Guards, Bounds, and Generalized Semantics
Johan van Benthem

Abstract:
The Guarded Fragment can be viewed in two ways: as a large decidable
fragment of first-order logic over standard models, or as a reflection
of a generalized semantics for the complete first-order language.  We
prove that the two perspectives are mutually reducible. We also show
how this duality sheds new light on fixed-point logics. Next, we
relate the use of guards to 'persistent' formulas from the earlier
literature, which have only bounded quantifiers, making second-order
logic effectively axiomatizable. Finally, we discuss some aspects of
guarding which have not been investigated yet.  This paper will appear
in an issue of the "Journal of Logic, Language and Information",edited
by Natasha Alechina, on 10 years of the Guarded Fragment.