Taken as a model for how groups should make collective judgments and decisions, the ideal of deliberative democracy is inherently ambiguous. Consider the idealised case where it is agreed on all sides that a certain conclusion should be endorsed if and only if certain premises are admitted. Does deliberative democracy recommend that members of the group debate the premises and then individually vote, in the light of that debate, on whether or not to support the conclusion? Or does it recommend (...) that members individually vote on the premises, and then let their commitment to the conclusion be settled by whether or not the group endorses the required premises? Deliberative-democratic theory has not addressed this issue, and this is a problem. The discursive dilemma of my title--a generalisation of the doctrinal paradox from analytical jurisprudence--shows that the procedures distinguished can come apart. Thus deliberative democrats must make up their minds on where they stand in relation to the issue; they cannot sit on the fence. This paper is an attempt to address the issue and look at the grounds on which it may be resolved. (shrink)

The aggregation of individual judgments on logically interconnected propositions into a collective decision on the same propositions is called judgment aggregation. Literature in social choice and political theory has claimed that judgment aggregation raises serious concerns. For example, consider a set of premises and a conclusion where the latter is logically equivalent to the former. When majority voting is applied to some propositions (the premises) it may give a different outcome than majority voting applied to another set of propositions (the (...) conclusion). This problem is known as the discursive dilemma (or paradox). The discursive dilemma is a serious problem since it is not clear whether a collective outcome exists in these cases, and if it does, what it is like. Moreover, the two suggested escape-routes from the paradox—the so-called premise-based procedure and the conclusion-based procedure—are not, as I will show, satisfactory methods for group decision-making. In this paper I introduce a new aggregation procedure inspired by an operator defined in artificial intelligence in order to merge belief bases. The result is that we do not need to worry about paradoxical outcomes, since these arise only when inconsistent collective judgments are not ruled out from the set of possible solutions. (shrink)

The contemporary theory of epistemic democracy often draws on the Condorcet Jury Theorem to formally justify the ‘wisdom of crowds’. But this theorem is inapplicable in its current form, since one of its premises – voter independence – is notoriously violated. This premise carries responsibility for the theorem's misleading conclusion that ‘large crowds are infallible’. We prove a more useful jury theorem: under defensible premises, ‘large crowds are fallible but better than small groups’. This theorem rehabilitates the importance of deliberation (...) and education, which appear inessential in the classical jury framework. Our theorem is related to Ladha's (1993) seminal jury theorem for interchangeable (‘indistinguishable’) voters based on de Finetti's Theorem. We also prove a more general and simpler such jury theorem. (shrink)

According to a theorem recently proved in the theory of logical aggregation, any nonconstant social judgment function that satisfies independence of irrelevant alternatives (IIA) is dictatorial. We show that the strong and not very plausible IIA condition can be replaced with a minimal independence assumption plus a Pareto-like condition. This new version of the impossibility theorem likens it to Arrow’s and arguably enhances its paradoxical value.

This paper develops a distinction between "substantive agreement" and "meta-agreement" and explores the significance of this distinction for democracy and social choice.

Knowledge in a Social World offers a philosophy for the information age. Alvin Goldman explores new frontiers by creating a thoroughgoing social epistemology, moving beyond the traditional focus on solitary knowers. Against the tides of postmodernism and social constructionism Goldman defends the integrity of truth and shows how to promote it by well-designed forms of social interaction. From science to education, from law to democracy, he shows why and how public institutions should seek knowledge-enhancing practices. The result is a bold, (...) timely, and systematic treatment of the philosophical foundations of an information society. (shrink)

