Neighborhood Models and Their Applications

January 2006
Institute of Logic, Language and Computation
Universiteit van Amsterdam


Stay tuned for information about an ESSLLI 2007 course on neighborhood models. Information will be available Spring 2007.

Intstructor: Dr Eric Pacuit

Class Meetings: We will meet around 5 times (5 lectures). The exact times are announced below. Please email me if you are planning on attending.

Classroom: See below.

Prerequisites: Basic knowledge of modal logic.

Teaching Goal: We will introduce neighborhood semantics for modal logic and discuss some applications. The main goal of the corse is to understand the basic techniques and results of neighborhood semantics for modal logics and to understand the exact relationship between the standard relational semantics and neighborhoood semantics for modal logics.

Content: Dana Scott and Richard Montague (influenced by a paper written by McKinsey and Tarski in 1944) proposed independently in 1970 a new semantic framework for the study of modalities, which today is known as neighborhood semantics. The semantic framework permits the development of elegant models for the family of classical modal logics, including many interesting non-normal modalities from Concurrent Propositional Dynamic Logic, to Coalitional Logic to various monadic operators of high probability used in various branches of game theory. After introducing the basic tools and techniques for neighborhood semantics, we will study applications to game theory, first-order modal logic and logics of high probability.

Examination: Some homework and a short paper. The homework will be worth 25% of the grade and the paper 75% of the grade. Details to be provided later.

Paper: In order to receive credit for this project, you must write a research paper. The paper is due Feb. 6, 2006. The length of the paper depends on the type of paper you plan on writing. An expository paper should be around 10 - 15 pages and survey at least 3 research papers. Research papers can be shorter, but must contain new results. The following is a list of some suggested topics (of course, feel free to come up with your own). Please send me your chosen topic by the end of this week.

  • Survey the main results in topologic
  • Survey topological semantics for modal logic
  • Study/survey logics for reasoning about agent's powers in a game
  • Survey game logic and/or coalitional logic
  • Compare topological semantics for modal logic and neighborhood models
  • Find a tableaux (or Gentzen) system for classical systems of modal logic
  • Study incompleteness with respect to neighborhood semantics
  • Argue for/against neighborhood semantics as a solution to the "logical omniscience" problem
  • Survey logics of high probability
  • Systematically study the model theory of modal logic with respect to neighborhood models
  • ...
Literature: The main text of the course will be Modal Logic: an Introduction by B. Chellas (Cambridge University Press, 1980) chapters 7 - 9 and course notes that will be handed be available in January. We will also rely on results from:
  1. Gasquet, O and Herzig, A. 'From Classical to Normal Modal Logics', in "Proof Theory of Modal Logics", Kluwer Academic Publishers, 1996.
  2. Gerson, M. 'The inadequacy of neighborhood semantics for modal logic,' Journal of Symbolic Logic, bf 40, No 2, 141--8, 1975.
  3. Hansen, H. H. Monotonic modal logics, Master's thesis, ILLC, 2003.
  4. Kracht, M and Wolter, F. 'Normal monomodal logics can simulate all others', Journal of Symbolic Logic 64 (1999).
For examples of applications, students may want to consult
  1. Arló Costa, H and Pacuit, E.. 'First-order classical modal logic', forthcoming in Studia Logica. See also 'First-order classical modal logic: Applications to Epistemic Logic and Probability', Proceedings of TARK 2005.
  2. Parikh, R. 'The logic of games and its applications,' In M. Karpinski and J. van Leeuwen, editors, Topics in the Theory of Computation, Annals of Discrete Mathematics 24. Elsevier, 1985. [pdf]
  3. Pauly. M. 'A modal logic for coalitional power in games,' Journal of Logic and Computation, 12(1):149--166, 2002.
  4. Epistemic Logic and the Theory of Games and Decisions, edited by M. O. Bacharach, L. A. Gerard-Varet, P. Mongi, and H. S. Shin
  5. Vardi, M. Y. 'A model-theoretical analysis of monotonic knowledge,' IJCAI'85, 509--512, 1985.
Of historical interest are:
  1. Montague, R. Universal Grammar, Theoria 36, 373-- 98, 1970.
  2. Scott, D. 'Advice in modal logic,' K. Lambert (Ed.) Philosophical Problems in Logic, Dordrecht, Netherlands: Reidel, 143--73, 1970.
  3. Segerberg. K. An Essay in Classical Modal Logic, Number13 in Filosofisska Studier. Uppsala Universitet, 1971.Gasquet, O and Herzig,

Course Schedule (tenative)

Fri., 13 Jan.
11 - 13
Introduction, Basic Results and Techniques
Wed. 18 Jan.
11 - 13
Examples, Topological Semantics, Completeness
Wed. 25 Jan.
11 - 13
Model Theory of Non-Normal Modal Logics I
Schedule a Meeting
Feb. 7
(Logic Tea)
Neighborhood Models for First-Order Modal Logic