Belief merging and the discursive dilemma: an argument-based account to paradoxes of judgment aggregation

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.     

The Logic of Group Decisions: Judgment Aggregation

Judgment aggregation studies how individual opinions on a given set of propositions can be aggregated to form a consistent group judgment on the same propositions. Despite the simplicity of the problem, seemingly natural aggregation procedures fail to return consistent collective outcomes, leading to what is now known as the doctrinal paradox. The first occurrences of the paradox were discovered in the legal realm. However, the interest of judgment aggregation is much broader and extends to political philosophy, epistemology, social choice theory, and computer science. The aim of this paper is to provide a concise survey of the discipline and to outline some of the most pressing questions and future lines of research.     

Aggregating Sets of Judgments: Two Impossibility Results Compared1

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.     

Applications of the MACBETH Approach in the Framework of an Additive Aggregation Model

In measurement theory terminology, MACBETH is an interactive approach for mapping into a real scale the various degrees to which the elements of a finite set possess a property P. The originality of MACBETH's questioning procedure is the possibility of establishing a constructive path towards cardinal measurement in both quantitative and substantive meaningful terms, avoiding the operational problems recognized as a weakness of other procedures. The use of the notion of semantic absolute judgements plays a key role here and the simplicity, interactivity and constructiveness of our approach insert it in the modern paradigms of decision aid. This article illustrates the usefulness of MACBETH as a tool to facilitate decision support, by describing its application in two real public decision situations where we acted as facilitators. Although the cases correspond to quite different decision contexts and problematics and our interventions have addressed diverse issues, they have in common the fact that we conducted both evaluation processes in the framework of an additive multicriteria aggregation procedure. In the first case MACBETH has been used to support the construction of (local) cardinal value functions, while the second illustrates how our approach can serve as a weighting procedure to determine the scaling constants of an additive aggregation model. © 1997 John Wiley & Sons, Ltd.     

Logical Constraints on Judgement Aggregation

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.     

Research note: The accuracy of group majorities for disjunctive and conjunctive decision tasks

For complex group decision tasks which can be described in terms of establishing the truth value of conjunctive (and) or disjunctive (or) composite propositions, the advantage for group judgmental accuracy of disaggregating the decision task into separate individual evaluations of the truth value of each component atomic proposition in the composite is investigated. For true propositions it is shown that disaggregation is preferable for conjunctive tasks, but not for disjunctive tasks; while the reverse is true for false propositions.     

Democratic Answers to Complex Questions – An Epistemic Perspective

This paper addresses a problem for theories of epistemic democracy. In a decision on a complex issue which can be decomposed into several parts, a collective can use different voting procedures: Either its members vote on each sub-question and the answers that gain majority support are used as premises for the conclusion on the main issue (premise based-procedure, pbp), or the vote is conducted on the main issue itself (conclusion-based procedure, cbp). The two procedures can lead to different results. We investigate which of these procedures is better as a truth-tracker, assuming that there exists a true answer to be reached. On the basis of the Condorcet jury theorem, we show that the pbp is universally superior if the objective is to reach truth for the right reasons. If one instead is after truth for whatever reasons, right or wrong, there will be cases in which the cbp is more reliable, even though, for the most part, the pbp still is to be preferred.     

Cutting it (too) fine

It is widely held that propositions are structured entities. In The Nature and Structure of Content (2007), Jeff King argues that the structure of propositions is none other than the syntactic structure deployed by the speaker/hearers who linguistically produce and consume the sentences that express the propositions. The present paper generalises from King’s position and claims that syntax provides the best in-principle account of propositional structure. It further seeks to show, however, that the account faces serve problems pertaining to the fine individuation of propositions that the account entails. The ‘fineness of cut’ problem has been raised by Collins (The unity of linguistic meaning, 2007) and others. King (Philos Stud 163(3):763–781, 2013) responds to these complaints in ways this paper rebuts. Thus, the very idea of structured propositions is brought into doubt, for the best in-principle account of such structure appears to fail.     

