The Problem of Vagueness

This chapter introduces the philosophical concept of vagueness and explains its significance for contemporary philosophy. The concept is seen to give rise to two main problems: the ‘soritic problem’ of finding a solution to the paradoxes of vagueness; and the ‘semantic problem’ of finding a satisfactory semantics and logic for vague language. It discusses three of the main attempts to deal with these problems – Supervaluationism, Degree theory, and Epistemicism. It indicates why none of these theories has been regarded as satisfactory and it concludes with a general impossibility result which seems to rule out any satisfactory account of the concept.

Automated Search for Impossibility Theorems in Social Choice Theory: Ranking Sets of Objects

We present a method for using standard techniques from satisfiability checking to automatically verify and discover theorems in an area of economic theory known as ranking sets of objects. The key question in this area, which has important applications in social choice theory and decision making under uncertainty, is how to extend an agent's preferences over a number of objects to a preference relation over nonempty sets of such objects. Certain combinations of seemingly natural principles for this kind of preference extension can result in logical inconsistencies, which has led to a number of important impossibility theorems. We first prove a general result that shows that for a wide range of such principles, characterised by their syntactic form when expressed in a many-sorted first-order logic, any impossibility exhibited at a fixed (small) domain size will necessarily extend to the general case. We then show how to formulate candidates for impossibility theorems at a fixed domain size in propositional logic, which in turn enables us to automatically search for (general) impossibility theorems using a SAT solver. When applied to a space of 20 principles for preference extension familiar from the literature, this method yields a total of 84 impossibility theorems, including both known and nontrivial new results.

On the Difficulties of Disclosure Prevention in Statistical Databases or The Case for Differential Privacy

In 1977 Tore Dalenius articulated a desideratum for statistical databases: nothing about an individual should be learnable from the database that cannot be learned without access to the database. We give a general impossibility result showing that a natural formalization of Dalenius’ goal cannot be achieved if the database is useful. The key obstacle is the side information that may be available to an adversary. Our results hold under very general conditions regarding the database, the notion of privacy violation, and the notion of utility. Contrary to intuition, a variant of the result threatens the privacy even of someone not in the database. This state of affairs motivated the notion of differential privacy [15, 16], a strong ad omnia privacy which, intuitively, captures the increased risk to one’s privacy incurred by participating in a database.