The notion originates in [[Feferman - Two notes on abstract model theory. II. Languages for which, the set of valid sentences is semi-invariantly implicitly definable]]. The definition in [[Barwise, Feferman (eds.) - Model-theoretic logics]] is: ![[Väänänen - Set-theoretic definability of logics#1.1. Adequacy to Truth]] In [[Kennedy, Magidor, Väänänen - Inner models from extended logics - Part 1]] another definition is presented: ![[Kennedy, Magidor, Väänänen - Inner models from extended logics - Part 1#^74a248]] >[!note] Remark >Seems that it is not entirely equivalent, as in the original definition you might need to require further things from the admissible sets (e.g. that they are large enough, as is noted there regarding second-order logic).