Neighborhood Semantics for Modal Logic

Lecturer: Eric Pacuit (website)
Meeting Times: 9am - 10:30am
Location: Tübingen, Germany

Course Description

Neighborhood models are generalizations of the standard relational models for modal logic invented independently by Dana Scott and Richard Montague in 1970. Neighborhood models provide simple semantics for the family of classical modal logics, including many interesting non-normal modalities from Concurrent Propositional Dynamic Logic to Coalitional Logic to weak systems of beliefs used in various branches of game theory. A general criticism of neighborhood models is that they are not well-motivated. They do provide a semantics for weak systems of modal logic, but do they do so in a principled way? There is certainly some truth to this criticism. Nonetheless, recent work has demonstrated the usefulness and interest of neighborhood semantics. Neighborhood structures naturally show up when studying cooperative and non-cooperative game theory. Furthermore, neighborhood semantics can be given an epistemic interpretation as the evidence that an agent has accepted at a given state. Finally, one can learn something about normal systems of modal logic by looking at how these systems behave in a more general semantics. In this course, I will introduced the basic techniques and results of neighborhood semantics for modal logic and explain the exact relationship between the standard relational semantics and neighborhood semantics for modal logics. The main objective is to demonstrate precisely where neighborhood models fit within the large family of semantic frameworks for modal logic and discuss both the pitfalls and potential uses of these very general structures.

Day 1: Introduction and Motivation I


Topics: Introduce both relational and neighborhood semantics for modal logic; Discuss key motivating for studying non-normal modal logics: Logics of ability, Logics for group decision making, Logic of classical deductive closure, Deontic logic paradoxes, the problem of logical omniscience and knowledge closure; Normal and non-normal modal logics

Reading material: Chapter 1 (Introduction and Motivation)

Day 2: Introduction and Motivation II


Topics: Neighborhood frames/models; Examples of logics using neighborhood structures: A logic of knowledge, evidence and belief, Coalitional Logic, Subset Space Logic (Topologic); Neighborhood semantics in the broader logical landscape: relationship with relational structures, topological models, n-ary relational models

Reading material: Chapter 1 (Introduction and Motivation) and Chapter 2 (Core Theory)

Day 3: Core Theory I


Topics: Relationship with plausibility structures; Completeness and incompleteness of non-normal modal logics

Reading material: Chapter 2 (Core Theory)

Day 4: Core Theory II


Topics: Decidability, Complexity, Definability, Model Theory (bisimulations, relationship with first-order logic)

Reading Material

Day 5: Core Theory II


Topics: First-order modal logic on neighborhood structures, Game Logic, Dynamics on neighborhood structures

Reading Material