# Stratified institutions and elementary
homomorphisms. (*abstract*)

- For conventional logic institutions, when one extends the
sentences to contain open sentences, their satisfaction is
then parameterized. For instance, in the first-order logic,
the satisfaction is parameterized by the valuation of unbound
variables, while in modal logics it is further by possible
worlds.

This paper proposes a uniform treatment
of such parameterization of the satisfaction relation within the
abstract setting of logics as institutions, by defining the
new notion of stratified institutions. In this new framework,
the notion of elementary model homomorphisms is defined
independently of an internal stratification or elementary
diagrams. At this level of abstraction, a general Tarski
style study of connectives is developed. This is an abstract
unified approach to the usual Boolean connectives, to
quantifiers, and to modal connectives. A general theorem
subsuming Tarski's elementary chain theorem is then proved for
stratified institutions with this new notion of connectives.

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