Neighborhood Semantics for Modal Logic

Lecturer: Eric Pacuit (website)
Location: Tshinghua University, Beijing

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 many interesting non-normal modalities. 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. This course will introduce the basic techniques and results of neighborhood semantics for modal logic and explain the exact relationship between the standard relational models, neighborhood models and topological models 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. The course will be based on a new book "Neighborhood Semantics for Modal Logic" by the instructor, but background material of current research papers will also be discussed.

Lecture 1: Introduction and Motivation


Date: Wednesday, November 7, 15:30 - 17:30

Reading: Sections 1.2, 1.2.1 1.2.2, and 1.3

Suggested Exercises: Appendix A: Exercises 89 & 91, Sections 1.1 & 1.2: Exercises 1 - 7
(Note: exercise numbers are from the online draft of the book)

Lecture 2: Examples and Core Theory


Date: Friday, November 9, 14:00-16:00

Reading: Sections 1.4.3, 1.4.4, 1.4.5, 2.1, 2.2.1, 2.2.2, and 2.3

Suggested Exercises: Sections 1.4.3 & 1.4.4: Exercises 20 - 24; Section 2.2.1: Exercises: 33 - 35
(Note: exercise numbers are from the online draft of the book)

Lecture 3: Core Theory and Extensions


Date: Wednesday, November 14, 15:30 - 17:30

Reading: Sections 2.1, 2.2.1, 2.2.2, 23, 2.3.2, 2.3.3, 2.5 , 2.6.2, & 3.5

Suggested Exercises: Section 2.3: Exercises: 39 - 43, Section 3.5.1, Exercise 82 - 85
(Note: exercise numbers are from the online draft of the book)