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Judgment aggregation functions and ultraproducts∗

Frederik Herzberg† ‡§

Abstract

The relationship between propositional model theory and social decision making

via premise-based procedures is explored. A one-to-one correspondence between

ultrafilters on the population set and weakly universal, unanimity-respecting, sys-

tematic judgment aggregation functions is established. The proof constructs an

ultraproduct of profiles, viewed as propositional structures, with respect to the ul-

trafilter of decisive coalitions. This representation theorem can be used to prove

other properties of such judgment aggregation functions, in particular sovereignty

and monotonicity, as well as an impossibility theorem for judgment aggregation in

finite populations. As a corollary, Lauwers and Van Liedekerke’s (1995) represen-

tation theorem for preference aggregation functions is derived.

Key words: Judgment aggregation function; ultraproduct; ultrafilter.

Journal of Economic Literature classification: D71.

AMS Mathematics Subject Classification: 91B14; 03C20.

1Introduction

Ultrafilters have an almost four-decades long history of successful application in the

theory of preference aggregation. Initiated by Fishburn (1970) and Hansson (1971,

Postscript 1976), a seminal contribution was made by Kirman and Sondermann (1972)

whose results motivated numerous other papers in this area; see Monjardet (1983) for

a survey.

A quarter-century later, Lauwers and Van Liedekerke (1995) provided an axiomatic

foundation for the ultrafilter method in the theory of preference aggregation: They con-

structed a one-to-one correspondence between preference aggregation functions satis-

fying Arrovian axioms and ultrafilters on the population set. The bijection is given by

therestriction oftheultraproduct1—withrespecttotheultrafilter ofdecisivecoalitions

— of the family of individual preference orderings to the original set of alternatives.

∗I would like to thank Dr. Daniel Eckert and Dr. Franz Dietrich for helpful comments on a previous

version of this paper.

†Department of Mathematics, University of California, Berkeley, CA 94720-3840, United States of

America.

‡Institut für Mathematische Wirtschaftsforschung, Universität Bielefeld, Universitätsstraße 25, D-33615

Bielefeld, Germany.

§This work was supported by a research grant from the German Academic Exchange Service (DAAD).

1Ultraproducts were first studied extensively by Ło´ s (1955). Expositions of ultraproducts can be found in

model-theoretic monographs such as Chang and Keisler (1973). The existence of non-principal ultrafilters

(i.e. non-dictatorial large coalitions) on infinite sets was established by Ulam (1929) and Tarski (1930), under

the assumption of the Axiom of Choice. (Strictly speaking, the Ultrafilter Existence Theorem only needs the

Boolean Prime Ideal Theorem, which is a weaker set-theoretic axiom than the Axiom of Choice, as Halpern

and Le! vy (1971) have shown. See also Halpern (1964) and Banaschewski (1983) for related results.)

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Ultrafilters have now also entered the theory of judgment aggregation, through pa-

pers by Gärdenfors (2006), Daniëls (2006), Dietrich and Mongin (2007), Eckert and

Klamler (2008),2and some of the results in this article have already been discovered,

via different methods, by Dietrich and Mongin (2007) (see Remark 12 below). It is

not surprising that the ultrafilter method is an appealing tool in judgment aggregation

theory, since ultrafilters are a means of relating the propositional logical structure of

the electorate’s a! genda with the algebraic structure of the voting coalitions.

In this paper, we use the methodology of Lauwers and Van Liedekerke (1995) to

find an axiomatisation of the relation between propositional model theory and social

decision making via premise-based procedures in general, and between judgment ag-

gregation functions and ultrafilters in particular. We prove that a given judgment ag-

gregation function maps profiles to ultraproducts of profiles — with respect to the ul-

trafilter of decisive coalitions — if and only if it satisfies the axioms of weak univer-

sality, respect for unanimous decisions as well as systematicity. This correspondence

between ultrafilters and certain judgment aggregation functions will be used to prove

several other properties of these aggregation functions, as well as an impossibility the-

orem for judgment aggregation in finite populations. We show that the main theorem

of Lauwers and Van Liedekerke (1995) is contained in our results.

In fact, even the converse is true: Many of our results could also be obtained in

a lengthy indirect argument via suitable corollaries of the findings of Lauwers and

Van Liedekerke (1995) through replacing the binary preference predicate on alterna-

tives by a unary truth predicate on propositions. However, the technical translation

effort required would be substantial, and the resulting arguments would therefore ulti-

mately not be significantly shorter than the direct derivation in this paper; furthermore,

it would conceal the simplicity of the ultraproduct construction for propositional struc-

turesandwouldthusdetractfromtheveryintuitivecorresepondencebetweenjudgment

aggregation functions (ultraproducts) and families of decisive coalitions (ultrafilters).

The paper is self-contained and only assumes basic familiarity with propositional

logic on the part of the reader.