Suppose that the members of a group each hold a rational set of judgments on some interconnected questions, and imagine that the group itself has to form a collective, rational set of judgments on those questions. How should it go about dealing with this task? We argue that the question raised is subject to a difficulty that has recently been noticed in discussion of the doctrinal paradox in jurisprudence. And we show that there is a general impossibility theorem that that (...) difficulty illustrates. Our paper describes this impossibility result and provides an exploration of its significance. The result naturally invites comparison with Kenneth Arrow's famous theorem (Arrow, 1963 and 1984; Sen, 1970) and we elaborate that comparison in a companion paper (List and Pettit, 2002). The paper is in four sections. The first section documents the need for various groups to aggregate its members' judgments; the second presents the discursive paradox; the third gives an informal statement of the more general impossibility result; the formal proof is presented in an appendix. The fourth section, finally, discusses some escape routes from that impossibility. (shrink)

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.. (shrink)

The aggregation of individual judgments over interrelated propositions is a newly arising field of social choice theory. I introduce several independence conditions on judgment aggregation rules, each of which protects against a specific type of manipulation by agenda setters or voters. I derive impossibility theorems whereby these independence conditions are incompatible with certain minimal requirements. Unlike earlier impossibility results, the main result here holds for any (non-trivial) agenda. However, independence conditions arguably undermine the logical structure of judgment aggregation. I therefore (...) suggest restricting independence to premises, which leads to a generalised premise-based procedure. This procedure is proven to be possible if the premises are logically independent. (shrink)

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 Rubinsteinís group identification problem. (shrink)

Standard impossibility theorems on judgment aggregation over logically connected propositions either use a controversial systematicity condition or apply only to agendas of propositions with rich logical connections. Are there any serious impossibilities without these restrictions? We prove an impossibility theorem without requiring systematicity that applies to most standard agendas: Every judgment aggregation function (with rational inputs and outputs) satisfying a condition called unbiasedness is dictatorial (or effectively dictatorial if we remove one of the agenda conditions). Our agenda conditions are tight. (...) When applied illustratively to (strict) preference aggregation represented in our model, the result implies that every unbiased social welfare function with universal domain is effectively dictatorial. (shrink)

The new …eld of judgment aggregation aims to …nd collective judgments on logically interconnected propositions. Recent impossibility results establish limitations on the possibility to vote independently on the propositions. I show that, fortunately, the impossibility results do not apply to a wide class of realistic agendas once propositions like “if a then b” are adequately modelled, namely as subjunctive implications rather than material implications. For these agendas, consistent and complete collective judgments can be reached through appropriate quota rules (which decide (...) propositions using acceptance thresholds). I characterise the class of these quota rules. I also prove an abstract result that characterises consistent aggregation for arbitrary agendas in a general logic. (shrink)

The new field 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 unified 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. (shrink)

The widely discussed "discursive dilemma" shows that majority voting in a group of individuals on logically connected propositions may produce irrational collective judgments. We generalize majority voting by considering quota rules, which accept each proposition if and only if the number of individuals accepting it exceeds a given threshold, where different thresholds may be used for different propositions. After characterizing quota rules, we prove necessary and sufficient conditions on the required thresholds for various collective rationality requirements. We also consider sequential (...) quota rules, which ensure collective rationality by adjudicating propositions sequentially and letting earlier judgments constrain later ones. Sequential rules may be path-dependent and strategically manipulable. We characterize path-independence and prove its essential equivalence to strategy-proofness. Our results shed light on the rationality of simple-, super-, and sub-majoritarian decision-making. (shrink)

All existing impossibility theorems on judgment aggregation require individual and collective judgment sets to be consistent and complete, arguably a demanding rationality requirement. They do not carry over to aggregation functions mapping profiles of consistent individual judgment sets to consistent collective ones. We prove that, whenever the agenda of propositions under consideration exhibits mild interconnections, any such aggregation function that is "neutral" between the acceptance and rejection of each proposition is dictatorial. We relate this theorem to the literature.

