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Intensional Logic/Non-Standard/Hintikka: intensional logic uses non-standard semantics (i.e., the individual domain is not fixed.). Therefore, it is not necessary to limit the domain of possible worlds of their framework. It would also be inappropriate for other reasons:
Epistemic logic: here the restriction would mean that everyone knows the identity of all individuals in the possible world. This would lead to omniscience.
Problem: this would make the situation even stranger: there must then often be epistemic alternatives to the world w0, which are not alethischen (logical) alternatives! This contradicts the natural assumption that:
Logical possibility forms the broadest class of possibilities.
Jaakko and Merrill B. Hintikka
The Logic of Epistemology and the Epistemology of Logic Dordrecht 1989
J. Hintikka/M. B. Hintikka
Untersuchungen zu Wittgenstein Frankfurt 1996