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Full Idea
In a modal logic in which names are non-vacuous and rigid, not only must existence and uniqueness in a definition be shown to hold necessarily, it must be shown that the definiens is satisfied by the same object across possible worlds.
Gist of Idea
A definition needs to apply to the same object across possible worlds
Source
Anil Gupta (Definitions [2008], 2.4)
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.12