2 Judgment sets

Where possible, we follow the terminology of List and Puppe (2007) and the notation

of Eckert and Klamler (2008).

Let N be a finite or infinite set, the population set. Its elements are also called

individuals. Consider a set Y of at least two propositional variables, and let Z be the set

of all propositions, in the sense of propositional calculus, with propositional variables

from Y . Z is called agenda with base Y and should be conceived as the agenda of a

premise-based procedure of social decision making. Let T be a consistent subset of Z

(i.e. T ?? ⊥), called the population’s (unanimous) theory. The set X =?

A fully rational judgment set in X given T is a subset A ⊂ X which is complete in

X (i.e. for all p ∈ Y either p ∈ A or ¬p ∈ A) and consistent with T (i.e. A ∪ T ?? ⊥).

The set of fully rational judgment sets in X given T shall be denoted by D.

p∈Y{p,¬p}

is called the atomic agenda.

2Eckert (2008) argues that this recent interest is only a rediscovery, as the very first application of the

ultrafilter method in social choice theory is in fact due to Guilbaud (1952) (English translation (2008) by

Monjardet), who proved an Arrow-like theorem for the aggregation of logically interconnected propositions.

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Through Henkin’s (1949) method3, every A ∈ D can be uniquely extended to a set

j(A) such that

• j(A) extends A in Z, i.e. A ⊆ j(A) ⊂ Z,

• j(A) is complete in Z, i.e. for all p ∈ Z either p ∈ j(A) or ¬p ∈ j(A),

• j(A) is consistent, i.e. j(A) ?? ⊥, and

• j(A) contains T, i.e. T ⊂ j(A).

j(A) is obtained as the maximally consistent superset of A ∪ T. In particular, A

itself is consistent. j(A) will be called the judgment completion (or conclusive com-

pletion) of A. From the perspective of propositional model theory, j(A) corresponds

to an interpretation of Z.

The following observations about j are almost trivial, but will be helpful later on.

Herein, we denote by − the negation operator on X, defined via −p = ¬p and −¬p =

p for all p ∈ Y .

Remark 1 (Inverse of j). A = j(A) ∩ X for all A ∈ D. Hence, j is injective.

Proof. By definition, A ⊆ j(A) and A ⊆ X, hence A ⊆ j(A) ∩ X. If there existed

some p ∈ j(A) ∩ X \ A, then also −p ∈ A ⊆ j(A) since A is complete in X. This

contradicts the consistency of j(A).

Remark 2 (Decuctive closedness). For every A ∈ D, the set j(A) is decuctively

closed, i.e. ∀p ∈ Z (j(A) ? p ⇒ p ∈ j(A)).

Proof by contraposition. If p ?∈ j(A), then j(A) ? ¬p as j(A) is complete. But j(A)

is also onsistent. Therefore, j(A) ?? p.

The following Remark 3 corresponds to Tarski’s (1933) definition of truth:

Remark 3 (à la Tarski). For all p,q ∈ Z and A ∈ D:

1. ¬p ∈ j(A) if and only if p ?∈ j(A).

2. p ∧ q ∈ j(A) if and only if both p ∈ j(A) and q ∈ j(A).

3. p ∨ q ∈ j(A) if and only if p ∈ j(A) or q ∈ j(A).

Proof. Let p,q ∈ Z and A ∈ D.

1. “⇒”. j(A) is consistent. “⇐”. j(A) is complete.

2. j(A) is deductively closed by Remark 2.

3. Combine De Morgan’s laws with the first two parts of the Remark.

Remark 4. Through writing A(p) = 1 instead of p ∈ A, one could also view A as a

map A : Y → 2 such that

T ∪ A−1{1} ∪?¬A−1{0}??? ⊥

Every such map A can be extended to a homomorphism of Boolean algebras j(A) :

Z → 2 such that T ⊆ j(A)−1{1}.

3The method of extending a consistent set of propositions to a maximally consistent set was used in

Henkin’s (1949) famous proof of the completeness proof of first-order logic.

(wherein ¬B is shorthand for?¬p : p ∈ A−1{0}?for any B ⊆ Z, and 2 = {0,1}).

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3Coalitions and judgment aggregation functions

The elements of DNare referred to as profiles. For any p ∈ Z and A ∈ DN, the

p-supporting coalition in A is denoted by A(p) := {i ∈ N : p ∈ j (A(i))}.4In this

section, we study the properties of the map A : p ?→ A(p).

First, the map A : p ?→ A(p) allows us to translate the Boolean operations on

coalitions into logical operations on propositions in Z:

Remark 5. For all A ∈ DN, the map A : DN→ P(N),

algebra homomorphism:

p ?→ A(p) is a Boolean

1. A(?) = N and A(⊥) = ∅.

2. ∀p ∈ Z

3. ∀p,q ∈ Z

Proof.1. For all i ∈ N, j (A(i)) is complete and consistent, hence ? ∈ j (A(i))

and ⊥ ?∈ j (A(i)).