Which rules for aggregating judgments on logically connected propositions are manipulable and which not? In this paper, we introduce a preference-free concept of non-manipulability and contrast it with a preference-theoretic concept of strategy-proofness. We characterize all non-manipulable and all strategy-proof judgment aggregation rules and prove an impossibility theorem similar to the Gibbard--Satterthwaite theorem. We also discuss weaker forms of non-manipulability and strategy-proofness. Comparing two frequently discussed aggregation rules, we show that “conclusion-based voting” is less vulnerable to manipulation than “premise-based voting”, (...) which is strategy-proof only for “reason-oriented” individuals. Surprisingly, for “outcome-oriented” individuals, the two rules are strategically equivalent, generating identical judgments in equilibrium. Our results introduce game-theoretic considerations into judgment aggregation and have implications for debates on deliberative democracy. (shrink)

The ``doctrinal paradox'' or ``discursive dilemma'' shows that propositionwise majority voting over the judgments held by multiple individuals on some interconnected propositions can lead to inconsistent collective judgments on these propositions. List and Pettit (2002) have proved that this paradox illustrates a more general impossibility theorem showing that there exists no aggregation procedure that generally produces consistent collective judgments and satisfies certain minimal conditions. Although the paradox and the theorem concern the aggregation of judgments rather than preferences, they invite comparison (...) with two established results on the aggregation of preferences: the Condorcet paradox and Arrow's impossibility theorem. We may ask whether the new impossibility theorem is a special case of Arrow's theorem, or whether there are interesting disanalogies between the two results. In this paper, we compare the two theorems, and show that they are not straightforward corollaries of each other. We further suggest that, while the framework of preference aggregation can be mapped into the framework of judgment aggregation, there exists no obvious reverse mapping. Finally, we address one particular minimal condition that is used in both theorems – an independence condition – and suggest that this condition points towards a unifying property underlying both impossibility results. (shrink)

This paper generalises the classical Condorcet jury theorem from majority voting over two options to plurality voting over multiple options. The paper further discusses the debate between epistemic and procedural democracy and situates its formal results in that debate. The paper finally compares a number of different social choice procedures for many-option choices in terms of their epistemic merits. An appendix explores the implications of some of the present mathematical results for the question of how probable majority cycles (as in (...) Condorcet's paradox) are in large electorates. (shrink)

Under the independence and competence assumptions of Condorcet’s classical jury model, the probability of a correct majority decision converges to certainty as the jury size increases, a seemingly unrealistic result. Using Bayesian networks, we argue that the model’s independence assumption requires that the state of the world (guilty or not guilty) is the latest common cause of all jurors’ votes. But often – arguably in all courtroom cases and in many expert panels – the latest such common cause is a (...) shared ‘body of evidence’ observed by the jurors. In the corresponding Bayesian network, the votes are direct descendants not of the state of the world, but of the body of evidence, which in turn is a direct descendant of the state of the world. We develop a model of jury decisions based on this Bayesian network. Our model permits the possibility of misleading evidence, even for a maximally competent observer, which cannot easily be accommodated in the classical model. We prove that (i) the probability of a correct majority verdict converges to the probability that the body of evidence is not misleading, a value typically below 1; (ii) depending on the required threshold of ‘no reasonable doubt’, it may be impossible, even in an arbitrarily large jury, to establish guilt of a defendant ‘beyond any reasonable doubt’. (shrink)

Condorcet’s voting paradox shows that pairwise majority voting may lead to cyclical majority preferences. In a famous paper, Sen identified a general condition on a profile of individual preference orderings, called triplewise value-restriction, which is sufficient for the avoidance of such cycles. This note aims to make Sen’s result easily accessible. We provide an elementary proof of Sen's possibility theorem and a simple reformulation of Sen’s condition. We discuss how Sen’s condition is logically related to a number of precursors. Finally, (...) we state a necessary and sufficient condition for the avoidance of cycles, and suggest that, although there is still some logical space between that condition and Sen’s sufficient condition, Sen’s condition cannot be further generalized in an appealing way. (shrink)

Taken as a model for how groups should make collective judgments and decisions, the ideal of deliberative democracy is inherently ambiguous. Consider the idealised case where it is agreed on all sides that a certain conclusion should be endorsed if and only if certain premises are admitted. Does deliberative democracy recommend that members of the group debate the premises and then individually vote, in the light of that debate, on whether or not to support the conclusion? Or does it recommend (...) that members individually vote on the premises, and then let their commitment to the conclusion be settled by whether or not the group endorses the required premises? Is deliberative democracy to enforce the discipline of reason at the individual level, as in the first possibility, or at the collective level, as in the second? (shrink)

