TRENDS IN LOGIC XV, 2015: Logics for Social Behaviour

   Social choice concerns the aggregation of local data---the preferences of agents or voters---into global data which is consistent with the given local data. The negative results of social choice theory set limits on the conditions under which such aggregation can be performed. Quantum contextuality, a key non-classical feature of quantum mechanics, arises precisely from situations where local data cannot be combined globally.
The mathematical language of sheaf theory can be used to talk about the passage from local to global. Sheaf cohomology can be used to characterise the obstructions to such passages. As this general setting reveals, there are also striking connections with logical paradoxes, database theory, and a range of other topics.

   (Joint work with Tudor Protopopescu)
The fundamental difference between constructive and classical knowledge lies in their relationship to their respective notions of truth. The classical truth of a proposition is necessary for it to be known. This yields the principle of the factivity of knowledge, a.k.a reflection (K is the knowledge modality): KA -> A. According to the constructive semantics, a proposition is true if proved. The constructive (intuitionistic) truth of a proposition is sufficient for knowledge because constructive truth contains proof and every proof is also a verification. We argue that on this view, co-reflection A -> KA is valid and reflection KA -> A is not; the latter is a distinctly classical principle, too strong as the intuitionistic truth condition on knowledge which is more adequately expressed by other modal means, e.g., ~A -> ~KA `false is not known'.
These and other observations led to an intuitionistic epistemic logic IEL which can serve as a foundation for constructive knowledge maintenance, evidence tracking, distributed-knowledge authorization systems, etc. Accidentally, IEL also describes the system of constructive propositions in type theories by Martin-Lof and Voevodsky et al.

   What is Parametricity? Many have heard of it but more people ought to understand more about it! Parametricity, also known as logical relations, is a fundamental logical concept widely used within programming languages to capture the idea that programs map related inputs to related outputs. Originally formulated by John Reynolds in the 1970s, parametrictiy has been a key tool ever since. In this talk, I'll explain some work we have been doing in Strathclyde which tries to synthesise parametricity with the higher dimensional techniques appearing in Homotopy Type Theory.

   (Based on joint work with Franz Dietrich)
What is the relationship between degrees of belief and all-or-nothing beliefs? Can the latter be expressed as a function of the former, without running into paradoxes? We reassess this 'belief-binarization' problem from the perspective of judgment-aggregation theory. Although some similarities between belief binarization and judgment aggregation have been noted before, the literature contains no general study of the implications of aggregation-theoretic impossibility and possibility results for belief binarization. We seek to fill this gap. This paper is organized around a baseline impossibility theorem, which we use to map out the space of possible solutions to the belief-binarization problem. The theorem shows that, except in simple cases, there exists no belief-binarization rule satisfying four initially plausible desiderata. A surprising finding is that this result is a direct corollary of the judgment-aggregation variant of Arrow's classic impossibility theorem.

   Aggregation rules have been studied extensively for a number of formal entities such as: preference relations and utility functions (in social choice theory), probability measures (in the formal management literature), binary judgments (in the more recently developed judgment aggregation theory). By constrast, aggregation rules for classifications are relatively little known and little explored, although some easy examples would suggest paying more attention to them. Consider, e.g., a vocational guidance committee that should dispatch applicants among professional categories and makes its decisions by aggregating the members' proposed dispatching. Or consider a state in which the population is officially categorized according to some criterion like ethnic origin or religion, and which decides to consult the population itself on how this categorization should be done. In either case the problem arises of aggregating classifications, with the added complexity in the latter case that the same group is the object of the classification and proposes it. We will review the short existing literature on this aggregation problem by connecting it to the better known problem of judgment aggregation, and we will in particular explain an impossibility theorem by Maniquet and Mongin (2014) that borrows from recent mathematical work on nonbinary judgment aggregation.

