee’s urelement set axiom, stating that the
urelements form a set, fails and fails badly in any language
which has a set theory like ZFC and a quantiﬁer ranging
over absolutely everything. (‘All sets great and small: And
I do mean ALL’, p. 477)
But perhaps this is an axiom we can do without.

Unfortunately, Bounded IE is also incompatible with ZFCU
ZFCU (without addition): Any set-many urelements form a set.
One axiom that provides sets of urelements is Pairing.
Pairing
For any pair of objects0 some set0 has them as its elements0
∀x0∀y0∃s0∀z0(z0 ∈0 s0 ↔ z0 = x0 ∨ z0 = y0)
The friend of Bounded IE must reject Pairing
Pairing, (I) Comprehensive1 Domain0, (II) Collapse ⊥
By (II) and Cantor’s thm. there are more sets1 than sets0.
By (I) and Pairing, x → {x} is a one-one function mapping any set1 to
a set0. So there are no more sets1 than sets0. Contradiction!
Loss of applicative generality:
e.g. predicate extensions cannot be encoded as sets.

End point: trade oﬀ
Expressive
generality
Applicative
generality
Absolutism Liberal Illiberal
Can quantify over
Can’t apply set theory
on top of any domain
Relativism Illiberal Liberal
Can apply set theory
on top of any domain
Bounded IE Illiberal Illiberal
Must give up standard
set theory.

Against bounded-indefinite-extensibility

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