AV 03.12
This course provides a comprehensive overview of the general theory of neighborhood semantics for modal logic, primarily based on my 2017 book Neighborhood Semantics for Modal Logic, and discusses several interesting new applications. Neighborhood models generalize the standard relational, or Kripke, models for modal logic. They were independently invented by Dana Scott and Richard Montague. In fact, the idea of 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 relational semantics was to provide a semantics for a wider class of modal logics, such as the so-called non-normal modal logics.
To define a neighborhood model, the analyst must specify, at each state, the propositions (i.e., sets of possible worlds) considered "necessary" at that state. Then, is true at a state if the truth set of (the set of states where 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 appear 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 examining how these systems behave in a more general semantics.
The main objective of the course is to demonstrate precisely where neighborhood models fit within the extensive family of semantic frameworks for modal logic and to 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 timely and of interest to many ESSLLI participants.
First, students will be introduced to the mathematical theory of modal logic, such as that found in the seminal textbook Modal Logic by Blackburn, de Rijke, and Venema, which is important for all ESSLLI students. Second, the mathematical ideas discussed in this course are useful in other research areas familiar to many ESSLLI students. Finally, this course will discuss interpretations of neighborhood semantics for modal logic that are of particular interest to many students at ESSLLI.
Slides
See https://pacuit.org/modal/neighborhoods/ for a brief introduction to neighborhood semantics, including a short video lecture, and a link to the book Neighborhood Semantics for Modal Logic.