Measuring voting power for dependent voters through causal models

We construct a new measure of voting power that yields reasonable measurements even if the individual votes are not cast independently. Our measure hinges on probabilities of counterfactuals, such as the probability that the outcome of a collective decision would have been yes, had a voter voted yes rather than no as she did in the real world. The probabilities of such counterfactuals are calculated on the basis of causal information, following the approach by Balke and Pearl. Opinion leaders whose votes have causal influence on other voters’ votes can have significantly more voting power under our measure. But the new measure of voting power is also sensitive to the voting rule. We show that our measure can be regarded as an average treatment effect, we provide examples in which it yields intuitively plausible results and we prove that it reduces to Banzhaf voting power in the limiting case of independent and equiprobable votes.

     

Instability Of Ex Post Aggregation In The Bolker–Jeffrey Framework And Related Instability Phenomena

Suppose n Bayesian agents need to make a decision as a group. The groupas a whole is also supposed to be a Bayesian agent whose probabilities andutilities are derived or aggregated in reasonable ways from the probabilitiesand utilities of the group members. The aggregation could beex ante, i.e., interms of expected utilities, or it could be ex post, i.e., in terms of utilitiesonly, or in terms of utilities and probabilities separately. This study exploresthe ex post approach. Using the Bolker/Jeffrey framework, we show thatex post aggregation is subject to an instability phenomenon. That is, it mayhappen that the group preference between actions ``flips back and forth' dependingon the level of detail in which the decision problem is described. Structurally verysimilar phenomena also occur elsewhere in social choice theory, in statistics (Simpson'sParadox), and in voting theory (Ostrogorski's Paradox).     

De se belief and rational choice

The Sleeping Beauty puzzle has dramatized the divisive question of how de se beliefs should be integrated into formal theories of rational belief change. In this paper, I look ahead to a related question: how should de se beliefs be integrated into formal theories of rational choice? I argue that standard decision theoretic frameworks fail in special cases of de se uncertainty, like Sleeping Beauty. The nature of the failure reveals that sometimes rational choices are determined independently of one’s credences in the kinds of ‘narrow’ de se propositions that Sleepy Beauty has set in relief. Consequently, in addition to pinpointing a failure of standard decision theoretic frameworks, this result casts doubt on a large class of strategies for determining principles for the rationally updating de se beliefs in cases like Sleeping Beauty, and also calls into question the importance of making such a determination at all.     

The grounding problem and presentist explanations

Opponents of presentism have often argued that the presentist has difficulty in accounting for what makes (presently) true past-tensed propositions (TptP) true in a way that is compatible with her metaphysical view of time and reality. The problem is quite general and concerns not only strong truth-maker principles, but also the requirement that truth be grounded in reality. In order to meet the challenge, presentists have proposed many peculiar present aspects of the world as grounds for truths concerning the past, such as uninstantiated haecceities, Meinongian non-existents, ersatz times, and dispositional and distributional properties. The main problem with all such solutions is that any explanation of what grounds a TptP that involves the past is eo ipso a better explanation than any that involves only the present. Thus, the quest for an account of grounding for TptP that is compatible with the presentist ontology and ideology is doomed to be explanatorily deficient with respect to eternalism. In a recent article, Ben Caplan and David Sanson have claimed that presentists should change their strategy and, rather than seeking for exotic grounds for TptP, should adopt a more liberal view of explanation. That is, they should allow themselves to resort to “past directed” explanations, even if they do not accept the past in their ontology and ideology. I argue that such a proposal is not compatible with the tenet that there is a substantial distinction between the ideology of such a version of presentism and that of eternalism. Therefore, the presentist cannot endorse such “deflationist” explanations as an easy way out to the problem of the grounding of TptP.     

