Neighborhood Semantics for Modal Logic

Course Information
  • Lecturer: Eric Pacuit (website)
  • Venue: Leuven, Belgium


  • Dates: August 5 - August 9, 2024
  • Meeting Times: TBA
  • Neighborhood models are generalizations of the standard relational, or Kripke, models for modal logic invented independently by Dana Scott and Richard Montague. (In fact, the idea for neighborhood semantics for modal logic is already implicit in the seminal work of McKinsey and Tarski on topological semantics for modal logic). The original motivation for generalizing the relational semantics was to provide a semantics for a wider class of modal logics (such as the so-called non-normal modal logics). The underlying idea for neighborhood models comes from point-set topology: In a topology, a neighborhood of a point xx is any set AA containing xx such that you can "wiggle" xx without leaving AA. What exactly it means to "wiggle" xx depends on the underlying topology. Using this idea, a general definition for truth of a modal formula φ\Box\varphi runs as follows: φ\Box\varphi is true at a state ww just in case the {\em truth set} of φ\varphi (i.e., the set of states where φ\varphi is true) is a neighborhood of ww. Different semantics for modal logic can be classified in terms of what it means to be a "neighborhood" of a state. For instance, in topological semantics, a neighborhood is defined with respect to some given topology. In the standard relational semantics for modal logic, a neighborhood of ww is any set containing the states that are immediately connected to ww via some relation on the set of states. The most general approach are neighborhood models where a neighborhood of ww is any element of a distinguished collection of sets.

    In order to define a neighborhood model, the analyst must write down, at each state, the propositions (i.e., sets of possible worlds) that are considered "necessary" at that state. Then, φ\Box\varphi is true at a state if the truth set of φ\varphi (the set of states where φ\varphi is true) is a member of this distinguished collection of sets. A general criticism of neighborhood models is that they are not well-motivated: It is ad hoc to simply assert that certain subsets and not others are in the neighborhood of a given state. 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 provide an comprehensive overview of the general theory of neighborhood semantics for modal logic and discuss a number of interesting new applications based primarily on my 2017 book Neighborhood Semantics for Modal Logic. The main objective of the course 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. Although neighborhood semantics have been around for many years, this course will be both timely and of interest to many ESSLLI participants. First of all, students will be introduced to the mathematical theory of modal logic (e.g., as found in the seminal text book Modal Logic by Blackburn, de Rijke and Venema), something which is important for all ESSLLI students. Second, the mathematical ideas that will be discussed in this course are useful in other research areas familiar to many ESSLLI students (for example, I will explain how to derive an ordering from a collection of sets, an idea used by Kratzer when she derives orderings from an order base, and similar ideas are found in Lewis' models of counterfactuals). Finally, I will discuss in interpretations of neighborhood semantics for modal logic of interest to many students at ESSLLI:

    • Coalitional logic for reasoning about what groups of players can "effectively force" in games (Section 1.4.3 of Neighborhood Semantics for Modal Logic)
    • A modal logic of evidence and beliefs (Section 1.4.4 of {\em Neighborhood Semantics for Modal Logic}), and other epistemic interpretations (Fan 2021; Fan and van Ditmarsch 2015; Klein, Roy, and Gratzl 2017; Li and Wang 2021)
    • Neighborhood semantics and generalized quantifiers (Westerstahl and van Benthem 2021)

Day 1    Introduction, motivation and background
  • Slides
Day 2    Core theory I: Completeness and relationship with normal modal logic
  • Slides
Day 3    Core theory II: Incompleteness and decidability
  • Slides
Day 4    Advanced topics: Bisimulations and expressivity
  • Slides
Day 5    Extending neighborhood semantics
  • Slides