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
A	—	Agenda	Propositions
J	—	Judgment set	Individual view
F	—	Aggregation	Collective function
¬	U+00AC	Negation	Not
∧	U+2227	Conjunction	And
n	—	Individuals	Number of agents

Metatheory. Impossibility: generalizes Arrow's theorem. Escape conditions: characterization theorems. Complexity: aggregation problems NP-hard in some cases. Logic-based: connects to propositional logic. Distance-based aggregation: possible world proximity.

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

Limitations. Impossibility limits perfect solutions. Agenda dependence. Strategic behavior not fully modeled. Computational cost for large agendas. No unique solution method. Philosophical debates ongoing.