There are commonly thought to be two kinds of set-theoretic semantics for second-order logic: the standard one, where relation (and function) variables range over the entire power set of a model domain and the Henkin one, where relation variables range over a fixed collection of relations which need not contain every relation. The dominant plural based semantics for basic plural logic or PFO is standard in a similar way: here, plural variables range over every sub plurality of the model domain. More recently (Florio and Linnebo 2016) and (Florio and Linnebo 2021) have formulated a plural based Henkin semantics for PFO, where plural variables take their values from the instances of a plural concept.
However, there is a third kind of set based semantics for second-order logic for which no one has produced a plural based analogue or so it seems. First-order set based semantics for second-order logic uses a notion of model, which consists of a first-order domain and a possibly unrelated domain of primitive relations. Additionally, a first-order model has, for every positive integer n, an predication relation of arity n, which relates n-tuples of first-order objects and n-place relations. Basically first-order semantics amounts to looking at second-order logic as a many-sorted first-order logic and it is equivalent to Henkin semantics (See Gilmore: The monadic theory of types in the lower predicate calculus. Summaries of talks presented at the Summer Institute for Symbolic Logic, Cornell University, 1957)
My question is: Is this there any research out there on first-order plural based semantics for plural logic? If not, might this be due to the fact that (as it seems to me) the distinction between first-order set based models and Henkin based set models cannot be reproduced in the plural case, so that plural logic has only two possible semantics, namely the standard and the Henkin one?
