Claim-strength problems are a distinctive class of allocation problems in which the currency of claims is different from that of the estate that is to be divided. We study mixtures (i.e. convex combinations) of three basic allocation rules for claim-strength problems: the proportional rule, the uniform allocation rule, and the plurality allocation rule. We observe that any such mixture satisfies a generalized transfer axiom in addition to a number of invariance properties. We establish a fundamental representation theorem: taken jointly, these axioms fully characterize the class of all mixtures of the three base rules. This result is tight. From it, we derive characterizations of more specific classes and the three basic rules themselves. We moreover show that within the class of mixtures, the only three rules that satisfy a familiar consistency axiom are the base rules.

Recent years have seen an increase in the use of online deliberation platforms (DPs). One of the main objectives of DPs is to enhance democratic participation, by allowing citizens to post, comment, and vote on policy proposals. But in what order should these proposals be listed? This paper makes a start with the principled evaluation of sorting methods on DPs. First, we introduce a conceptual framework that allows us to classify and compare sorting methods in terms of their purpose and the parameters they take into account. Second, we observe that the choice for a sorting method is often ad hoc and rarely justified. Third and last, we criticise sorting by number of approvals ('likes'), a method that is very common in practice. On the one hand, we show that if approvals are used for sorting, this should be done in an integrated way, also taking into account other parameters. On the other hand, we argue that even if proposals are on a par in terms of those other parameters, there are other, more appropriate ways to sort proposals in light of the approvals they have received.

We explore logic as a method to model social phenomena, giving rise to what we term social logic. After introducing and defining this concept, we identify main examples of social logic, ranging from key precursors that have enabled researchers to start modeling complex social phenomena and concepts using the tools of logic, to more central examples of recent trends in social logic. We then reflect on logic as a modeling tool, discussing more generally why and how it is a relevant and valuable method in the social realm. Finally, we outline some promising directions for future research in social logic.

This paper makes a twofold contribution to the study of expressivity. First, we introduce and study the novel concept of conditional expressivity. Taking a universal logic perspective, we characterize conditional expressivity both syntactically and semantically. We show that our concept of conditional expressivity is related to, but different from, the concept of explicit definability in Beth's definability theorem. Second, we use the concept to explore inferential relations between collective deontic admissibility statements for different groups. Negative results on conditional expressivity are stronger than standard (unconditional) inexpressivity results: we show that the well-known inexpressivity results from epistemic logic on distributed knowledge and on common knowledge only concern unconditional expressivity. By contrast, we prove negative results on conditional expressivity in the deontic logic of collective agency. In particular, we consider the full formal language of the deontic logic of collective agency, define a natural class of sublanguages of the full language, and prove that a collective deontic admissibility statement about a particular group is conditionally expressible in a sublanguage from the class if and only if that sublanguage includes a collective deontic admissibility statement about a supergroup of that group. Our negative results on conditional expressivity may serve as a proof of concept for future studies.

This white paper outlines a long-term scientific vision for the development of digital-democracy technology. We contend that if digital democracy is to meet the ambition of enabling a participatory renewal in our societies, then a comprehensive multi-methods research effort is required that could, over the years, support its development in a democratically principled, empirically and computationally informed way. The paper is co-authored by an international and interdisciplinary team of researchers and arose from the Lorentz Center Workshop on “Algorithmic Technology for Democracy” (Leiden, October 2022).

We study original position arguments in the context of social choice under ignorance. First, we present a general formal framework for such arguments. Next, we provide anaxiomatic characterization of social choice rules that can be supported by original position arguments. We illustrate this characterization in terms of various well-known social choicerules, some of which do and some of which do not satisfy the axioms in question. Depending on the perspective one takes, our results can be used to argue against certainrules, against Rawlsian theories of procedural fairness, or in support of richer, multidimensional models of individual choice.

John Rawls famously argued that the Difference Principle would be chosen by any rational agent in the original position. Derek Parfit and Philippe Van Parijs have claimed, contra Rawls, that it is not the Difference Principle which is implied by Rawls’ original position argument, but rather the more refined Lexical Difference Principle. In this paper, we study both principles in the context of social choice under ignorance. First, we present a general format for evaluating original position arguments in this context. We argue that in this format, the Difference Principle can be specified in three conceptually distinct ways. We show that these three specifications give the same choice recommendations, and can be grounded in an original position argument in combination with the well-known maximin rule. Analogously, we argue that one can give at least four plausible specifications of the Lexical Difference Principle, which however turn out to give different recommendations in concrete choice scenarios. We prove that only one of these four specifications can be grounded in an original position argument. Moreover, this one specification seems the least appealing from the viewpoint of distributive justice. This insight points towards a general weakness of original position arguments.

AbstractWe prove neighbourhood canonicity and strong completeness for the logics $\mathbf {EK}$ and $\mathbf {ECK}$ , obtained by adding axiom (K), resp. adding both (K) and (C), to the minimal modal logic $\textbf {E}$ . In contrast to an earlier proof in [10], ours is constructive. More precisely, we construct minimal characteristic models for both logics and do not rely on compactness of first order logic. The proof involves a specific circumscription technique and quite some set-theoretic maneuvers to establish that the models satisfy the appropriate frame conditions. After giving both proofs, we briefly spell out how they generalize to four stronger logics and to the extensions of the resulting six logics with a global modality.

