Indexed actuality (Modal operators, logic)

The word 'actually' often refers to what is in fact the case, but it also o ften points to what would have been the case in a possible situation that i s being envisaged. To capture such nuances, the formal languages discussed in the paper add subscripts to modal operators; in the model theory the sub scripts 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.