   (Based on joint works with Guido Boella and Leon van der Torre)
The focus of this talk is the social/organizational structure of a multiagent system, and in particular norms and normative behavior. Normative systems must be able to evolve over time, for example due to actions creating or removing norms in the system. The only formal framework to evaluate and classify normative system change methods is the so-called AGM framework of theory change, which has originally been developed as a framework to describe and classify both belief and normative system change. However, it has been used for belief change only, since the beliefs or norms are represented as propositional formulas. We therefore take AGM theory change as a framework to evaluate the dynamics of rule based systems, to replace propositional formulas in the AGM framework of theory change by pairs of propositional formulas, representing the rule based character of norms, and to add several principles from the input/output logic framework. In this new framework, we show that results from belief base dynamics can be transferred to rule base dynamics, but that a similar transfer of AGM theory change to rule change is much more problematic.

   (Joint work with Vassili Vergopoulos of the Paris School of Economics)
Individuals and societies must often make difficult decisions, which are fraught with uncertainty. How should an agent decide when faced with such uncertainty? This is the subject of a branch of theoretical economics called Decision Theory.
Bernoulli (1738) claimed that we should choose the alternative which yields the highest expected utility. But what justifies this methodology? Savage (1954) showed that, if our decision-making process satisfies certain axioms (encoding basic properties of "consistency" and "rationality"), then it must maximize expected utility. Savage's Theorem is considered the foundational result of modern Decision Theory.
Savage posited a set S of possible "states of nature" and a set X of possible "outcomes". He supposed that each alternative defined a function (an "act") mapping states to outcomes. His theorem constructs a probability measure on S and a utility function on X. However, this approach raises at least three issues.
1. Savage assumed that S and X were arbitrary sets, and acts were arbitrary functions. But what if S and X are topological spaces, and acts must be continuous? What if S and X are differentiable manifolds, and acts must be differentiable? We would like a single theory which works in all of these environments (and others).
2. In many applications, it is unrealistic to suppose that we can enumerate all possible states of nature or all possible outcomes "in advance". Thus, there is growing interest in developing decision theory without an explicit specification of S or X.
3. At different times, the same agent might be faced with many different sources of uncertainty (i.e. different instances of S) and many different menus of outcomes (different instances of X), in different combinations. We would like a single holistic description of the agent's decisions over all of these possible decision problems.
In this talk, we will reformulate decision theory using the tools of category theory, and derive a version of Savage's theorem which addresses all three of these issues.

   (The results reported on in this lecture are based on on-going joint work with A. Baltag and R. Boddy)
In this presentation I focus on the `epistemic potential' of a group of agents, i.e. the knowledge (or beliefs) that the group may come to possess if all its members join their forces and share their individual information. Among the different notions of group knowledge studied in the literature, which one can give us a good measure of a group's epistemic potential? A first candidate is `distributed knowledge', which can in principle be converted into actual individual knowledge by means of simple inter agent communication. However in practice there are many factors which may prevent the full actualization of distributed knowledge. These factors include the group's dynamics, the structure of the social network, the individuals' different epistemic interests and agendas, etc. When we take these realistic conditions into account, a more accurate formalization of a group's potential knowledge can be developed. I show that in interrogative scenarios allowing inter-agent communication as the group's main knowledge-aggregation method, the group's true epistemic potential may well turn out to be very different from both distributed knowledge and from common knowledge (lying instead somewhere in between these extremes).

   There is an increasing demand in management science for more formal foundations. The main motivation of this demand is to make theory more robust, allowing the assumptions and results of different studies from different perspectives to be compared more systematically, and to provide a more solid foundation for different tools of quantitative analysis. However, next to this highly important and – as yet underutilized role, logic could be of use to management science by opening up or at least pointing at new and promising approaches and topics, in particular those connected to multi-agent decision making. This includes not just the explicit processes of collective decision-making, for instance in a hiring committee, an award jury or of a group of voters in a political context, but also implicit processes, such as the growth of a “community consensus” among expert reviewers or consumers. A proper understanding of these processes requires, among others, a deeper and sharper analysis of Matthew effects, self-reinforcing and multidirectional effects, of strategic choices that take the probabilities others are likely to perceive into account, of the impact of categorization and the dynamics of category emergence, of the structuring of decisions and the framing of hypotheses.