Modern belief revision theory is based to a large extent on partial meet contraction that was introduced in the seminal article by Carlos Alchourrón, Peter Gärdenfors, and David Makinson that appeared in 1985. In the same year, Alchourrón and Makinson published a significantly different approach to the same problem, called safe contraction. Since then, safe contraction has received much less attention than partial meet contraction. The present paper summarizes the current state of knowledge on safe contraction, provides some new results (...) and offers a list of unsolved problems that are in need of investigation. (shrink)

Recent work has formally linked the traditional axiomatisation of incomplete preferences à la Anscombe-Aumann with the theory of desirability developed in the context of imprecise probability, by showing in particular that they are the very same theory. The equivalence has been established under the constraint that the set of possible prizes is finite. In this paper, we relax such a constraint, thus de facto creating one of the most general theories of rationality and decision making available today. We provide the (...) theory with a sound interpretation and with basic notions, and results, for the separation of beliefs and values, and for the case of complete preferences. Moreover, we discuss the role of conglomerability for the presented theory, arguing that it should be a rationality requirement under very broad conditions. (shrink)

We study the model-theoretic aspects of a probability logic suited for talking about measure spaces. This nonclassical logic has a model theory rather different from that of classical predicate logic. In general, not every satisfiable set of sentences has a countable model, but we show that one can always build a model on the unit interval. Also, the probability logic under consideration is not compact. However, using ultraproducts we can prove a compactness theorem for a certain class of weak models.

There are two general conceptions on the relationship between probability and logic. In the first, these systems are viewed as complementary—having offsetting strengths and weaknesses—and there exists a fusion of the two that creates a reasoning system that improves upon each. In the second, probability is viewed as an instance of logic, given some sufficiently broad formulation of it, and it is this that should inform the development of more general reasoning systems. These two conceptions are in conflict with each (...) other, where the root issue of contention is the proper abstraction of the concept of logical consequence. In this work, we put forth a proposal on this abstraction based on an extension of the subset relation through the use of projections, which in turn, allows for the formalization of valid inferences to more general settings. Our proposal results in a formalism that encompasses probability and classical logic, and importantly, does so with minimal machinery. This formalism makes assertions about the relationship between these two systems that are explicit, and suggests a path forward in the development of alternatives to them. (shrink)

The volume analyses and develops David Makinson’s efforts to make classical logic useful outside its most obvious application areas. The book contains chapters that analyse, appraise, or reshape Makinson’s work and chapters that develop themes emerging from his contributions. These are grouped into major areas to which Makinsons has made highly influential contributions and the volume in its entirety is divided into four sections, each devoted to a particular area of logic: belief change, uncertain reasoning, normative systems and the resources (...) of classical logic. Among the contributions included in the volume, one chapter focuses on the “inferential preferential method”, i.e. the combined use of classical logic and mechanisms of preference and choice and provides examples from Makinson’s work in non-monotonic and defeasible reasoning and belief revision. One chapter offers a short autobiography by Makinson which details his discovery of modern logic, his travels across continents and reveals his intellectual encounters and inspirations. The chapter also contains an unusually explicit statement on his views on the role of logic in philosophy. (shrink)