Judgment aggregation in non-classical logics (original) (raw)
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Judgment aggregation without full rationality
Social Choice and Welfare, 2008
Several recent results on the aggregation of judgments over logically connected propositions show that, under certain conditions, dictatorships are the only propositionwise aggregation functions generating fully rational (i.e., complete and consistent) collective judgments. A frequently mentioned route to avoid dictatorships is to allow incomplete collective judgments. We show that this route does not lead very far: we obtain oligarchies rather than dictatorships if instead of full rationality we merely require that collective judgments be deductively closed, arguably a minimal condition of rationality, compatible even with empty judgment sets. We derive several characterizations of oligarchies and provide illustrative applications to Arrowian preference aggregation and Kasher and Rubinstein's group identi…cation problem.
Belief Aggregation for Non-Standard Reasoners
2017
The goal of this paper is to study belief aggregation in non-normal modal logic. The paper has two parts. First we look abstractly at aggregated beliefs in neighborhood semantics, defined in analogy with distributed knowledge in epistemic logic. We find that for doxastic logics weaker than K several different versions of aggregated belief can be defined. We provide sound and complete axiomatizations for four of them as well as a proof system in sequent calculus. Then we turn to a probabilistic foundations for these aggregation operations. We study the logic of individual and group beliefs resulting from using the Lockean thesis on (non-standard) probabilities based on Belnap-Dunn four valued logic. Aggregated beliefs in Neighborhood Semantics Our starting point is a non-normal model of (categorical, i.e. non-graded) individual beliefs using neighborhood frames (see e.g. [5], [3]): Definition 1. A neighborhood frame F is a pair 〈W,ni〉, where W is a set of possible worlds and ni : W →...
A generalised model of judgment aggregation
Social Choice and Welfare, 2007
The new …eld of judgment aggregation aims to merge many individual sets of judgments on logically interconnected propositions into a single collective set of judgments on these propositions. Judgment aggregation has commonly been studied using classical propositional logic, with a limited expressive power and a problematic representation of conditional statements ("if P then Q") as material conditionals. In this methodological paper, I present a simple uni…ed model of judgment aggregation in general logics. I show how many realistic decision problems can be represented in it. This includes decision problems expressed in languages of classical propositional logic, predicate logic (e.g. preference aggregation problems), modal or conditional logics, and some multi-valued or fuzzy logics. I provide a list of simple tools for working with general logics, and I prove impossibility results that generalise earlier theorems.
On the logic of preference and judgment aggregation
Autonomous Agents and Multi-Agent Systems, 2011
Agents that must reach agreements with other agents need to reason about how their preferences, judgments, and beliefs might be aggregated with those of others by the social choice mechanisms that govern their interactions. The emerging field of judgment aggregation studies aggregation from a logical perspective, and considers how multiple sets of logical formulae can be aggregated to a single consistent set. As a special case, judgment aggregation can be seen to subsume classical preference aggregation. We present a modal logic that is intended to support reasoning about judgment aggregation scenarios (and hence, as a special case, about preference aggregation): the logical language is interpreted directly in judgment aggregation rules. We present a sound and complete axiomatisation. We show that the logic can express aggregation rules such as majority voting; rule properties such as independence; and results such as the discursive paradox, Arrow's theorem and Condorcet's paradox-which are derivable as formal theorems of the logic. The logic is parameterised in such a way that it can be used as a general framework for comparing the logical properties of different types of aggregation-including classical preference aggregation. As a case study we present a logical study of, including a formal proof of, the neutrality lemma, the main ingredient in a well-known proof of Arrow's theorem.
Premise Independence in Judgment Aggregation
Dagstuhl Seminars, 2007
Judgment aggregation studies how agent opinions on logically in- terconnected propositions can be mapped into a collective judg- ment on the same propositions, and is plagued by impossibility re- sults. In this paper we study the central notion of independence in these impossibility results. First, we argue that the distinction be- tween the premises and conclusions play an important role
Arrow’s theorem in judgment aggregation
Social Choice and Welfare, 2007
In response to recent work on the aggregation of individual judgments on logically connected propositions into collective judgments, it is often asked whether judgment aggregation is a special case of Arrowian preference aggregation. We argue for the converse claim. After proving two impossibility theorems on judgment aggregation (using "systematicity" and "independence" conditions, respectively), we construct an embedding of preference aggregation into judgment aggregation and prove Arrow's theorem (stated for strict preferences) as a corollary of our second result. Although we thereby provide a new proof of Arrow's theorem, our main aim is to identify the analogue of Arrow's theorem in judgment aggregation, to clarify the relation between judgment and preference aggregation, and to illustrate the generality of the judgment aggregation model. JEL Classi…cation: D70, D71
Judgment Aggregation in Abstract Dialectical Frameworks
Advances in Knowledge Representation, Logic Programming and Abstract Argumentation: Essays Dedicated to Gerhard Brewka on the Occasion of his 60th Birthday, 2015
Abstract dialectical frameworks (ADFs) are a knowledge representation formalism introduced as a generalisation of Dung’s abstract argumentation frameworks (AFs) by Gerhard Brewka and co- authors. We look at a judgment aggregation problem in ADFs, namely the problem of aggregating a profile of complete interpretations. We generalise the family of interval aggregation methods, studied in the AF case in our previous work, to the ADF case. Along the way we define the notions of down-admissible and up-complete interpretations, that were already previously defined for the AF case by Caminada and Pigozzi. These aggregation methods may open the way to define interesting new semantics for ADFs, such as a generalisation to the ADF case of the ideal semantics for AFs.
Propositionwise judgment aggregation: the general case
Social Choice and Welfare, 2012
In the theory of judgment aggregation, it is known for which agendas of propositions it is possible to aggregate individual judgments into collective ones in accordance with the Arrow-inspired requirements of universal domain, collective rationality, unanimity preservation, non-dictatorship and propositionwise independence. But it is only partially known (e.g., only in the monotonic case) for which agendas it is possible to respect additional requirements, notably non-oligarchy, anonymity, no individual veto power, or extended unanimity preservation. We fully characterize the agendas for which there are such possibilities, thereby answering the most salient open questions about propositionwise judgment aggregation. Our results build on earlier results by Nehring and Puppe (Strategy-proof social choice on single-peaked domains: possibility, impossibility and the space between, 2002), Nehring
Journal of Philosophical Logic, 1999
Paraconsistent logic is an area of philosophical logic that has yet to find acceptance from a wider audience. The area remains, in a word, disreputable. In this essay, we try to reassure potential consumers that it is not necessary to become a radical in order to use paraconsistent logic. According to the radicals, the problem is the absurd classical account of contradiction: Classically inconsistent sets explode only because bourgeois classical semantics holds, in the face of overwhelming evidence to the contrary, that both A and ∼A cannot simultaneously be true! We suggest (more modestly) that there is, at least sometimes, something else worth preserving, even in an inconsistent, unsatisfiable premise set. In this paper we present, in a new guise, a very general version of this "preservationist" approach to paraconsistency.