Projects in Previous Years

1st Semester 2005/06: Neighbourhood Models

Dr Eric Pacuit

If you are interested in this project, please contact the instructor by e-mail.
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.
Basic knowledge of modal logic.
Some homework and a short paper.
The main text of the course will be "Modal logic an introduction" by B. Chellas (Cambridge University Press, 1980) chapters 7 - 9 . 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: applications in logics of knowledge and probability', forthcoming in Studia Logica.
  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.
  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 abalysis 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.