An Institution-independent Generalization of Tarski's Elementary Chain Theorem

We prove an institutional version of Tarski's elementary chain theorem applicable to a whole plethora of ‘first-order-accessible’ logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulae by means of classical first-order connectives and quantifiers. These include the unconditional equational, positive, ${\left ( \prod \bigcup \sum\right )}^{0}_{n}$ and full first-order logics, as well as less conventional logics, used in computer science, such as hidden or rewriting logic.