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An Indexed Approach to Modalities
Abstract:
The shape of the rules of standard proof-theoretical systems for modal
logics is often an example of what a good rule should not be. The
intensionality of the modal connectives forces the rules either to be
non symmetric or to involve a global constraint on the shape of the
deduction.
To restore symmetry and locality we propose the framework of
2-sequents. This proposal is based on a notion of "indexed systems"
(the idea goes back to Kripke and has been developed by many authors),
where "indexes" (natural numbers) are used to mimic the interplay of
variables and quantifiers in first-order logic. Different modal and
linear logics are obtained with simple side conditions on indexes.
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