We study classical modal logics with pooling modalities , i.e. unary modal operators that allow one to express properties of sets obtained by the pointwise intersection of neighbourhoods. We discuss salient properties of these modalities, situate the logics in the broader area of modal logics (with a particular focus on relational semantics), establish key properties concerning their expressive power, discuss dynamic extensions of these logics and provide reduction axioms for the latter.

Rawls famously claimed that choices based on the Difference Principle coincide with the choices of any rational individual in the Original Position. In this paper, we develop a logic in which we can express and prove Rawls’ thesis in its object language. Starting from a standard semantics of choice under uncertainty, we enrich our models in order to represent uncertainty about one’s position. We then introduce a sound and strongly complete logic that allows us to speak about agents’ positions and their derived utilities, and that can express changes in the uncertainty about those positions using dynamic operators. Finally, we show how this logic allows us to define various types of obligation based on a Rawlsian notion of procedural fairness.

Methodological individualists often claim that any social phenomenon can ultimately be explained in terms of the actions and interactions of individuals. Any Nagelian version of methodological individualism requires that there be bridge laws that translate social statements into individualistic ones. We show that Nagelian individualism can be put to logical scrutiny by making the relevant social and individualistic languages fully explicit and mathematically precise. In particular, we prove that the social statement that a group of (at least two) agents performs a deontically admissible group action cannot be expressed in a well-established deontic logic of agency that models every combination of actions, omissions, abilities, and obligations of finitely many individual agents.

General obligations such as 'every driver has to give way to a driver coming from the right' are central in legal reasoning, but have been mostly overlooked in deontic logic. We claim that a simple extension of Standard Deontic Logic to the predicative level is insufficient to capture general obligations. Instead, we argue for an explicit representation of bearers (and counterparties) of obligations as terms in a quantified deontic logic. To achieve this we develop a term-modal counterpart of Standard Deontic Logic and give a sound and strongly complete axiomatization for it. We go on to show that this logic is not only suitable for capturing reasoning with general obligations, but also with (multital and paucital) Hohfeldian rights relations and rules of rights.

We introduce classical term-modal logics and argue that they are useful for modelling agent-relative notions of obligation, evidence and abilities, and their interaction with properties of and relations between the agents in question. We spell out the semantics of these logics in terms of neighborhood models, provide sound and strongly complete axiomatizations and establish the decidability of specific (agent-finite) variants.

The problem of no hands concerns the existence of so-called responsibility voids: cases where a group makes a certain decision, yet no individual member of the group can be held responsible for this decision. Criteria-based collective decision procedures play a central role in philosophical debates on responsibility voids. In particular, the well-known discursive dilemma has been used to argue for the existence of these voids. But there is no consensus: others argue that no such voids exist in the discursive dilemma under the assumption that casting an untruthful opinion is eligible. We argue that, under this assumption, the procedure used in the discursive dilemma is indeed immune to responsibility voids, yet such voids can still arise for other criteria-based procedures. We provide two general characterizations of the conditions under which criteria-based collective decision procedures are immune to these voids. Our general characterizations are used to prove that responsibility voids are ruled out by criteria-based procedures involving an atomistic or monotonic decision function. In addition, we show that our results imply various other insights concerning the logic of responsibility voids.

We use a deontic logic of collective agency to study reducibility questions about collective agency and collective obligations. The logic that is at the basis of our study is a multi-modal logic in the tradition of stit (‘sees to it that’) logics of agency. Our full formal language has constants for collective and individual deontic admissibility, modalities for collective and individual agency, and modalities for collective and individual obligations. We classify its twenty-seven sublanguages in terms of their expressive power. This classification enables us to investigate reducibility relations between collective deontic admissibility, collective agency, and collective obligations, on the one hand, and individual deontic admissibility, individual agency, and individual obligations, on the other.

We study a STIT logic for two agents i and j, augmented with three deontic constants 1, 1i, 1j . The constants express, respectively, optimality for the group {i, j}, optimality for agent i, and optimality for agent j. An action X is optimal for an (individual or group) agent if and only if X is not strongly dominated by another action X ′ that is available to that same agent. We propose an axiomatization for this logic and we study its expressive power. In particular, we show that the deontic constants are not interdefinable, but nevertheless display significant interaction properties.

Adaptive Logics (ALs) are a viable and useful formal tool to handle various issues in deontic logic. In this paper, we motivate, explain, illustrate, and discuss the use of ALs in deontic logic. Published work on deontic ALs focusses mainly on conflicttolerant deontic logics (logics that can accommodate conflicting obligations) and - to a lesser extent - on problems concerning factual and deontic detachment. So does the present paper. Near the end of the paper, however, we also indicate some of the possibilities that the adaptive logic framework creates for tackling other types of problems within deontic logic.