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Input and Output in Judgment Aggregation

This work by Gabriella Pigozzi and Leon van der Torre explores the critical distinction between input and output in judgment aggregation. They detail a framework to analyze the discursive dilemma, collective reasoning, and individual judgments, while addressing key concepts such as premise independence and voting on rules. The paper also presents a running example to illustrate these principles and discusses the implications of independence in judgment aggregation procedures. This exploration seeks to avoid impossibility results prevalent in social choice theory and emphasizes the need for a refined understanding of judgment processes.

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Input and Output in Judgment Aggregation

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  1. Input and Output inJudgment Aggregation Gabriella Pigozzi Leon van der Torre Individual and Collective Reasoning University of Luxembourg

  2. Claim • Working assumptions: judgment aggregation  preference aggregation judgment aggregation  probabilistic reasoning • Claim: Input and output must be distinguished in judgment aggregation. Pigozzi & van der Torre Input and Output in Judgment Aggregation

  3. Outline • Discursive dilemma • Judgment aggregation framework • Running example • Conclusion independence • Towards new operators • A side issue: voting on rules? • Conclusions Pigozzi & van der Torre Input and Output in Judgment Aggregation

  4. 1. Discursive Dilemma P = Valid contract; Q = Breach; R = Defendant liable • Paradox: inconsistent collective judgment • Discursive dilemma: premises versus conclusions [Kornhauser & Sager, 1986] Pigozzi & van der Torre Input and Output in Judgment Aggregation

  5. Discursive Dilemma P = Candidate is worthy of tenure on teaching Q = Candidate is worthy of tenure on research R = The candidate is worthy of tenure tout court [Bovens & Rabinowicz, 2006] Pigozzi & van der Torre Input and Output in Judgment Aggregation

  6. Two Levels of Input/Output in JA 1. Input and output as premises and conclusion (PQ, R): if input PQ, then output R Pigozzi & van der Torre Input and Output in Judgment Aggregation

  7. Two Levels of Input/Output in JA 2. Input and output as individual and collective Profile Collective judgment (profile, collective judgment) Pigozzi & van der Torre Input and Output in Judgment Aggregation

  8. 2. Judgment Aggregation List & Pettit 2002: There exists no aggregation procedure satisfying the following conditions: • Universal Domain (UD): admissible inputs are any logically possible profile of individual sets of judgments. • Anonymity: all individuals have equal weight. • Systematicity: the aggregation procedure treats all propositions in an evenhanded way. Inspired by social choice theory: independence conditions and impossibility results Pigozzi & van der Torre Input and Output in Judgment Aggregation

  9. Mongin’s Impossibility Theorem Mongin (2006): UD + IIPA + Unanimity  Dictator IIPA Pigozzi & van der Torre Input and Output in Judgment Aggregation

  10. 3. Running Example • 3 agents • Agenda contains: The literals: P, Q, R, P, Q, R The rule RPQ: PQR, PR, QR More constraints: PR, QR, PQR • 26 = 64 inputs, 22 outputs, 6 dilemmas • Dagstuhl winner: in case of no conflict between premise and conclusion based procedures, adopt the majority rule. Pigozzi & van der Torre Input and Output in Judgment Aggregation

  11. 4. Conclusion Independence • Likewise for premise independence CIIPA Pigozzi & van der Torre Input and Output in Judgment Aggregation

  12. 5. Premise-Based Procedure • The procedure: P, Q := majority; R := PQ. • No conclusion independence (nor dictator) • Premise independence (and anonymity) • (If S=PQR, then “dilemma of the Paretian Rational”) Pigozzi & van der Torre Input and Output in Judgment Aggregation

  13. Conclusion-Based Procedure Revised • The revised procedure: • P:= R or P1  P2  P3; Q, R := majority • No premise independence (nor dictator) • Conclusion independence (and anonymity) Pigozzi & van der Torre Input and Output in Judgment Aggregation

  14. Conclusion Dictator • Conclusion dictator procedure: P:= R1 or P1  P2  P3; Q := Q1, R := R1 • No premise independence (nor dictator/anonymity) • Conclusion independence Pigozzi & van der Torre Input and Output in Judgment Aggregation

  15. 6. Voting on Rules • Doctrinal paradox no violations, in JA possible • Advantages analogous to “logic by translation” • What does it mean to vote for or against a rule? • ab: "If carbon dioxide emissions are above threshold x, then there will be global warming” [Dietrich and List, 2005] • False ab: "carbon dioxide emissions are above threshold x, and there will be no global warming" Pigozzi & van der Torre Input and Output in Judgment Aggregation

  16. 7. Conclusions • Two levels of input/output reasoning in JA: • Relating premises and conclusion • Relating individual and collective judgments • Escape routes from impossibility results: • Weakened notions of independence • Mixed premise and conclusion based operators • Not found in preference aggregation • (or in probabilistic reasoning) Pigozzi & van der Torre Input and Output in Judgment Aggregation

  17. Dilemma of the Paretian Rational P = Duty; Q = Negligence; R = Causation; S = Damages [Nehring, 2006] Pigozzi & van der Torre Input and Output in Judgment Aggregation

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