2. For all i ∈ N, j (A(i)) is complete and consistent, hence ¬p ∈ j (A(i)) if and

only if p ?∈ j (A(i)).

3. For all i ∈ N, j (A(i)) is deductively closed (Remark 2), hence p,q ∈ j (A(i))

if and only if p ∧ q ∈ j (A(i)). This proves A(p ∧ q) = A(p) ∩ A(q). The

formula A(p ∨ q) = A(p) ∪ A(q) follows via De Morgan’s laws combined with

the already established parts 1 and 2 of the Remark.

A(¬p) = N \ A(p).

A(p ∧ q) = A(p) ∩ A(q),A(p ∨ q) = A(p) ∪ A(q).

Almost needless to say, the whole population supports its unanimous theory, re-

gardless of the population’s profile.

Remark 6. For all p ∈ Z with T ? p and every A ∈ DN, one has A(p) = N.

Proof. Letp ∈ Z withT ? p, letA ∈ DN, andleti ∈ N. Sincej (A(i))isdeductively

closed (Remark 2) and contains T, we must have p ∈ j (A(i)).

This implies that the map A : p ?→ A(p) is well-defined on equivalence classes

with respect to provable logical equivalence under T:

Remark 7. For all p,q ∈ Z with T ? (p ↔ q) and every A ∈ DN, one has A(p) =

A(q).

Proof. Let p,q ∈ Z with T ? (p ↔ q) and A ∈ DN. By Remark 6, A(p ↔ q) = N.

Combining this with the definition of p ↔ q (i.e. p ↔ q = (¬p ∨ q) ∧ (p ∨ ¬q)) and

Remark 5, we obtain

N = A((¬p ∨ q) ∧ (p ∨ ¬q)) = ((N \ A(p)) ∪ A(q)) ∩ (A(p) ∪ (N \ A(q))),

which via De Morgan’s laws can be simplified to

∅ = (A(p) \ A(q)) ∪ (A(q) \ A(p)).

The right-hand side is the symmetric difference between A(p) and A(q) This proves

A(p) = A(q).

4We may assume N ∩ Z = ∅ to avoid amiguity.

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An aggregation function is a map f from a subset Dfof DNto D.

Consider now the following axioms:

(A1) Universality. Df= DN.

(A1’) Weak Universality. There exist p,q ∈ Y and A1,A2,A3∈ D such that

– p,q ∈ A1,

– p,¬q ∈ A2,

– ¬p,q ∈ A3, and

– {A1,A2,A3}N⊆ Df.

(A2) Respect for Unanimity. For all A ∈ Dfand for all p ∈ Z, if p ∈ j ◦ f (A), then

A(p) ?= ∅.

(A3) Systematicity. For all A,A?∈ Dfand for all p,p?∈ Z with A(p) = A?(p?), one

has p ∈ j ◦ f (A) if and only if p?∈ j ◦ f?A??.

Remark 8. If there exist p,q ∈ Y such that p∧q, p∧¬q and ¬p∧q are each consistent

with T, then (A1) implies (A1’).

Finally, the set of decisive coalitions is

Ff:= {A(p) : A ∈ Df,p ∈ Z,p ∈ j ◦ f (A)}.

Remark 9. Let A ∈ Dfand p ∈ Z, and suppose f satisfies (A3). Then, A(p) ∈ Ffif

and only if p ∈ j ◦ f (A).

Proof. “⇒”. If p ?∈ j ◦ f (A), then (A3) yields that p??∈ j ◦ f?A??for all p?∈ Z and

A?∈ Dfsatisfying A(p) = A?(p?). Hence A(p) ?∈ Ff. “⇐”. Definition of Ff.

4Ultrafilters and ultraproducts

In this section, we review ultrafilters and define ultraproducts of profiles. We define an

ultrafilter on N as a collection G of subsets of N which is

• non-trivial, i.e. ∅ ?∈ G,

• maximal, i.e. for all U ⊆ N, either U ∈ G or N \ U ∈ G, and

• closed under finite intersections, i.e. U ∩ U?∈ G for all U,U?∈ G.

These properties ensure that there is a one-to-one correspondence between ultra-

filters on N and {0,1}-valued finitely additive measures on P(N): Given any such

measure µ, the corresponding ultrafilter is just the collection of sets of µ-measure 1.

Often, ultrafilters are defined as being also closed under supersets, thus being spe-

cial filters5per definitionem. This part of the definition is, in fact, redundant, as was

stressed e.g. by Lauwers and Van Liedekerke (1995):

Remark 10. Every ultrafilter G is closed under supersets, i.e. if U?⊇ U ∈ G, then

U?∈ G. Hence, ultrafilters are filters.

5A filter on N is a non-trivial collection of subsets of N which is closed under finite intersections and

supersets.

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