Changing a semantics: opportunism or courage?
Hajnal Andreka, Johan van Benthem, Nick Bezhanishvili, István Németi

Abstract:
The generalized models for higher-order logics introduced by Leon
Henkin, and their multiple offspring over the years, have become a
standard tool in many areas of logic. Even so, discussion has
persisted about their technical status, and perhaps even their
conceptual legitimacy. This paper gives a systematic view of
generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and
results about their role in algebraic representation, calibrating
provability, lowering complexity, understanding fixed-point logics,
and achieving set-theoretic absoluteness. We also show how thinking
about Henkin’s approach to semantics of logical systems in this
generality can yield new results, dispelling the impression of
adhocness.