Indexed actuality

Journal of Philosophical Logic 30 (4):355-393 (2001)
Abstract
The word 'actually' often refers to what is in fact the case, but it also often points to what would have been the case in a possible situation that is being envisaged. To capture such nuances, the formal languages discussed in the paper add subscripts to modal operators; in the model theory the subscripts allow an actuality operator to turn the evaluation of a formula to a world introduced by a preceding possibility or necessity operator having the same subscript. The paper covers both propositional and predicate logic and proves the completeness of axiomatizations that extend standard modal systems beginning with K
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