Beyond subgoaling: A dynamic knowledge generation framework for creative problem solving in cognitive architectures

In this paper we propose a computational framework aimed at extending the problem solving capabilities of cognitive artificial agents through the introduction of a novel, goal-directed, dynamic knowledge generation mechanism obtained via a non monotonic reasoning procedure. In particular, the proposed framework relies on the assumption that certain classes of problems cannot be solved by simply learning or injecting new external knowledge in the declarative memory of a cognitive artificial agent but, on the other hand, require a mechanism for the automatic and creative re-framing, or re-formulation, of the available knowledge. We show how such mechanism can be obtained trough a framework of dynamic knowledge generation that is able to tackle the problem of commonsense concept combination. In addition, we show how such a framework can be employed in the field of cognitive architectures in order to overcome situations like the impasse in SOAR by extending the possible options of its subgoaling procedures.     

Team performance and collective efficacy in the dynamic psychology of competitive team: A Bayesian network analysis

The purpose of this paper was to discover the relationships among 22 relevant psychological features in semi-professional football players in order to study team’s performance and collective efficacy via a Bayesian network (BN). The paper includes optimization of team’s performance and collective efficacy using intercausal reasoning pattern which constitutes a very common pattern in human reasoning. The BN is used to make inferences regarding our problem, and therefore we obtain some conclusions; among them: maximizing the team’s performance causes a decrease in collective efficacy and when team’s performance achieves the minimum value it causes an increase in moderate/high values of collective efficacy. Similarly, we may reason optimizing team collective efficacy instead. It also allows us to determine the features that have the strongest influence on performance and which on collective efficacy. From the BN two different coaching styles were differentiated taking into account the local Markov property: training leadership and autocratic leadership.     

If structured propositions are logical procedures then how are procedures individuated?

This paper deals with two issues. First, it identifies structured propositions with logical procedures. Second, it considers various rigorous definitions of the granularity of procedures, hence also of structured propositions, and comes out in favour of one of them. As for the first point, structured propositions are explicated as algorithmically structured procedures. I show that these procedures are structured wholes that are assigned to expressions as their meanings, and their constituents are sub-procedures occurring in executed mode (as opposed to displayed mode). Moreover, procedures are not mere aggregates of their parts; rather, procedural constituents mutually interact. As for the second point, there is no universal criterion of the structural isomorphism of meanings, hence of co-hyperintensionality, hence of synonymy for every kind of language. The positive result I present is an ordered set of rigorously defined criteria of fine-grained individuation in terms of the structure of procedures. Hence procedural semantics provides a solution to the problem of the granularity of co-hyperintensionality.

     

Propositions,Properties and Relations: Wittgenstein's “Notes on Logic” and the Tractatus

Frege famously argued that truth is not a property or relation. In the “Notes on Logic” Wittgenstein emphasised the bi‐polarity of propositions which he called their sense. He argued that “propositions by virtue of sense cannot have predicates or relations.” This led to his fundamental thought that the logical constants do not represent predicates or relations. The idea, however, has wider ramifications than that. It is not just that propositions cannot have relations to other propositions but also that they cannot have relations to anything at all. The paper explores the consequences of this insight for the way in which we should read the Tractatus. In the “Notes on Logic” the insight led to Wittgenstein's emphasis on “facts” in any attempt to understand the nature of symbolism. This emphasis is continued in the Tractatus. It is central to his view that propositions are facts which picture facts which prevent us from construing such picturing as a relation between what pictures and what is pictured. It illuminates the importance of context principle with regard to the distinction between showing and saying to which Wittgenstein attached so much importance and it underlies the non‐relational view of psychological propositions which he advocates. Finally, if propositions by virtue of sense cannot have predicates or relations the paradox at the end of a work which consist largely of propositions about propositions becomes intelligible.     