Social choice theory is the study of collective decision processes and procedures. It is not a single theory, but a cluster of models and results concerning the aggregation of individual inputs (e.g., votes, preferences, judgments, welfare) into collective outputs (e.g., collective decisions, preferences, judgments, welfare). Central questions are: How can a group of individuals choose a winning outcome (e.g., policy, electoral candidate) from a given set of options? What are the properties of different voting systems? When is a voting system (...) democratic? How can a collective (e.g., electorate, legislature, collegial court, expert panel, or committee) arrive at coherent collective preferences or judgments on some issues, on the basis of its members' individual preferences or judgments? How can we rank different social alternatives in an order of social welfare? Social choice theorists study these questions not just by looking at examples, but by developing general models and proving theorems. (shrink)

Wulf Gaertner provides a comprehensive account of an important and complex issue within social choice theory: how to establish a social welfare function while restricting the spectrum of individual preferences in a sensible way. Gaertner's starting point is K. J. Arrow's famous 'Impossibility Theorem', which showed that no welfare function could exist if an unrestricted domain of preferences is to be satisfied together with some other appealing conditions. A number of leading economists have tried to provide avenues out of this (...) 'impossibility' by restricting the variety of preferences: here, Gaertner provides a clear and detailed account, using standardized mathematical notation, of well over forty theorems associated with domain conditions. Domain Conditions in Social Choice Theory will be an essential addition to the library of social choice theory for scholars and their advanced graduate students. (shrink)

Logical puzzles like the doctrinal paradox raise the problem of how to aggregate individual judgements into a collective judgement, or alternatively, how to merge collectively inconsistent knowledge bases. In this paper, we view judgement aggregation as a function on propositional logic valuations, and we investigate how logic constrains judgement aggregation. In particular, we show that there is no non-dictatorial decision method for aggregating sets of judgements in a logically consistent way if the decision method is local, i.e., only depends on (...) the individual judgements on the proposition under consideration. (shrink)

I model sequential decisions over multiple interconnected propositions and investigate path-dependence in such decisions. The propositions and their interconnections are represented in propositional logic. A sequential decision process is path-dependent if its outcome depends on the order in which the propositions are considered. Assuming that earlier decisions constrain later ones, I prove three main results: First, certain rationality violations by the decision-making agent—individual or group—are necessary and sufficient for path-dependence. Second, under some conditions, path-dependence is unavoidable in decisions made by (...) groups. Third, path-dependence makes decisions vulnerable to strategic agenda setting and strategic voting. I also discuss escape routes from path-dependence. My results are relevant to discussions on collective consistency and reason-based decision-making, focusing not only on outcomes, but also on underlying reasons, beliefs, and constraints. (shrink)

Drawing on the so-called “doctrinal paradox”, List and Pettit (2002) have shown that, given an unrestricted domain condition, there exists no procedure for aggregating individual sets of judgments over multiple interconnected propositions into corresponding collective ones, where the procedure satisfies some minimal conditions similar to the conditions of Arrow’s theorem. I prove that we can avoid the paradox and the associated impossibility result by introducing an appropriate domain restriction: a structure condition, called unidimensional alignment, is shown to open up a (...) possibility result, similar in spirit to Black’s median voter theorem. Specifically, I prove that, given unidimensional alignment, propositionwise majority voting is the unique procedure for aggregating individual sets of judgments into collective ones in accordance with the above mentioned minimal conditions. (shrink)

An aggregation procedure merges a list of objects into a representative object. This paper considers the problem of aggregating n rows in an n-by-m matrix into a summary row, where every entry is an element in an algebraic field. It focuses on consistent aggregators, which require each entry in the summary row to depend only on its column entries in the matrix and to be the same as the column entry if the column is constant. Consistent aggregators are related to (...) additive, linear and.. (shrink)