In this paper the language of
first-order modal logic is enriched with an operator @ ('actually')
such that, in any model, the evaluation of a formula @A at a
possible world depends on the evaluation of A at the actual
world. The models have world-variable domains. All the logics that are
discussed extend the classical predicate calculus, with or without
identity, and conform to the philosophical principle known as serious
actualism. The basic logic relies on the system K, whereas others
correspond to various properties that the actual world may have. All
the logics are axiomatized.