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Neighbourhood semantics is a natural generalization of Kripke semantics
and topological semantics. Despite this, discussion of neighbourhood semantics
is usually left out of basic courses of modal logic. However, a number of modal
systems (primarily non-normal ones) are incomplete with respect to the more
common Kripke semantics, but complete in the neighbourhood case. Moreover,
there are important modal systems (for example, the Gödel-Löb provability logic
$GL$) that are strongly complete in the neighbourhood case, but not in the case of
Kripke. In our course, we plan to tell the main theorems and facts about
neighbourhood semantics, analyze specific examples of interesting
neighborhood-complete logics, and also talk about normal logics for which
neighborhood semantics gives interesting non-trivial results. Students are
required to have a good knowledge of classical propositional logic. Familiarity
with modal logic is desirable, but not required.
Course plan:
- Basic definitions: neighbourhood semantics, Kripke semantics, topological
semantics.
- Definable properties of neighbourhood frames, bisimulations, truth-
preserving operations.
- Normal and non-normal modal logics. A general soundness theorem.
- Construction of canonical models. Completeness for the logics $E, EC, EN, EM, K$.
- Filtrations and decidability of modal logics.
- The standard translation into first-order logic.
- The logic $S4$ and its neighbourhood frames as topological spaces. Extensions of
the logic $wK4$ and derivational semantics as a special case of neighbourhood
semantics.
- Neighborhood semantics of modal logics $GL$ and $S4CI$.
- Construction of a neighbourhood frame from a Kripke frame (construction of
paths with stops).
- Products of Kripke frames and neighborhood frames. Axiomatization and the
completeness theorem for products of logics from a set ${D, T, D4, S4}$.
- Axiomatization and the completeness theorem for $K \times K$.
 RSS: Forthcoming seminars
 Lecturers
Kudinov Andrey Valer'evich
Shamkanov Daniyar Salkarbekovich
Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center |
