Judgment Aggregation

Origin. List and Pettit developed judgment aggregation (2002). Extends Arrow's social choice to logic. Aggregate individual judgments on propositions. Impossibility results: no perfect aggregation. Foundation for collective reasoning and deliberation.

Models. Aggregating logical judgments. Individuals: each has judgment set (consistent, complete). Aggregation: combine into collective judgment. Doctrinal paradox: majority on premises ≠ majority on conclusion. Systematic constraints on aggregation.

Formalism.

Agenda: A ⊆ L: set of propositions and their negations. Closed under negation: φ ∈ A iff ¬φ ∈ A.

Judgment set: J ⊆ A: subset of agenda. Complete: ∀φ ∈ A. φ ∈ J or ¬φ ∈ J. Consistent: no contradiction derivable from J.

Aggregation function: F: (J₁,...,Jₙ) → J Maps profile of individual judgments to collective judgment.

Doctrinal paradox: Premises: p, q. Conclusion: p ∧ q. Individual 1: p, q, (p ∧ q) — all true. Individual 2: p, ¬q, ¬(p ∧ q). Individual 3: ¬p, q, ¬(p ∧ q). Majority: p, q, ¬(p ∧ q) — inconsistent!

Impossibility theorem (List & Pettit): If A has certain structure, no aggregation F satisfies:

Universal domain
Collective rationality (consistent, complete)
Systematicity (proposition-by-proposition)
Anonymity
Non-dictatorship

Escape routes: Relax completeness (abstention). Domain restriction. Premise-based vs conclusion-based.

Symbols.

Symbol	Unicode	Name	Meaning
⊆	U+2286	Subset	Agenda inside the language; judgment set inside the agenda
∈	U+2208	Membership	Proposition belongs to an agenda or a judgment set
∀	U+2200	Universal quantifier	Completeness ranges over the whole agenda
¬	U+00AC	Negation	Agenda is closed under it; the rejected proposition
∧	U+2227	Conjunction	Connective forming the conclusion in the doctrinal paradox
→	U+2192	Rightwards arrow / Function	Type of the aggregation function
≠	U+2260	Not equal to	Majority on the premises differs from majority on the conclusion
φ	U+03C6	Greek small letter phi	Arbitrary proposition of the agenda
₁	U+2081	Subscript one	Index of the first individual
ₙ	U+2099	Latin subscript small letter n	Index of the last individual
p	U+0070	Latin small letter p	First premise in the doctrinal paradox
q	U+0071	Latin small letter q	Second premise in the doctrinal paradox
A	—	Agenda	Propositions voted on, closed under negation
L	—	Language	Logic the agenda is drawn from
J	—	Judgment set	Propositions one individual accepts
F	—	Aggregation function	Maps a profile of judgment sets to a collective one
n	—	Individuals	Number of agents

Metatheory. The doctrinal paradox is not an artefact of the example. List and Pettit (2002) proved that for any agenda containing two atomic propositions together with a truth-functional combination of them, no aggregation function satisfies universal domain, anonymity, systematicity and collective rationality; Pauly and van Hees (2006) traded anonymity for unanimity and obtained the sharper conclusion that the only such functions are dictatorships. The general question is which agendas force this, and it has been answered by combinatorial conditions on the agenda rather than by conditions on the rule: Dietrich and List (2007), with Nehring and Puppe and with Dokow and Holzman independently, characterized the agendas on which universal domain, independence, unanimity and collective rationality entail dictatorship in terms of total blockedness together with an even-negatability condition. Arrow's theorem then falls out as the special case in which the agenda consists of pairwise ranking propositions, which is what makes judgment aggregation the more general theory. Endriss, Grandi and Porello (2012) showed winner determination to be polynomial for the quota rules and the premise-based procedure but Θ₂ᵖ-complete for the distance-based procedure.

Applies to. Multi-agent systems. Legal decision (courts). Committee decisions. Collective belief. Expert panels. Democratic theory. AI collective reasoning.

Limitations. The impossibility results are not obstacles to be engineered around: on a totally blocked agenda every rule satisfying universal domain, independence, unanimity and collective rationality is a dictatorship, so any escape gives up a condition the theory itself counts as desirable. Dropping completeness admits abstention and leaves the collective silent on propositions every individual has decided; dropping independence makes the verdict on one proposition depend on judgments about others, so premise-based and conclusion-based procedures return different answers on the same profile with nothing internal to adjudicate. Outcomes are sensitive to the agenda, which the theory takes as given rather than derives, so the same underlying question yields different collective judgments under different formulations. Individuals are assumed to hold complete and consistent judgment sets, and misreporting is not excluded — Dietrich and List (2007) showed strategy-proofness to require exactly independence and monotonicity, the conditions the impossibility results already indict.