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Stratified institutions and elementary homomorphisms. (English) Zbl 1189.03038
Summary: For conventional logic institutions, when one extends the sentences to contain open sentences, their satisfaction is then parameterized. For instance, in first-order logic, satisfaction is parameterized by the valuation of unbound variables, while in modal logics it is further parameterized 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 homomorphism 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.

MSC:
03B70 Logic in computer science
03C95 Abstract model theory
03G30 Categorical logic, topoi
68Q65 Abstract data types; algebraic specification
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