Frederik Van De Putte: Pointwise intersection in neighbourhood semantics

Lecture by Frederik Van De Putte (Ghent), organized by Grolog

We study the logic of neighbourhood models with pointwise intersection, as a means to interpret non-normal multi-modal logics. Pointwise intersection takes us from a set of neighbourhood sets N_i (one for each member i of a set G) to a new neighbourhood set N_G, which in turn allows us to interpret the operator G. Here, X is in the neighbourhood for G if and only if X equals the intersection of some Y = {Y_i | i is a member of  G}.

We argue that the notion of pointwise intersection has various applications in epistemic and doxastic logic, deontic logic, coalition logic, and evidence logic. We then establish sound and strongly complete axiomatizations for the weakest modal logic characterized by pointwise intersection, and for a number of variants. We briefly spell out the way the canonical model for such systems is constructed. Finally, we discuss potential generalizations of our results to (a) multisets of indexes and (b) other operations on (tuples) of neighbourhood sets (union, negation, concatenation, etc). (Joint work with Dominik Klein.)

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