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.
"First-Order Modal Logic with an 'Actually' Operator." Notre Dame J. Formal Logic 46 (4) 381 - 405, 2005. https://doi.org/10.1305/ndjfl/1134397658