Against bounded-indefinite-extensibility

Absolute generality Paradox lost Paradox regained Against the third way

Against Bounded Indefinite Extensibility

James Studd

Language, Truth & Logic WorkshopPrinceton

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: YesSome domain comprises absolutely everything.

Relativist about quantifiers: NoNo domain comprises absolutely everything.

(First approximation: notoriously difficult to formulate.)

James Studd Against bounded indefinite extensibility 1/26


AbsolutismSome 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 extensibilityBut there is also a well known argument against absolutism.

‘Set’ seems to be ‘indefinitely extensible’Our intuitive understanding of ‘set’ seems to preclude pinningdown an absolutely comprehensive extension for it.

Given any initial domain—including M—we appear able tospecify sets which demonstrably lie outside it.

Zermelo-Russell ArgumentConsider 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 LIntuitively: 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 ArgumentLet 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.James Studd Against bounded indefinite extensibility 4/26


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 rLß,∈ 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 interpretationLß,∈ 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 ⊂ . . .James Studd Against bounded indefinite extensibility 5/26


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 generalityRelativist 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.James Studd Against bounded indefinite extensibility 6/26


Absolutist: Illiberal about applicative generalityCannot 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 seriouslyError theory: semanticists don’t employ HOL.



End point: trade off.

Expressive generality Applicative generality

Absolutism Liberal IlliberalCan quantify overabsolutely every set

Can’t apply set theory ontop of any domain

Relativism Illiberal LiberalCan’t quantify overabsolutely every set

Can apply set theory ontop 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 ArgumentLet’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 definedover that totality, then of course there will be some definitesubset of objects of the totality that satisfy the predicate‘F(ξ)’. . . What there is no warrant for is the assumption that theobjects so denoted must belong to the totality with which westarted. . . It is of no use to say that we assumed that that totalitycomprised all objects whatever, because we have no ground forsupposing that there is any totality closed under the operations ofmapping arbitrary predicates defined over it on to classes. . . infact. . . 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 SeparationFor any definite totality T , and predicate φ(x), there is a setwhose 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 TotalityNo 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 objectOption 2: definite totality = ‘plurality’



Option 1: definite totality = setlike objectOption 1a: definite totality = set.

(P1a) SeparationFor any set s and predicate φ(x), there is a set whose elementscomprise 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 SetNo 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 absolutistsTo quantify over some things presupposes that there be some oneset-like collection comprising those things.

Consider what it implies: that we cannot speak of the cookies inthe jar unless they constitute a set; . . . I do not mean to imply thatthere is no set the members of which are the cookies in the jar,. . . The point is rather that the needs of quantification are alreadyserved by there being simply the cookies in the jar,. . . noadditional 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 1: definite totality = setlike objectOption 1a fails.

Option 1a: definite totality = set(C1a) No comprehensive set: no threat to absolutism.Some things may comprise absolutely everything even ifno 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 ifno class has these things as its elements.

Other variants of option 1 face similar difficulties.



Option 2: definite totality = ‘plurality’‘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 SeparationFor any zero or more things, and predicate φ(x), there is a set whoseelements are those of them that satisfy φ(x).∀xx∃s∀x(x ∈ s↔ x ≺ xx ∧ φ)

(C2) No Comprehensive DomainNo 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 ComprehensionAny 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 + Separationremain inconsistent even if we restrict Comprehension topredicative 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 collectionThe conclusion is no threat to absolutism.

Option 2: definite totality = ‘plurality’The premiss of the argument is inconsistent.

Can the relativist make a better case from the paradoxes?



Outline






Prior problem: stating relativismDavid Lewis offers a two-sentence repudiation of relativism.

Maybe [the relativist] replies that some mystical censor stops usfrom 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 thesisthat, for any discussion, there are things that lie outside theuniverse of discourse of that discussion is a position that cannotbe coherently maintained. Consider the discussion we are havingright now. We cannot coherently claim that there are things thatlie outside the universe of our discussion, for any witness to thetruth of that claim would have to lie outside the claim’s universeof discourse.(‘Everything’, p. 55)



Comprehensive Domain trivially trueSome zero or more things comprise everything.∃xx∀z(z ≺ xx)

No Comprehensive Domain trivially falseNo 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 settingSuppose 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 languagexi, 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 IncreaseEverything1 that is one of some things0 is a thing0.∀xx0(∀y1 ≺ xx0)∃x0(x0 = y1)



Paradox regimented(I) Some Domain0 is Comprehensive1Some zero or more things0 comprise everything1.∃xx0∀x1(x1 ≺ xx0)

(II) Pluralities0 Collapse into Sets1For any sets0, some set1 has them as its elements1.∀ss0∃s1(∀x1(x1 ∈1 s1 ↔ x1 ≺ ss0))

(III) Proper Sets1 are Proper Objects1Everything1 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)(I) Some Domain0 is Comprehensive1.


〈M0, S0,E0〉, 〈M1, S1,E1〉, 〈M2, S2,E2〉, . . .



Bounded indefinite extensibility (Bounded IE)(I) Some Domain0 is Comprehensive1.


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 appealExpressive generality? Quantification over M is available.Applicative generality? Collapse is available.



Outline






Trouble with expressive generalityBounded IE: we can’t generalise over absolutely every set.

The Axiom of FoundationEvery 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 3 a 3 a 3 · · ·a 3 b 3 c 3 · · ·

Bounded IE: FoundationiPredicate ß has Si as its extension.

Foundationi fails to rule out infinite descending ∈-chains in Si+1



Trouble with applicative generalitySets of non-sets are necessary for applied set theory: ZFCU.

McGee’s Urelement Set AxiomSome 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 mostof the ordinals1 cannot be pure sets0.. . . It followsthat. . . Vann McGee’s urelement set axiom, stating that theurelements form a set, fails and fails badly in any languagewhich has a set theory like ZFC and a quantifier rangingover absolutely everything. (‘All sets great and small: AndI do mean ALL’, p. 477)

But perhaps this is an axiom we can do without.James Studd Against bounded indefinite extensibility 24/26


Unfortunately, Bounded IE is also incompatible with ZFCUZFCU (without addition): Any set-many urelements form a set.

One axiom that provides sets of urelements is Pairing.

PairingFor 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 7→ {x} is a one-one function mapping any set1 toa 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

Expressivegenerality

Applicativegenerality

Absolutism Liberal IlliberalCan quantify overabsolutely every set

Can’t apply set theoryon top of any domain

Relativism Illiberal LiberalCan’t quantify overabsolutely every set

Can apply set theoryon top of any domain

Bounded IE Illiberal IlliberalCan’t quantify overabsolutely every set

Must give up standardset theory.


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