Intuition and The Junctures of Judgment in Decision Procedures for Clinical Ethics

Moral decision procedures such as principlism or casuistry require intuition at certain junctures, as when a principle seems indeterminate, or principles conflict, or we wonder which paradigm case is most relevantly similar to the instant case. However, intuitions are widely thought to lack epistemic justification, and many ethicists urge that such decision procedures dispense with intuition in favor of forms of reasoning that provide discursive justification. I argue that discursive justification does not eliminate or minimize the need for intuition, or constrain our intuitions. However, this is not a problem, for intuitions can be justified in easy or obvious cases, and decision procedures should be understood as heuristic devices for reaching judgments about harder cases that approximate the justified intuitions we would have about cases under ideal conditions, where hard cases become easy. Similarly, the forms of reasoning which provide discursive justification help decision procedures perform this heuristic function not by avoiding intuition, but by making such heuristics more accurate. Nonetheless, it is possible to demand too much justification; many clinical ethicists lack the time and philosophical training to reach the more elaborate levels of discursive justification. We should keep moral decision procedures simple and user-friendly so that they will provide what justification can be achieved under clinical conditions, rather than trying to maximize our epistemic justification out of an overstated concern about intuition.     

Reasoning defeasibly about probabilities

In concrete applications of probability, statistical investigation gives us knowledge of some probabilities, but we generally want to know many others that are not directly revealed by our data. For instance, we may know prob(P/Q) (the probability of P given Q) and prob(P/R), but what we really want is prob(P/Q&;R), and we may not have the data required to assess that directly. The probability calculus is of no help here. Given prob(P/Q) and prob(P/R), it is consistent with the probability calculus for prob(P/Q&;R) to have any value between 0 and 1. Is there any way to make a reasonable estimate of the value of prob(P/Q&;R)? A related problem occurs when probability practitioners adopt undefended assumptions of statistical independence simply on the basis of not seeing any connection between two propositions. This is common practice, but its justification has eluded probability theorists, and researchers are typically apologetic about making such assumptions. Is there any way to defend the practice? This paper shows that on a certain conception of probability—nomic probability—there are principles of “probable probabilities” that license inferences of the above sort. These are principles telling us that although certain inferences from probabilities to probabilities are not deductively valid, nevertheless the second-order probability of their yielding correct results is 1. This makes it defeasibly reasonable to make the inferences. Thus I argue that it is defeasibly reasonable to assume statistical independence when we have no information to the contrary. And I show that there is a function Y(r, s, a) such that if prob(P/Q) = r, prob(P/R) = s, and prob(P/U) = a (where U is our background knowledge) then it is defeasibly reasonable to expect that prob(P/Q&;R) = Y(r, s, a). Numerous other defeasible inferences are licensed by similar principles of probable probabilities. This has the potential to greatly enhance the usefulness of probabilities in practical application.     

Bayesian Group Agents and Two Modes of Aggregation

Suppose we have a group of Bayesian agents, and suppose that theywould like for their group as a whole to be a Bayesian agent as well. Moreover, suppose that thoseagents want the probabilities and utilities attached to this group agent to be aggregated from theindividual probabilities and utilities in reasonable ways. Two ways of aggregating their individual data areavailable to them, viz., ex ante aggregation and ex post aggregation. The former aggregatesexpected utilities directly, whereas the latter aggregates probabilities and utilities separately.A number of recent formal results show that both approaches have problematic implications. This studydiscusses the philosophical issues arising from those results. In this process, I hope to convincethe reader that these results about Bayesian aggregation are highly significant to decision theorists, butalso of immense interest to theorists working in areas such as ethics and political philosophy.     

Collective Informed Consent and Decision Power

It has been suggested that, in addition to individual level decision-making, informed consent procedures could be used in collective decision-making too. One of the main criticisms directed at this suggestion concerns decision-making power. It is maintained that consent is a veto power concept and that, as such, it is not appropriate for collective decision-making. This paper examines this objection to collective informed consent. It argues that veto power informed consent can have some uses in the collective level and that when it is not appropriate the decision power a concerned party ought to have in connection with an arrangement should be made relative to the interest she has at stake in it. It concludes that the objection examined does not undermine collective informed consent.